diff --git "a/2tE1T4oBgHgl3EQflgTe/content/tmp_files/load_file.txt" "b/2tE1T4oBgHgl3EQflgTe/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/2tE1T4oBgHgl3EQflgTe/content/tmp_files/load_file.txt" @@ -0,0 +1,959 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf,len=958 +page_content='Molecular and solid-state topological polaritons via optical saturation Sindhana Pannir-Sivajothi,1 Nathaniel P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Stern,2 and Joel Yuen-Zhou1, ∗ 1Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, USA 2Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA Strong coupling between electronic excitations in materials and photon modes results in the formation of hybrid quasiparticles called polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Polariton systems often display larger nonlinearities than their photonic counterparts due to their material component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In this work, we theoretically investigate how to optically control the topological properties of molecular and solid-state exciton-polariton systems by exploiting one such nonlin- earity: saturation of electronic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We study an optically pumped film of porphyrin molecules strongly coupled to the photon modes of a perylene filled Fabry-Perot cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, optical pumping with circularly polarized light breaks time-reversal symmetry instead of the frequently used large magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We can op- tically tune properties such as the Berry curvature and Chern numbers of the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Importantly, while optical pumping does lead to non-zero Chern invariants, unidirectional edge states do not emerge in our system as the bulk-boundary correspondence is not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Finally, we illustrate the broad applicability of our scheme by computing the Berry curvature of two other systems with slightly modified level structures that lead to different nonlinear behavior when placed in a microcavity and pumped with circularly polarized light: (a) monolayer MoS2 and (b) Ce:YAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This work demonstrates a versatile platform to control topological properties of hybrid light-matter systems to enrich the toolbox of optoelectronic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' INTRODUCTION Exciton-polaritons are hybrid excitations that exist in sys- tems where photonic modes couple strongly with optical tran- sitions in materials and their coupling strength exceeds losses [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Electronic strong coupling (ESC), where the optical tran- sitions correspond to semiconductor excitons or molecular electronic transitions, has been observed in a wide variety of inorganic and organic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' While some polariton sys- tems, such as GaAs and CdTe quantum wells in microcavi- ties [1, 2], often require cryogenic temperatures for operation, due to their small exciton binding energies, organic materials [3] along with others such as GaN [4], ZnO [5], perovskites [6, 7], and transition metal dichalcogenides (TMD) [8, 9] can FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Illustration of the system under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Porphyrin (molecules at the center) and perylene (green blocks) placed within a Fabry- Perot cavity and pumped with circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ∗ joelyuen@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='edu achieve ESC at room temperature when placed in Fabry-Perot cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In particular, organic exciton-polaritons have re- ceived attention for their ability to modify chemical reactiv- ity [10], demonstrate polariton condensation at room temper- ature [11, 12], improve photoconductivity [13], and display topological properties [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Exciton-polariton systems are versatile platforms for topo- logical applications as their hybrid nature provides the unique opportunity to take advantage of the nonlinearities and mag- netic response of the material component while still enjoy- ing benefits of the coherence properties of the photonic part [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In the presence of photonic lattices, they also offer the possibility of unidirectional transport of energy through edge states that are robust to disorder [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' A few approaches are frequently used to achieve topological exciton-polariton bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In one of the approaches, the non-trivial topology re- sides in the winding light-matter coupling rather than individ- ual photon or exciton components [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, it is limited in application due to the requirement of large mag- netic fields to break time-reversal symmetry (TRS) and low temperatures to achieve Zeeman splitting in the exciton com- ponent which exceeds the exciton linewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In another ap- proach, TRS is preserved and a quantum spin hall insulator analogue is created in a polariton system [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This approach does not require a large magnetic field, however, there, a topo- logical polariton system is created by coupling a topologically non-trivial photonic lattice with a topologically trivial exciton system and the interesting topology is almost entirely encoded in the photonic component of the polariton [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Both the approaches mentioned above were experimentally realized in polariton lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' More recently, polaritons in Fabry-Perot cavities have emerged as a viable platform for topological po- laritonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Several experiments have demonstrated measure- ment and control of the Berry curvature of exciton-polariton and photon bands in these systems [23–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Our work will focus on these Fabry-Perot cavity systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In this work, we theoretically propose a scheme for gener- ating topological polaritons that combines advantages of both arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='03287v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='chem-ph] 9 Jan 2023 2 the approaches mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, the light-matter cou- pling contains the non-trivial topology instead of the individ- ual photon or exciton components and optical pumping with circularly polarized light breaks TRS instead of a large mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Breaking TRS in a molecular system using the helicity of light is an idea that has been demonstrated in sev- eral other contexts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' it has been used to achieve all-optical non- reciprocity [27, 28] and theoretical results suggest that it can also induce optical-activity in achiral molecules [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Addi- tionally, a similar idea that relies on breaking TRS using cir- cularly polarized light has been previously proposed for po- lariton lattices by Bleu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We focus on the topological properties of polaritons formed by the coupling of Frenkel excitons hosted in organic semi- conductors with photon modes in a Fabry-Perot cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, optical pumping with circularly polarized light saturates cer- tain electronic transitions and breaks TRS in the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' this results in non-zero Chern numbers of polariton bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We exploit the primary nonlinearity of organic exciton- polaritons, saturation [11], to generate topological exciton- polariton bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Our scheme relies on the contraction of Rabi splitting due to saturation, and we find modified Berry curva- ture and Chern number of the bands under circularly polarized pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Berry curvature of the more photonic sections of the bands computed in our work can be experimentally measured using pump-probe spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Furthermore, the applicability of our scheme is not limited to organic polariton systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' It only requires certain key ingredients: transitions that can be selectively excited with circularly polarized light, saturation effects, and Rabi splitting contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To highlight this, we compute the Berry curvature of two other systems un- der strong coupling and optical pumping: (a) Ce:YAG and (b) monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Our work provides a viable strategy to in- duce non-reciprocal behavior in standard microcavity polari- tons, leading to the optical tuning of isolators and circulators [27], as well as fabrication of elliptically-polarized lasers and condensates [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' RESULTS Model In our theoretical study, we consider a Fabry-Perot cavity containing a thin film of porphyrin molecules at the center and a bulk perylene crystal filling the rest of the volume (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The porphyrin and perylene molecules are not treated on an equal footing in our model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' while the molecular transitions of porphyrin are considered explicitly in the Hamiltonian, those of the perylene crystal are not, and they can be accounted for through effective cavity modes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This is a valid ap- proximation because we focus on photon modes with fre- quencies close to those of electronic transitions in porphyrin (∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='81eV) [32, 33] and far off-resonant from the transitions of perylene (∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='98eV) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, the birefringent perylene crystal plays the role of providing anisotropy and emergent optical activity to the cavity modes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We model each porphyrin molecule as a three-level elec- 1 |G⟩ |−!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' "#⟩ |+!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' "#⟩ 𝝁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝝁" a b FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (a) Illustration of circularly polarized light exciting a met- alloporphyrin molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (b) Three-level model of porphyrin with a ground state |G⟩ and two degenerate excited states |+mol⟩,|−mol⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The transition dipole moment for a transition from |G⟩ to |±mol⟩ is µµµ± = µ0(ˆx±iˆy)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The number of yellow circles at each state rep- resents the fraction of molecules in that state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, the ratio of the fraction of molecules in the ground, fG, and |±mol⟩ excited states, f±, is fG : f+ : f− = 3 : 1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Such population ratios can be achieved through pumping with circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' tronic system with a ground state |G⟩ and two excited states |+mol⟩ and |−mol⟩ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 2b) [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In the absence of a magnetic field, the two excited states are degenerate and the energy difference between the ground and excited states is ¯hωe = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='81eV [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The transition dipole moments for transi- tions from |G⟩ to |+mol⟩ and |−mol⟩ are µµµ+ = µ0(ˆx+iˆy)/ √ 2 and µµµ− = µ0(ˆx−iˆy)/ √ 2, respectively, with µ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='84D [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, ˆx and ˆy are unit vectors along the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Using circular polarized light, the |+mol⟩ or |−mol⟩ states can be selectively excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In our model, we consider a thin film of metalloporphyrins or metallophtalocyanines arranged in a square lattice with nearest neighbor spacing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The choice of lattice is irrele- vant because later we will take the continuum limit a → 0 as we are only interested in length scales much larger than the intermolecular spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Each molecule is labeled with the index m = (mx,my), where mx,my ∈ Z and the molecule’s position is given by rm = mxaˆx + myaˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' States of the mth molecule are then written as |m,G⟩, |m,+mol⟩ and |m,−mol⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The creation operator ˆσ† m,± = |m,±mol⟩⟨m,G| ⊗n̸=m In ex- cites the mth molecule from |m,G⟩ to |m,±mol⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, In = |n,G⟩⟨n,G| + |n,+mol⟩⟨n,+mol| + |n,−mol⟩⟨n,−mol| is the identity operator for nth molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' These molecular operators satisfy commutation relations (a generalization of the commu- tation relations of paulion operators [38, 39]), � ˆσn,±, ˆσ† m,± � = δm,n(1− ˆσ† n,∓ ˆσn,∓ −2 ˆσ† n,± ˆσn,±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (1) We model the effective photon modes of a Fabry-Perot cav- ity filled with perylene as in Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [25] For the photon modes of a Fabry-Perot cavity, the component of wave vec- tor orthogonal to the mirrors kz = 2nπ/L is quantized, where L is the effective distance between the mirrors of the cav- ity and n is the mode index [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For a given n, the modes are labeled by the in-plane wave vector k = kxˆx + kyˆy and polarization α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' the creation operators associated with these 00003 modes are ˆa† k,α and they satisfy bosonic commutation rela- tions � ˆak,α, ˆa† k′,α′ � = δα,α′δk,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As a result of in-plane trans- lational invariance of a cavity, k can take any value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', kx,ky ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Throughout this work, we specify the cavity mode polarization in the circularly polarized basis α = ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Hamiltonian of the full system is ˆH = ˆHmol + ˆHcav + ˆHcav−mol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (2) where ˆHmol =∑ m � ¯hωe ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + ¯hωe ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � ˆHcav =∑ k �� E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2e−i2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2ei2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav−mol =∑ m ∑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α − ˆµµµm · ˆEk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) ≈∑ m ∑ k eik·rm �NxNy � (µµµ+ ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +(µµµ− ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +(µµµ+ ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='−) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− +(µµµ− ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='−) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (3) Above, ˆHmol describes the porphyrin molecules, ˆHcav the ef- fective cavity modes (including contributions from the pery- lene crystal), and ˆHcav−mol the coupling between the por- phyrin molecules and effective cavity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, φ is the angle between the in-plane wave vector and the x-axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', cosφ = kx/|k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Within ˆHcav, β specifies the TE-TM splitting, β0 quantifies the linear birefringence of the perylene crystal which splits the H-V modes, and ζ describes the emergent optical activity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Additionally, E0 is the frequency of the cavity modes at |k| = 0 in the absence of the perylene crys- tal (β0 = 0 and ζ = 0), and m∗ is the effective mass of the photons in the absence of perylene (β0 = 0 and ζ = 0) and TE-TM splitting (β = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The term ˆHmol describes an Nx ×××Ny array of porphyrin molecules with periodic boundary condi- tions along both the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We have made the electric dipole approximation and the rotating-wave approxi- mation in ˆHcav−mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, ˆµµµm is the electric dipole operator associated with the mth molecule and ˆEk,α(r,z) is the electric field operator of the mode with polarization α and in-plane wave vector k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In addition, µµµα′ · Jk,α is the collective cou- pling strength of the cavity mode labeled by k,α and the |G⟩ to ��α′ mol � transition of the molecules (see Supplementary sec- tion S1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The photon modes of an empty cavity experience TE-TM splitting due to polarization dependent reflection from the mir- rors [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' While the TE-TM splitting lifts the degeneracy be- tween photon modes at |k| ̸= 0, photon modes of both polar- izations remain degenerate at |k| = 0 due to rotational symme- try of the cavity mirrors about the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, for Berry curvature and Chern invariant to be well-defined, we need the photon/polariton bands to be separated in energy at all k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' to achieve this, we include the perylene crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The anisotropy and emergent optical activity of the perylene crystal lifts the degeneracy between the photon modes at all k [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To compute the Berry curvature and Chern number, we fo- cus on the first excitation manifold which is spanned by states |m,±mol⟩ = ˆσ† m,± |vac⟩ and |k,±cav⟩ = ˆa† k,± |vac⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, |vac⟩ is the absolute ground state of the system where the pho- ton modes are empty and all molecules are in their ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Rewriting the Hamiltonian with operators ˆσk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' where ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α = 1 √ NxNy ∑k∈BZ eik·rm ˆσk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α and restricting ourselves to the first excitation manifold,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' we find ˆH(k) = ⟨k| ˆH |k⟩ to be ˆH(k) = ˆHmol(k)+ ˆHcav(k)+ ˆHcav−mol(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (4) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHmol(k) =¯hωe |+mol⟩⟨+mol|+ ¯hωe |−mol⟩⟨−mol|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav(k) = � E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � |+cav⟩⟨+cav| + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � |−cav⟩⟨−cav| + � −β0 +β|k|2e−i2φ� |+cav⟩⟨−cav| + � −β0 +β|k|2ei2φ� |−cav⟩⟨+cav|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav−mol(k) =Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · � µµµ+ |+mol⟩+ µµµ− |−mol⟩ � ⟨+cav| +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · � µµµ+ |+mol⟩+ µµµ− |−mol⟩ � ⟨−cav| +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (5) Here, k lies within the first Brillouin zone determined by the porphyrin lattice kx,ky ∈ [−π/a,π/a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As we are only inter- ested in length scales much larger than a, we take the contin- uum limit a → 0 while keeping µ0/a a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Therefore, terms such as the collective light-matter coupling strength, Jk,α · µµµα′, remain constant in this limit (see Supplementary section S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Moreover, upon taking the continuum limit, ˆH(k) does not change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' only the range of k becomes infinitely large, kx,ky ∈ R, that is, our system acquires complete translational invariance in the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For such continuous systems, since kx,ky ∈ R is unbounded, we need to map (kx,ky) onto a sphere which is a closed and bounded surface using stere- ographic projection before we compute Chern numbers [42] (see Supplementary section S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' When we diagonalize the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5, we ob- tain four bands which we label with l = 1,2,3,4 in increas- ing order of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3a we plot the Berry curvature, Ω1(k), of the lowest band l = 1, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3e we plot the ky = 0 slice of the band structure of the two bands lowest in energy, l = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As expected, in the absence of optical pumping, this system preserves TRS, which can be verified 4 a e f g h b c d 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝑓" = 0 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓" = 0 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝑓" = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓" = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 S3 𝐶!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝐶" = 0 𝐶!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 1 𝐶" = −1 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝑓" = 0 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓" = 0 𝐶!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = −1 𝐶" = 1 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝑓" = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝐶!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝐶" = 0 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓" = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 Ω1 (𝜇m2) Ω1 (𝜇m2) Ω1 (𝜇m2) Ω1 (𝜇m2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (a-d) Berry curvature of the lowest energy band, Ω1(k), and (e-h) a slice of the band structure at ky = 0 of the lower two bands, under different levels of optical pumping which create populations: (a,e) f+ = f− = 0, (b,f) f+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3, f− = 0, (c,g) f+ = 0, f− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3, and (d,h) f+ = f− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (e-h) The colors of the band indicate the value of the Stokes parameter, S3(k), which measures the degree of circular polarization of a mode (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Chern numbers C1 and C2 of the bands are also specified and are non-zero under time-reversal symmetry (TRS) breaking, that is, when f+ ̸= f−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We used parameters β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1eV, β = 9×10−4eVµm2, ζ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5×10−3eVµm, m∗ = 125¯h2eV−1µm−2, E0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='80eV and ¯hωe = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='81eV (see Supplementary section S4 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' using the condition on Berry curvature Ωl(k) = −Ωl(−k), and the Chern numbers of the all the bands Cl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Also, note that, the smallest splitting between the lower two bands within −13µm−1 < kx,ky < 13µm−1 is ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='8meV which is larger than the linewidth of the transition in porphyrin at 4K (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5meV) [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Optical pumping Optical pumping can saturate the electronic transitions of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This leads to reduction in the effective light-matter coupling strength, and, therefore, Rabi splitting contraction [11, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For instance, when the pump excites a fraction of molecules, fE, to the excited state and the remaining popu- lation stays in the ground state, fG, it results in Rabi splitting contraction proportional to √fG − fE = √1−2fE [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In our system, when the molecules are optically pumped, a fraction, f+, of the molecules occupy the |+mol⟩ state, an- other fraction, f−, occupy the |−mol⟩ state, and the remaining fraction, fG, are in the ground state |G⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Rabi contraction corresponding to the |G⟩ to |+mol⟩ transition should then be proportional to √fG − f+ which equals √1− f− −2 f+ since fG+ f++ f− = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Similarly, the contraction should be propor- tional to √1− f+ −2 f− for the |G⟩ to |−mol⟩ transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This difference in light-matter coupling when f+ ̸= f− effectively introduces 2D chirality into the system [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To derive an effective Hamiltonian under optical pumping, we use Heisenberg equations of motion and make a mean- field approximation following the approach of Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [47] (Supplementary section S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We then obtain the effective Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff(k) = ˆHeff mol(k)+ ˆHeff cav(k)+ ˆHeff cav−mol(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (6) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff mol(k) =¯hωe |+mol⟩′ ⟨+mol|′ + ¯hωe |−mol⟩′ ⟨−mol|′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav(k) = � E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � |+cav⟩′ ⟨+cav|′ + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � |−cav⟩′ ⟨−cav|′ + � −β0 +β|k|2e−i2φ� |+cav⟩′ ⟨−cav|′ + � −β0 +β|k|2ei2φ� |−cav⟩′ ⟨+cav|′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav−mol(k) =Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · �� 1− f− −2 f+µµµ+ |+mol⟩′ + � 1− f+ −2 f−µµµ− |−mol⟩′ � ⟨+cav|′ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · �� 1− f− −2 f+µµµ+ |+mol⟩′ + � 1− f+ −2 f−µµµ− |−mol⟩′ � ⟨−cav|′ +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (7) Here, the states |γ⟩′ are different from states |γ⟩ in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5, where γ = ±mol,±cav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As expected, the light-matter coupling terms are scaled by factors √1− f∓ −2 f± which is a consequence of the commutation relation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 1 (see Supplementary sec- tion S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' If the pump pulse is circularly polarized, f+ ̸= f−, the Rabi contraction factor that multiplies the light-matter coupling dif- fers for transitions to the |+mol⟩ and |−mol⟩ states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' as a re- sult, time-reversal symmetry is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Consequently, when f+ > f−, we find that bands 1 and 2 have non-zero Chern numbers +1 and -1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Under the opposite condition, f+ < f−, the Chern numbers reverse sign as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' When f+ = f−, TRS is preserved, and all bands have Chern 5 a c d b 𝒇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝟑 𝒇" = 𝟎 𝒇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝟑 𝒇" = 𝟎 S3 Band 1 Band 2 𝒇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 𝟎 𝒇" = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝟑 𝒇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 𝟎 𝒇" = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝟑 Band 1 Band 2 S3 S3 S3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Stokes parameter, S3(k), which is a measure of the degree of circular polarization of a mode (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 8), under pumping with (a,c) σ+ polarized light which creates populations f+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3, f− = 0 and (b,d) σ− polarized light which creates populations f+ = 0, f− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 of the two lowest energy bands (Band 1 and 2 as indicated in the inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We used parameters β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1eV, β = 9×10−4eVµm2, ζ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5 × 10−3eVµm, m∗ = 125¯h2eV−1µm−2, E0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='80eV and ¯hωe = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='81eV (see Supplementary section S4 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' number 0 as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3e and 3h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3b-c, we plot the computed Berry curvature when f+ ̸= f− and due to broken TRS, we find Ωl(k) ̸= −Ωl(−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We also plot the Stokes parameter, S3(k), for bands 1 and 2, under pumping with circularly polarized light, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Stokes parameter, S3(k), provides information on the de- gree of circular polarization of the photonic component of an exciton-polariton band and is calculated as S3(k) = |b+,cav(k)|2 −|b−,cav(k)|2 |b+,cav(k)|2 +|b−,cav(k)|2 (8) where the eigenvectors of the band are ��ul,k � = b+,cav(k)|+cav⟩ + b−,cav(k)|−cav⟩ + b+,mol(k)|+mol⟩ + b−,mol(k)|−mol⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In the absence of pumping, we find that within a band, one half of the modes are predominantly σ+ polarized and the other half are σ− polarized (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Once TRS is broken with circularly polarized optical pumping, a large number of modes within each band become overwhelmingly of the same polarization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3f-g and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In experiments, the Berry curvature of photon bands in a Fabry-Perot cavity can be extracted from the components of the Stokes vector [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, in the case of exciton- polariton bands, the Berry curvature of only sections of the band that are predominantly photonic and have negligible molecular character [49] can be measured experimentally as, to the best of our knowledge, it is difficult to obtain the phase relationship between the photonic and molecular components, unless light-matter cross-correlation functions are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Therefore, in our case, the Berry curvature of only parts of the band that are mostly photonic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3a-d can be measured using pump-probe spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This measurement should be feasible as long as the time delay between the pump and probe pulses is shorter than the time the system takes to depo- larize and reach a state with f+ = f−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As the depolarization timescale for porphyrins ranges from 210 fs to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='6 ps, this measurement should be viable [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As the Chern numbers of bands 1 and 2 are modified through pumping with circularly polarized light, if we per- form a calculation where a region of the system is pumped with σ+ polarized light ( f+ ̸= 0 and f− = 0) and an adjacent region is pumped with σ− polarized light (f+ = 0 and f− ̸= 0), we expect edge states at the boundary between these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, as our Hamiltonian does not contain couplings be- tween neighboring molecules, and the position of a molecule does not enter the Hamiltonian anywhere except through the phase of the light-matter coupling eik·rm, the standard bulk- boundary correspondence is no longer applicable and we do not observe edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We do not include plots for these cal- culations in this work and leave it an open question whether there is an analogous statement for bulk-boundary correspon- dence in these types of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' On the other hand, for exciton-polariton systems where nearest-neighbor couplings are present, edge states have been predicted and observed [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Other systems To emphasize that our scheme of saturating electronic tran- sitions with circularly polarized light to modify topological properties is not limited to organic exciton-polariton systems, we compute the Berry curvature of two other polariton sys- tems where porphyrin is replaced with (i) Ce:YAG and (ii) MoS2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5a and 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Other materials can also be used in place of porphyrins, as long as they have transitions that can be selectively excited with circularly polarized light and these transitions have large enough transition dipole moments that they can couple strongly to the photon modes of a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In Yttrium Aluminum garnet (YAG) doped with Cerium, Ce3+ ions replace some Y3+ and Ce3+ has transitions that can be selectively excited with circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, each Ce3+ has two possible ground states, one with the elec- tron in spin up |4 f(1) ↑⟩, and the other with it in spin down |4 f(1) ↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Similarly, it has a degenerate pair of excited spin states |5d(1) ↑⟩ and |5d(1) ↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The |4 f(1) ↓⟩ ↔ |5d(1) ↑⟩ transition has ∼ 400 times larger oscillator strength for ex- citation with σ+ polarized light than with σ− polarized light, therefore, we take the transition dipole moment to be µµµ+ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5b) [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Similarly, we take the transition dipole to be µµµ− for the |4f(1) ↑⟩ ↔ |5d(1) ↓⟩ transition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The transitions in Ce:YAG do couple to photon modes, however, to the best of our knowledge, strong coupling has not been reported in the literature [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nevertheless, strong light-matter cou- pling has been achieved with a similar system: Nd3+ doped YSO and YVO crystals [54, 55], and based on our calcula- tions, with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1µm thick sample of Ce:YAG at concentration 1% Ce3+ (relative to Y3+), we should be able to attain strong 6 Ce:YAG a b d f e c |4f(1)↓⟩ |5d(1)↑⟩ |5d(1)↓⟩ 𝝁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝝁" |4f(1)↑⟩ 𝑓↓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='4 𝑓↑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='6 𝑓# = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝝁" 𝝁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' K K’ Ω1 (𝜇m2) Ω1 (𝜇m2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (a) Illustration of Ce:YAG (salmon block) and perylene (green blocks) within a Fabry-Perot cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (b) Atomic levels of Ce3+ ions embedded in Yttrium Aluminum garnet (YAG) where the yellow circles indicate the fraction f↓ of Ce3+ ions in the |4f(1) ↓⟩ state and the fraction f↑ in the |4f(1) ↑⟩ state after optical pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The transition dipoles µµµ± = µ0(ˆx ± iˆy)/ √ 2 are also indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (c) Berry curvature of the lowest energy band, Ω1(k), under pumping with circularly polarized which creates populations f↓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='4 and f↑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (d) Illustration of monolayer MoS2 and perylene (green blocks) within a Fabry-Perot cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (e) Illustration of A-excitons in the K and K’ valleys of monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (f) Berry curvature of the lowest energy band, Ω1(k), under pumping with circularly polarized which creates exciton populations fK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 and fK′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We used parameters β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1eV, β = 9×10−4eVµm2, ζ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5×10−3eVµm, m∗ = 125¯h2eV−1µm−2, (c) E0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='50eV, ¯hωe = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='53eV and (f) E0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='80eV, ¯hωe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='855eV (see Supplementary section S4 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' coupling with photon modes in a Fabry-Perot cavity (see Sup- plementary section S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Under thermal equilibrium, the populations of the |4f(1) ↑⟩ and |4 f(1) ↓⟩ states are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, under pumping with pulses of σ+ polarization, in the presence of a small magnetic field ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='049T, the population of |4 f(1) ↑⟩ will exceed that of |4 f(1) ↓⟩ because population is selectively removed from |4f(1) ↓⟩ and added to |5d(1) ↑⟩ by the circularly polarized pulses, but decay from the excited |5d(1) ↑⟩ state to the two ground states has equal probability [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In principle, a mag- netic field is not required;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' however, as we do not know the spin relaxation time in the absence of the magnetic field, we report the magnetic field used in the experimental study [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Under optical pumping with circularly polarized light, the 5d states will have very small populations which we take to be zero, while the |4f(1) ↓⟩ and |4 f(1) ↑⟩ states will have unequal populations f↓ and f↑, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' here, f↓ + f↑ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Op- tically pumped Ce:YAG can then be modeled using the effec- tive Hamiltonian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 6 and 7, with |±mol⟩′ → |5d(1) ↑ / ↓⟩ and √1− f∓ −2 f± → � f↓/↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The large spin relaxation time of ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5 ms makes this system particularly well-suited for our scheme because it maintains f↓ ̸= f↑, and hence non-zero Chern invariants, for an extended period of time [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5c we plot Berry curvature of the lowest band of a perylene filled cavity strongly coupled with Ce:YAG, where f↓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='4 and f↑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='6 (see Supplementary section S4 for values of other parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' TMDs, such as single-layer MoS2, display optically con- trollable valley polarization and could also be used in place of porphyrins [57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Due to lack of inversion symme- try in these systems, the K and K’ valleys are inequivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' this results in optical selection rules that allow selective cre- ation of excitons at K and K’ valleys with σ+ and σ− polar- ized light, respectively [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Additionally, strong light- matter coupling has been observed when monolayer MoS2 is placed within a Fabry-Perot cavity [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This system has depolarization times of ∼ 200fs - 5ps making it possible to measure Berry curvature using pump-probe spectroscopy be- fore depolarization occurs [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We model this exciton- polariton system (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5d) using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 6 and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 7 (we focus on the A-exciton, see Supplementary section S4 for parame- ters) with |+mol⟩ → |K⟩, |−mol⟩ → |K′⟩ and √1− f∓ −2 f± → �1−2 fK/K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5f we plot the Berry curvature of the lowest band when fK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 and fK′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Unfortunately, sig- nificant Rabi contraction upon optical pumping has not been experimentally observed in these systems which will make it challenging to observe Berry curvature as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5f since our model relies on saturation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, for exciton polaritons formed from monolayer TMDs, even if Rabi con- traction through resonant optical pumping may not produce the intended effect, off-resonant optical pumping can break the degeneracy of excitons in the K and K’ valleys through optical stark effect [64], and this may have interesting conse- quences for the Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Additionally, if bilayer MoS2 is used in place of monolayer MoS2, effects on the Berry cur- vature described in our work may be more pronounced as bi- layer MoS2 hosts interlayer excitons which possess large op- tical nonlinearities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' specifically, they display saturation and 7 Rabi contraction under strong coupling [65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Finally, so far we have only considered replacing porphyrin with a different material, such as MoS2 or Ce:YAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In addi- tion to this, perylene can also be replaced with other suitable materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In our work, we choose to use a cavity filled with perylene because we do not want degeneracy at any k within the photon bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Other systems also satisfy this requirement and could be used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For instance, we could use an elec- trically tunable, highly anisotropic, liquid-crystal cavity with well separated H and V polarized photon modes [24, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' A perovskite cavity is another potential candidate due to its high anisotropy, and optical pumping may help lift the degeneracy of polariton modes in this system [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Additionally, other photonic structures can also be used instead of a cavity, as long as the photon bands are not degenerate at any k and have non-zero light-matter coupling at all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' CONCLUSION In summary, we show that TRS can be broken in organic exciton-polariton systems through selectively saturating elec- tronic transitions with a circularly polarized pump and that the resulting bands possess non-zero Chern invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In particu- lar, we demonstrate this theoretically for a Fabry-Perot cavity filled with porphyrin and perylene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Berry curvature of the more photonic parts of the bands of this system can be measured experimentally using pump-probe spectroscopy, as long as the time delay is shorter than the depolarization time for porphyrin (210fs-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='6ps) [50], and this will reveal non-zero Berry curvature and Chern number under circularly polarized pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Our scheme relies on Rabi contraction from satu- ration of optical transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' It is important to note that edge states do not emerge in our system despite non-zero Chern in- variants as our model does not contain sufficient positional information about the molecules or the unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Bleu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [30] have previously proposed breaking TRS in inorganic exciton-polariton systems through pumping with circularly polarized light, however, their work relies on polariton con- densation and having patterned lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Finally, we demon- strate that saturating electronic transitions to modify topol- ogy is not limited to organic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To illustrate this, we calculate the Berry curvature and Chern numbers of exciton- polariton bands of two other systems under optical pumping: (a) Ce:YAG and (b) monolayer MoS2, and find similar results as the organic exciton-polariton case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In view of recent devel- opments on electrically tuning the Berry curvature of liquid- crystal and perovskite filled cavities [24, 26], our work pro- vides an additional control knob to optically tune the Berry curvature of exciton-polariton systems using circularly polar- ized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Additionally, ultrafast control of topological prop- erties of systems with light may find use in nonreciprocal and nonlinear optoelectronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ACKNOWLEDGEMENTS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' acknowledges support from NSF Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' CA- REER CHE 1654732 for the development of the model and calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The conceptualization of the molecular and solid-state systems was guided by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' as part of the Center for Molecular Quantum Transduction (CMQT), an Energy Frontier Research Center funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Depart- ment of Energy, Office of Science, Basic Energy Sciences un- der Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' DE-SC0021314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' thanks Kai Schwen- nicke and Stephan van den Wildenberg for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' CODE AVAILABILITY Code available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='com/SindhanaPS/Topological_Polaritons_Submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' REFERENCES [1] Claude Weisbuch, Mr Nishioka, A Ishikawa, and Y Arakawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity.' 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Diego, La Jolla, California 92093, USA 2Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' LIGHT-MATTER COUPLING The light-matter coupling part of the total Hamiltonian under the electric dipole approximation is, ˆHcav−mol =∑ m ∑ k,α − ˆµµµm · ˆEk,α(rm,0), =∑ m ∑ k,α − � ∑ α′=± (µµµα′ ˆσ† m,α′ + µµµ∗ α′ ˆσm,α′) � ˆEk,α(rm,0), (S1) where µµµα′ = µµµm,α′ = � m,α′ mol �� ˆµµµ |m,G⟩ is independent of m since we assume that all porphyrin molecules lie flat in the x-y plane and are oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The electric field operator of the mode labeled by k and α is ˆEk,α(r,z) = � ¯hωk,α 2Vεε0 � f∗ k,α(r,z) ˆa† k,α +fk,α(r,z) ˆak,α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S2) Here, V = LxLyLz is the volume of the box we consider, where as mentioned in the main manuscript, we apply periodic boundary conditions along the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' From here on, we will call the in-plane area of the box A = LxLy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, fk,α(r,z) is the mode profile and it satisfies[1] � dr � Lz 0 dzf∗ k,α(r,z)fk,α(r,z) = LzA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S3) For the TE and TM modes[2], fk,TE(r,z) =eik·r√ 2sin � nzπ Lz � z+ Lz 2 �� ˆφφφ, fk,TM(r,z) =eik·r � 2 |k|2 + � nzπ Lz �2 ��nzπ Lz � sin � nzπ Lz � z+ Lz 2 �� ˆρρρ −i|k|cos � nzπ Lz � z+ Lz 2 �� ˆz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S4) We make the rotating-wave approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav−mol =∑ m ∑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α − � ∑ α′=± (µµµα′ ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ + µµµ∗ α′ ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′) � �� ¯hωk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α 2Vεε0 � f∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α +fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ≈ ∑ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ∑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α − � ¯hωk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α 2Vεε0 � µµµα′ ·fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α + µµµ∗ α′ ·f∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = ∑ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ∑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α � eik·rm �NxNy (µµµα′ ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α + e−ik·rm �NxNy (µµµ∗ α′ ·J∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α) ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S5) where Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α = −�NxNy � ¯hωk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α 2Vεε0 e−ik·rfk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) and µµµα′ ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α is the collective light-matter coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The annihilation operators of photon modes polarized along the horizontal (H) or x-axis and vertical (V) or y-axis are ˆak,H and ˆak,V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' They are related to α = ± polarized modes through ˆak,± = 1 √ 2( ˆak,H ∓i ˆak,V)[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In addition, we assume ∗ joelyuen@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='03287v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='chem-ph] 9 Jan 2023 2 that they are related to the TM and TE modes through ˆak,TM = cosφ ˆak,H +sinφ ˆak,V and ˆak,TE = −sinφ ˆak,H +cosφ ˆak,V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Using this, we obtain the relationship between ˆak,TE, ˆak,TM and ˆak,+, ˆak,− modes to be, ˆak,TM = 1 √ 2 � eiφ ˆak,+ +e−iφ ˆak,− � , ˆak,TE = 1 √ 2 � ieiφ ˆak,+ −ie−iφ ˆak,− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S6) It is important to note that, based on these relationships and S4, the α =H/V modes are not completely linearly polarized and the α = ± modes are not completely circularly polarized when |k| becomes comparable with nzπ/Lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We also find, Jk,+ = eiφ √ 2 � Jk,TM +iJk,TE � , Jk,− =e−iφ √ 2 � Jk,TM −iJk,TE � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S7) To keep the collective coupling strength µµµα′ ·Jk,α constant while taking the a → 0 limit, we take the magnitude of the collective transition dipole of the bright state �NxNyµ0 over square root of the quantization area of the photon mode √ A to be a constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' that is, we keep √ρAµ0 = µ0/a a constant, where ρA = NxNy/A is the areal density of quantum emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Jk,α =−√ρA � ¯hωk,α 2Lzεε0 e−ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='rfk,α(r,0) =− 1 a � ¯hωk,α 2Lzεε0 e−ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='rfk,α(r,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S8) S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' CHERN NUMBER CALCULATION a b (kmax,kmax) (kmax,-kmax) (-kmax,-kmax) (-kmax,kmax) x y FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (a) This is a cartoon figure that demonstrates the way Berry flux and Chern number are computed in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The small squares are the plaquettes over which Berry flux is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The blue arrows specify the orientation used for Berry flux computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Note that the direction is opposite for the small squares and the large square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (b) Same as (a), but placed on a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, it is more clear that the direction of the arrow for the large square indicates the way Berry flux is computed for the giant plaquette covering the rest of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For the Chern invariant to be an integer, it is important that the Berry curvature is integrated over a closed and bounded surface [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For periodic systems with a finite period, the Brillouin zone is a torus which satisfies this requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, 3 for a continuous system, (kx,ky) lies on an unbounded plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' for such systems, Silveirinha[5] proposed mapping this infinitely large plane onto a sphere to compute the Chern number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This is the procedure we follow in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We discretize k-space and compute the Berry flux in each plaquette within a square-shaped region in k-space, −kmax ≤ kx,ky ≤ kmax [4, 6] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S1a and S1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The entire region that satisfies the condition kx,ky > kmax or kx,ky < −kmax is taken as a single giant plaquette (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S1b), and the Berry flux within this region is computed by taking the Berry phase along the boundary of the plaquette but in a direction opposite to that used to compute Berry flux for plaquettes within the square −kmax ≤ kx,ky ≤ kmax as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S1a and S1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To ensure that we obtain a converged Chern number, we calculate the Chern number for different kmax and find that, for our system, once kmax ≳ 100µm−1, the Chern number converges to C1 = ±1,C2 = ∓1,C3 = 0, and C4 = 0 when f+ ̸= f− with |f+ − f−| ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Smaller differences between f+ and f−, |f+ − f−| ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='11 require larger kmax for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This is not a problem for the f+ = f− case because the Chern invariant will always be zero due to time-reversal symmetry Ωl(k) = −Ωl(−k), and we can use kmax ≈ 100µm−1 to compute it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' OPTICAL PUMPING The number of excitations in the system Nex = ∑k,α a† k,αak,α + ∑n,α σ† n,ασn,α is a conserved quantity of this Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Therefore, when we have f+ fraction of molecules in the |+mol⟩ state and f− in the |−mol⟩ state, we will only have to look at the ( f+ + f−)Nth excitation manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Unfortunately, the dimensions of the Hilbert space of this manifold scale as � N (f++f−)N � , and this quickly becomes computationally intractable as the system size, N, increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Using mean-field theory, we reduce this many-body problem to a one-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' That is, we derive an effective Hamiltonian for a single excitation in the mean-field of the remaining ( f+ + f−)N excitations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' in this way, we reduce the dimensions of the Hilbert space to that of the first excitation manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To do this, we follow a procedure similar to that used by Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [7] and write the Heisenberg equations of motion (EOM) for the operators ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± and ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' i¯hd ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± dt = � ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHmol � + � ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav � + � ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav−mol � =¯hωe ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± + 1 �NxNy ∑ k eik·rn � (1− ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='∓ ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='∓ −2 ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±) � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ± ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ± ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � − ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='∓ ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ∓ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ∓ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' i¯hd ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± dt = � ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHmol � + � ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav � + � ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav−mol � = � E0 + ¯h2|k|2 2m∗ ±ζ|k|cosφ � ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± + � −β0 +β|k|2e∓i2φ� ˆa∓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='k + 1 �NxNy ∑ m eik·rm � J∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± · µµµ∗ + ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +J∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± · µµµ∗ − ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S9) We make a mean-field approximation to linearize these EOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For instance, we use mn ≈ ¯mn, that is, ˆσ† n,+ ˆσn,+ ˆak,+ = � ⟨ ˆσ† n,+ ˆσn,+⟩+ ˆσ† n,+ ˆσn,+ −⟨ ˆσ† n,+ ˆσn,+⟩ � ˆak,+ =⟨ ˆσ† n,+ ˆσn,+⟩ ˆak,+ +( ˆσ† n,+ ˆσn,+ −⟨ ˆσ† n,+ ˆσn,+⟩)⟨ ˆak,+⟩ ≈⟨ ˆσ† n,+ ˆσn,+⟩ ˆak,+, (S10) where ⟨ ˆO⟩ = Tr � ˆρ0 ˆO � with ˆρ0 ≈ ∏m ˆρm ∏k ∏α=+,− ˆρα,k[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, we assume that after dephasing of the molecular amplitudes, ˆρm = fG |m,G⟩⟨m,G|+ f+ |m,+mol⟩⟨m,+mol|+ f− |m,−mol⟩⟨m,−mol|, ˆρα,k = |k,αcav,0⟩⟨k,αcav,0|, and, therefore, ⟨ ˆak,+⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The EOM then become i¯hd ˆσn,± dt ≈¯hωe ˆσn,± + 1 �NxNy (1− f∓ −2 f±)∑ k eik·rn � Jk,+ · µµµ± ˆak,+ +Jk,− · µµµ± ˆak,− � , i¯hd ˆak,± dt = � E0 + ¯h2|k|2 2m∗ ±ζ|k|cosφ � ˆak,± + � −β0 +β|k|2e∓i2φ� ˆa∓,k + 1 �NxNy ∑ m eik·rm � J∗ k,± · µµµ∗ + ˆσm,+ +J∗ k,± · µµµ∗ − ˆσm,− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S11) 4 We define rescaled operators ˆσ′ n,± = ˆσn,±/√1− f∓ −2 f± and rewrite the EOM, i¯hd ˆσ′ n,± dt ≈¯hωe ˆσ′ n,± + 1 �NxNy � 1− f∓ −2f±∑ k eik·rn � Jk,+ · µµµ± ˆak,+ +Jk,− · µµµ± ˆak,− � , i¯hd ˆak,± dt = � E0 + ¯h2|k|2 2m∗ ±ζ|k|cosφ � ˆak,± + � −β0 +β|k|2e∓i2φ� ˆa∓,k + �NxNy ∑ m eik·rm �� 1− f− −2f+J∗ k,± · µµµ∗ + ˆσ′ m,+ + � 1− f+ −2 f−J∗ k,± · µµµ∗ − ˆσ′ m,− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S12) From these EOM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' along with the fact that ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± act effectively as bosonic operators in mean-field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' � ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆσ′† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ � = 1− ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='−−2 ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ 1−f−−2 f+ ≈ ˆI and � ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆσ′† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � = − ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ 1−f−−2 f+ ≈ ˆ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' where ˆI and ˆ0 are the identity and zero operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' we can construct an effective Hamiltonian ˆHeff = ˆHeff mol + ˆHeff cav + ˆHeff cav−mol in ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± and ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff mol =∑ n � ¯hωe ˆσ′† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + ¯hωe ˆσ′† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav =∑ k � E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2e−i2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2ei2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav−mol = 1 �NxNy ∑ m ∑ k eik·rm � � 1− f− −2 f+ � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ+ ˆσ′† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ+ ˆσ′† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � + � 1− f+ −2f− � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ− ˆσ′† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ− ˆσ′† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− �� +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', (S13) which is the mean-field Hamiltonian when the system has f+, f− excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Writing this effective Hamiltonian in k-space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff mol =∑ k � ¯hωe ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆσ′ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + ¯hωe ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆσ′ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav =∑ k � E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2e−i2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2ei2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav−mol =∑ k � � 1− f− −2f+ � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ+ ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ+ ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � + � 1− f+ −2f− � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ− ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ− ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− �� +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S14) We define states |k,±mol⟩′ and |k,±cav⟩′ corresponding to operators ˆσ′† k,± and ˆa† k,±, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Writing the Hamiltonian ˆHeff(k) = ⟨k| ˆHeff |k⟩ in the above basis we obtain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff(k) = ˆHeff mol(k)+ ˆHeff cav(k)+ ˆHeff cav−mol(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S15) 5 where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff mol(k) =¯hωe |+mol⟩′ ⟨+mol|′ + ¯hωe |−mol⟩′ ⟨−mol|′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav(k) = � E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � |+cav⟩′ ⟨+cav|′ + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � |−cav⟩′ ⟨−cav|′ + � −β0 +β|k|2e−i2φ� |+cav⟩′ ⟨−cav|′ + � −β0 +β|k|2ei2φ� |−cav⟩′ ⟨+cav|′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav−mol(k) =Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · �� 1− f− −2 f+µµµ+ |+mol⟩′ + � 1− f+ −2 f−µµµ− |−mol⟩′ � ⟨+cav|′ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · �� 1− f− −2 f+µµµ+ |+mol⟩′ + � 1− f+ −2f−µµµ− |−mol⟩′ � ⟨−cav|′ +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S16) S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' PARAMETERS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Perylene filled cavity We take parameters for the perylene filled cavity β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1eV, β = 9×10−4eVµm2, ζ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5×10−3eVµm, m∗ = 125¯h2eV−1µm−2, and Lz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='745µm, where these are similar to those used to model the experiments of Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [9] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3, 4, and 5 in main manuscript).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' On the other hand, we modify E0 and nz such that they make the photon modes in our model near resonant with the transition that is strongly coupled to the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For instance, we take E0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='80eV and nz = 11 for porphyrin (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3 and 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' E0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='50eV and nz = 9 for Ce:YAG (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5b-c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' and E0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='80eV and nz = 5 for MoS2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5e-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We assume that perylene has a similar effect on these different photon modes, as it does on modes with E0 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='27eV at k = 0 in experiments[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This may not necessarily be true, however, as we consider a perylene filled cavity only to achieve frequency separation of photon modes with different polarization, and this can instead be easily achieved with an electrically tunable liquid crystal cavity [10], replacing a perylene filled cavity with a liquid-crystal cavity will not modify the underlying physics of the phenomenon we are interested in, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', the idea of using saturation to break TRS will remain intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Porphyrin, Ce:YAG, and monolayer MoS2 We take areal density ρA = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='55 × 105µm−2 (∼ 2000 molecules in 75nm ××× 75nm)[11], relative permittivity ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5[12], frequency ¯hωe = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='8056eV and transition dipole µ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1184au × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5417D/au = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='84D [13] for the porphyrin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Also, we consider 100 such porphyrin films stacked one over the other along the z direction within the cavity to achieve strong light-matter coupling, Nz = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Therefore, the effective areal density of molecules ρ′ A = NzρA will be used instead of ρA while computing Jk,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' These are the parameters used to generate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Similarly, using density ρYAG = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='11g cm−3, molar mass MYAG = 738 g mol−1, number of Y3+ per unit cell nY3+ = 3, and concentration of Ce3+ (relative to Y3+) 1% = 10−2 [14], we obtain the effective areal density of Ce3+ ions in a L′ z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1µm thick layer of Ce:YAG to be ρ′ A = 10−2L′ znY3+ρYAGNA/MYAG = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='25 × 107µm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This will be used while computing Jk,α in place of ρA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We use relative permittivity ε = 12[15] and frequency ¯hωe = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='53eV (489nm[16]) for the transition in a Ce:YAG crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Using the oscillator strength of this transition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='286[16], we calculate the transition dipole µ0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='46D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' These are the parameters used to generate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For monolayer MoS2, we consider A-excitons at ¯hωe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='855eV[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' From Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [17], we take the Rabi splitting at resonance, and use µ0√ρA � ¯hωe/2Lzεε0 ≈ 39meV/2 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5meV in our calculations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' SUPPLEMENTARY REFERENCES [1] Fabre, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & Treps, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Modes and states in quantum optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Reviews of Modern Physics 92, 035005 (2020).' 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