Cavity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics
David Hagenmüller, Stefan Schütz, Johannes Schachenmayer, Claudiu Genes, Guido Pupillo
CCavity-assisted mesoscopic transport of fermions:Coherent and dissipative dynamics.
David Hagenm¨uller
IPCMS (UMR 7504) and ISIS (UMR 7006), University of Strasbourg and CNRS,67000 Strasbourg, France
Stefan Sch¨utz
IPCMS (UMR 7504) and ISIS (UMR 7006), University of Strasbourg and CNRS,67000 Strasbourg, France
Johannes Schachenmayer
IPCMS (UMR 7504) and ISIS (UMR 7006), University of Strasbourg and CNRS,67000 Strasbourg, France
Claudiu Genes
Max Planck Institute for the Science of Light, Staudtstraße 2, D-91058 Erlangen,Germany
Guido Pupillo
IPCMS (UMR 7504) and ISIS (UMR 7006), University of Strasbourg and CNRS,67000 Strasbourg, France
Abstract.
We study the interplay between charge transport and light-matterinteractions in a confined geometry, by considering an open, mesoscopic chain of two-orbital systems resonantly coupled to a single bosonic mode close to its vacuum state.We introduce and benchmark different methods based on self-consistent solutions ofnon-equilibrium Green’s functions and numerical simulations of the quantum masterequation, and derive both analytical and numerical results. It is shown that in thedissipative regime where the cavity photon decay rate is the largest parameter, thelight-matter coupling is responsible for a steady-state current enhancement scalingwith the cooperativity parameter. We further identify different regimes of interestdepending on the ratio between the cavity decay rate and the electronic bandwidth.Considering the situation where the lower band has a vanishing bandwidth, we showthat for a high-finesse cavity, the properties of the resonant Bloch state in the upperband are transfered to the lower one, giving rise to a delocalized state along the chain.Conversely, in the dissipative regime with low cavity quality factors, we find that thecurrent enhancement is due to a collective decay of populations from the upper to thelower band. a r X i v : . [ qu a n t - ph ] J a n avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. Contents1 Introduction 3 g = 0) . . . . . . . . . . . . . . . . . . 203.2 Spectral broadening and polaritons . . . . . . . . . . . . . . . . . . . . . 223.3 Comparison between the different methods . . . . . . . . . . . . . . . . . 243.4 Dissipative regime κ/W (cid:29) κ/W (cid:28) N . . . . . . . . . . . . . . . . . . . . . . . . . 38 avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics.
1. Introduction
Investigating how the transport of excitations can be modified by the coupling to lightis a topic of considerable fundamental and practical interest [1–4]. Recent studies havepredicted drastic modifications of the transport of electron-hole pairs called excitonswhen interacting with photons in confined geometries such as cavities [5] or plasmonicresonators [6]. The modification of exciton transport in a cavity can be understoodusing the Tavis-Cummings model [7] (TC), which describes the collective behavior of N dipoles (two-level systems) resonantly coupled to a single bosonic mode. As localizedexcitons hop toward their nearest neighboring sites, the exciton propagation from oneside of the cavity to the other can be bypassed by exchanging energy with polaritonmodes delocalized over the entire cavity mode volume. This energy transfer can beinterpreted as a long-range dipole-dipole-type interaction mediated by the cavity [8].Studies of charge transport modifications induced by the coupling to bosonicfields in condensed matter systems have traditionally focused on electron-phononinteractions [9]. In polar semiconductors, the latter provide a screening of the electronmotion by the lattice polarization [10–12] (polaron), which is responsible for increasingthe electron effective mass and reducing the mobility [9]. Electron-phonon coupling inmetals is known to lead to different instabilities at sufficiently low temperature, such asBCS electron pairing leading to superconductivity [13–15] and Peierls-type instabilitiesresponsible for a metal-insulator phase transition in one-dimensional systems [16, 17].The crucial difference between electron-photon and electron-phonon coupling stemsfrom the possibility of low-energy electron scattering with both vanishing and largemomenta (of the order of the Fermi wavevector k F ) in the latter case. In particular,the aforementioned instabilities occur due to large-momentum ( ∼ k F ) scattering acrossthe Fermi surface, within a narrow energy band of the order of the Debye frequency.Conversely, light-matter coupling typically involves quasi-vertical electronic excitationsacross a band-gap, resulting in the absence of both low-energy and large-momentumexcitations. In the macroscopic limit, this usually provides a decoupling between low-energy charge transport and light-matter coupling occuring at finite frequencies [18].An emerging topic of interest is the modification of material properties using anexternal electromagnetic radiation [19], and in particular the possibility of light-inducedsuperconductivity in the ultraviolet [20] and terahertz portions of the spectrum [21–27],as well as the emergence of zero-resistance states in quantum Hall systems subjected tomicrowave radiation [28–31]. On the other hand, the study of light-matter interactionsin confined geometries is attracting increasing attention in various fields, such asin quantum optics [32–40], quantum chemistry [41–45], and condensed matter [46–53], opening the way to investigate the rich interplay between many-body physicsand strong light-matter interactions [54, 55]. In the case of charge transport, largeconductivity enhancements ( ∼ one order of magnitude) have been recently reported avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. N sites with twoorbitals, providing complementary and original methods to investigate this system. Wecharacterize different regimes of transmission involving either a dissipative or a coherentdynamics, and identify the presence of collective effects and electronic correlationsdepending on the strength of the light-matter coupling. Our model might find directapplications in several fields, such as transport in organic semiconductors [56], quantumdot arrays [58–64], and nanowires [65–67], as well as for quantum simulations usingultracold atoms [68, 69] or superconducting qubits [70–72] in the microwave domain. We consider a 1D chain of N sites with two electronic orbitals of energy ω α ( (cid:126) = 1),where α = 1 , a) ]. Eachorbital α on site j is coupled to its nearest neighbors j ± t α , resultingin two bands in a tight-binding picture. In the following, we will always consider thesituation where the upper band is much broader than the lower one ( t (cid:29) t ). Dependingon N , the upper electronic bandwidth varies between 2 t ( N = 2) and 4 t ( N → ∞ ),and will be denoted as W (respectively W for the lower band). Electrons are consideredas spin-less. The edges of the chain are connected to a source and a drain (leads) witha large bias voltage across, such that the Fermi level of the source (the drain) is higher(lower) than any other energy scale in the system. This allows for injection/extractionin both orbitals at a rate Γ α . Although different injection/extraction rates are kept forthe sake of generality, we will only discuss the results obtained for Γ = Γ ≡ Γ. Allenergies are in units of ω (set to 1), which is assumed to be the largest parameter.The on-site transition between lower and upper orbitals with energy ω = ω − ω isresonantly coupled (with a coupling strength g ) to a single cavity mode with decay rate κ . Letting the contributions from the leads and the extra-cavity photonic environmentaside for now, the 1D chain Hamiltonian can be written as H S = H e + H c + H t + H I ,where: H e = (cid:88) α N (cid:88) j =1 ω α c † α,j c α,j H c = ω c a † a, avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. c α,j and c † α,j respectively annihilate and create an electron in the orbital α onsite j , and satisfy the anti-commutation relations { c α,i , c † α (cid:48) ,j } = δ α,α (cid:48) δ i,j . On the otherhand, a and a † denote the bosonic annihilation and creation operators of a photon inthe cavity mode with energy ω c , and satisfy the commutation relation [ a, a † ] = 1. Thenearest-neighbor hopping in both orbitals is described by the contribution: H t = − (cid:88) α t α (cid:32) N − (cid:88) j =1 c † α,j +1 c α,j + N (cid:88) j =2 c † α,j − c α,j (cid:33) , (1)and the light-matter coupling by the term: H I = g N (cid:88) j =1 (cid:16) c † ,j c ,j + c † ,j c ,j (cid:17) A, (2)with A = a + a † . In the absence of Eq. (1), and if one restricts the orbital occupationto one per site, H S corresponds to the TC Hamiltonian [7] with counter-rotating terms,where the bosonic field A is coupled to the collective pseudo-spin operator: S x = 12 √ N N (cid:88) j =1 (cid:16) c † ,j c ,j + c † ,j c ,j (cid:17) . In the case of the TC model, the size of the electronic part of the Hilbert spaceis 2 N [see Fig. 1 b) ], and one can use boson mapping techniques [73] to find thespectrum of H S . The cavity field thus interacts with a collective mode formed of acoherent superposition of N single-spin excitations, with an enhanced coupling strengthΩ = g √ N called vacuum Rabi frequency [74]. In particular, the strong coupling regimeof cavity QED [74] is achieved when Ω > κ , allowing a quasi-reversible energy transferbetween the collective dipole and the cavity field, and providing vacuum Rabi oscillationsat a frequency Ω. Instead of the bare cavity resonance, the cavity spectrum featurestwo polariton resonances separated by a splitting 2Ω.In the presence of H t , however, charge transport can occur due to the couplingbetween the two quantum states associated with each local pseudo-spin and the twonew states with both orbitals either occupied or empty [see Fig. 1 b) ]. Our modelthus exhibits a larger Hilbert space (4 N ) compared to the TC model, and features amore complex physics. Introducing the electron density operator in the orbital α asˆ n αj = c † α,j c α,j , one realizes that the total density at a given site j is conserved bythe light-matter coupling Hamiltonian, namely [ H I , (cid:80) α ˆ n αj ] = 0. This means that incontrast to exciton transport, the cavity-induced modification of charge transport canonly occur through the interplay between H I and H t . avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. b) S O U R C E D R A I N a)c) DissipativeCoherent NGFs Effective QME I nd i v i dua l C o ll e c t i v e Figure 1. a)
1D chain of N two-orbital systems, each one consisting of a lower orbital( α = 1, red) and an upper one ( α = 2, blue). The first and last sites j = 1 and j = N are coupled to two leads with equal injection/extraction rate Γ. The transition withenergy ω between lower and upper orbitals is resonantly coupled (with strength g )to a single cavity mode, with decay rate κ . t α is the hopping rate between neighboringsites in the band α (we always consider the case t (cid:29) t ). b) TC model: The Hilbertspace associated with a given site is spanned by the two quantum states represented onthe left side, providing a 2 N -states basis for the whole chain. Right-side: The hoppingHamiltonian H t provides a coupling of these states to two new states with both orbitalseither occupied or empty. The chain is thus spanned by a 4 N -states basis. c) Sketchshowing the different regimes investigated, together with the applicability domains ofthe different methods used in this article. W and δω denote the upper electronicbandwidth and the typical separation between two adjacent Bloch states in the upperband, respectively. NGFs stands for Non-equilibrium Green’s functions, and QMEfor Quantum Master Equation. The full QME is valid everywhere on the diagram.The dashed line corresponds to g / ( κ Γ) = 1 (the left-hand side is the cooperativity),separating the perturbative regime (above the line) from the non-perturbative regime(below the line). The horizontal line κ = W represents the separation between thedissipative regime κ (cid:29) W and the coherent regime κ (cid:28) W . While for g < δω , thecoupling to light always involves a single Bloch state (“individual dressing regime”),a collective coupling of many Bloch state arises when g > δω (“collective dressingregime”). Note that since δω → N → ∞ , the coupling istherefore always collective in this case. avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. The main results of the paper are summarized in the following: • In Sec. 2.1, we explain how to compute the relevant physical observables (current,populations, electron and photon density of states) using different theoreticalmethods that are presented in detail. In Sec. 2.2, we introduce a frequency-domainmethod based on the self-consistent solutions of Non-equilibrium Green’s Functions(NGFs), valid in the perturbative regime where the cooperativity g / ( κ Γ) < c) ]. The results obtained with this methodare benchmarked with a suitable Quantum Master Equation (QME) presented inSec. 2.3, exact in the rotating-wave approximation [75] and as long as the Markovianapproximation for the system-baths coupling holds true, but nevertheless limitedto a small number of sites. In Sec. 2.4, we show that in the dissipative regime κ/W (cid:29) ∝ g /κ . • We present our results in Sec. 3, by first discussing the physical properties of thesystem in the absence of light-matter coupling (Sec. 3.1), and explaining how theelectron density of states (DOS) is broadened by light-matter interactions in theperturbative regime (Sec. 3.2). We further explain how polariton modes arise fromthe dressing of the photon GF by the electron-hole polarization. In the asymmetricsituation where t (cid:29) t , we show that the light-matter coupling is responsiblefor opening a new transmission channel in the lower band, which leads to anenhancement of the steady-state current. In Sec. 3.3, we compare the currentenhancement predicted by the different numerical methods, identifying the regimesof interest based on the ratio between the upper electronic bandwidth W ∼ t andthe cavity photon decay rate κ . In the dissipative regime κ/W (cid:29)
1, we find thatthe current enhancement scales with the cooperativity. • We further investigate the dissipative regime in Sec. 3.4 [upper part on Fig. 1 c) ]. In particular, we derive an analytical formula for the current enhancementvalid for small coupling strengths (Sec. 3.4.1), and characterize the presence ofcollective effects by computing the different observables numerically in Sec. 3.4.2.We show that a collective coupling of many Bloch states to the cavity modeoccurs when g > δω , namely when the coupling strength is larger than the typicalenergy separation between two adjacent Bloch states in the upper band. In thisdissipative, collective “dressing” regime, the current enhancement stems from aglobal transfer of populations from the upper to the lower band, with only marginalpropagation through the lower band. For large coupling strengths (Sec. 3.4.3), weshow that the current enhancement saturates to about twice its value for g = 0, andthat the collective coupling is responsible for the existence of non-local electroniccorrelations. • The “coherent” regime obtained for κ/W (cid:28) g < δω ,only one given resonant Bloch state is individually coupled to the cavity mode [left avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. c) ], which is refered to as “individual dressing regime”. Afterhaving characterized the latter by computing the different observables in Sec. 3.5.1,we show that a transfer of spectral weight ∼
10% occurs from the upper to the lowerband, resulting in a new state with energy ∼ ω delocalized across the whole chain(Sec. 3.5.2). In this case, the current enhancement can be interpreted as stemmingfrom coherent dynamics sustained by the absorption and emission of cavity photons.Ultimately, for N (cid:29)
1, or when the coupling strength becomes larger than the upperelectronic bandwidth, we expect to recover a collective coupling of the Bloch statesto the cavity mode [right bottom part on Fig. 1 c) ]. • Concluding remarks concerning the cavity photons population and the scaling ofthe current with the chain length N are presented in Sec. 3.6, and perspectives aredrawn in Sec. 4.
2. Methods
This section is structured as follows. In Sec. 2.1, we introduce the steady-state currentflowing through the chain in the presence of light-matter coupling, showing that thisobservable can be calculated by using the QME and NGFs formalisms, depending onwhether the problem is formulated in real time or in the frequency domain, respectively.In Sec. 2.2, we focus on the NGFs method, introducing the total Hamiltonian includingthe contributions from the environment, and explain how to compute the current bysolving a set of self-consistent equations for electron and photon Green’s functions (GFs).In Sec. 2.3, we introduce a suitable QME to compute the steady-state current, exact butlimited to a small number of sites. In Sec. 2.4, we introduce an effective master equationvalid in the dissipative regime where the fast cavity field evolution can be adiabaticallyeliminated, resulting in an effective QME involving only electronic degrees of freedom.
In the frequency domain, the steady-state current can be put in a form reminiscent ofthe Landauer formula [76] for equilibrium mesoscopic systems [77]: J = J s − J d (cid:88) α e Γ α (cid:90) dω π T α ( ω ) , (3)where J s ( J d ) is the steady-state current flowing through the source (drain), ω thefrequency, and e the electron charge. In the high-bias regime, the transmission spectrum T α ( ω ) is expressed in terms of the electron GFs G rα and G <α defined in Sec. 2.2.2: T α ( ω ) = Tr (cid:2) − σ ◦ (cid:61) G rα ( ω ) + (cid:0) σ N − σ (cid:1) ◦ (cid:61) G <α ( ω ) (cid:3) , (4)where underlined quantities denote N × N matrices, ◦ is the element-wise Hadamardproduct, (cid:61) stands for imaginary part, and Tr denotes the sum over all matrix elements. avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. J = J s − J d (cid:88) α e Γ α (cid:104) − ˆ n α (cid:105) + (cid:104) ˆ n αN (cid:105) ) , (5)and can be directly computed by simulating the time-evolution of the joint densityoperator for the chain and the cavity field. In Sec. 2.3, we explain how this can be doneby using a suitable QME. We also show that the cavity mode can be adiabaticallyeliminated in the dissipative regime (lossy cavity), resulting in an effective QMEinvolving only electronic degrees of freedom (Sec. 2.4). In this section, we focus on the NGFs method. In Sec. 2.2.1, we introduce the totalHamiltonian including the contributions from the environment. In Sec. 2.2.2, we presentthe derivation of the steady-state current written in terms of electron GFs, and explainhow to compute this current by solving a set of self-consistent equations for electron andphoton Green’s functions in Sec. 2.2.3. This method is based on a generalization of themodel presented in the Chapter 12 of [77], with a treatment of light-matter couplingsimilar to the one presented in [78] for electron-phonon interactions.
In the framework of the NGFs formalism, the environmentis described by Hamiltonian terms. In total, one can write H = H S + H L + H P , wherethe chain Hamiltonian H S is given in Sec. 1.2.The two leads injecting and extracting electrons are described by the contribution: H L = (cid:88) α (cid:88) η = s,d (cid:88) q ω q b † α, q ,η b α, q ,η + (cid:88) α (cid:88) η = s,d (cid:88) j, q λ α,j, q ,η (cid:16) c α,j b † α, q ,η + b α, q ,η c † α,j (cid:17) , with coupling constants λ α,j, q ,s = (cid:40) λ α, q for j = 10 for j (cid:54) = 1 λ α,j, q ,d = (cid:40) λ α, q for j = N j (cid:54) = N. (6)The operators b † α, q ,η ( b α, q ,η ) create (annihilate) a fermion in the state ( α, q ) withenergy ω q in the lead η , and obey fermionic commutation relations. The photonic bathresponsible for cavity photon losses is described by the Hamiltonian: H P = (cid:88) p ω p a † p a p + (cid:88) p µ p A p A, (7) avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. a † p and a p denote the extra-cavity photon operators (obeying bosoniccommutation rules) with corresponding energy ω p , and A p = a p + a † p . The cavityphotons-bath coupling strength is denoted as µ p . The continous variables q and p are arbitrary quantities respectively associated with the electronic (leads) and photonicbaths.The chain operators can be expanded in the Bloch states basis as c α,j = (cid:80) Nk =1 ϕ jk ˜ c α,k , with ϕ jk = (cid:114) N + 1 sin (cid:18) πjkN + 1 (cid:19) , (8)such that the contribution H e + H t takes the diagonal form (cid:80) α,k ω α,k ˜ c † α,k ˜ c α,k with ω α,k = ω α − t α cos( πk/ ( N + 1)). The Hamiltonian H S can thus be partitioned intoa diagonal part H = H e + H t + H c with known eigenstates, and the light-matterinteraction Eq. (2) treated perturbatively. In the steady-state, the charge current J η flowing throughthe lead η is given by the continuity equation J η = − e∂ t (cid:104) N η (cid:105) = − ie (cid:104) [ H, N η ] (cid:105) , with J s = − J d . Here, (cid:104)· · · (cid:105) denotes the statistical average with respect to the densityoperator (cid:37) of the whole system (chain+environment), whose evolution is governed by thetotal Hamiltonian H . N η = (cid:80) α, q b † α, q ,η b α, q ,η is the number of electrons in the lead η . Asdetailed in Appendix A, the steady-state current can be put in the form given by Eqs. (3)and (4). The matrix elements of σ j entering Eq. (4) are given by σ jk,k (cid:48) = ϕ jk ϕ jk (cid:48) , andthe matrix elements of the so-called retarded and “lesser” electron GFs are respectivelydefined (in the frequency domain) as: G rα,k,k (cid:48) ( ω ) = − i (cid:90) + ∞ dτ e iωτ (cid:104){ ˜ c α,k ( τ ) , ˜ c † α,k (cid:48) (0) }(cid:105) G <α,k,k (cid:48) ( ω ) = i (cid:90) + ∞−∞ dτ e iωτ (cid:104) ˜ c † α,k (cid:48) (0)˜ c α,k ( τ ) (cid:105) , where {· · · } denotes the anticommutator. On the other hand, the time-ordered electronGF is given by the expression: G α,k,k (cid:48) ( τ − τ (cid:48) ) = − i (cid:104)T ˜ c α,k ( τ )˜ c † α,k (cid:48) ( τ (cid:48) ) (cid:105) = − i (cid:104)T ˜ c α,k ( τ )˜ c † α,k (cid:48) ( τ (cid:48) ) e − i (cid:82) dτ H ( τ ) (cid:105) (cid:104) e − i (cid:82) dτ H ( τ ) (cid:105) , (9)where T denotes the time-ordered product for fermions, and (cid:104)· · · (cid:105) refers to thestatistical average with respect to the density operator (cid:37) of the whole system(chain+environment), whose evolution is governed by the free Hamiltonian ( H without H I and the interaction terms entering H L and H P ). The first contribution of Eq. (4)involves the trace of the electron spectral function A ( α ) ( ω ) = − (cid:61) G rα ( ω ) in the band α .Physically, the quantity (cid:80) k,k (cid:48) A ( α ) k,k (cid:48) ( ω ) corresponds to the normalized electron DOS in avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. α . The spectral function normalization (cid:82) dωA ( α ) k,k (cid:48) ( ω ) = 2 πδ k,k (cid:48) implies that theeffect of light-matter interactions on the steady-state current is entirely determined bythe second term in Eq. (4), which is proportional to the trace of the “lesser” electronGF. The latter can be used to compute the steady-state electron population in realspace as [79]: n αj = (cid:104) ˆ n α,j (cid:105) = (cid:88) k,k (cid:48) ϕ jk ϕ jk (cid:48) (cid:90) dω π (cid:61) G <α,k,k (cid:48) ( ω ) . Using the spectral function normalization and inverting the previous equation, thesteady-state current Eq. (3) takes the form given in Eq. (5): J = (cid:88) α e Γ α − n α + n αN ) (cid:32) = (cid:88) α e Γ α n αN (cid:33) , (10)showing that the latter only depends on the electron populations at the edges of thechain. We now explain in detailthe procedure to compute the electron GFs entering the expression of the transmissionfunction Eq. (4). It can be shown (see Appendix A) that retarded and advanced electronGFs obey a Dyson equation of the form: G βα ( ω ) = (cid:0) ( G βα ( ω )) − − Σ βα ( ω ) (cid:1) − , (11)with β = r, a , while “lesser” and “greater” GFs are obtained from the Keldysh equation: G γα ( ω ) = G rα ( ω )Σ γ ( ω ) G aα ( ω ) , (12)with γ = <, > for lesser and greater. The matrix elements of the unperturbed GFs G α ( ω )(evaluated in the absence of light-matter coupling and interactions with the leads) areall proportional to δ k,k (cid:48) : G <α,k,k (cid:48) ( ω ) = − iπδ k,k (cid:48) δ ( ω − ω α,k ) n α,k G >α,k,k (cid:48) ( ω ) = 2 iπδ k,k (cid:48) δ ( ω − ω α,k ) (cid:0) − n α,k (cid:1) G aα,k,k (cid:48) ( ω ) = δ k,k (cid:48) ω − ω α,k − i + , (13)and G rα,k,k (cid:48) = ( G aα,k,k (cid:48) ) ∗ , where n α,k = (cid:104) ˜ c † α,k ˜ c α,k (cid:105) is the population of the Bloch states( α, k ) in the initial, non interacting ground state. avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. ≶ α ( ω ) = Σ ≶ I,α ( ω ) +Σ ≶ L,α , where Σ ( ω (cid:48) )Σ >I,α ( ω ) = ig (cid:88) α (cid:48) (1 − δ α,α (cid:48) ) (cid:90) dω (cid:48) π G >α (cid:48) ( ω + ω (cid:48) ) D < ( ω (cid:48) ) (14)represent the electron SE corrections [represented by the diagram in Fig. 2 a) ] due tothe light-matter coupling, stemming from the emission/absorption of cavity excitations[poles of D ( ω )] when electrons undergo optical transitions between the two bands. Thecontributions: Σ
Electron SE diagram ∼ g corresponding to the SCBA. b) Example ofvertex corrections diagram ∼ g where different photon lines cross eachother, that arenot taken into account in the SCBA. c) Bubble diagram for the photon SE ∼ g inthe SCBA. Electron GFs are represented as double straight lines while photon GFsare represented as double wiggly lines. d) Self-consistent algorithm used to computeelectron and photon GFs. We proceed by successive iterations starting from the non-interacting electron GFs (left box with Σ = 0) until convergence is reached. γ = <, > stands for “lesser” and “greater” GFs, respectively. H [ χ α ]( ω ) = 1 π p.v (cid:90) dω (cid:48) χ α ( ω (cid:48) ) ω − ω (cid:48) denotes the Hilbert transform, and p.v the Cauchy principal value. As a causal function,the real and imaginary parts of Σ rα ( ω ) are related to each other by Kramers-Kronigrelations, as it can be checked directly from Eq. (16). The advanced SE is given byΣ aα ( ω ) = (Σ rα ( ω )) † . The function χ α ( ω ) describes the broadening of Bloch states inducedby the coupling to the leads and to the cavity mode , while the real part of Σ rα ( ω ) provides ashift of the Bloch state energies ω α,k . The retarded and “lesser” photon GFs are definedas: D r ( ω ) = − i (cid:90) + ∞ dte iωt (cid:104) [ A ( t ) , A (0)] (cid:105) .D < ( ω ) = − i (cid:90) + ∞−∞ dte iωt (cid:104) A ( t ) A (0) (cid:105) , (17)with similar definitions for D a ( ω ) and D > ( ω ). As for electrons, one can show (seeAppendix A) that D r and D a satisfy the Dyson equation: avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. D β ( ω ) = (cid:18)(cid:16) D β ( ω ) (cid:17) − − Π β ( ω ) (cid:19) − , (18)while D > and D < are obtained from the Keldysh equation: D γ ( ω ) = D r ( ω )Π γ ( ω ) D a ( ω ) . (19)The expressions of the non-interacting (in the cavity vacuum state) photon GFs D ( ω ) are given by: D < ( ω ) = − iπδ ( ω + ω c ) D > ( ω ) = − iπδ ( ω − ω c ) D a ( ω ) = 2 ω c ( ω − i + ) − ω c , (20)and D r = ( D a ) ∗ .In the SCBA, the “lesser” and “greater” photon SEs can again be decomposed asΠ ≶ ( ω ) = Π ≶ I ( ω ) + Π ≶ P ( ω ), where the light-matter contributionΠ α (cid:48) ( ω (cid:48) )Π >I ( ω ) = − ig (cid:88) α,α (cid:48) (1 − δ α,α (cid:48) ) Tr (cid:90) dω (cid:48) π G >α ( ω + ω (cid:48) ) G <α (cid:48) ( ω (cid:48) ) (21)can be identified with the polarization function associated with the transition dipolemoments, which provides a dressing of the bare cavity photon GF D . The polarizationis represented by the bubble diagram shown on Fig. 2 c) . On the other hand, thecoupling between the cavity mode and the photon bath is described by the “exact” SEcontribution: Π
P ( ω ) = − iκθ ( ω ) . (22)Here, we have assumed a vanishing mean population of extra-cavity photons,namely (cid:104) a † p a p (cid:105) ≈
0. Similarly to electrons, the retarded and advanced photon SEscan be computed from the equation Π r ( t ) = θ ( t ) (Π > ( t ) − Π < ( t )), by introducing aphotonic broadening function similar to Eq. (16). The retarded photon GF can be usedto define the normalized cavity photon DOS as: A c ( ω ) = − (cid:61) D r ( ω ) , (23) avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. D < ( ω ) is related to the mean cavity photon numberin the steady state (up to small squeezing terms) as ¯ n ≡ (cid:104) a † a (cid:105) = − (cid:0)(cid:82) dω π (cid:61) D < ( ω ) + 1 (cid:1) .Above, we have shown that electron/photon SEs and GFs are related to each otherby a closed set of integro-differential equations. The numerical procedure to solve theseequations self-consistently is sketched on Fig. 2 d) : One substitutes the fully interactingelectron GFs in Eq. (21) with the non-interacting ones Eq. (13), to compute the first-order “lesser” and “greater” photon SEs. From the latter, one deduces the retarded andadvanced photon SEs and then computes the first-order photon GFs using Eqs. (18),(19), and (20). These photon GFs combined with the non-interacting electron GFsEq. (13) are then used to compute the first-order electron SEs from Eqs. (14) and (16),which in turn can be substituted in the Dyson and Keldysh equations (11) and (12) toobtain the first-order electron GFs. The whole cycle is repeated until convergence. In this section, we introduce the full QME relevant to investigate the system (Sec. 2.3.1),and show how it can be used to compute the steady-state current in Sec. 2.3.2.
The time evolution of the joint density operator ρ for the 1D chain and the cavity mode is given by the QME: ∂ τ ρ = − i [ H S , ρ ] + L ρ + L N ρ + L ph ρ. (24)Here, the commutator − i [ H S , ρ ] describes the coherent dynamics due to theHamiltonian H S = H e + H t + H I + H c introduced in Sec. 1.2. In the previous section,we have seen that while counter-rotating terms are formally included in the couplingHamiltonian H I , they do not play any role in the absence of vertex corrections. Here, wedirectly use the rotating-wave approximation and consider the coupling Hamiltonian: H I = g N (cid:88) j =1 (cid:16) c † ,j c ,j a + a † c † ,j c ,j (cid:17) , (25)instead of Eq. (2). The additional terms in the right-hand side of Eq. (24) are due tothe coupling of the chain to the external degrees of freedom. The injection of electronsat the first site is described by the term [93]: L ρ = (cid:88) α Γ α D [ c † α, ] ρ. Similarly, the extraction of electrons at the last site is given by: avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. L N ρ = (cid:88) α Γ α D [ c α,N ] ρ, while the action of D [ A ] on ρ is defined by the Lindblad superoperator [94, 95]: D [ A ] ρ = −{ A † A, ρ } + 2 AρA † . Assuming the extra-cavity photon bath close to its vacuum state, the cavity photondecay is described by the term: L P ρ = κ D [ a ] ρ. Since we are interested in the steady-state current flowingthrough the chain, we now explain how the latter can be computed from the QME (24).The time evolution of the expectation value of a generic observable A is given by theequation: ∂ τ (cid:104) A (cid:105) = Tr( A∂ τ ρ ) , (26)where the trace Tr denotes the sum over the diagonal elements in matrix representation.Using Eq. (24), one can show that the expectation value of the total charge operator Q S = e (cid:80) α,j ˆ n αj evolves according to: ∂ τ (cid:104) Q S (cid:105) = J s + J d , (27)where the currents flowing through the source and the drain (leads) are respectivelyexpressed as: J s = (cid:88) α e Γ α (cid:104) − ˆ n α (cid:105) = (cid:88) α e Γ α (cid:0) ˆ n α D [ c † α, ] ρ (cid:1) J d = − (cid:88) α e Γ α (cid:104) ˆ n αN (cid:105) = (cid:88) α e Γ α (cid:0) ˆ n αN D [ c α,N ] ρ (cid:1) . (28)The last equalities in the right-hand side of both lines can be derived by using thecyclic properties of the trace and fermionic commutation relations. In the steady-state,since ∂ τ ρ = 0, we have ∂ τ (cid:104) Q S (cid:105) = 0 and from Eq. (27), J s = − J d . It is straightforwardto check that the steady-state current calculated from Eq. (28) corresponds to Eq. (10)of Sec. 2.2. We numerically solve for the steady-state, either by computing the timeevolution of Eq. (24) using the Runge-Kutta method (fourth order), or by looking forthe null eigenvector of the Liouvillian L (with ∂ τ ρ = L ρ ) written in a matrix form [96].Details on how to implement fermionic operators in matrix representation can be foundin [97]. avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. In this section, we consider the dissipative regime obtained when the photon decay rate κ is larger than any other energy scale except ω . We show that the fast cavity fieldevolution can be adiabatically eliminated in this regime, resulting in an effective QMEinvolving only electronic degrees of freedom.It is convenient to introduce the density operator ˜ ρ = U ρU † in the rotating framedefined by the unitary operator: U ( τ ) = exp (cid:2) i ( H e + ω a † a ) τ (cid:3) . The time evolution of the operator ˜ ρ is then derived as: ∂ τ ˜ ρ = − i [ (cid:101) H, ˜ ρ ] + L ˜ ρ + L N ˜ ρ + L P ˜ ρ, (29)with the Hamiltonian (cid:101) H = H t + (cid:101) H c + H I . The (rescaled) cavity Hamiltonian (cid:101) H c = − ∆ a † a contains the detuning ∆ = ω − ω c between the transition and the cavity mode frequencies ω and ω c . Despite the factthat we will only discuss results obtained in the resonant case ∆ = 0, we perform theadiabatic elimination in the general situation for the sake of completeness. The adiabaticelimination procedure using projectors [98, 99] is detailed in Appendix B and outlinedin the following. We first recast the right-hand side of Eq. (29) as: ∂ τ ˜ ρ = L e ˜ ρ + (cid:101) L c ˜ ρ + L I ˜ ρ, in terms of the purely electronic part L e ˜ ρ = − i [ H t , ˜ ρ ] + L ˜ ρ + L N ˜ ρ , the photonic part (cid:101) L c ˜ ρ = L c ˜ ρ + κa ˜ ρa † , as well as the interaction part L I ˜ ρ = − i [ H I , ˜ ρ ]. The photonic part (cid:101) L c ˜ ρ contains the contribution: L c ˜ ρ = (cid:16) i ∆ − κ (cid:17) a † a ˜ ρ + (cid:16) − i ∆ − κ (cid:17) ˜ ρa † a, which generally gives rise to damped oscillations for the relaxation of the cavity field.When the cavity decay rate κ is much larger than the rates governing the electrondynamics (i.e. t α and Γ α ), one can separate the fast cavity dynamics from theelectronic one occuring on a comparably long time-scale. In the presence of light-matter interactions, such a separation is still possible whenever the light-matter couplingstrength g is sufficiently weak. In Appendix B, we present a detailed derivation of avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. t α /κ and/or Γ α /κ areneglected), and where the light-matter interaction is treated to second order. Moreover,we restrict our discussion to the case where the cavity field remains close to its vacuumstate, which is consistent with the large damping rate κ . In this limit, the time evolutionof ˆ ρ , the full density operator projected onto the cavity vacuum, is governed by theeffective QME: ∂ τ ˆ ρ = L e ˆ ρ − i (cid:2) g ∆∆ + ( κ/ S + S − , ˆ ρ (cid:3) − g κ/ + ( κ/ (cid:0) S + S − ˆ ρ + ˆ ρS + S − − S − ˆ ρS + (cid:1) . (30)Here, S + = (cid:80) j c † ,j c ,j ( S − = ( S + ) † ) denotes a collective raising (lowering) operatorfor the electrons from the lower (upper) to the upper (lower) band. In the resonant case( ω = ω c ), the time evolution of ˆ ρ can be simplified: ∂ τ ˆ ρ = L e ˆ ρ + L Γ c ˆ ρ, (31) where light-induced interactions between electrons are entirely cast into the dissipator: L Γ c ˆ ρ = − c (cid:0) S + S − ˆ ρ + ˆ ρS + S − − S − ˆ ρS + (cid:1) , (32)with Γ c = g /κ . This shows that in the dissipative regime where κ is the largestparameter, light-matter interactions are governed by the parameter Γ c . We remarkthat such a term also appears in the case of pseudo-spins (e.g. in a two-level atomicdescription) coupled to a cavity mode with strong dissipation [98, 100, 101]. Introducingthe local raising operators s + j = c † ,j c ,j and the corresponding lowering operators s − j = c † ,j c ,j , the dissipator Eq. (32) can be rewritten as: L Γ c ˆ ρ = − c N (cid:88) j =1 (cid:0) s + j s − j ˆ ρ + ˆ ρs + j s − j − s − j ˆ ρs + j (cid:1) − c N (cid:88) i,ji (cid:54) = j (cid:0) s + j s − i ˆ ρ + ˆ ρs + j s − i − s − i ˆ ρs + j (cid:1) . (33)Here, both local and non-local coupling terms can be identified, and correspond tothe first and second terms in the right-hand side of Eq. (33), respectively. In spin-cavity setups, non-local terms can induce spin-spin correlations and ultimately leadto synchronization and superradiance [98, 102, 103]. In our situation, they give rise tonon-local exchange of interband excitations, which are partly taken into account in theSCBA, as discussed in Sec. 2.2.3. Regarding charge transport, the dissipator Eq. (32)induces a global (collective) population transfer of electrons from the upper to the lower avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. α by N α = (cid:80) j ˆ n αj ,its time evolution due to light-matter interactions in the dissipative regime is: ∂ τ (cid:104) N (cid:105) = Tr( N L Γ c ˆ ρ ) = 4Γ c (cid:104) S + S − (cid:105) ∂ τ (cid:104) N (cid:105) = Tr( N L Γ c ˆ ρ ) = − c (cid:104) S + S − (cid:105) , which provides ∂ τ (cid:104) N (cid:105) = − ∂ τ (cid:104) N (cid:105) , and demonstrates the population exchange betweenthe two bands. Moreover, the rate associated with this population transfer can be relatedto the mean intra-cavity photon number, approximated as (cid:104) a † a (cid:105) (cid:39) (4Γ c /κ ) (cid:104) S + S − (cid:105) inthe adiabatic limit [98, 102]. The change of the first band population thus takes thesimple form ∂ τ (cid:104) N (cid:105) = κ (cid:104) a † a (cid:105) : The population transfer from the upper to the lowerband is accompanied by the creation of photons which are then dissipated with the rate κ . In Sec. 3.4.1, we derive an analytical estimate for the current enhancement in thedissipative regime, by calculating the time evolution of expectation values (cid:104) c † α,i c α,j (cid:105) withsuitable approximations.
3. Results
In this section, we present both analytical and numerical results using the QME andNGFs methods. In Sec. 3.1, we first discuss the situation without light-matter coupling,by computing the steady-state current, the electron density profile in both bands, aswell as the time evolution of the electron spectral function. In Sec. 3.2, we explain howlight-matter interactions lead to a broadening of the electron DOS, and show that thelatter scales with the cooperativity. We further explain how polariton modes arise fromthe dressing of the photon GF by the electron-hole polarization. In Sec. 3.3, we comparenumerical results for the steady-state current obtained with the different methods, anddistinguish between two regimes characterized by the ratio between the cavity photondecay rate κ and the upper electronic bandwidth W . In Sec. 3.4, we investigate thedissipative regime κ/W (cid:29) g exceeds the energyspacing between adjacent Bloch states in the upper band. First, an analytical expressionof the steady-state current valid for small coupling strength is given in Sec. 3.4.1,while numerical calculations using both NGFs and QMEs are presented in Sec. 3.4.2.In Sec. 3.4.3, we show that non-local electronic correlations occur for large couplingstrengths in this regime. The “coherent” regime κ/W (cid:28) g is smaller than the energy spacing between adjacent Bloch states in the upper band.Concluding remarks concerning the cavity photon population and the chain length aregiven in Sec. 3.6. avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. g = 0 ) Here, we discuss the system properties in the absence of light-matter coupling, bycomputing the steady-state current, the electron density spatial profile, as well as thetime evolution of the electron spectral function.In the absence of light-matter coupling ( g = 0), the two bands are independent andthe eigenstates of the chain consist of two identical sets of the N Bloch states defined inEq. (8). The only finite SE contribution Eq. (15) is due to the coupling to the leads, andis proportional to the decay rate Γ α of the Bloch states in the band α . The transportproperties of the chain are only driven by the ratio between Γ α and t α , and the steady-state current does not depend on the chain length N . Using the spectral function sumrule (cid:82) dωA α,k,k (cid:48) ( ω ) = 2 πδ k,k (cid:48) in Eq. (3), the steady-state current J (0) α flowing throughthe band α can be written as: J (0) α = e Γ α (cid:18) (cid:90) dω π Tr (cid:2)(cid:0) σ N − σ (cid:1) ◦ (cid:61) G <α ( ω ) (cid:3)(cid:19) . (34)One can then use Eqs. (11), (12), (15), and (16) with e.g. N = 2, and obtain thecurrent as: J (0) α = e Γ α /
21 + (cid:16) Γ α t α (cid:17) , (35)in agreement with the results of [93]. The electron populations at the edges of the chainfollow from Eq. (10): n αN = 1 − n α = 12 + Γ α t α . Two different regimes of transport can be distinguished. When t α (cid:28) Γ α , transportis inhibited due to the small lifetime of Bloch states compared to the typical hoppingtime. Bloch states are thus not well resolved and the steady-state current is given by J (0) α ∼ et α / Γ α ≈
0. In this situation, the first and last sites are respectively fullyoccupied and completely empty, i.e. n α ≈ n αN ≈
0. The opposite regime t α (cid:29) Γ α features single-electron transport through well-resolved Bloch states. In thisregime, the current J (0) α ≈ e Γ α / α , and the first and lastsites are half-filled, namely n α = n αN = 0 . As already mentioned, we only consider the situation where Γ = Γ ≡ Γ and t (cid:29) Γ (cid:29) t . The different transport regimes can be identified in the transmissionspectrum T ( ω ), which is represented on Fig. 3 for g = 0. In the vicinity of the upperorbital ω ≈ ω , the relation t (cid:29) Γ leads to N well-resolved peaks (Bloch states) avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. − − − − − − − Site positionSite position d)b) F r equen cy T r an s m i ss i on T r an s m i ss i on F r equen cy c)a) E l e c t r on popu l a t i on s E l e c t r on popu l a t i on s e) T i m e P o s i t i on f) T i m e P o s i t i on Figure 3. a) - c) (Log-scale) Transmission spectrum T ( ω ) versus frequency for g = 0,in the vicinity of a) the upper orbital energy ω = 0 . c) the lowerorbital energy ω = − . b) - d) Spatial profiles of the electron density n αj for g = 0, in b) the upper orbitals (blue squares), and d) the lower orbitals (redsquares). e) - f ) Spectral function A ( α ) j ,j ( τ ) for g = 0 as a function of position and time,obtained after injection of a particle at site j = 1 and time τ = 0 in e) the upperorbital, and f ) the lower orbital. Time is in units of the hopping rates in the lower andupper bands, respectively. Parameters are N = 10, t = 10 − , Γ = 10 − , and t = 0 . of width ∼ Γ /N , distributed over the bandwidth W [Fig. 3 a) ]. Moreover, all sitesare half-filled [Fig. 3 b) ], and the partial current obtained by integrating T ( ω ) in thevicinity of ω is J (0)2 ≈ e Γ /
2. In the vicinity of the lower orbital ω ≈ ω , however, thedynamics does not involve well-resolved Bloch states since t (cid:28) Γ. This results in anumber of peaks smaller than N within the bandwidth W [Fig. 3 c) ], a half-filling ofall sites except for the first and last ones [Fig. 3 d) ], and therefore a very small current J (0)1 /e Γ ≈ t / Γ) (cid:28) avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. A ( α ) j ,j ( τ ) defined as the Fourier transform (overboth space and time variables) of the function A ( α ) k,k (cid:48) ( ω ) introduced in Sec. 2.2.2: A ( α ) j ,j ( τ ) = 2 (cid:60)(cid:104){ c α,j ( τ ) , c † α,j (0) }(cid:105) . Physically, this function can be interpreted as follows: Considering an electroninjected in the steady-state at site j and time τ = 0 in the level α , its wavefunction willbe decomposed over the different sites under the time evolution governed by the totalHamiltonian (including interactions with the leads). The function A ( α ) j ,j ( τ ) correspondsto the overlap between this wavefunction at later time τ > τ at an other site j , and provides information on what the wavefunctionof an electron (or a hole) injected at a given site at τ = 0 looks like after a certain time τ . This function is represented on Fig. 3 e) and f ) , considering excitations propagatingin the upper and the lower orbitals, respectively. In the former case, the dynamics of aparticle injected in the upper level of site j = 1 at τ = 0 involves a decomposition overthe different well-resolved Bloch states of the upper band, resulting in the propagationof this particle throughout the chain [Fig. 3 e) ]. On the other hand, since t (cid:28) Γ,propagation in the lower band is hampered and most of the spectral weight stayslocalized at the injection site before being damped after a typical time ∼ / Γ [Fig. 3 f ) ].In the following, we investigate how this physical picture is modified when the couplingto the cavity mode is turned on.
In this section, we compute the first-order GFs and SEs, and explain how the electronDOS is broadened by the presence of light-matter interactions. In particular, we showthat this broadening scales with the cooperativity in the dissipative regime, and explainhow the dressing of the photon GF by the electron-hole polarization results in theappearance of polariton states.When g (cid:54) = 0, electrons can undergo interband transitions concurrently with theaborption/emission of cavity photons with energy ω ∈ [ ω − t , ω + 2 t ] (for t (cid:28) t ).This leads to the hybridization of the two bands, and provides a modification of theelectron DOS and the transmission spectrum. In Sec. 2.2, we have seen that the couplingbetween the two electronic bands and the cavity field is a self-consistent problem. Theelectron dynamics is affected by the electromagnetic field through emission/absorptionof cavity photons, and the cavity field is in turn dressed by its interactions with theelectron-hole polarization. The simplest approximation consists in neglecting the self-consistency § and calculate the first-order ( ∝ g ) electron SE induced by the coupling to § We find that this approximation gives a correct description only for very small values of the ratioΓ c / Γ. If it is not the case, neglecting the self-consistency leads to spurious limiting behaviors as a resultof the breaking of conservation laws such as the continuity equation for the current. avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. D r ( ω ) = 2 ω c ω − ω c + iκω c sgn( ω ) , (36)where sgn denotes the sign function. We point out that considering only the couplingof cavity photons to the external electromagnetic environment when calculating thephoton SE is expected to be valid in the dissipative regime, where κ is the largestenergy scale. At resonance ω c = ω , the first-order broadening function defined inSec. 2.2.3 is calculated as: χ α,k,k (cid:48) ( ω ) = (cid:88) α (cid:48) κg ω (1 − δ α,α (cid:48) ) δ k,k (cid:48) (( ω − ω α (cid:48) ,k ) − ω ) + ( κω ) (cid:16) (1 − n α (cid:48) k ) θ ( ω − ω α (cid:48) k ) + n α (cid:48) k θ ( ω α (cid:48) ,k − ω ) (cid:17) , (37)where n αk is the population of the Bloch state ( α, k ) in the initial ground state, withoutany interactions. This broadening function is diagonal with respect to k . Consideringa Bloch state k in the lower band α = 1, its light-induced broadening depends on thefilling of the state k in the upper band α (cid:48) = 2. When n k = 1, the associated electron canundergo a transition from the upper to the lower band by emitting a photon with energy ω ,k − ω ,k . For ω = ω ,k , and κ/W (cid:29) ω ,k − ω ≈ ω ,which simply yields: χ ,k ( ω ) ≈ c , (38)where Γ c = g /κ has been introduced in Sec. 2.4. Since the SE broadening due to thecoupling to the leads is ∝ Γ, one finds that in the dissipative regime, the light-inducedrelative broadening of the electron DOS is driven by the ratio Γ c / Γ , which plays the roleof a cooperativity parameter . Moreover, the validity domain of the NGFs method islimited to the perturbative (“quasiparticle”) regime with Γ c / Γ (cid:46) t (cid:29) t , when the coupling strength g becomes eventually largerthan the typical energy spacing between two adjacent Bloch states in the upper band,a collective coupling of the different Bloch states to the cavity mode arises, and theelectron-hole polarization given by Eq. (21) can no longer be neglected. In order to seehow this collective coupling is related to the polarization dressing of the photon GF, onecan compute the first-order retarded photon GF in the absence of cavity losses. We thusproceed in an opposite way to the one used previously, by neglecting the contributiondue to the coupling to extra-cavity photons Eq. (22), and replacing the fully interacting avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. r ( ω ) = (cid:88) k g ( n k − n k ) ( ω ,k − ω )( ω + i + ) − ( ω ,k − ω ) . (39)Furthermore, if we also assume t (cid:28) ω , namely neglecting the upper bandwidthwith respect to the transition frequency, Eq. (39) takes the form of the usual interbandpolarization [104–106] (which enters the definition of the dielectric permittivity [107])involving a collective response of the electron states:Π r ( ω ) = 2Ω n ω ( ω + i + ) − ω . (40)At this level of approximation, the collective vacuum Rabi frequency is defined asΩ n = g (cid:112)(cid:80) k n k − n k , and depends on the initial population imbalance between thetwo bands. Replacing Eq. (40) in the Dyson equation (18), the first-order retardedphoton GF can be written as: (cid:101) D r ( ω ) = 2 ω ( ω − ω ) (cid:2) ( ω + i + ) − ω (cid:3) (cid:2) ( ω + i + ) − ω − (cid:3) , (41)at resonance ( ω c = ω ). This function exhibits poles at the polariton frequencies ω ± = (cid:112) ω + 2 ω Ω n . Note that taking the cavity decay rate κ into account wouldturn the latter into quasi-modes with imaginary frequency. The effect of this collectivedressing of the photon GF on the electron spectral broadening can then be studied bycomputing the electron SE Eq. (14) together with Eqs. (41) and (13). While the resultdepends on the initial populations n αk at this level of approximation, it is not the casewhen the self-consistency is taken into account, namely when using the fully interactingelectron GFs in the photon SE Eq. (21). This will be studied numerically in Sec. 3.4.2. In this section, we benchmark the different methods used to compute the steady-state current, and show that the light-matter coupling is responsible for a currentenhancement driven by the cooperativity parameter in the dissipative regime. We alsocompute numerically the broadening function introduced in Sec. 2.2 using self-consistentNGFs.Introducing J (0) = J (0)1 + J (0)2 the overall steady-state current in the absence oflight-matter coupling ( g = 0) [see Eq. (35)], we now study numerically the relativecurrent enhancement ∆ J = ( J/J (0) ) − g and κ . This is shown on Fig. 4, for an example in the regime t (cid:29) Γ (cid:29) t with N = 3, t = 10 − , Γ = 10 − , and t = 10 − ( W ≈ . avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. -4 -2 a) b) c)d) Full QMEEffective QME NGFs P ho t on de c a y r a t e C oup li ng s t r eng t h -2 e) C oup li ng s t r eng t h C oup li ng s t r eng t h Figure 4.
Relative current enhancement ∆ J (see text) versus coupling strength g and photon decay rate κ (log-scale), obtained from a) the effective QME (31), b) the full QME (24), and c) the NGFs method. The diagonal dashed line g /κ = cstis a guide to the eye, and the horizontal solid line corresponds to κ = W . d) - e) Relative current enhancement ∆ J versus cooperativity Γ c / Γ, for two different valuesof κ represented by the horizontal dashed lines in the upper panels. d) κ = 0 . κ/W ≈ e) κ = 8 × − ( κ/W ≈ . c / Γ = 1. Parameters are N = 3, t = 10 − ,Γ = 10 − , t = 10 − . For the full QME method, the maximum number of photons inthe Hilbert space is set to 3. Panels a) , b) , and c) correspond respectively to the results obtained from theeffective QME (31), the full QME (24), and the NGFs method. We observe anenhancement of the steady-state current with respect to the non-interacting case g = 0 ,as the coupling strength is increased for a given decay rate κ . Furthermore, thisenhancement is substantially larger in the high-finesse cavity regime with small κ .Note that the full QME result is exact (assuming the Markovian approximation forthe system-lead coupling) as long as counter-rotating terms can be neglected in thecoupling Hamiltonian Eq. (2), which is here assumed in all cases.As already discussed in Sec. 2.4, the effective QME result only depends on theparameter Γ c = g /κ , which explains that the lines with constant current enhancementon panel a) scale linearly with log g and log κ over the whole range of parameters.Nevertheless, we point out that this result is only valid in the dissipative regime where κ/W (cid:29)
1. This is shown on panels b) and c) , where the full QME and NGFs resultsfeature the same scaling law as the effective QME result for κ/W (cid:29)
1. However, a avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. κ/W (cid:28)
1, indicating the emergence of a new regimewith different physical properties than the ones discussed in Secs. 2.4 and 3.2. We willshow later on that this regime can be characterized by a coherent dynamics stemmingfrom the hybridization of only one Bloch state in the upper band with the states of thelower band.The relative current enhancement ∆ J is represented on Figs. 4 d) and e) as afunction of the cooperativity Γ c / Γ, for two different values of κ (horizontal dashed linesin the upper panels). The results obtained with the effective QME, the full QME, andthe NGFs method are represented as light-blue triangles, black circles, and magentasquares, respectively. In the dissipative regime [Fig. 4 d) ], all methods coincide forΓ c / Γ (cid:28) c / Γ becomes larger than 1, discrepancies betweenthe NGFs and the full QME results increase, while the effective QME and full QMEresults are still in a surprisingly good agreement, given that the former is expected tobe valid only for small coupling strengths. In the “coherent” regime with κ/W (cid:28) e) ], while the effective QME fails to reproduce the full QME result even in theperturbative regime, NGFs provide a surprisingly good approximation of the currenteven far away from the perturbative regime Γ c / Γ (cid:29)
1. However, while the currentenhancement obtained from the two master equation methods always increases with g ,this qualitative trend is not reproduced by the NGFs method when Γ c / Γ (cid:38) P ho t on de c a y r a t e C oup li ng s t r eng t h Figure 5.
Relative broadening of the electron DOS log( χ/χ ) (see text) versuscoupling strength g and photon decay rate κ (log-scale), obtained from the NGFsmethod (self-consistent calculation). The dashed line represents the equation Γ c / Γ = 1.Parameters are identical to that of Fig. 4.
Furthermore, it is interesting to compare the current enhancement represented onFig. 4 c) with the cavity-induced broadening of the electron DOS [see Secs. 2.2.3 and 3.2]computed self-consistently with the NGFs method. We denote by χ ≡ χ ,k ( ω ,k ) thebroadening function of the resonant Bloch states with quasi-momentum k = ( N + 1) / ω = ω ,k . This quantity represents thelinewidth of the electron DOS (with a Lorentzian lineshape) in the lower band for g (cid:54) = 0.For g = 0, the linewidth χ is only determined by the retarded SE due to the couplingto the leads Σ rL, ∝ Γ. The relative broadening log( χ/χ ) is shown on Fig. 5, as a avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. g and κ (log-scale). As for the current enhancement, we observe that thelines with constant relative broadening scale with the cooperativity Γ c / Γ for κ/W (cid:29) κ/W (cid:28) κ/W (cid:29) κ/W (cid:29)
1, we nowpresent numerical calculations using both NGFs in the frequency domain, and QMEmethods in the time domain. In Sec. 3.4.1, we show that an analytical expression ofthe current valid for small coupling strengths can be derived starting from the effectiveQME, confirming the scaling of the current enhancement with the cooperativity. InSec. 3.4.2, we present numerical results for the transmission spectrum and the cavityphoton DOS, evidencing the presence of a collective coupling to light when the couplingstrength is larger than the typical separation between two adjacent Bloch states. Forlarge coupling strengths, we show that the current enhancement saturates to about twiceits value for g = 0, and that the system features non-local electron-electron correlationswhen one goes beyond the perturbative regime Γ c / Γ > In Sec. 2.1, we have seen that thesteady-state current is directly related to the populations of the first/last site in bothorbitals. Hence, the former can be obtained by computing the expectation values (cid:104) c † α,i c α,j (cid:105) , with i = j = 1 or i = j = N . In the absence of light-matter interactions( g = 0), the expectation values (cid:104) c † α,i c α,j (cid:105) evolve as: ∂ t (cid:104) c † α,i c α,j (cid:105) = it α N − (cid:88) (cid:96) =1 (cid:104) c † α,i (cid:0) δ (cid:96) +1 ,j c α,(cid:96) + δ (cid:96),j c α,(cid:96) +1 (cid:1) (cid:105) + h.c. − Γ α (cid:16) δ i + δ j + δ iN + δ jN (cid:17) (cid:104) c † α,i c α,j (cid:105) + Γ α δ i δ j , (42)with h.c. the hermitian conjugate, and where we have used ∂ t (cid:104) c † α,i c α,j (cid:105) = Tr( c † α,i c α,j L e ˆ ρ )according to Eq. (26). Equation (42) forms a closed set of linear differential equations.When solving these equations in the case of uncoupled bands, and plugging the solutionin Eq. (5), one recovers the overall steady-state current which is the sum of the individualcurrents [see Eq. (35)] flowing through the two bands.We now explain how to modify the set Eq. (42) in the presence of light-matterinteractions in the dissipative regime. In the resonant case ∆ = 0, we see from Eq. (31)that we need to compute the additional contribution: ∂ t (cid:104) c † α,i c α,j (cid:105) = Tr( c † α,i c α,j L Γ c ˆ ρ ) , avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. L Γ c ˆ ρ given by Eq. (32). We obtain: ∂ t (cid:104) c † ,i c ,j (cid:105) = 2Γ c (cid:104) c † ,i c ,j S − + S + c † ,i c ,j (cid:105) ∂ t (cid:104) c † ,i c ,j (cid:105) = − c (cid:104) c † ,i c ,j S − + S + c † ,i c ,j (cid:105) . (43)The differential equations (43) now contain four-operator products, in contrastto the non-interacting case with only quadratic operators. Full computation of theexpectation values thus involves higher-order correlation functions, and the NGFsmethod can be efficiently used in this case (see Sec. 2.2.3). In order to get a first estimateof the current enhancement, starting from Eq. (43) for the time evolution of (cid:104) c † α,i c α,j (cid:105) ,we only focus on the populations ( i = j ) and discard non-local contributions arisingfrom the second term in the right-hand-side of Eq. (33). We have verified numericallythat the effective QME with and without non-local coupling terms in the dissipatorgives comparable results for small g , as we will discuss in more detail in Sec. 3.4.3. If,in addition, we factorize the expectation value of four-operator products as: (cid:104) ˆ n i ˆ n i (cid:105) (cid:39) (cid:104) ˆ n i (cid:105)(cid:104) ˆ n i (cid:105) , (44)Eq. (43) provides: ∂ τ (cid:104) ˆ n i (cid:105) = 4Γ c (cid:104) ˆ n i (cid:105) (cid:0) − (cid:104) ˆ n i (cid:105) (cid:1) ∂ τ (cid:104) ˆ n i (cid:105) = − c (cid:104) ˆ n i (cid:105) (cid:0) − (cid:104) ˆ n i (cid:105) (cid:1) . (45)At this level of approximation, the cavity mode induces a local population transferfrom the upper to the lower orbitals at each site. Solving the differential equations (42)together with the contribution Eq. (45) stemming from the light-matter coupling, onecan check numerically that the steady-state current is nearly independent of the chainlength N as long as t (cid:28) Γ. We therefore restrict the calculation to the case N = 2. Forthe sake of simplicity, we limit the derivation to the case t = 0 and Γ = Γ ≡ Γ. Thetime-evolution of the mean population n in the lower band at the first site is obtainedas: ∂ τ n = − Γ n + Γ + 4Γ c n (1 − n ) . In the steady-state, ∂ τ n = 0, which provides the solution n = 1. Furthermore,as a solution of the equations ∂ τ (cid:104) c † c (cid:105) = − Γ (cid:104) c † c (cid:105) and ∂ τ (cid:104) c † c (cid:105) = − Γ (cid:104) c † c (cid:105) , thelower orbital coherence (cid:104) c † c (cid:105) = (cid:104) c † c (cid:105) vanishes in the steady-state. The remainingset of equations can be rewritten as: avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. ∂ τ n = − Γ n − t C + Γ ∂ τ n = − Γ n + 2 t C − c n (1 − n ) ∂ τ n = − Γ n + 4Γ c n (1 − n ) ∂ τ C = − Γ C + t ( n − n ) , (46)where we have introduced the imaginary part of the upper orbital coherence C = (cid:61)(cid:104) c † c (cid:105) . The latter is related to the local current in the upper band between thefirst and the second site. Setting the left-hand side of Eq. (46) to zero, one can computethe overall steady-state current J = e Γ( n + n ) as: J = e Γ t / t (1 + φ ) / / , (47)with φ = 4Γ c n + Γ4Γ c ( n + 1) + Γ . Since the population n >
0, the function φ is positive and has an upper bound 1.This value is reached, for instance, in the absence of light-matter coupling Γ c = 0. In thiscase, one can verify that the current coincides with Eq. (35). Further inspection showsthat whenever Γ c (cid:54) = 0 , φ < , resulting in an enhancement of the steady-state current. Nevertheless, we expect the result Eq. (47) to be a reasonable approximation only forsmall Γ c (small coupling strength), as pointed out before. It is therefore convenient toexpand Eq. (47) to the lowest non-vanishing order in Γ c , which provides: J = J (0) (1 + ∆ J ) + O (Γ c ) , where J (0) is the overall steady-state current for g = 0, and the relative currentenhancement introduced in Sec. 3.3:∆ J = 2 t t + Γ / (cid:18) Γ c Γ (cid:19) . (48)In this regime, the current enhancement is induced by a population transfer fromthe upper to the lower band. Indeed, the upper band population at the last site n = n (0)22 (cid:18) − t + Γ / t + Γ / (cid:18) Γ c Γ (cid:19)(cid:19) + O (Γ c ) avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. c . Here, n (0)22 = t / t +Γ / denotes the population in the upperband at the last site for g = 0. As Γ c increases, the population in the lower band at thelast site increases as: n = 2 t t + Γ / c Γ , and vanishes for g = 0 (as long as t = 0). Importantly, the overall population at thelast site increases with Γ c , which explains the observed current enhancement [57]. In theprevious derivation, we only considered the local terms in the dissipator Eq. (32), whichis valid for small coupling strengths, and further discarded the contributions of theseterms to the time-evolution of the intraband coherence ∂ C /∂t . Taking them explicitelyinto account, the last equation of motion in Eq. (46) is modified as: ∂ τ C = − Γ C + t ( n − n ) − c (2 − n − n ) C , (49)where we have used a factorization procedure similar to Eq. (44) for the four-operatorproducts entering the last term in the right-hand-side of Eq. (49). This term describesan additional damping of the intraband coherence due to the light-matter coupling.Since the intraband coherence is proportional to the local current, we therefore expectthis correction to lead to a smaller current enhancement. Moreover, we have checkednumerically that it can even lead to a reduction of the overall current when t (cid:28) Γ. Inthis article, we only focus on the regime t (cid:29) Γ, where one can show that the effect ofthe additional term in Eq. (49) becomes negligible for the relative current enhancement.In this case, Eq. (48) simply reduces to∆ J = 2Γ c / Γ . Here, we clearly confirm the relevance of the cooperativity Γ c / Γ, as found inSecs. 3.2 and 3.3. Nevertheless, we point out that non-local contributions entering thedissipator Eq. (33) have not been taken into account in this derivation. Therefore, thisanalytical estimation is unable to describe any collective effects arising from these non-local terms, namely long-range electronic correlations due to the collective coupling tothe cavity mode [102, 103] . As we will see in the next section, these terms play a crucialrole beyond the perturbative regime Γ c / Γ > avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. In the dissipative regime κ/W (cid:29)
1, allBloch states are comprised within the cavity linewidth, which allows a collective couplingto arise when the coupling strength is larger than the typical separation between twoadjacent Bloch states, namely g > δω = ω ,k +1 − ω ,k (cid:107) . This regime is refered toas “collective dressing regime” [57]. Frequency domain calculations using the NGFsmethod are shown on Fig. 6, for an example with N = 11, t = 5 × − , Γ = 5 × − , t = 5 × − , κ = 0 . κ/W ≈ g = 2 . × − (Γ c / Γ ≈ . a) B l o c h w a v e v e c t o r F r equen cy C a v i t y m ode -0.8 -0.6 -0.4 -0.210 -5 -1 F r equen cy F r equen cy T r an s m i ss i on T r an s m i ss i on c) d) C a v i t y pho t on D O S F r equen cy b) Figure 6. a)
Sketch of the energy bands in the dissipative regime with t = 5 × − ,and κ = 0 . b) (Log-scale) Cavity photon DOS A c ( ω ). c) (Log-scale) Transmissionspectrum T ( ω ) in the vicinity of the lower orbital energy ω = − . d) (Log-scale)Transmission spectrum T ( ω ) in the vicinity of the upper orbital energy ω = 0 . g = 0, while the red, blue, and green lines correspond to g = 2 . × − . The chain length is N = 11, and the other parameters are t = 5 × − and Γ = 5 × − . In this regime with large photon damping, the interband transitions with frequencies ω ,k − ω between the states of the quasi-flat lower band and the upper band Bloch states[Fig. 6 a) ] are all quasi-resonant to the broad bare cavity mode of width κ [thin blackline on Fig. 6 b) , hardly visible], resulting in a collective coupling of the Bloch states tothe cavity mode when g > δω (in this case δω (cid:46) . × − ). The photon DOS [Eq. (23)]is shown as a thick green line on Fig. 6 b) for g = 2 . × − . The central region of width (cid:107) This feature can be qualitatively understood from the spectrum of an effective, bosonic TChamiltonian (obtained from the TC model, by considering the leading-order of the Holstein-Primakoff [108] expansion of spin operators in terms of bosons) H TC = ω c a † a + (cid:80) Nk =1 ( ω ,k − ω ) b † k b k + g (cid:80) Nk =1 ( b † k a + b k a † ), where b k , b † k are bosonic operators associated with the N interband transitions. avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. W ≈ t and centered at ω consists of N − c) and d) display the transmission spectrum given by Eq. (4) for g = 0(black line), g = 2 . × − (colored lines), in the vicinity of the lower and upperorbital energies ω and ω , respectively. The key feature is that the narrow transmissionassociated with the partial current J ∼ et / Γ flowing through the lower band for g = 0is broadened by a quantity ∼ Γ c , giving rise to the current enhancement (∆ J ≈ . ω is only slightly reduced withrespect to the case g = 0.The cavity photon DOS calculated with the full QME and the NGFs methodsis represented on Fig. 7 a) - b) , in the perturbative regime (Γ c / Γ = 0 .
1) and at largecoupling strength (Γ c / Γ = 145), respectively. The former case [Fig. 7 a) ] correspondsto Fig. 6 b) for N = 3 instead of N = 11, but with the same other parameters. Here,the two methods are in good agreement, showing the validity of the NGFs method inthe perturbative regime Γ c / Γ (cid:28)
1. On the other hand, when g > δω ( δω (cid:46) .
03 in thiscase), the collective coupling gives rise to two polariton peaks separated by a splittingwhich we define as Ω S > g [Fig. 7 b) ]. c) C oup li ng s t r eng t h a) F r equen cy C a v i t y pho t on D O S b) F r equen cy C a v i t y pho t on D O S Figure 7. a) - b) (Log-scale) Cavity photon DOS A c ( ω ) for N = 3, computed with thefull QME method using the quantum regression theorem (thin black line) and the NGFsmethod (thick green line). a) Small coupling strength g = 2 . × − (Γ c / Γ ≈ . b) Large coupling strength g = 8 . × − (Γ c / Γ ≈ c) Polariton half-splitting Ω S (solid line) and vacuum Rabi frequency Ω n (dashed line) as a function of g for N = 11.Other parameters are the same as in Fig. 6, and the maximum number of photons inthe calculation using the full QME is set to 2. Importantly, the result obtained with the NGFs method features a small photonspectral weight in the central region of width W , and inaccurately predicts that thisphoton spectral weight vanishes in the limit of large coupling strengths. This explainswhy the current enhancement computed with NGFs decreases in this region, as observedin [57]. Indeed, when no photon weight is present in the range [ ω − t , ω + 2 t ], avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. g is increased while the twopolariton peaks further split. This is consistent with the observation of Sec. 3.3 that the(exact) current enhancement computed with the full QME always increases with g . Therelative current enhancements obtained in the case of small and large coupling strengthscorrespond respectively to: ∆ J ≈ .
13 for both methods [Fig. 7 a) ], and ∆ J ≈ . J ≈ . b) ]. Note that the asymmetry of thepolariton peaks computed with NGFs is inherited from the asymmetry of the first-orderphoton GF Eq. (36), only valid in the perturbative regime, namely when the SE dueto the coupling to extra-cavity photons is much smaller than the bare cavity photonenergy κ (cid:28) ω c .On Fig. 7 c) , we compare the polariton half-splitting Ω S with the vacuum Rabifrequency defined as Ω n = g √ N − N (see Sec. 3.2), where N α = (cid:80) j n α,j is obtainedfrom the steady-state population imbalance between the two bands for g (cid:54) = 0 [106].As already mentioned in [57], we observe that these two quantities coincide in thedissipative regime, thereby connecting to the physics of the TC model [7] where therelevant coupling strength is not g but the collective coupling constant Ω n . Importantly,since sites with both orbitals occupied (or empty) are not effectively coupled to light,we always find Ω n < g √ N , in contrast to the TC model [see Sec. 1.2].As a side comment, we have already mentioned that the full QME method predictsthat the total current admits an upper limit < e Γ as g → ∞ . We find numerically thatfor t (cid:29) Γ (cid:29) t , and for the two values κ = 8 × − (coherent regime) and κ = 0 . e Γ (twice thecurrent for g = 0) when counter-rotating terms are included in the coupling HamiltonianEq. (25). This shows that higher-order correlations such as the one depicted on Fig. 2 b) are important to determine the full current enhancement in the limit of large couplingstrengths. These effects will be further investigated in a future work. As already mentioned in Sec. 2.4, an interesting pointis the existence of non-local electron-electron correlations for large coupling strengths.This can be seen on Fig. 8, where we have represented the steady-state current computedwith different approximations as a function of the cooperativity. Fig. 8 a) displays acomparison between the full QME results with fermions (filled circles) and hard-corebosons (empty circles). The discrepancy observed for Γ c / Γ > b) , we have represented the results obtained with the effective QME, using either thefull dissipator of Eq. (33) (filled triangles), or only the local terms in the right-handside of the same equation (empty triangles). We remark that these local terms wouldcorrespond to a situation in which each site is individually coupled to its own lossycavity (with decay rate κ and coupling strength g ). avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. -4 -2 -4 -2 b)a) Figure 8.
Steady-state current
J/e
Γ as a function of the cooperativity Γ c / Γ (log-scale) for N = 3. a) Calculation using the full QME with fermions (filled circles)and hard-core bosons (empty circles). b) Same quantity computed with the effectiveQME, using the full dissipator Eq. (33) (filled triangles), or only the local terms (emptytriangles). Other parameters are identical to that of Fig. 6, and the maximum numberof photons in the calculation using the full QME is set to 2.
Moreover, we observe on Fig. 8 b) that non-local terms play an important roleas one moves away from the perturbative regime. This points to the existence of non-local electronic correlations for Γ c / Γ >
1, that are obtained only when considering thatall sites are coupled to the same cavity mode. These correlations can be understoodby transforming the light-matter coupling Hamiltonian Eq. (2) in terms of a two-bodyretarded interaction between electrons, as in the case of electron-phonon interactionsfor BCS superconductivity [109]. This is done by rearranging the real-time electronGF given by Eq. (9), after having replaced H by the light-matter coupling HamiltonianEq. (2) without counter-rotating terms. The latter is then rewritten exactly as: H I ( τ ) = 12 (cid:88) i (cid:54) = j g (cid:90) dτ (cid:48) D r ( τ − τ (cid:48) ) c † ,i ( τ (cid:48) ) c † ,j ( τ ) c ,j ( τ ) c ,i ( τ (cid:48) ) + h . c ., where D r ( τ − τ (cid:48) ) is the real-time retarded photon GF entering Eq. (17). In this form, thelight-matter coupling can be interpreted as a retarded dipole-dipole interaction, wherean interband excitation is created on site i and destroyed at later time on site j . Thisplays the role of a retarded, long-wavelength interaction mediated by the cavity mode,which induces long-range correlations ∝ g . Note that the overall correlations can bedescribed by including the static Coulomb repulsion, which will not be addressed in thispaper. κ/W (cid:28) κ/W (cid:28)
1, and in particular on the“individual dressing regime” occuring when the coupling strength g is smaller than thetypical separation between two adjacent Bloch states in the upper band [57]. Afterhaving characterized the transmission spectrum and the cavity photon DOS using theNGFs method in Sec. 3.5.1, we compare the electron density profiles along the chain avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. Frequency domain calculations usingthe NGFs method are presented on Fig. 9, for an example with N = 11, t = 5 × − ,Γ = 5 × − , t = 0 . κ = 10 − ( κ/W ≈ . × − ), and g = 2 . × − (Γ c / Γ ≈ δω (cid:46) .
05, and each transition between the states of the lower band (notresolved) and the different Bloch states of the upper band can therefore be addressedindividually by the narrow cavity mode.
As already mentioned, we focus on the situationwhere the cavity mode is resonant with the transition between the flat lower band andthe Bloch state lying in the center of the upper band ( k = ( N + 1) / for N odd). Thelatter corresponds to a spatial half-period of two sites with the maximum Bloch velocity2 t [Fig. 9 a) ]. a) B l o c h w a v e v e c t o r F r equen cy C a v i t y m ode -0.7 -0.6 -0.5 -0.4 -0.310 -5 -5 F r equen cy T r an s m i ss i on d) F r equen cy T r an s m i ss i on c) -5 b) C a v i t y pho t on D O S F r equen cy Figure 9. a)
Sketch of the energy bands in the individual dressing regime with t = 0 . κ = 10 − . δω denotes the typical energy spacing between adjacent Blochstates in the upper band. b) (Log-scale) Cavity photon DOS A c ( ω ). c) (Log-scale)Transmission spectrum T ( ω ) in the vicinity of the lower orbital energy ω = − . d) (Log-scale) Transmission spectrum T ( ω ) in the vicinity of the upper orbital energy ω = 0 .
5. The black lines correspond to g = 0, while the red, blue, and green linescorrespond to g = 2 . × − . The other parameters are identical to that of Fig. 6( N = 11, t = 5 × − , and Γ = 5 × − ). avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. b) in this regime.The bare cavity mode centered at ω (black line) is dressed by the resonant interbandtransition resulting in a broadened cavity resonance, as well as small satellite peaksoriginating from the dressing of the detuned interband transitions (green line). Thetransmission spectrum T α ( ω ) is shown on Fig. 9 c) and d) , for g = 0 (thin black line), g = 2 . × − (colored lines), in the vicinity of ω and ω , respectively. Similarly as inthe dissipative regime, we observe a broad peak centered at ω [Fig. 9 c) ], responsible forthe current enhancement (∆ J ≈ . ω , the peak corresponding tothe resonant Bloch state in the upper band is reduced compared to the case g = 0 [hardlyvisible on Fig. 9 d) ]. In this regime, the light-matter coupling is clearly dominated bythis resonant Bloch state (note the log-scale). The other small peaks are reminiscent ofthe off-resonant Bloch states that are only weakly coupled to the cavity field. The particular bandhybridization occuring in the individual dressing regime can be further investigatedby computing the electron populations along the chain (NGFs method), and compareit to the population profile in the dissipative regime. This is represented on Fig. 10 for g = 2 . × − and N = 11. P opu l a t i on Position b)a) P opu l a t i on Position
Figure 10. a) - b) Spatial profile of the electron population n αj for g = 2 . × − and N = 11. Populations in the lower ( α = 1) and upper ( α = 2) orbitals are respectivelydepicted as red and blue squares. a) Dissipative regime with t = 5 × − and κ = 0 . b) Individual dressing regime with t = 0 . κ = 10 − . Other parametersare identical to that of Fig. 6. First, in the dissipative regime [Fig. 10 a) ], the current enhancement associatedwith the new transmission channel in the vicinity of ω can be interpreted as a transferof population from the upper to the lower band [see Sec. 3.4.1]. On Fig. 10 a) , we observethat the lower orbital populations strongly increase when g (cid:54) = 0, while the upper bandpopulations slightly decrease. In this regime, large photonic losses are responsible for aglobal (collective) transfer of populations down to the lower band. On the other hand,the population n N in the lower level of the last site is depopulated due to the couplingto the drain. Importantly, for g (cid:54) = 0, n α,N (cid:54) = 1 − n α, (as it was the case for g = 0),and the partial currents J and J resulting from the integration of T ( ω ) and T ( ω ) do avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. e Γ n ,N and e Γ n ,N as one could have naively expectedfrom Eq. (10). This implies that for g (cid:54) = 0, J and J can not be interpreted as twoindependent currents respectively flowing through the lower and the upper orbitals, asa result of band hybridization.In the individual dressing regime [Fig. 10 b) ], however, the density profiles exhibitssmall oscillations with a period of two sites consistent with the resonant coupling ofthe central Bloch state ( k = ( N + 1) / d) , but with alarger effective hopping t (cid:48) reducing (increasing) the population of the first (last) site. Inthis case, the current enhancement can be associated with a coherent hopping dynamics,sustained by the absorption and emission of cavity photons.
To further evidence the existence of a coherent dynamics in the individual dressingregime, we compare the spectral function A (1) j ,j ( τ ) introduced in Sec. 3.1 in the collective(dissipative) and the individual dressing regimes. This function is computed usingNGFs, and shown on Fig. 11, with j = 1, g = 2 . × − , N = 11, and the sameother parameters as in Fig. 6. In the dissipative regime [Fig. 11 a) ], a particle injectedat the first site for g (cid:54) = 0 stays essentially localized, and no propagation occurs throughthe lower band whatsoever, not even with the small hopping rate t as in the case g = 0[Fig. 3 e) ]. In this case, the dynamics consists of a collective damping of populationsfrom the upper to the lower band, involving localized states (superpositions of differentBloch states). Pictorially, the large photon damping rate constantly projects the systemonto its initial state (similarly to the quantum Zeno effect [110, 111]), thereby preventingthe hopping through the chain to occur. . . . . . . . . . . . . . . . . . . C oup li ng s t r eng t h T i m e P o s i t i on b) c) T i m e P o s i t i on a) Figure 11. a) - b) Contour plot of the spectral function A (1) j ,j ( τ ) of the lower bandas a function of position and time. An electron is injected in the lower level at site j = 1 and time τ = 0. The chain length is N = 11 and the coupling strength g = 2 . × − . a) Dissipative regime with t = 5 × − , and κ = 0 . b) Individualdressing regime with t = 0 . κ = 10 − . A transfer of spectral weight occurs ata time T represented as a vertical line. c) The time T is represented as a function ofthe coupling strength. Other parameters are identical to that of Fig. 6. In the individual dressing regime [Fig. 11 b) ], however, we observe a small transferof spectral weight ≈
10% occuring after a time T with a period of two sites, which showsthat the properties of the resonant upper band Bloch state are transfered in the lower avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. ∼ ω delocalized across the whole chain.In this sense, this corresponds to an effective hopping mechanism restoring propagationin the quasi-blocked lower band. However, one can not a priori write an Hamiltonianterm which reproduces this single-Bloch-state dynamics, as nearest neighbors hoppingin a 1D chain involves the complete set of the chain Bloch states. Finally, we find thatthe spectral weight transfer induced by the coupling to the cavity mode occurs at a time T ∼ /g , which corresponds to the time to emit a photon concurrently with the transferof populations to the lower band [Fig. 11 c) ].Similarly as in the dissipative regime, we find that two polariton peaks appear inthe cavity photon DOS (outside the upper electronic bandwidth), when g exceeds thetypical energy separation δω between two adjacent Bloch states in the upper band. Inthe individual dressing regime, however, Ω S (cid:54) = Ω n indicating that the dynamics doesnot involve a collective response of the Bloch states [57]. Ultimately, for κ (cid:28) W and g (cid:29) δω , all Bloch states are coupled to the cavity mode, and we expect to recover thephysics of the collective dressing regime [see Fig. 1 c) ]. We point out, however, thata quantitative study is difficult as neither the effective QME nor the NGFs methodare valid in this strongly non-perturbative regime. The full QME is the only suitablemethod, but identifying collective effects is hard since this method is in any case limitedto small N . N As concluding remarks, we have checked that the mean cavity photon number in thesteady-state ¯ n = (cid:104) a † a (cid:105) (see end of Sec. 2.2) remains small even for large coupling strength(¯ n (cid:46) − in the dissipative regime, and ¯ n (cid:46) N P inthe bath, in which case we find that the current enhancement depends on the rescaledcoupling strength g √ N P . On the other hand, order-of-magnitude current enhancementscan occur when considering different injection/extraction rates Γ (cid:54) = Γ for the two bands(e.g. for Γ (cid:29) Γ ), as well as a small photon population N P (cid:46)
1. In this case, stillconsidering t (cid:28) Γ and t (cid:29) Γ , the small injection/extraction rate in the upper bandprovides a strong reduction of the bare current ≈ e Γ / g = 0, leading tocurrent enhancements only limited by the ratio Γ / Γ when g (cid:54) = 0 [57].For given g and κ , we find that the saturation value for the steady-state currentdecreases sublinearly when increasing the chain length N , restricting the scope of ourstudy to mesoscopic systems. In addition, the current typically exhibits small oscillationsbetween odd and even values of N , with slightly larger values for N odd. The existenceof a Bloch state resonant with the cavity mode for N odd leads to a slightly largercurrent enhancement than for N even, where the two closest states to the upper bandcenter ω are only quasi-resonant with the cavity mode. In the limit N (cid:29)
1, theseparation between adjacent Bloch states close to the upper band center is ∼ πt /N . avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. κ is thus ∼ N κ/t , and one can conclude that the individual band dressing in the large N regime is only limited by the cavity quality factor. Ultimately, when N κ/t (cid:29) , thesystem always enters the collective dressing regime.
4. Conclusion
In conclusion, we have studied in detail the interplay between the transport of fermionsthrough a 1D mesoscopic chain of two-orbital systems and light-matter coupling toa single cavity mode close to its vacuum state. We have derived both analytical andnumerical results using complementary methods based on Keldysh and QME techniques,providing new perspectives for the investigation of many-body fermionic systems coupledto confined photons. We have compared the steady-state current obtained with thesedifferent methods, and shown that light-matter coupling leads to a current enhancement.Depending on the ratio between the cavity photon decay rate and the upper electronicbandwidth, different regimes have been identified and discussed. In the dissipativeregime, we have derived an analytical formula for the current enhancement valid for smallcoupling strengths, showing that the current enhancement scales with the cooperativity.We have characterized the presence of a collective coupling of all the Bloch statesto the cavity mode, when the coupling strength is larger than the typical energyseparation between two adjacent Bloch states in the upper band. In this case, thecurrent enhancement is shown to stem from a global transfer of populations from theupper to the lower band, with only marginal propagation through the latter. In thecoherent regime, however, we have shown that when the coupling strength is smallerthan the typical energy separation between two adjacent Bloch states in the upper band,only the resonant Bloch state is “individually” coupled to the cavity mode. Moreover,a small transfer of spectral weight occurs from the upper to the lower band, resultingin a new state with energy ∼ ω delocalized across the whole chain. In this case,the current enhancement has been interpreted as stemming from a coherent hoppingdynamics sustained by the absorption and emission of cavity photons. Ultimately, whenthe coupling strength becomes larger than the upper electronic bandwidth, or when thesystem size becomes large, we expect to recover the collective dressing regime.In a realistic situation, additional random potentials due to disorder and impuritieswill affect transport properties through the chain. In the presence of light-mattercoupling at optical frequencies, orbitals are separated by a large gap ∼ g = 0. Possible extensions ofthis model include considering a frequency-dependent leads coupling and/or cavitydecay rate to study how non-Markovian (memory) effects affect our results. Further avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. g (cid:54) = 0. In this situation, charge transport can be reduced as thesystem exhibits interference between the different quantum paths connecting the sameorbital at two distant sites for some specific coupling g . It would thus be interesting tostudy how this competes with the time-reversed loop trajectories leading to Andersonlocalization in random lattices [112]. Our model might find direct applications in severalfields, such as transport in organic semiconductors [56] and quantum dot arrays [58, 59,63, 64], which have recently been coupled to surface plasmon resonators [41, 42, 56] andmicrowave cavities [60–62]. Acknowledgements
We are grateful to Stefano Azzini, Thibault Chervy, Roberta Citro, Thomas Ebbesen,Cyriaque Genet, Emanuele Orgiu, and Paolo Samor`ı for fruitful discussions. Workin Strasbourg was supported by the ERC St-Grant ColDSIM (No. 307688), withadditional funding from Rysq and ANR-FWF grant BLUESHIELD. C.G. acknowledgessupport from the Max Planck Society and from the COST action NQO 1403 (Nano-scaleQuantum Optics). This work is supported by IdEx Unistra (project STEMQuS) withfunding managed by the French National Research Agency as part of the “Investmentsfor the future program”.
Appendix A. Keldysh formalism
In this appendix, we propose a detailed derivation of the results presented in Sec. 2.2.We first write the steady-state current in terms of electron GFs, and then show thatelectron and photon GFs can be computed by solving a closed set of equations involvingelectron and photon SEs. We consider (cid:126) = 1, and use the short-hand notations ∂ τ ≡ ∂∂τ and δ f ( τ ) ≡ δδf ( τ ) , for function and functional derivatives, respectively. Steady-state current.
As seen in Sec. 2.2, the steady-state current J η flowingthrough the lead η is proportional to the commutator between the total Hamiltonian H and the number of electrons in the lead η . A direct calculation of this commutatorallows us to express J η in terms of a GF which describes the correlations between theleads and the chain: J η = − e (cid:88) α,k (cid:88) q ϕ j η k λ α, q (cid:90) dω π (cid:60) (cid:2) G <α,k, q ,η ( ω ) (cid:3) , (A.1)where (cid:60) stands for real part, λ α, q is defined in Eq. (6), and G <α,k, q ,η ( ω ) denotes theFourier transform of the “lesser” mixed system-leads GF G <α,k, q ,η ( τ − τ (cid:48) ), which can beobtained from the time-ordered GF: G α,k, q ,η ( τ − τ (cid:48) ) = − i (cid:104)T ˜ c α,k ( τ ) b † α, q ,η ( τ (cid:48) ) (cid:105) . (A.2) avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. T denotes the time-ordered product for fermions. Taking the time derivative ∂ τ (cid:48) ofEq. (A.2), and computing the different commutators entering the Heisenberg equation ∂ τ (cid:48) b † α, q ,η ( τ (cid:48) ) = i [ H, b † α, q ,η ]( τ (cid:48) ), the equation of motion of G α,k, q ,η ( τ − τ (cid:48) ) is derived as:( − i∂ τ (cid:48) − ω q ) G α,k, q ,η ( τ − τ (cid:48) ) = − λ α, q (cid:88) k (cid:48) ϕ j η k (cid:48) G α,k,k (cid:48) ( τ − τ (cid:48) ) , (A.3)where G α,k,k (cid:48) ( τ − τ (cid:48) ) = − i (cid:104)T ˜ c α,k ( τ )˜ c † α,k (cid:48) ( τ (cid:48) ) (cid:105) is the time-ordered GF of the chain, referedto as the “electron GF”. Equation (A.3) can be formally solved in the frequency domainas: G α,k, q ,η ( ω ) = − λ α, q (cid:88) k (cid:48) ϕ j η k (cid:48) G α,k,k (cid:48) ( ω ) G q ,η ( ω ) , (A.4)where G α,k,k (cid:48) ( ω ) and G q ,η ( ω ) denote the Fourier transforms of the electron GF and thelead GF G q ,η ( τ − τ (cid:48) ) = − i (cid:104)T b α, q ,η ( τ ) b † α, q ,η ( τ (cid:48) ) (cid:105) , and (cid:104)· · · (cid:105) refers to the quantum averagein the ground state of the Hamiltonian H without the interaction terms H I , H L , and H P . One can then use the Langreth rules [77] in Eq. (A.4) to compute the “lesser” GF: G <α,k, q ,η ( ω ) = − λ α, q (cid:88) k (cid:48) ϕ j η k (cid:48) G rα,k,k (cid:48) ( ω ) G < q ,η ( ω ) − λ α, q (cid:88) k (cid:48) ϕ j η k (cid:48) G <α,k,k (cid:48) ( ω ) G a q ,η ( ω ) , (A.5)with r and a for retarded and advanced GFs, respectively. Using the results: G < q ,η ( ω ) = 2 iπδ ( ω − ω q ) n η ( ω ) G a q ,η ( ω ) = 1 ω − ω q − i + , (A.6)where 0 + denotes an infinitesimal positive quantity and n η ( ω ) is the Fermi occupationnumber of the lead η , we substitute Eq. (A.5) in the expression of the current Eq. (A.1),and convert the summation over q into a frequency integral (cid:80) q → (cid:82) ∞ dωρ ( ω ), where ρ ( ω ) represents the electron density of states in the leads. Introducing the tunnellingrate between the chain and the leads as Γ α = 2 πρ ( ω ) λ α ( ω ) (assumed to be energyindependent), we finally recover Eqs. (3) and (4). Note that we have assumed n s ( ω ) = 1and n d ( ω ) = 0 ∀ ω (high-bias regime). Dyson equation for electrons GFs.
In order to compute the transmissionspectrum Eq. (4), we now need an equation of motion of the time-ordered electron GFs.As before, we compute the time derivative ∂ τ G α,k,k (cid:48) ( τ − τ (cid:48) ), use the Heisenberg equation ∂ τ ˜ c α,k ( τ ) = i [ H, ˜ c α,k ]( τ ), and obtain:( i∂ τ − ω α,k ) G α,k,k (cid:48) ( τ − τ (cid:48) ) = δ k,k (cid:48) δ ( τ − τ (cid:48) ) + g (cid:88) α (cid:48) (1 − δ α,α (cid:48) ) F α (cid:48) ,k,α,k (cid:48) ( τ − τ (cid:48) ) − (cid:88) q ,η λ α, q ϕ j η k G q ,η,α,k (cid:48) ( τ − τ (cid:48) ) , (A.7) avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. G q ,η,α,k ( τ − τ (cid:48) ) = − i (cid:104)T b α, q ,η ( τ )˜ c † α,k ( τ (cid:48) ) (cid:105) is a mixed system-leads GF similar to theone defined in Eq. (A.2), and F α (cid:48) ,k,α,k (cid:48) ( τ − τ (cid:48) ) = − i (cid:104)T ˜ c α (cid:48) ,k ( τ )˜ c † α,k (cid:48) ( τ (cid:48) ) A ( τ ) (cid:105) is a higher-order correlation function mixing the electronic and photonic degrees of freedom. First,the equation of motion for the Fourier transform G q ,η,α,k ( ω ) is derived similarly as beforeand reads: G q ,η,α,k ( ω ) = − λ α, q (cid:88) k (cid:48) ϕ j η k (cid:48) G q ,η ( ω ) G α,k (cid:48) ,k ( ω ) . (A.8)Secondly, the correlation function F α (cid:48) ,k,α,k (cid:48) ( τ − τ (cid:48) ) can be written in terms of single-particle GFs by considering a term H (cid:48) = J A in the Hamiltonian, where J denotes avanishing current source [78]. Taking the functional derivative δ J ( τ ) G α (cid:48) ,k,α,k (cid:48) ( τ − τ (cid:48) ),where G α (cid:48) ,k,α,k (cid:48) ( τ − τ (cid:48) ) is given by: G α (cid:48) ,k,α,k (cid:48) ( τ − τ (cid:48) ) = − i (cid:104)T ˜ c α (cid:48) ,k ( τ )˜ c † α,k (cid:48) ( τ (cid:48) ) e − i (cid:82) dτ H ( τ ) (cid:105) (cid:104) e − i (cid:82) dτ H ( τ ) (cid:105) , (A.9)we obtain: F α (cid:48) ,k,α,k (cid:48) ( τ − τ (cid:48) ) = ig (cid:88) k ,k (cid:90) { dτ } G α (cid:48) ,k,k ( τ − τ )Λ α (cid:48) ,k ,α,k ( { τ } ) D ( τ − τ ) G α,k ,k (cid:48) ( τ − τ (cid:48) ) , (A.10)where { τ } ≡ τ , τ , τ , (cid:82) { dτ } ≡ (cid:82) dτ (cid:82) dτ (cid:82) dτ . The time-ordered photon GF is definedas: D ( τ − τ ) = δ J ( τ ) (cid:104) A ( τ ) (cid:105) = − i (cid:104)T A ( τ ) A ( τ ) (cid:105) , and the so-called vertex function as:Λ α (cid:48) ,k ,α,k ( τ , τ , τ ) = − g δ (cid:104) A ( τ ) (cid:105) G − α (cid:48) ,k ,α,k ( τ − τ ) . It can be shown that this vertex function satisfies a self-consistent equation [78].The SCBA consists in considering only the leading term (undressed vertex) of this self-consistent equation, which provides:Λ α (cid:48) ,k (cid:48) ,α,k ( τ, τ (cid:48) , τ (cid:48)(cid:48) ) = (1 − δ α (cid:48) ,α ) δ k,k (cid:48) δ ( τ − τ (cid:48) ) δ ( τ − τ (cid:48)(cid:48) ) . (A.11)Higher order corrections in Λ correspond to the so-called vertex correctionsassociated with crossed diagrams [78] such as the one sketched on Fig. 2 b) , which areneglected in the SCBA. Using Eqs. (A.8), (A.10), and (A.11), the equation of motion(A.7) written in the frequency domain takes the form: avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. (cid:88) k (cid:0) ( G α,k,k ( ω )) − − Σ α,k,k ( ω ) (cid:1) G α,k ,k (cid:48) ( ω ) = δ k,k (cid:48) , with the SCBA self-energy:Σ α,k,k (cid:48) ( ω ) = ig (1 − δ α,α (cid:48) ) (cid:90) dω (cid:48) π G α (cid:48) ,k,k (cid:48) ( ω + ω (cid:48) ) D ( ω (cid:48) ) + (cid:88) q ,η λ α, q ϕ j η k ϕ j η k (cid:48) G q ,η ( ω ) , (A.12)and the non-interacting time-ordered GF G α,k,k (cid:48) ( ω ). Still considering the high-biasregime, we now use the Langreth rules together with Eq. (A.6), and convert thesummation over q in Eq. (A.12) into a frequency integral. This leads to the expressionsof the “lesser” and “greater” electron SEs given by Eqs. (14) and (15). Dyson equation for photons GFs.
The equation of motion for the time-orderedphoton GF D ( ω ) can be derived by taking the second time derivative of the cavity vectorpotential A ( t ), and then use the Heisenberg equation ∂ τ A ( τ ) = i [ H, A ]( τ ) two times ina row. As in the previous section, we consider a vanishing source term H (cid:48) = J A inthe Hamiltonian H . The functional derivative of the ground-state expectation of theobtained equation with respect to J ( τ (cid:48) ) yields the following equation of motion for D ( τ − τ (cid:48) ): (cid:18) − ∂ τ ω c − ω c (cid:19) D ( τ − τ (cid:48) ) = δ ( τ − τ (cid:48) ) − ig (cid:88) α,α (cid:48) (cid:88) k (1 − δ α,α (cid:48) ) δ J ( τ (cid:48) ) G α,k,α (cid:48) ,k ( τ, τ + )+ (cid:88) p µ p D p ( τ − τ (cid:48) ) , (A.13)where the time τ + = τ + 0 + , and the mixed GF D p ( τ − τ (cid:48) ) = − i (cid:104)T A p ( τ ) A ( τ (cid:48) ) (cid:105) describes correlations between the cavity mode and the electromagnetic environment.The equation of motion for D p can be derived similarly as before (by calculating itssecond time derivative): (cid:0) − ∂ τ − ω p (cid:1) D p ( τ − τ (cid:48) ) = 2 ω p µ p D ( τ − τ (cid:48) ) , which is solved in the frequency domain as D p ( ω ) = µ p D p ( ω ) D ( ω ). Here, D p ( ω ) is theFourier transform of the (time-ordered) extra-cavity photon GF − i (cid:104)T A p ( τ ) A − p ( τ (cid:48) ) (cid:105) .Using Eq. (A.9), the second term in the right-hand side of Eq. (A.13) can be written inthe form: δG α,k,α (cid:48) ,k ( τ, τ + ) δ J ( τ (cid:48) ) = g (cid:88) k ,k (cid:90) { dτ } G α,k,k ( τ − τ )Λ α,k ,α (cid:48) ,k ( { τ } ) D ( τ − τ (cid:48) ) G α (cid:48) ,k ,k ( τ − τ + ) , (A.14) avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. α,k,α (cid:48) ,k (cid:48) ( τ, τ (cid:48) , τ (cid:48)(cid:48) ) = (1 − δ α,α (cid:48) ) δ k,k (cid:48) δ ( τ − τ (cid:48) ) δ ( τ − τ (cid:48)(cid:48) ), which we substitutein Eq. (A.14) to put the equation of motion (A.13) into the form: (cid:0) D − ( ω ) − Π( ω ) (cid:1) D ( ω ) = 1with the cavity photon SE:Π( ω ) = − ig (cid:88) α,α (cid:48) Tr (1 − δ α,α (cid:48) ) (cid:90) dω (cid:48) π G α ( ω + ω (cid:48) ) G α (cid:48) ( ω (cid:48) ) + (cid:88) p µ p D p ( ω ) , (A.15)and the bare cavity photon GF D ( ω ). The summation over the continuous index p can again be converted into a frequency integral, namely (cid:80) p → (cid:82) ∞ dωρ ( ω ), where ρ ( ω ) denotes the extra-cavity photon density of states. We introduce the cavity photondecay rate as κ = 2 πρ ( ω ) µ ( ω ) (assumed to be frequency-independent), and use theLangreth rules in Eq. (A.15). Assuming a vanishing mean population in the photonbath, i.e. (cid:104) a † p a p (cid:105) = 0, one can compute the (non-interacting) extra-cavity photon GFsas D > p ( ω ) = − iπδ ( ω − ω p ) and D < p ( ω ) = − iπδ ( ω + ω p ), and show that the “lesser”and “greater” photon SEs correspond to Eqs. (21) and (22). Appendix B. Elimination of the cavity field
In this appendix, we show that ˆ ρ – the projection of the density operator ˜ ρ (in therotating frame) onto the cavity vacuum state – evolves according to Eq. (30) in thedissipative regime. Dissipative regime.
We consider the case when the cavity decay rate κ is largecompared to the other rates. In particular, κ is larger than the injection/extraction ratesΓ α and tunneling rates t α governing the uncoupled evolution of the electronic degreesof freedom, and larger than the coupling strength g between electronic and bosonicvariables. This choice has two main consequences: • We expect the strongly damped cavity field to stay close to its vacuum state (steadystate for g = 0). • The electrons’ observables evolve on a much longer time-scale than the oneassociated with the cavity field.The last point allows us to adiabatically eliminate the light field from the overalldynamics. For this purpose, we first define the electron reduced density operator:˜ ρ el = Tr F [ ˜ ρ ] = (cid:88) n (cid:104) n | ˜ ρ | n (cid:105) = (cid:88) n ˜ ρ nn , avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. F [ A ] = (cid:80) n (cid:104) n | A | n (cid:105) denotes the trace of the observable A over the cavity field, and | n (cid:105) with n = 0 , , , · · · is the photonic part of the (Fock) state containing n photons. As alreadymentioned, we assume that the light field is close to its vacuum state, i.e. ˜ ρ el (cid:39) ˜ ρ . Inthe following, we derive a closed time-evolution for the relevant part ˜ ρ of the reduceddensity operator [98, 99]. Projectors and coupled differential equations.
We introduce the projectors P and Q with P ˜ ρ = (cid:104) | ˜ ρ | (cid:105) | (cid:105) (cid:104) | = ˜ ρ | (cid:105) (cid:104) | ≡ ˆ ρ,Q ˜ ρ = (cid:88) n,mn,m (cid:54) =0 (cid:104) n | ˜ ρ | m (cid:105) | n (cid:105) (cid:104) m | = (cid:88) n,mn,m (cid:54) =0 ˜ ρ nm | n (cid:105) (cid:104) m | ≡ ˇ ρ. Using the decomposition (see Sec. 2.4): ∂ τ ˜ ρ = ( L e + L c + L I + I c ) ˜ ρ, with I c = κa ˜ ρa † , together with the property P + Q = 1, the coupled differential equationsfor ˆ ρ and ˇ ρ can be written as: ∂ τ ˆ ρ = P L e ˆ ρ + P ( L I + I c ) ˇ ρ, (B.1) ∂ τ ˇ ρ = Q L I ˆ ρ + Q ( L e + L c + L I + I c ) ˇ ρ. (B.2)The formal solution of Eq. (B.2) is given by:ˇ ρ ( τ ) = e Q ( L c + L e ) δτ ˇ ρ ( τ ) + (cid:90) ττ dτ (cid:48) e Q ( L c + L e )( τ − τ (cid:48) ) V ( τ (cid:48) ) , (B.3)with V ( τ (cid:48) ) = Q L I ˆ ρ ( τ (cid:48) ) + Q ( L I + I c ) ˇ ρ ( τ (cid:48) ) , and δτ = τ − τ . The formal solution Eq. (B.3) can be plugged into Eq. (B.1), andkeeping terms up to second order in L I , we obtain: ∂ τ ˆ ρ (cid:39) P L e ˆ ρ + P L I (cid:90) ττ dτ (cid:48) e Q ( L c + L e )( τ − τ (cid:48) ) Q L I ˆ ρ ( τ (cid:48) )+ P I c (cid:90) ττ dτ (cid:48) e Q ( L c + L e )( τ − τ (cid:48) ) Q L I (cid:90) τ (cid:48) τ dτ (cid:48)(cid:48) e Q ( L c + L e )( τ (cid:48) − τ (cid:48)(cid:48) ) Q L I ˆ ρ ( τ (cid:48)(cid:48) ) , (B.4) avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. ρ ( τ ) = 0 (cavity intially prepared in its vacuum state). Time-scale separation and integration.
We first focus on the second term inthe right-hand side of Eq. (B.4), which, after change of variables, reads: P L I (cid:90) δτ dτ (cid:48) e Q ( L c + L e ) τ (cid:48) Q L I ˆ ρ ( τ − τ (cid:48) ) . (B.5)Letting the operator Q L I act on ˆ ρ , one obtains: Q L I ˆ ρ ( τ − τ (cid:48) ) = − igS − ˜ ρ ( τ − τ (cid:48) ) | (cid:105) (cid:104) | + h.c. , (B.6)with the collective lowering operator S − = (cid:80) j c † ,j c ,j , ( S + = ( S − ) † ). Subsequently,according to Eq. (B.5), we apply the free evolution exp( Q ( L c + L e ) τ (cid:48) ) to the previousexpression Eq. (B.6): (cid:90) δτ dτ (cid:48) e Q ( L c + L e ) τ (cid:48) Q L I ˆ ρ ( τ − τ (cid:48) ) (cid:39) (cid:90) δτ dτ (cid:48) ( − ig ) S − ˜ ρ ( τ − τ (cid:48) ) | (cid:105) (cid:104) | e ( i ∆ − κ ) τ (cid:48) + h.c. , (B.7)where we have used e Q ( L c + L e ) τ (cid:48) ≈ e Q L c τ (cid:48) in the integrand. This approximation is justifiedin the dissipative regime where | i ∆ − κ/ | (cid:29) Γ α , t α . Corrections to the previousapproximation could be taken into account, e.g. by using partial integration. Theyare expected to scale with t α / | i ∆ − κ/ | and Γ α / | i ∆ − κ/ | and are small whenever thelight-field evolves on a much shorter time-scale than the electronic degrees of freedom.The time-scale separation allows us to further neglect the variation of ˜ ρ during therelaxation time ∼ /κ of the cavity, namely:˜ ρ ( τ − τ (cid:48) ) e ( i ∆ − κ ) τ (cid:48) ≈ ˜ ρ ( τ ) e ( i ∆ − κ ) τ (cid:48) . (B.8)We point out that since the evolution of ˜ ρ is governed by both the electronicterm L e ˜ ρ in Eq. (B.4), and the photon-mediated effective dynamics which we aim atcalculating, checking the assumption Eq. (B.8) will be required (for consistency) at theend of the calculation. Under these approximations, the second term in the right-handside of Eq. (B.4) takes the form: P L I (cid:90) δτ dτ (cid:48) e Q L c τ (cid:48) Q L I ˆ ρ ( τ ) = − i Γ ∆ (cid:16) S + S − ˆ ρ ( τ ) (cid:16) − e ( i ∆ − κ ) δτ (cid:17) − h.c. (cid:17) − κ (cid:16) S + S − ˆ ρ ( τ ) (cid:16) − e ( i ∆ − κ ) δτ (cid:17) + h.c. (cid:17) , (B.9)where Γ ∆ = g ∆2∆ + κ / and Γ κ = g κ + κ . We now turn to the third term in the right-handside of Eq. (B.4). After change of variables, integration provides: avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. P I c (cid:90) δτ dτ (cid:48) e Q L c τ (cid:48) Q L I (cid:90) δτ − τ (cid:48) dτ (cid:48)(cid:48) e Q L c τ (cid:48)(cid:48) Q L I ˆ ρ ( τ ) = 4Γ κ S − ˆ ρ ( τ ) S + (cid:16) e − κδτ − e − κδτ cos(∆ δτ ) (cid:17) , (B.10)where we have used ˜ ρ ( τ − τ (cid:48) − τ (cid:48)(cid:48) ) (cid:39) ˜ ρ ( τ ) and made use of similar considerations asfor the second term in the right-hand side of Eq. (B.4). Collecting the two contributionsEqs. (B.9) and (B.10), and substituting them in Eq. (B.4), we finally obtain: ∂ τ ˆ ρ = P L e ˆ ρ − i Γ ∆ (cid:16) S + S − ˆ ρ ( τ ) (cid:16) − e ( i ∆ − κ ) δτ (cid:17) − h.c. (cid:17) − κ (cid:16) S + S − ˆ ρ ( τ ) (cid:16) − e ( i ∆ − κ ) δτ (cid:17) + h.c. (cid:17) + 4Γ κ S − ˆ ρ ( τ ) S + (cid:16) e − κδτ − e − κδτ cos(∆ δτ ) (cid:17) . (B.11)This result shows that the photon-mediated dynamics of the electrons scales withΓ ∆ and Γ κ . Those rates should be small compared to κ in order to use the time-scaleseparation, and in particular the approximation Eq. (B.8). This provides an upperbound for the coupling strength g . We point out that neglecting further correctionsscaling with Γ ∆ and Γ κ in Eq. (B.8) is consistent with the approximation of keepingterms only up to second order in L I in Eq. (B.4). In the regime κδτ (cid:29)
1, the effectivemaster equation (30) can be finally derived from Eq. (B.11): ∂ τ ˆ ρ ≡ L red ˆ ρ = L e ˆ ρ − i Γ ∆ [ S + S − , ˆ ρ ] − κ (cid:0) S + S − ˆ ρ + ˆ ρS + S − − S − ˆ ρS + (cid:1) . (B.12) Coarse graining and discussion.
In the regime of parameters considered here,a coarse-grained time scale ∆ τ satisfying: κ − (cid:28) ∆ τ (cid:28) t − α , Γ − α , (B.13)can be introduced. Using Eqs. (B.11) and (B.13), one can show that the evolution ofthe reduced density operator on the time scale ∆ τ is:ˆ ρ ( τ + ∆ τ ) − ˆ ρ ( τ )∆ τ = τ +∆ τ (cid:90) τ dτ (cid:48) ∂ τ (cid:48) ˆ ρ ( τ (cid:48) )∆ τ (cid:39) L red ˆ ρ ( τ ) , (B.14)with τ ≥ τ , and where consistently with the approximations used in Eqs. (B.7) and(B.8), contributions ∼ ( κ ∆ τ ) − and ∼ Γ α ∆ τ, t α ∆ τ have been neglected. The effectivemaster equation (30) is well-established on such a footing, and it is not suitable todescribe the dynamics occuring on time-scales smaller than 1 /κ . In addition, ∆ τ hasto be small compared to the time-scale associated with the photon-mediated dynamics, avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. τ ( L red − L e ) ˆ ρ ( τ ) is negligible. Together with the condition ∆ τ (cid:29) κ − , thisprovides an upper bound for the coupling strength g , whose exact form depends on thestates that are involved in the dynamics, and whether collective effects play a role ornot. In coupled spin-cavity systems (with ∆ = 0), the condition: √ N g (cid:28) κ (B.15)has been considered sufficient, or even required [98]. We expect that the conditionEq. (B.15) is also sufficient in our fermionic model to use the time-scale separation. Asa matter of fact, since quantum states with both orbitals either empty or fully occupiedare not coupled to light, the collective coupling constant in our open fermionic modelis < g √ N (see Sec. 3.4.2), which places us on the safe side regarding Eq. (B.15). Notethat the equations of motion (B.12) and (B.14) are given in the adiabatic limit, andretardation effects between cavity and electronic dynamics [103] are neglected. Weconclude this appendix by a short discussion on how to compute the mean photonnumber of the cavity mode, when the latter can be considered as close to its vacuumstate. Photon number.
In the adiabatic limit considered above, and for ∆ = 0, it canbe shown that the mean photon number is well-approximated by the formula [98, 102]: (cid:104) ˆ a † ˆ a (cid:105) (cid:39) g ( κ/ (cid:104) S + S − (cid:105) . (B.16)Drawing the conjecture (cid:104) S + S − (cid:105) = (cid:88) i (cid:104) s + i s − i (cid:105) + (cid:88) i (cid:54) = j (cid:104) s + i s − j (cid:105) ≤ (cid:88) i (cid:104) s + i s − i (cid:105) from numerical simulations, and using Eq. (B.16) together with (cid:104) s + i s − i (cid:105) = (cid:104) ˆ n i (1 − ˆ n i ) (cid:105) ≤ , we obtain (cid:104) ˆ a † ˆ a (cid:105) (cid:54) Ng ( κ/ . Restricting the light-matter coupling strength to values suchthat the condition Eq. (B.15) is fulfilled, one can thus reasonably expect the cavity modeto stay close to its vacuum state, consistently with the time-scale separation argumentdiscussed before. avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. References [1] Plenio M B and Huelga S F 2008
New Journal of Physics New Journal of Physics New Journal of Physics Reports on Progress in Physics Phys. Rev. Lett. (19) 196403[6] Feist J and Garcia-Vidal F J 2015
Phys. Rev. Lett. (19) 196402[7] Tavis M and Cummings F W 1968
Phys. Rev. (2) 379–384[8] Zhong X, Chervy T, Wang S, George J, Thomas A, Hutchison J A, Devaux E, Genet C andEbbesen T W 2016
Angewandte Chemie International Edition Many-Particle Physics
Phys. Z. Sowjetunion US AEC Report AEC-tr-5575 [12] Fr¨ohlich H 1954
Advances in Physics Proceedings of the Royal Society of London A: Mathematical, Physical andEngineering Sciences
Phys. Rev. (4) 1189–1190[15] Bardeen J, Cooper L N and Schrieffer J R 1957
Phys. Rev. (5) 1175–1204[16] Peierls R 1996
Quantum Theory of Solids
International Series of Monographs on Physics(Clarendon Press) ISBN 9780192670175[17] Fr¨ohlich H 1954
Proceedings of the Royal Society of London A: Mathematical, Physical andEngineering Sciences
Phys. Rev. B (7) 075317[19] Lindner N H, Refael G and Galitski V 2011 Nature Physics Science (6223) 743–746[21] Mitrano M, Cantaluppi A, Nicoletti D, Kaiser S, Perucchi A, Lupi S, Di Pietro P, Pontiroli D,Ricc M, Clark S R, Jaksch D and Cavalleri A 2016
Nature (7591) 461–464[22] Rajasekaran S, Casandruc E, Laplace Y, Nicoletti D, Gu G D, Clark S R, Jaksch D and CavalleriA 2016
Nature Physics Phys. Rev. B (1) 014512[24] Sentef M A, Tokuno A, Georges A and Kollath C 2017 Phys. Rev. Lett. (8) 087002[25] Schlawin F, Dietrich A S D, Kiffner M, Cavalleri A and Jaksch D 2017
Phys. Rev. B (6) 064526[26] Kennes D M, Wilner E Y, Reichman D R and Millis A J 2017 Phys. Rev. B (5) 054506[27] Mazza G and Georges A 2017 Phys. Rev. B (6) 064515[28] Mani R G, Smet J H, von Klitzing K, Narayanamurti V, Johnson W B and Umansky V 2002 Nature
Phys. Rev. Lett. (4) 046807[30] Durst A C, Sachdev S, Read N and Girvin S M 2003 Phys. Rev. Lett. (8) 086803[31] Dmitriev I A, Vavilov M G, Aleiner I L, Mirlin A D and Polyakov D G 2005 Phys. Rev. B (11)115316[32] Tsintzos S I, Pelekanos N T, Konstantinidis G, Hatzopoulos Z and Savvidis P G 2008 Nature (7193) 372–375[33] Lagoudakis K G, Wouters M, Richard M, Baas A, Carusotto I, Andre R, Dang L S and Deveaud-Pledran B 2008
Nat. Phys. (9) 706–710[34] Kena-Cohen S and Forrest S R 2010 Nat. Photon. (6) 371–375[35] Cristofolini P, Christmann G, Tsintzos S I, Deligeorgis G, Konstantinidis G, Hatzopoulos Z,Savvidis P G and Baumberg J J 2012 Science avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. Y and Hofling S 2013
Nature (7449) 348–352[37] Javadi A, S¨ollner I, Arcari M, Hansen S L, Midolo L, Mahmoodian S, Kirˇsansk˙e G, PregnolatoT, Lee E H, Song J D, Stobbe S and Lodahl P 2015
Nature Communications Science
Phys. Rev. A (6) 063805[40] Manzoni M T, Chang D E and Douglas J S 2017 Nature Communications (1) 1743[41] Shalabney A, George J, Hutchison J A, Pupillo G, Genet C and Ebbesen T W 2015 NatureCommunications The Journal of PhysicalChemistry Letters Phys. Rev. Lett. (23) 238301[44] Chikkaraddy R, de Nijs B, Benz F, Barrow S J, Scherman O A, Rosta E, Demetriadou A, FoxP, Hess O and Baumberg J J 2016
Nature (7610) 127–130[45] Galego J, Garcia-Vidal F J and Feist J 2016
Nature Communications Phys.Rev. Lett. (21) 3906–3909[47] Kasprzak J, Richard M, Kundermann S, Baas A, Jeambrun P, Keeling J M J, Marchetti F M,Szymanska M H, Andre R, Staehli J L, Savona V, Littlewood P B, Deveaud B and Dang L S2006 Nature (7110) 409–414[48] Amo A, Sanvitto D, Laussy F P, Ballarini D, Valle E d, Martin M D, Lemaitre A, Bloch J,Krizhanovskii D N, Skolnick M S, Tejedor C and Vina L 2009
Nature (7227) 291–295[49] Deng H, Haug H and Yamamoto Y 2010
Rev. Mod. Phys. (2) 1489–1537[50] Carusotto I and Ciuti C 2013 Rev. Mod. Phys. (1) 299–366[51] Kavokin A and Lagoudakis P 2016 Nat. Mater. (6) 599–600[52] Maghrebi M F, Yao N Y, Hafezi M, Pohl T, Firstenberg O and Gorshkov A V 2015 Phys. Rev.A (3) 033838[53] Zhu G, Suba¸sı Y, Whitfield J D and Hafezi M 2017 ArXiv e-prints ( Preprint )[54] Smolka S, Wuester W, Haupt F, Faelt S, Wegscheider W and Imamoglu A 2014
Science (6207)332–335[55] Cotlet¸ O, Zeytinoˇglu S, Sigrist M, Demler E and Imamoˇglu A 2016
Phys. Rev. B (5) 054510[56] Orgiu E, George J, Hutchison J, Devaux E, Dayen J F, Doudin B, F Stellacci F, Genet C,Schachenmayer J, Genes C, Pupillo G, Samori P and Ebbesen T W 2015 Nature Materials Phys. Rev. Lett. (22)223601[58] Kagan C R and Murray C B 2015
Nat. Nano. Appl. Phys. Lett. Phys. Rev. Lett.
Physical Review B Science
Phys. Rev.B (19) 195307[64] Moldoveanu V, Gudmundsson V and Manolescu A 2007 Phys. Rev. B (8) 085330[65] Rurali R 2010 Rev. Mod. Phys. Rev. Mod. Phys. avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. [67] Gudmundsson V, Gainar C, Tang C S, Moldoveanu V and Manolescu A 2009 New Journal ofPhysics Science
Preprint http://science.sciencemag.org/content/337/6098/1069.full.pdf )[69] Laflamme C, Yang D and Zoller P 2017
Phys. Rev. A (4) 043843[70] Devoret M H and Martinis J M 2004 Quantum Information Processing Nature
Nature Physics Rev. Mod. Phys. (2) 375–558[74] Raimond J M, Brune M and Haroche S 2001 Rev. Mod. Phys. (3) 565–582[75] Scully M O and Zubairy M S 1997 Quantum Optics: (Cambridge: Cambridge University Press)ISBN 9780511813993[76] Landauer R 1987
Zeitschrift f¨ur Physik B Condensed Matter Quantum Kinetics in Transport and Optics of Semiconductors (Springer-Verlag Berlin Heidelberg) ISBN 9783540735649[78] Engelsberg S and Schrieffer J R 1963
Phys. Rev. (3) 993–1008[79] Pourfath M 2014
The Non-Equilibrium Green’s Function Method for Nanoscale Device Simulation (Springer-Verlag Wien) ISBN 9783709117996[80] Ciuti C, Bastard G and Carusotto I 2005
Phys. Rev. B (11) 115303[81] Rza˙zewski K, W´odkiewicz K and ˙Zakowicz W 1975 Phys. Rev. Lett. (7) 432–434[82] Yamanoi M 1976 Physics Letters A
437 – 439 ISSN 0375-9601[83] Knight J M, Aharonov Y and Hsieh G T C 1978
Phys. Rev. A (4) 1454–1462[84] Bialynicki-Birula I and Rza¸˙zewski K 1979 Phys. Rev. A (1) 301–303[85] Gaw¸edzki K and Rza¸´zewski K 1981 Phys. Rev. A (5) 2134–2136[86] Keeling J 2007 Journal of Physics: Condensed Matter Nature Communication [88] Viehmann O, von Delft J and Marquardt F 2011 Phys. Rev. Lett. (11) 113602[89] Hagenm¨uller D and Ciuti C 2012
Phys. Rev. Lett. (26) 267403[90] De Liberato S and Ciuti C 2013
Phys. Rev. Lett. (13) 133603[91] Vukics A, Grießer T and Domokos P 2014
Phys. Rev. Lett. (7) 073601[92] Bamba M and Ogawa T 2014
Phys. Rev. A (6) 063825[93] Medvedyeva M V and Kehrein S 2013 ArXiv e-prints ( Preprint )[94] Benenti G, Casati G, Prosen T, Rossini D and ˇZnidariˇc M 2009
Phys. Rev. B (3) 035110[95] Medvedyeva M V, ˇCubrovi´c M T and Kehrein S 2015 Phys. Rev. B (20) 205416[96] Navarrete-Benlloch C 2015 ArXiv e-prints ( Preprint )[97] Moritz G 2007
On a New Solution to the Electron Correlation Problem in Quantum Chemistry:The Density Matrix Renormalization Group Algorithm
Ph.D. thesis ETH Zurich[98] Bonifacio R, Schwendimann P and Haake F 1971
Phys. Rev. A (1) 302–313[99] Sch¨utz S, Habibian H and Morigi G 2013 Phys. Rev. A (3) 033427[100] Bullough R K 1987 Hyperfine Interactions Philosophical Transactions of the Royal Society of London A: Mathematical,Physical and Engineering Sciences
Phys. Rev. A (6) 063827[103] J¨ager S B, Xu M, Sch¨utz S, Holland M J and Morigi G 2017 Phys. Rev. A (6) 063852[104] Wendler L and Kraft T 1996 Phys. Rev. B (16) 11436–11456[105] Lee S C and Galbraith I 1999 Phys. Rev. B (24) 15796–15805[106] Todorov Y, Andrews A M, Colombelli R, De Liberato S, Ciuti C, Klang P, Strasser G and SirtoriC 2010 Phys. Rev. Lett. (19) 196402[107] Hopfield J J 1958
Phys. Rev. (5) 1555–1567[108] Holstein T and Primakoff H 1940
Phys. Rev. (12) 1098–1113[109] Bruus H and Flensberg K 2004 Many-body quantum theory in condensed matter physics - an avity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics. introduction (United States: Oxford University Press)[110] Misra B and Sudarshan E C G 1977 Journal of Mathematical Physics Phys. Rev. A (2) 023825[112] Anderson P W 1958 Phys. Rev.109