Coupled exciton-photon Bose condensate in path integral formalism
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r Coupled exciton-photon Bose condensate in path integral formalism
A. A. Elistratov ∗ and Yu. E. Lozovik Institute for Nanotechnology in Microelectronics RAS, 119334 Leninskiy ave. 32a, Moscow, Russia Institute for Spectroscopy RAS, 142190 Troitsk, Moscow, Russia
We study the behavior of exciton polaritons in an optical microcavity with an embedded semicon-ductor quantum well. We use two-component exciton-photon approach formulated in terms of pathintegral formalism. In order to describe spatial distributions of the exciton and photon condensatedensities, the two coupled equations of the Gross-Pitaevskii type are derived. For a homogeneoussystem, we find the noncondensate photon and exciton spectra, calculate the coefficients of transfor-mation from the exciton-photon basis to the lower-upper polariton basis, and obtain the exciton andphoton occupation numbers of the lower and upper polariton branches for nonzero temperatures.For an inhomogeneous system, the set of coupled equations of the Bogoliubov-de-Gennes type isderived. The equations govern the spectra and spatial distributions of noncondensate photons andexcitons.
PACS numbers: 03.75.Kk, 03.75.Mn, 05.30.Jp, 67.85.De, 71.36.+c
I. INTRODUCTION
In the last two decades there has been extensivework on Bose-Einstein condensation in the system oftwo-dimensional exciton polaritons in optical micro-cavites (see the reviews and the references therein).The profound investigation of properties of the polari-ton system has taken place . Bose condensationhas been demonstrated and many features of thepolariton Bose condensate such as superfluidity ,vortices and solitons formation were studied,and have been shown to have specific character in thenon-equilibrium spatially inhomogenious system of exci-ton polaritons. Recently, novel phenomena such as bal-listic transport over macroscopic distances , sponta-neous oscillations , Josephson-like effects , op-tical Aharonov-Bohm and spin Hall effects are ac-tively discussed.Most authors of theoretical works use, explicitly ornot, the conventional approach consisting of the follow-ing steps: one writes out the Hamiltonian of the systemincluding exciton and photon creation and annihilationoperators, performs the Hopfield canonical transforma-tion from the exciton-photon to the lower-upper polari-ton basis, neglects the upper polariton branch as emptyat low temperatures, and regards polaritons on the lowerpolariton branch as a dilute Bose gas weakly interactingvia the exciton component. Such an approach allowingthe straightforward application of the Bogoliubov the-ory has led to a great amount of important results (seethe broad review of the approach in Ref. ).However, this approach has two serious shortages.The first one is the neglection of the upper polaritonbranch. It is quite justified at the temperature range ofa few degrees Kelvin which takes place in up-to-date ex-periments. Though, as many authors have noted, Bosecondensation of polaritons is so attractive due to ex-tremely high transition temperature being the effect ofvery small polariton effective mass. Already for nitrogentemperatures, the upper polariton branch occupation is not fully negligible. Second and more serious shortage ofthe conventional approach is that the canonical transfor-mation from the photon-exciton to the polariton basisis performed in the momentum space and hence it re-quires the spatial uniformity or quasi-uniformity of thesystem.With the up-to-date traps having dimensions of theorder of dozen microns , one can perform the canon-ical transformation at each point in space in the frameof the quasi-classical approximation. However, latestexperiments have studied systems already with microndimensions and it is likely that the trend of minia-turization will go further. Besides, the Bose condensedsystem can exhibit topological defects such as vorticesand solitons . The exciton and photon healinglengths, which determine the spatial scales of these de-fects, are of the order of micron. Thus, starting from themicron scale the conventional approach becomes non-applicable. Moreover, polariton as a well defined quasi-particle at this scale does not make sense.In this paper, we study the behavior of exciton po-laritons in the framework of a two-component exciton-photon approach. The exciton and photon componentsare described as independent quantum fields interact-ing via Rabi splitting. Bose condensation leads to theappearance of the two coupled exciton and photon con-densates. In the mean field approximation this approachhas been used, for example, in Refs. . The au-thors of Refs. have gone beyond the mean field ap-proximation and calculated the energy spectrum of non-condensate photons and excitons.We analyze the equilibrium Bose condensed polari-ton system without pumping and photon decay fromthe cavity at nonzero, but sufficiently small tempera-ture. These effects were studied in the two-componentapproach in Ref. . Moreover, we do not take into ac-count the spin degree of freedom of polaritons.The approach is formulated in terms of path inte-gral formalism for the following reasons. First, this for-malism allows the study of both condensate and non-condensate parts of the system from the unified pointof view. The second advantage is the use of coherentstates, which are the natural choice for an explorationof quantum properties of light emitted from the polari-ton system. After all, the method involves Matsubaratechnique and hence it is proved to be effective for astudy of the polariton behavior at nonzero tempera-tures. The application of the path integral method to aweakly interacting condensed Bose gas can be found inthe excellent review by Stoof .The paper is organized as follows. In Section II wedescribe the geometry of the polariton system and in-troduce the exciton and photon one-particle eigenstates.In Section III we write the partition function of the sys-tem as a path integral, which contains an action fullydescribing our system. The exciton and photon quan-tum fields in the path integral formalism are c-numberfields written in the basis of coherent states. In SectionIV we turn to the Bose condensed polariton system andwrite out both exciton and photon fields as a sum of theorder parameter and fluctuations around it. The partof the action, which does not contain fluctuations, isthe Pitaevskii-like functional. Its minimization leads tothe two coupled equations of the Gross-Pitaevskii type.These equations give the description of the system of thetwo coupled condensates in the mean field approxima-tion. We analyze the properties of the obtained equa-tions and find the conditions of equivalence between theone-component and two-component approaches. In or-der to explore the noncondensate part of the system, inSection V we analyze the quadratic in fluctuations partof the action. The most important result here is thecanonical transformation from the exciton-photon basisto the lower-upper polariton basis. This transforma-tion incorporates both the Hopfield polariton transfor-mation and the Bogoliubov transformation for weakly-interacting Bose gas. Moreover, we obtain nonconden-sate particles energy spectra and calculate exciton andphoton occupation numbers for lower and upper polari-ton branches at nonzero temperatures. In Section VI wederive the set of coupled equations of the Bogoliubov-de-Gennes type. This equations govern the energy spectraand spatial distributions of the noncondensate excitonsand photons in a spatially inhomogeneous system. II. GEOMETRY AND ONE-PARTICLEEIGENSTATES
We study a system of exciton polaritons in a semicon-ductor optical microcavity with an embedded quantumwell. In the simple case when the quantum well pos-sesses in-plane translational symmetry, energy spectrumof excitons in the region of small in-plane momenta hasthe form ε ex = ε ex (0) + p m ex , (1) where ε is the dielectric constant of the medium, ε ex (0) =2 m ex e /ε ¯ h is the 2D exciton binding energy, m ex = m e + m h is the 2D exciton mass, and m e and m h are theeffective masses of an electron and a hole, respectively.An external confining potential V ex ( x ) for excitonsin the quantum well can be created. Experimentally,there are several methods to realize the confinement ofexcitons. One of them is the exciton energy shiftingusing a stress-induced band-gap shift .We assume one-particle eigenstates describing non-interacting excitons confined by the external poten-tial V ex ( x ) to be found from the time-independentSchr ¨ o dinger equation (cid:18) − ¯ h ▽ m ex + V ex ( x ) − ε ex n (cid:19) χ n ( x ) = 0 . (2)The set of solutions χ n ( x ) is orthonormalized Z d x ¯ χ n ( x ) χ m ( x ) = δ nm (3)and full X n χ n ( x ) ¯ χ n ( x ′ ) = δ ( x − x ′ ) . (4)In the case of the planar microcavity the photons inthe region of small in-plane momenta have the followingdispersion: ε ph = ¯ hc √ ε q k ⊥ + k ≈ π ¯ hcL √ ε n + p m ph . (5)Here m ph = π ¯ h √ ε/cL is the effective photon mass. Weconsider the lowest state n = 1 .There are several experimental approaches to realizethe confinement of photons . It is, for example, cre-ating a special dielectric permittivity profile inside thecavity which would lead to photon localization, or mak-ing a trap for photons based on special shaping of themicrocavity width L ( x ) (up to creation of a finite systemlimited by mirrors from all sides). In this case the effec-tive photon mass m ph and the first term in (5) becomefunctions of coordinates. As a result, the following one-particle problem arises for photons in the microcavitywith a nonconstant width L ( x ) : − ¯ h ∇ (cid:18) m ph ( x ) ∇ ψ n ( x ) (cid:19) + (cid:18) π ¯ hcL ( x ) √ ε − ε ph n (cid:19) ψ n ( x ) = 0 . (6)We will assume microcavities to possess in-plane trans-lational symmetry, and will not consider microcavitieswith photon confinement, i.e. m ph ( x ) = m ph = const .The one-particle problem for photons reduces to (cid:18) − ¯ h ▽ m ph − ε ph n (cid:19) ψ n ( x ) = 0 . (7)Similarly to the above exciton one-particle eigenstates,we assume the set of solutions of this problem to beorthonormalized and full Z d x ¯ ψ n ( x ) ψ m ( x ) = δ nm , (8) X n ψ n ( x ) ¯ ψ n ( x ′ ) = δ ( x − x ′ ) . (9) III. ACTION AND FREE-FIELD GREEN’SFUNCTIONS
In the functional approach, the grand-canonical func-tion of the exciton-photon system is the functional in-tegral Z = Z Dχ D ¯ χ Dψ D ¯ ψ e − h S [¯ χ,χ, ¯ ψ,ψ ] , (10)where the action S consists of four terms S [ ¯ χ, χ, ¯ ψ, ψ ] = S ex [ ¯ χ, χ ] + S ph [ ¯ ψ, ψ ]+ S RABI [ ¯ χ, χ, ¯ ψ, ψ ] + S int [ ¯ χ, χ ] , (11)describing excitons, photons, Rabi splitting and interex-citon interaction, respectively: S ex [ ¯ χ, χ ] = Z ¯ hβ dτ Z d x ¯ χ ( x , τ ) × (cid:18) ¯ h ∂∂τ − ¯ h ▽ m ex + V ex ( x ) + E − µ (cid:19) χ ( x , τ ) , (12) S ph [ ¯ ψ, ψ ] = Z ¯ hβ dτ Z d x ¯ ψ ( x , τ ) × (cid:18) ¯ h ∂∂τ − ¯ h ▽ m ph − µ (cid:19) ψ ( x , τ ) , (13) S RABI [ ¯ χ, χ, ¯ ψ, ψ ] = Z ¯ hβ dτ Z d x × ¯ h Ω2 (cid:2) ¯ ψ ( x , τ ) χ ( x , τ ) + ψ ( x , τ ) ¯ χ ( x , τ ) (cid:3) , (14) S int [ ¯ χ, χ ] = − h Z ¯ hβ dτ Z d x d x ′ ¯ χ ( x , τ ) ¯ χ ( x ′ , τ ) × V ( x − x ′ ) χ ( x ′ , τ ) χ ( x , τ ) . (15)Here χ ( x , τ ) and ψ ( x , τ ) are the field operators writtenin the basis of coherent states and hence they constitutec-number functions connected with χ n ( x ) и ψ n ( x ) viarelations χ ( x , τ ) = X n χ n ( τ ) χ n ( x ) , ψ ( x , τ ) = X n ψ n ( τ ) ψ n ( x ) (16) with the coefficients of expansion χ n ( τ ) and ψ n ( τ ) de-pendent on time.We include the detuning E between the exciton spec-trum ε ex p = E + p / m ex and photon one ε ph p = p / m ph into S ex and consider E > .Further we will assume the exciton interaction to havethe approximate form V ( x − x ′ ) = V δ ( x − x ′ ) .The partition function S ex [ ¯ χ, χ ] can be rewritten as − S ex [ ¯ χ, χ ]¯ h = Z ¯ hβ dτ Z d x Z ¯ hβ dτ ′ Z d x ′ ¯ χ ( x , τ ) G − ex (0) ( x , τ ; x ′ , τ ′ ) χ ( x ′ , τ ′ ) , (17)where G − ex (0) ( x , τ ; x ′ , τ ′ ) = − h (cid:18) ¯ h ∂∂τ − ¯ h ▽ m ex + V ex ( x ) + E − µ (cid:19) δ ( x − x ′ ) δ ( τ − τ ′ ) (18)or, equivalently, (cid:18) ¯ h ∂∂τ − ¯ h ▽ m ex + V ex ( x ) + E − µ (cid:19) × G ex (0) ( x , τ ; x ′ , τ ′ ) = − ¯ hδ ( x − x ′ ) δ ( τ − τ ′ ) . (19)Substituting δ ( x − x ′ ) from (4) and using the equality δ ( τ − τ ′ ) = 1¯ hβ X m e − iω m ( τ − τ ′ ) (20)we obtain G − ex (0) ( x , τ ; x ′ , τ ′ ) = − h X n,m ( − i ¯ hω m + ε exn + E − µ ) × χ n ( x ) ¯ χ n ( x ′ ) e − iω m ( τ − τ ′ ) ¯ hβ . (21)The solution of the equation (19) reads G ex (0) ( x , τ ; x ′ , τ ′ ) = X n,m − ¯ h − i ¯ hω m + ε exn + E − µ × χ n ( x ) ¯ χ n ( x ′ ) e − iω m ( τ − τ ′ ) ¯ hβ (22)Since the functions G − ex (0) and G ex (0) are mutually in-verse, they obey the relation G ex (0) G − ex (0) = I , where I is the unity matrix. In coordinate space, this relationhas the form Z ¯ hβ dτ ′′ Z d x ′′ G ex (0) ( x , τ ; x ′′ , τ ′′ ) × G − ex (0) ( x ′′ , τ ′′ ; x ′ , τ ′ ) = δ ( x − x ′ ) δ ( τ − τ ′ ) . (23)Similarly, S ph [ ¯ ψ, ψ ] can be written as − S ph [ ¯ ψ, ψ ]¯ h = Z ¯ hβ dτ Z d x Z ¯ hβ dτ ′ Z d x ′ × ¯ ψ ( x , τ ) G − ph (0) ( x , τ ; x ′ , τ ′ ) ψ ( x ′ , τ ′ ) , (24)where G − ph (0) ( x , τ ; x ′ , τ ′ ) = − h (cid:18) ¯ h ∂∂τ − ¯ h ▽ m ph − µ (cid:19) δ ( x − x ′ ) δ ( τ − τ ′ ) . (25)This equation is equivalent to (cid:18) ¯ h ∂∂τ − ¯ h ▽ m ph − µ (cid:19) G ph (0) ( x , τ ; x ′ , τ ′ ) = − ¯ hδ ( x − x ′ ) δ ( τ − τ ′ ) . (26)For the functions G ph (0) ( x , τ ; x ′ , τ ′ ) and G − ph (0) ( x , τ ; x ′ , τ ′ ) we have G − ph (0) ( x , τ ; x ′ , τ ′ ) = − h X n,m ( − i ¯ hω m + ε phn − µ ) × ψ n ( x ) ¯ ψ n ( x ′ ) e − iω m ( τ − τ ′ ) ¯ hβ , (27) G ph (0) ( x , τ ; x ′ , τ ′ ) = X n,m − ¯ h − i ¯ hω m + ε phn − µ × ψ n ( x ) ¯ ψ n ( x ′ ) e − iω m ( τ − τ ′ ) ¯ hβ , (28) Z ¯ hβ dτ ′′ Z d x ′′ G ph (0) ( x , τ ; x ′′ , τ ′′ ) × G − ph (0) ( x ′′ , τ ′′ ; x ′ , τ ′ ) = δ ( x − x ′ ) δ ( τ − τ ′ ) . (29) IV. COUPLED GROSS-PITAEVSKII-LIKEEQUATIONS
We now turn to the Bose condensed exciton-photonsystem. Let us introduce the time independent orderparameters χ ( x ) and ψ ( x ) as χ ( x , τ ) = χ ( x ) + χ ′ ( x , τ ) , ψ ( x , τ ) = ψ ( x ) + ψ ′ ( x , τ ) , (30)where χ ′ ( x , τ ) and ψ ′ ( x , τ ) are the fluctuations of theexciton and photon quantum fields around the order pa-rameters χ ( x ) and ψ ( x ) . We shall use the Bogoliubovapproximation assuming χ ′ ( x , τ ) ≪ χ ( x ) ψ ′ ( x , τ ) ≪ ψ ( x ) , i.e. we confine ourselves to the case of low tem-peratures. Substituting the definitions (30) into the action givenby (11) and rearranging the derived terms accordingto their orders in χ ′ ( x , τ ) and ψ ′ ( x , τ ) , we obtain thefollowing results.The zeroth-order part of the action is given by ¯ hβF P [ χ , ¯ χ ; ψ , ¯ ψ ] = ¯ hβ Z d x (cid:26) ¯ h m ex |▽ χ ( x ) | + ( V ex ( x ) + E − µ ) | χ ( x ) | + V | χ ( x ) | + ¯ h m ph |▽ ψ ( x ) | − µ | φ ( x ) | + ¯ h Ω2 (cid:2) ¯ ψ ( x ) χ ( x ) + ψ ( x ) ¯ χ ( x ) (cid:3)(cid:27) (31)and has the form of the Pitaevskii functional.According to the semiclassical method, the functionalreaches its minimum at functions χ ( x ) and ψ ( x ) , forwhich the functional variation vanishes. Equivalently,we can set the part of the action linear in χ ′ ( x , τ ) and ψ ′ ( x , τ ) equal to zero. As a result, we obtain the setof coupled stationary equations analogous to the Gross-Pitaevskii equation: (cid:18) − ¯ h ▽ m ex + V ex ( x ) + E − µ + V | χ ( x ) | (cid:19) χ ( x ) + ¯ h Ω2 ψ ( x ) = 0 , (cid:18) − ¯ h ▽ m ph − µ (cid:19) ψ ( x ) + ¯ h Ω2 χ ( x ) = 0 . (32)These equations provide a minimum of the functional(31) and describe the system of the two coupled con-densates in the mean field approximation.Let us discuss the derived equations by consideringfirst the homogeneous case V ex ( x ) = 0 . In the absenceof the Rabi splitting ¯ h Ω = 0 , the chemical potentialof excitons equals µ = E + V n ex , same as for theconventional one-component Bose condensed gas. Herewe introduced the exciton condensate density | χ | = n ex . It is known that the Rabi splitting leads to theappearance of the two branches in the energy spectrum,which are referred to as the lower and upper polaritonstates (below we denote them by superscripts “L” and“U”). The chemical potential is equal to µ ( L,U ) = 12 (cid:20) ( E + V n ex ) ∓ q ( E + V n ex ) + ¯ h Ω (cid:21) (33)and varies within the interval µ ( L ) ∈ [ µ ( L ) min , , µ ( U ) ∈ [ µ ( U ) min , ∞ ] , where µ ( L,U ) min = ( E ∓ q E + ¯ h Ω ) / .In both cases the solution of the set of equations (32)has the form χ = s V µ − E µ − ¯ h Ω / µ ,ψ = ¯ h Ω2 µ s V µ − E µ − ¯ h Ω / µ . (34)For the lower polariton state, the exciton and photoncondensate phases differ by π , since the chemical poten-tial is always negative. For the upper polariton state,the exciton and photon condensate phases coincide andthe chemical potential is positive.The energy of the homogeneous system per unit area,as it follows from (31), can be calculated as follows: E ( L,U ) = E n ex + 12 V ( n ex ) ∓ ¯ h Ω q n ex n ph , (35)where n ph = | ψ | is the photon condensate density.Thus, when the homogeneous exciton-photon systemis the condensate of lower polaritons, it has the mini-mal energy E ( L ) . When the system is the condensateof upper polaritons, it has the maximal energy E ( U ) .The difference between these energies per one particleis the energy of Rabi splitting ¯ h Ω (in the limit of lowdensity). The energies of the uniform states, for whichphases of the exciton and photon condensates differ by avalue ranging from to π , are situated within the inter-val [ E ( L ) , E ( U ) ] . Such states are time-dependent, sincethe exciton and photon condensates cyclically turn intoeach other, i.e. a kind of the internal Josephson effecttakes place. We shall discuss these states elsewhere .Here we confine ourselves only to the study of the lowerpolariton branch with the energy E ( L ) , omitting furtherthe superscript “ L ” for shortness.Substituting Eq. (34) into (35), we get the connectionbetween the energy and the chemical potential E = 12 V (cid:20) µ + 12 ¯ h Ω − E − ¯ h Ω µ E −
316 ¯ h Ω µ (cid:21) . (36)We introduce the polariton condensate density as thesum n P = n ex + n ph , (37)Substituting Eq. (34) into (37), we find the connec-tion between the polariton condensate density and thechemical potential n P = 1 V (cid:20) µ − E −
14 ¯ h Ω µ E −
116 ¯ h Ω µ (cid:21) . (38)With the help of Eqs. (36) and (38) we can easily provethat ∂E/∂n P = µ , i.e. the chemical potential is equal tothe change in the energy as one extra polariton is addedto the system. The fact that µ < means that the en-ergy always decreases. It is worth pointing out that the expressions (34) could be obtained from the condi-tion that the energy has a minimum at a fixed value ofchemical potential ( ∂E/∂n ex ) µ = ( ∂E/∂n ph ) µ = 0 . Asit is seen from (34), in the limit of large density µ → the polariton condensate becomes mainly photonic one,and the energy is now connected with the density bythe relation E = − (3 / h Ω /V ) / ( n P ) / , whichdiffers principally from the relation for the conventionalone-component Bose condensate E = (1 / V ( n ex ) .Let us next discuss the inhomogeneous case. Thelower equation in (32), describing the photon conden-sate, possesses the spatial scale which plays a role ofthe healing length and equals ξ ph = ¯ h p − m ph µ . (39)The length ξ ph has a minimal value in the limit of zerodensity and increases with the chemical potential.For the upper equation in (32), describing the excitoncondensate, the healing length equals ξ ex = ¯ h p m ex ( E − µ ) (40)and increases with the chemical potential as well.It is important to point out that the relation m ph /m ex ∼ − observed typically in experimentscauses the following inequality ξ ph ≫ ξ ex . (41)Let’s now study the conditions of the equivalence ofthe two-component exciton-photon and one-componentpolariton approaches. These approaches are obvi-ously equivalent in the homogeneous case, when ψ =(¯ h Ω / µ ) χ , n P = | ψ | + | χ | . We expect these localrelations to be valid also in the case of smooth densitieschange. We rewrite the lower equation in (32) in theintegral form using the fact that the MacDonald func-tion K ( x ) (see, for example, ) is the Green’s functionof the equation. Thus we have ψ ( r ) = − π ¯ h Ω2 µ Z K ( | r − r ′ | ) χ ( r ′ ) d r ′ . (42)Here we introduced the dimensionless vector r = x /ξ ph .Taking the asympotics K ( x ) ≈ p π/ xe − x valid for x → ∞ into account, we can expand the smooth func-tion χ ( r ′ ) in a power series in the vicinity of r ′ = r andkeep only the first terms χ ( r ′ ) ≈ χ ( r ) + ∇ χ ( r ) · s + 12 X i,j ∂ χ ( r ) ∂r i ∂r j s i s j , (43)where s = r ′ − r . We substitute this expansion into (42)and integrate it with respect to d s = s ds dϕ . Here it isconvenient to use the integral formula for the MacDon-ald function K ( x ) = Z ∞ e − xη p η − dη. (44)As a result, the gradient term and the term contain-ing the second mixed partial derivative vanish after theintegration with respect to the angle dϕ , and we obtain ψ ( x ) = ¯ h Ω2 µ χ ( x ) (cid:20) ξ ph ∆ χ ( x ) χ ( x ) (cid:21) , (45)where ∆ is the Laplace operator. Therefore the condi-tion that determines the applicability of the local rela-tions between particles densities is ∆ χ ( x ) χ ( x ) ≪ ξ ph (46)which means that the relative exciton density changemust be small on the scales comparable to the photonhealing length.In topological structures such as vortices and solitons,the exciton density changes considerably on the scalesof the exciton healing length. The same picture takesplace near the boundaries of the condensate confined bythe box potential. In all these cases the one-componentpolariton approach is not applicable.The extensively used Thomas-Fermi approximationconsists in neglecting the quantum pressure terms inthe equations (32) at sufficiently high densities of con-densate particles. The obtained estimate establishes therange of validity of Thomas-Fermi approximation.The solution of Eq. (32) for the harmonic trap ispublished in Ref. , the vortex solution is discussed inRef. . V. NONCONDENSATE PARTICLESA. Matrix Green’s function
We can represent the quadratic part of the action S quad as a quadratic form S quad = − ¯ h Z ¯ hβ dτ Z d x [ ¯ χ ′ , ¯ ψ ′ , χ ′ , ψ ′ ] · G − · χ ′ ψ ′ ¯ χ ′ ¯ ψ ′ , (47)where the Green’s function has a matrix structure − G = * χ ′ ψ ′ ¯ χ ′ ¯ ψ ′ · [ ¯ χ ′ , ¯ ψ ′ , χ ′ , ψ ′ ] + (48)and can be written as G − ( x , τ ; x ′ , τ ′ ) = G − ( x , τ ; x ′ , τ ′ ) −− h V | χ ( x ) | ¯ h Ω / V χ ( x ) h Ω / V ¯ χ ( x ) V | χ ( x ) | ¯ h Ω /
20 0 ¯ h Ω / × δ ( x − x ′ ) δ ( τ − τ ′ ) , (49) where G − = G − ex (0) G − ph (0) G − ex (0)
00 0 0 G − ph (0) . (50)Here G − ex (0) ( x , τ ; x ′ , τ ′ ) and G − ph (0) ( x , τ ; x ′ , τ ′ ) are theinverse exciton and photon Green’s functions given by(18) and (25), respectively.We can rewrite (49) as G − = G − − Σ , where Σ is a matrix formed by self-energies. Multiplying thisequation from the left by G (0) and from the right by G − we obtain the Dyson equation G = G (0) + G (0) ΣG .Some Green’s functions from G are presented in Fig.1. We will only consider the case of a uniform system V ex ( x ) = 0 . The exciton and photon single-particleenergies have the form ε exn = ε ex p = p m ex , ε phn = ε ph p = p m ph . (51)Substituting µ , Eqs. (18) and (25) into (49) gives − ¯ h G − ( p , k ) = E ex p − i ¯ hω k ¯ h Ω / V χ h Ω / E ph p − i ¯ hω k V ¯ χ E ex p + i ¯ hω k ¯ h Ω /
20 0 ¯ h Ω / E ph p + i ¯ hω k , (52)FIG. 1. Diagrammatic representation of some Dysonequations from G = G (0) + G (0) ΣG . Solid and dotted linesrepresent free-field exciton and photon Green’s functions,respectively. Double wiggly line represents exciton-excitoninteraction counted in the ladder approximation. Circlerepresents the Rabi splitting. (a) Exact exciton Green’sfunction. (b) Exact photon Green’s function. (c) Exactanomalous exciton-photon Green’s function. (d) Exactanomalous photon-exciton Green’s function. where we introduced the energies as E ex p = ε ex p + E − µ + 2 V n = ε ex p + 12 (cid:18) V n − q ( E + V n ) + ¯ h Ω (cid:19) + (cid:18) E V n (cid:19) , (53) E ph p = ε ph p − µ = ε ph p + 12 ( V n − q ( E + V n ) + ¯ h Ω (cid:19) − (cid:18) E V n (cid:19) . (54)Hereafter, we denote n ex by n and replace χ by √ n e iφ . B. Spectrum
The determinant of the matrix G − ( p , ω ) is zero atthe poles ω of the matrix G ( p , ω ) . Equating the deter-minant to zero yields ¯ h ω − (cid:18) E ex p + E ph p + ¯ h Ω − V n (cid:19) ¯ h ω + (cid:18) E ex p E ph p − ¯ h Ω (cid:19) − V n E ph p ! = 0 . (55)Solving Eq. (55) for ω , we obtain the polariton energyspectrum modified by the condensate: ¯ h ω ( L,U )2 p = 12 (cid:18) E ex p + E ph p − V n + ¯ h Ω (cid:19) ± (cid:20)(cid:16) − E ex p + E ph p + V n (cid:17) + ¯ h Ω (cid:16)(cid:0) E ex p + E ph p (cid:1) − V n (cid:17)(cid:21) . (56)In the absence of the condensate n = 0 , it follows fromEqs. (53), (54) that E ex p = ε ex p + E − µ , E ph p = ε ph p − µ ,and equation (56) gives ¯ h ω ( L,U )2 p = 12 (cid:18) ( E + ε ex p ) + ε ph p + ¯ h Ω µ − µ ( E + ε ex p + ε ph p ) (cid:19) ± (cid:0) E + ε ex p + ε ph p − µ (cid:1) × r(cid:16) E + ε ex p − ε ph p (cid:17) + ¯ h Ω , (57)which is equivalent to ¯ hω ( L,U ) p = 12 (cid:0) E + ε ex p + ε ph p (cid:1) ± r(cid:16) E + ε ex p − ε ph p (cid:17) + ¯ h Ω − µ = ε ( L,U ) p − µ. (58) One can see that the spectrum turns into the energydispersion of the lower and upper polaritons.In the absence of the Rabi splitting we have ¯ h ω p = 12 (cid:16) E ex p + E ph p − V n (cid:17) ± (cid:16) − E ex p + E ph p + V n (cid:17) , (59)which gives the photon energy spectrum ¯ hω p = ± E ph p = ± ( ε ph p − µ ) (60)and the exciton energy spectrum modified by the con-densate ¯ hω p = ± q E ex p − V n = ± q ε ex p + 2 V n ε ex p = ± ε B p . (61)The equation (61) is the Bogoliubov dispersion law.Here we assume µ = E + V n as discussed in Section4. For small momenta p → the lower branch takes thephonon-like form ¯ hω ( L ) p = v s p, (62)where v s is the sound velocity. Assuming that α = m ph /m ex = 0 , V n ≪ ¯ h Ω , which is commonly encoun-tered in practice, the velocity can be estimated as fol-lows: v s = s ¯ h Ω E + ¯ h Ω V n m ph (63)At p → the upper branch has the form ¯ hω ( U ) p = ∆ + p m ( U )0 , (64)where ∆ = q E + ¯ h Ω + 12 E q E + ¯ h Ω V n (65)in the limit V n ≪ ¯ h Ω , and m ( U )0 = " − E q E + ¯ h Ω − ¯ h Ω ( E + ¯ h Ω ) / V n m ph (66)in the approximation α = m ph /m ex = 0 , V n ≪ ¯ h Ω .In the limit of large momenta p → ∞ the lower polari-ton branch turns into the dispersion law of free excitons ¯ hω ( L ) p = p m ex + 12 (cid:18)q ( E + V n ) + ¯ h Ω + E + 3 V n (cid:19) . (67)If ¯ h Ω = 0 , the square root should be taken with thenegative sign; this corresponds to µ = E + V n .The upper branch for p → ∞ takes the similar form ¯ hω ( U ) p = p m ph + 12 (cid:18)q ( E + V n ) + ¯ h Ω − ( E + V n ) (cid:19) . (68)The outlined properties of the polariton spectrum mod-ified by the condensate are shown in Fig. 2. We seethat the energy and momentum of the polariton reso-nance increase with the condensate density. This resem-bles a positive detuning between the exciton and photonmodes. C. Polariton basis
Let us now diagonalize the matrix G − . The diag-onalization does not reduce to straightforward deter-mination of the eigenvalues because of opposite signsin front of i ¯ hω k in the diagonal elements of the ma-trix (52). An option is to consider the generalizedeigenvalue problem G − ξ = (¯ hω − i ¯ hω k ) W ξ , where W = diag (1 , , − , − (see, for example, ). However,we suggest a different approach: to multiply the twolower rows of G − by − and thus obtain the stan-dard eigenvalue problem ˜G − ξ = (¯ hω − i ¯ hω k ) ξ , whereFIG. 2. (Color online) Polariton energy spectrum modifiedby the condensate. The momentum is expressed in theunits of p m ph ¯ h Ω , the energy is expressed in the units of ¯ h Ω . (a) V n = 0 , (b) V n = 0 . h Ω , (c) V n = 0 . h Ω ,(d) V n = 0 . h Ω , (e) V n = 0 . h Ω . Inset: phonon-likespectra in the vicinity of zero momentum. ˜G − = WG − is a non-Hermitian operator with re-spect to the scalar product ( a, b ) = P α ¯ a α b α . Let usdefine the new scalar product as (( a, b )) = ( a, W b ) . (69)The Hermitian conjugation with respect to the old andnew scalar products is given by the formulae ( O a, b ) = ( a, O + b ) , (( O a, b )) = (( a, O L b )) . (70)One can obtain the following relationship between theoperators O L and O + : O L = WO + W . (71)The new scalar product is chosen in such a way that theequation ( ˜G − ) L = ˜G − holds. Therefore the operator ˜G − is Hermitian with respect to the scalar product (69)and its eigenvalues are real: ˜G − ξ = λξ , λ = ¯ λ , if thenorm of the vector ξ is nonzero: (( ξ, ξ )) = 0 .Let us perform the transformation from the exciton-photon basis ( χ, ψ, ¯ χ, ¯ ψ ) to the polariton basis ( P ( L ) , P ( U ) , ¯ P ( L ) , ¯ P ( U ) ) , where P ( L ) and P ( U ) describethe lower and upper polaritons, respectively. We shallcarry out the canonical transformation P ( L,U ) p = η ( L,U ) χ p χ p + η ( L,U ) ψ p ψ p + ¯ ν ( L,U ) χ p ¯ χ p + ¯ ν ( L,U ) ψ p ¯ ψ p (72)where η ( L,U ) χ p , η ( L,U ) ψ p , ν ( L,U ) χ p , and ν ( L,U ) ψ p are the unknowncoefficients to be found.It is clear that ¯ P ( L,U ) p = ¯ η ( L,U ) χ p ¯ χ p + ¯ η ( L,U ) ψ p ¯ ψ p + ν ( L,U ) χ p χ p + ν ( L,U ) ψ p ψ p (73)and therefore the two bases are related to each other bythe equation P ( L ) p P ( U ) p ¯ P ( L ) p ¯ P ( U ) p = η ( L ) χ p η ( L ) ψ p ¯ ν ( L ) χ p ¯ ν ( L ) ψ p η ( U ) χ p η ( U ) ψ p ¯ ν ( U ) χ p ¯ ν ( U ) ψ p ν ( L ) χ p ν ( L ) ψ p ¯ η ( L ) χ p ¯ η ( L ) ψ p ν ( U ) χ p ν ( U ) ψ p ¯ η ( U ) χ p ¯ η ( U ) ψ p χ p ψ p ¯ χ p ¯ ψ p . (74)Let us denote the matrix in the right-hand side by B.The diagonalization of a Hermitian matrix implies theuse of a unitary matrix, which columns are normalizedeigenvectors of the Hermitian matrix. Let us assumethat the columns of the matrix B + are the normalizedeigenvectors of the operator ˜G − , i.e. we have ˜G − B + = B + ˜G − d , (75)where ˜G − d = diag (cid:16) ¯ hω ( L ) p − i ¯ hω k , ¯ hω ( U ) p − i ¯ hω k , − ¯ hω ( L ) p − i ¯ hω k , − ¯ hω ( U ) p − i ¯ hω k (cid:17) . (76)The columns of the matrix B + are normalized with re-spect to the scalar product (( • )) , i.e. their componentssatisfy the equation (cid:12)(cid:12)(cid:12) η ( L,U ) χ p (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) η ( L,U ) ψ p (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ν ( L,U ) χ p (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ν ( L,U ) ψ p (cid:12)(cid:12)(cid:12) = 1 . (77)The components of the eigenvectors can be found fromthe equations ˜G − ξ = (¯ hω − i ¯ hω k ) ξ . We obtain thefollowing relationships between the coefficients of thetransformation: η ( L,U ) ψ p = − ¯ h Ω / E ph p − ¯ hω ( L,U ) p η ( L,U ) χ p ,ν ( L,U ) χ p = E ph p + ¯ hω ( L,U ) p ¯ h Ω / − ( E ex p + ¯ hω ( L,U ) p )( E ph p + ¯ hω ( L,U ) p ) , × V n e − iφ η ( L,U ) χ p ,ν ( L,U ) ψ p = − ¯ h Ω / h Ω / − ( E ex p + ¯ hω ( L,U ) p )( E ph p + ¯ hω ( L,U ) p ) , × V n e − iφ η ( L,U ) χ p . (78)Using (77) and (78) we can find η ( L,U ) χ p , η ( L,U ) ψ p , ν ( L,U ) χ p ,and ν ( L,U ) ψ p .We present another method for the calculation of thetransformation’s coefficients. From Eq. (75) we obtain ˜G − = B + ˜G − d ( B + ) − = B + ˜G − d ( B + ) L = B + ˜G − d WBW . (79)If the matrix G − = W ˜G − corresponded to the diag-onal matrix G d − = W ˜G − d , we would write G − = B − G d − WBW . (80)From this equation, it follows that G = B + G d B , (81)where G d = diag − i ¯ hω k + ¯ hω ( L ) p , − i ¯ hω k + ¯ hω ( U ) p , i ¯ hω k + ¯ hω ( L ) p , i ¯ hω k + ¯ hω ( U ) p ! . (82)Multiplying the matrices G d , B and B + accordingto (81), we express the components of the matrix G through the components of B . Thus, for example, G ( p , k ) = (cid:12)(cid:12)(cid:12) η ( L ) χ p (cid:12)(cid:12)(cid:12) − i ¯ hω k + ¯ hω ( L ) p + (cid:12)(cid:12)(cid:12) ν ( L ) χ p (cid:12)(cid:12)(cid:12) i ¯ hω k + ¯ hω ( L ) p + (cid:12)(cid:12)(cid:12) η ( U ) χ p (cid:12)(cid:12)(cid:12) − i ¯ hω k + ¯ hω ( U ) p + (cid:12)(cid:12)(cid:12) ν ( U ) χ p (cid:12)(cid:12)(cid:12) i ¯ hω k + ¯ hω ( U ) p , (83) G ( p , k ) = (cid:12)(cid:12)(cid:12) η ( L ) ψ p (cid:12)(cid:12)(cid:12) − i ¯ hω k + ¯ hω ( L ) p + (cid:12)(cid:12)(cid:12) ν ( L ) ψ p (cid:12)(cid:12)(cid:12) i ¯ hω k + ¯ hω ( L ) p + (cid:12)(cid:12)(cid:12) η ( U ) ψ p (cid:12)(cid:12)(cid:12) − i ¯ hω k + ¯ hω ( U ) p + (cid:12)(cid:12)(cid:12) ν ( U ) ψ p (cid:12)(cid:12)(cid:12) i ¯ hω k + ¯ hω ( U ) p . (84)On the other hand, we can find the matrix G directlyby inverting the matrix G − from Eq. (52). Thus, for G and G we have G ( p , k ) = − ¯ hD h E ex p ¯ h ω k + (cid:16) E ph p + ¯ h ω k + (¯ h Ω) (cid:19) i ¯ hω k + E ex p E ph p − E ph p (¯ h Ω) (cid:21) , (85) G ( p , k ) = − ¯ hD h E ph p ¯ h ω k + (cid:16) E ex p + ¯ h ω k + (¯ h Ω) − V n (cid:19) i ¯ hω k + E ex p E ph p − E ex p (¯ h Ω) − E ph p V n (cid:21) , (86)where D = ( i ¯ hω k − ¯ hω L p )( i ¯ hω k − ¯ hω U p ) × ( i ¯ hω k + ¯ hω L p )( i ¯ hω k + ¯ hω U p ) . (87)Bringing these equations to the form similar to thatof Eqs. (83) and (84), we obtain the coefficients η ( L,U ) χ p , η ( L,U ) ψ p , ν ( L,U ) χ p , and ν ( L,U ) ψ p .Both methods lead to the following expressions forthe coefficients of the transformation from the basis ofexcitons and photons to the basis of lower and upperpolaritons in the presence of the condensate:0 P ( L,U ) p = e iξ ( L,U ) q hω ( L,U ) p q ¯ h ω ( U )2 p − ¯ h ω ( L )2 p ×× vuut ¯ h Ω / − (cid:16) E ex p + ¯ hω ( L,U ) p (cid:17) (cid:16) E ph p + ¯ hω ( L,U ) p (cid:17) ¯ hω ( L,U ) p − E ph p (cid:20)(cid:16) ¯ hω ( L,U ) p − E ph p (cid:17) χ p + ¯ h Ω2 ψ p (cid:21) ++ V n e iφ vuut ¯ hω ( L,U ) p − E ph p ¯ h Ω / − (cid:16) E ex p + ¯ hω ( L,U ) p (cid:17) (cid:16) E ph p + ¯ hω ( L,U ) p (cid:17) (cid:20)(cid:16) ¯ hω ( L,U ) p + E ph p (cid:17) ¯ χ p − ¯ h Ω2 ¯ ψ p (cid:21) . (88)Here ξ ( L,U ) are the phases which up to an arbitraryphase ξ satisfy the equation ξ ( L ) = ξ ( U ) − π/ ξ .In a sense, these transformations incorporate both theHopfield transformation for polaritons and the Bogoli-ubov transformation for weakly-interacting Bose gas.For ¯ h Ω = 0 the expression (88) in the case of lowerpolariton yields P ( L ) p = − e iξ s ε ex p + V n + ε B p ε B p χ p + e iφ + iξ s ε ex p + V n − ε B p ε B p ¯ χ p . (89)The coefficients of χ p и ¯ χ p are the coefficients u p and v p of the Bogoliubov transformation. An upper polaritonturns into a photon: P ( U ) p = e iξ ψ p . For V n = 0 theexpression (88) yields P ( L,U ) p = ε ( L,U ) p − ε ph p r(cid:16) ε ph p − ε ( L,U ) p (cid:17) + ¯ h Ω / χ p + ¯ h Ω / r(cid:16) ε ph p − ε ( L,U ) p (cid:17) + ¯ h Ω / ψ p . (90)Here the phase ξ is assumed to be zero. The coefficientsof χ p and ψ p are the Hopfield coefficients.In Fig. 3 we plot the momentum dependences of thecalculated coefficients. It is seen that the coefficients forthe lower polariton similar to the Bogoliubov transfor-mation coefficients have a singularity at zero momen-tum. This is due to the fact that, by virtue of Eq. (62), ¯ hω ( L ) p in the denominator of Eq. (88) behaves like v s p .In addition, the presence of the condensate shifts thebalance between the excitons and photons to the pho-tons in both polariton branches. Apparently, it is a con-sequence of the positive detuning between the excitonand photon modes discussed in Section 5.2. It is worthpointing out that exciton-photon interaction causes anextremely slow decay of the coefficient ν ( L ) χ p . D. Occupation numbers
We can find the noncondensate exciton density takinginto account that n ex ′ = − G ( x , τ, x ′ , τ + ) . Using Eq.FIG. 3. (Color online) The transformation coefficientsfrom excitons and photons to lower and upper polaritons asfunctions of momentum. The momentum is expressed inthe units of p m ph ¯ h Ω . (a) V n = 0 , (b) V n = 0 . h Ω ,(c) V n = 0 . h Ω , (d) V n = 0 . h Ω , (e) V n = 0 . h Ω . n ex ′ = − X p =0 ,k G ( p , k ) == − X p =0 ,k (cid:12)(cid:12)(cid:12) η ( L ) χ p (cid:12)(cid:12)(cid:12) − i ¯ hω k + ¯ hω ( L ) p + (cid:12)(cid:12)(cid:12) ν ( L ) χ p (cid:12)(cid:12)(cid:12) i ¯ hω k + ¯ hω ( L ) p + (cid:12)(cid:12)(cid:12) η ( U ) χ p (cid:12)(cid:12)(cid:12) − i ¯ hω k + ¯ hω ( U ) p + (cid:12)(cid:12)(cid:12) ν ( U ) χ p (cid:12)(cid:12)(cid:12) i ¯ hω k + ¯ hω ( U ) p . (91)To evaluate the sum over Matsubara frequencies, we usethe formula lim η → hβ X k e iω k η iω k − ( ε − µ ) / ¯ h = − e β ( ε − µ ) − (92)and find that n ex ′ = X p =0 (cid:12)(cid:12)(cid:12) η ( L ) χ p (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ν ( L ) χ p (cid:12)(cid:12)(cid:12) e βω ( L ) p − (cid:12)(cid:12)(cid:12) ν ( L ) χ p (cid:12)(cid:12)(cid:12) + X p =0 (cid:12)(cid:12)(cid:12) η ( U ) χ p (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ν ( U ) χ p (cid:12)(cid:12)(cid:12) e βω ( U ) p − (cid:12)(cid:12)(cid:12) ν ( U ) χ p (cid:12)(cid:12)(cid:12) (93)Substituting the explicit expressions for the coefficients η ( L,U ) χ p , η ( L,U ) ψ p , ν ( L,U ) χ p , ν ( L,U ) ψ p , we can write this equationas follows: n ex ′ = 1 V X p =0 n exL ′ p + X p =0 n exU ′ p , (94)where the first term contains the sum over exciton oc-cupation numbers of the lower polariton branch n exL ′ p = − − E ph p + ¯ h ω L p − (¯ h Ω) / (cid:16) ¯ h ω L p − ¯ h ω U p (cid:17) + − E ex p E ph p + E ex p ¯ h ω L p + E ph p (¯ h Ω) / h ω L p − ¯ h ω U p × hω L p (cid:18)
12 + 1 e β ¯ hω L p − (cid:19) (95)and the second sum is over exciton occupation numbersof the upper polariton branch n exU ′ p = − E ph p + ¯ h ω U p − (¯ h Ω) / (cid:16) ¯ h ω L p − ¯ h ω U p (cid:17) − − E ex p E ph p + E ex p ¯ h ω U p + E ph p (¯ h Ω) / h ω L p − ¯ h ω U p × hω U p (cid:18)
12 + 1 e β ¯ hω U p − (cid:19) . (96) Similarly, for photons one has n ph ′ = − G ( x , τ, x ′ , τ + ) = − X p =0 ,k G ( p , k ) == − X p =0 ,k (cid:12)(cid:12)(cid:12) η ( L ) ψ p (cid:12)(cid:12)(cid:12) − i ¯ hω k + ¯ hω ( L ) p + (cid:12)(cid:12)(cid:12) ν ( L ) ψ p (cid:12)(cid:12)(cid:12) i ¯ hω k + ¯ hω ( L ) p + (cid:12)(cid:12)(cid:12) η ( U ) ψ p (cid:12)(cid:12)(cid:12) − i ¯ hω k + ¯ hω ( U ) p + (cid:12)(cid:12)(cid:12) ν ( U ) ψ p (cid:12)(cid:12)(cid:12) i ¯ hω k + ¯ hω ( U ) p == X p =0 (cid:12)(cid:12)(cid:12) η ( L ) ψ p (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ν ( L ) ψ p (cid:12)(cid:12)(cid:12) e βω ( L ) p − (cid:12)(cid:12)(cid:12) ν ( L ) ψ p (cid:12)(cid:12)(cid:12) + X p =0 (cid:12)(cid:12)(cid:12) η ( U ) ψ p (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ν ( U ) ψ p (cid:12)(cid:12)(cid:12) e βω ( U ) p − (cid:12)(cid:12)(cid:12) ν ( U ) ψ p (cid:12)(cid:12)(cid:12) (97)which leads to n ph ′ = 1 V X p =0 n phL ′ p + X p =0 n phU ′ p , (98)where n phL ′ p = − − E ex p + ¯ h ω L p − (¯ h Ω) / V n (cid:16) ¯ h ω L p − ¯ h ω U p (cid:17) + − E ex p E ph p + E ph p ¯ h ω L p + E ex p (¯ h Ω) / E ph p V n ¯ h ω L p − ¯ h ω U p × hω L p (cid:18)
12 + 1 e β ¯ hω L p − (cid:19) , (99) n phU ′ p = − E ex p + ¯ h ω U p − (¯ h Ω) / V n (cid:16) ¯ h ω L p − ¯ h ω U p (cid:17) − − E ex p E ph p + E ph p ¯ h ω U p + E ex p (¯ h Ω) / E ph p V n ¯ h ω L p − ¯ h ω U p × hω U p (cid:18)
12 + 1 e β ¯ hω U p − (cid:19) . (100)Here n phL ′ p and n phU ′ p are the photon occupation num-bers of the lower and upper polariton branches.Let us now analyze the results. In the absence of theRabi splitting the first term in Eq. (95) equals / andthe brackets’ coefficient in the second term turns into ( ε ex p + V n ) / (¯ hω p ) , where ¯ hω p is given by Eq. (61).Both terms in Eq. (96) are equal to zero, i.e. thereare no excitons in the upper branch. As a result, for2excitons we obtain the expression n ex ′ p = n exL ′ p + n exU ′ p == ε p + V n − ¯ hω p hω p + ε p + V n ¯ hω p e β ¯ hω p − (101)which is the standard momentum distribution of non-condensate particles in the Bogoliubov model.Both terms in (99) are equal to zero, i.e. there are nophotons in the lower polariton branch. The first termin (100) equals − / and the brackets’ coefficient equals . Thus for photons we obtain n ph ′ p = n phL ′ p + n phU ′ p = 1 e β ¯ hω p − (102)which is the standard distribution of particles in theideal Bose gas. In Fig. 4 we plot the noncondensateexciton and photon distributions at zero temperature.At small momenta the exciton distribution is linearlydecreasing starting from some finite value: πpn exL ′ p ≈ π V n v s (cid:18) − E ∆ (cid:19) − π (cid:18) − E ∆ (cid:19) (cid:20) − (cid:18) − E ∆ (cid:19) V n ∆ (cid:21) p, (103)but then it starts increasing with growth of momen-tum. Such behavior is a consequence of very slow dropof the coefficient ν ( L ) χ p (see the previous subsection). TheFIG. 4. (Color online) Exciton and photon momentumdistributions in the lower and upper polariton branches.Momentum is expressed in the units of p m ph ¯ h Ω .(a) V n = 0 , (b) V n = 0 . h Ω , (c) V n = 0 . h Ω ,(d) V n = 0 . h Ω , (e) V n = 0 . h Ω . The inset shows theexciton momentum distribution in the lower polaritonbranch at large momenta (of the order of p m ex /m ph ∼ ). further behavior of the exciton noncondensate distribu-tion is shown in the inset of Fig. 4,a. It is seen thatthe distribution drops off at very large momenta. It isworth noting that the population of the upper polari-ton branch is high (comparable with that of the lowpolariton branch). This population increases with thecondensate density. The maximum of the populationcorresponds to the region of the polariton resonance andshifts to large momenta with the growth of the conden-sate density. VI. COUPLEDBOGOLIUBOV-DE-GENNES-LIKE EQUATIONS
In order to investigate the inhomogeneous case weintroduce the operators ˆ K ex = − ¯ h ▽ m ex + V ex ( x ) − µ, (104) ˆ K ph = − ¯ h ▽ m ph − µ (105)and solve the following eigenvalue problem: ˆ K ex + 2 V | χ ( x ) | ¯ h Ω / V ( χ ( x )) h Ω / K ph V ( ¯ χ ( x )) K ex + 2 V | χ ( x ) | ¯ h Ω /
20 0 ¯ h Ω / K ph × u exn ( x ) u phn ( x ) v exn ( x ) v phn ( x ) = ¯ hω n − − u exn ( x ) u phn ( x ) v exn ( x ) v phn ( x ) . (106)Thus, we obtain a set of equations which is an analog ofthe Bogoliubov – de Gennes system of equations and de-scribes excitations in the system of coupled condensatesof photons and excitons: (cid:18) − ¯ h ▽ m ex + V ex ( x ) − µ + 2 V n ( x ) (cid:19) u exn ( x )+ V ( χ ( x )) v exn ( x ) + ¯ h Ω2 u phn ( x ) = ¯ hω n u exn ( x ) , (cid:18) − ¯ h ▽ m ph − µ (cid:19) u phn ( x ) + ¯ h Ω2 u exn ( x ) = ¯ hω n u phn ( x ) , (cid:18) − ¯ h ▽ m ex + V ex ( x ) − µ + 2 V n ( x ) (cid:19) v exn ( x )+ V ( ¯ χ ( x )) u exn ( x ) + ¯ h Ω2 v phn ( x ) = − ¯ hω n v exn ( x ) , (cid:18) − ¯ h ▽ m ph − µ (cid:19) v phn ( x ) + ¯ h Ω2 v exn ( x ) = − ¯ hω n v phn ( x ) . (107)3Similarly to Eq. (93), one can write an expression fornoncondensate excitons n ex ′ ( x ) = X n =0 ( | u exn ( x ) | + | v exn ( x ) | e βω n − | v exn ( x ) | ) . (108)and for photons n ph ′ ( x ) = X n =0 ( (cid:12)(cid:12) u phn ( x ) (cid:12)(cid:12) + (cid:12)(cid:12) v phn ( x ) (cid:12)(cid:12) e βω n − (cid:12)(cid:12) v phn ( x ) (cid:12)(cid:12) ) . (109)Functions u exn ( x ) , v exn ( x ) , u phn ( x ) and v phn ( x ) are nor-malized as Z d x (cid:16) | u exn ( x ) | + (cid:12)(cid:12) u phn ( x ) (cid:12)(cid:12) −| v exn ( x ) | − (cid:12)(cid:12) v phn ( x ) (cid:12)(cid:12) (cid:17) = 1 . (110)This equation is analogous to Eq. (77) and can be ob-tained in the same manner.The solutions of the set of equations (107) are out ofthe scope of the present discussion and will be publishedelsewhere. VII. CONCLUSIONS
In conclusion, the central result of the work is thecanonical transformation from the basis of excitons andphotons to the basis of lower and upper polaritons in thepresence of the condensate. The transformation coeffi-cients we obtained can be called the Hopfield coefficientsrenormalized by the condensate.The path integral method we used has essentiallymore abilities than were demonstrated in the work. Themethod opens an effective way to analyze the formationand dynamics of a non-equilibrium condensate. Thetwo-component approach based on the path integralmethod leads to the two-component diagram technique,which was briefly presented (Fig. 1), but not applied.All this aspects will be discussed in future works.
VIII. ACKNOWLEDGMENTS
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