5 E. Bloch states, the effects of s-wave scattering and that of the collective coupling For further analysis we suppose that V$_{ext}$ ( x ) ≡ 0, i.e. the only potential in Eq. (2c) is the periodic optical po- tential induced by the resonator field. In this case the Bloch functions are good candidates for the complete set of single particle wavefunctions: The left and right eigenvectors are normalized as usual: $^{(}$l ( k $^{)}$, r( l ) ) = δ$_{k,l}$ . With the help of Eqs. (10) and (11), the normal modes obey the following uncoupled equation of motion: with the transformed noise operator Q$_{k}$ = $^{(}$l ( k $^{)}$, Z $^{)}$. On integrating Eq. (14), the time dependence of the normal mode operators can be obtained: For a dynamically stable sytem, one needs to have eigen- values with negative imaginary parts. In this case fluc- tuations (the normal modes) decay to a steady-state. ψ$_{n,q}$ ( x ) = N e $^{iqx}$u$_{n,q}$ ( x ) , (16) ρ$_{k}$ ( t ) = e − iω$_{k}$t ρ$_{k}$ (0) +$^{∫}$ t 0 e − iω$_{k}$ ( t − t $^{′}$) Q$_{k}$ ( t ′ ) dt ′ . (15) i d d t ρ$_{k}$ ( t ) = ω$_{k}$ ρ$_{k}$ ( t ) + iQ$_{k}$ ( t ) , (14) u$_{n,q}$ ( x ) = 1 √$_{L}$ e ink$_{0}$$^{x}$, (17) with u$_{n,q}$ ( x ) being a periodic function of period λ/ 2. Here n is the so-called band index and q ∈ [0 , 4 π/λ ] is the quasi-momentum of the particle. N is the constant for normalization. If we impose the Born-von Karman boundary condition with p periods (the quantization vol- ume is p L ), then the normalization constant N = p − 1 / $^{2}$, and the functions u$_{n,q}$ ( x ) are normalized to unity inside a period L = λ/ 2. For practical purposes, we also use plane-waves as basis of the u$_{n,q}$ ( x ) functions (not depend- ing on the quasi-momentum q ) The symmetry property (9) has an important conse- quence on the spectrum of M . Namely, if ω = ϵ − iγ is an eigenvalue of M , than − ω ∗ = − ϵ − iγ is also an eigen- value of M . The modes corresponding to these eigenval- ues form a pair with positive and negative energies. There is an important issue concerning the stability of the normal modes: not all of them include the radiation components. For simplicity, assume that the external po- tential is even with respect to the center of the cavity. In this case the combined external and optical potentials is also even, and parity is a symmetry of the full system. In this case the condensate wavefunction is also even; and the matrix M commutes with the parity operator. The eigenfunctions of M can therefore be classified by their symmetry (being odd or even). If the condensate fluctua- tion parts (third and fourth components) of an eigenvec- tor r ( k ) are odd, than the result of the operator X acting on these components of r ( k ) is zero, since X contains an integration on the whole cavity axis and its kernel is even. For these modes there is no coupling term between the cavity field and the condensate part [see the first two rows of the matrix M in (8b)], and, since damping of the modes comes from the cavity decay, these modes remain undamped and just marginally stable. Consequently, if we want to describe the steady-state values of the correla- tions of fluctuations we have to omit those normal modes which do not couple to the radiation field (assuming that they are initially not populated). with k$_{0}$ = 2 k = 4 π/λ . (Of course the basis (17) is suit- able to express all other L -periodic functions as Fourier series.) The field operator of the atoms are expanded as with b$_{n,q}$ the annihilation operator of the plane-wave state ψ$_{n,q}$ . Here n goes over all integers and q = 4 πm/ ( λp ), with m ∈ { 0 , 1 , . . . p − 1 } . The equation of motion of the annihilation operators b$_{n,q}$ is obtained easily from Eq. (2c), by inserting the field operator (18) and using the orthogonality of the plane- wave states, Ψ( x ) = 1 √$_{L p}$ ∑ n,q b$_{n,q}$ e i ( nk$_{0}$ + q ) $^{x}$. (18) i ℏ d d t b$_{n,q}$ = ℏ $^{2}$( nk$_{0}$ + q ) 2 2 m b$_{n,q}$ + ℏ a † a U$_{0}$ ∑ n ′ L$_{n,n}$ ′ b$_{n}$ ′ $_{q}$+ 2 g λp ∑ n 1 n 2 n 3 q$_{1}$q$_{2}$q$_{3}$ b † n$_{1}$$_{q$_{1}$}$b$_{n}$$_{2}$$_{q}$$_{2}$ b$_{n}$$_{3}$$_{q}$$_{3}$δ$_{(}$$_{n}$$_{+}$$_{n}$$_{1}$$_{)}$$_{k}$$_{0}$$_{+(}$$_{q}$$_{+}$$_{q}$$_{1}$$_{)}$$_{,}$$_{(}$$_{n}$$_{2}$$_{+}$$_{n}$$_{3}$$_{)}$$_{k}$$_{0}$$_{+(}$$_{q}$$_{2}$$_{+}$$_{q}$$_{3}$$_{)}$. (19) L$_{n,n}$ ′ = 1 4$^{[}$ 2 δ$_{n,n}$ ′ + δ$_{n,n}$ $^{′}$+1 + δ$_{n,n}$ $^{′}$− 1 $^{]}$. (20) matrix The first term corresponds to the kinetic energy, which is diagonal for the plane wave states used. The matrix L$_{n,n}$ ′ appearing in the optical potential is a simple, tridiagonal