6 n q n = − 1 n = 1 n = 0 n = 2 n = − 2 transitions by the optical potential transitions by s-wave scattering q = 0 states are not p opulated for n = 0 FIG. 1: (color online) Schematic diagram of fluctuations cou- pling to the the homogeneous BEC in linear order. The grey circle represents the macroscopically occupied (BEC) state, while the empty circles represent states which are not macro- scopically occupied. The arrows show how these states can be populated in linear order by scattering from the conden- sate via the interaction with the photon field and also via the s-wave scattering. It is trivial that the kinetic energy is the lowest for n = 0 , q = 0. The appearance of the optical potential makes the Hamiltonian nondiagonal in plane-wave basis, however, it mixes only operators with the same quasi- momentum q . Coupled plane waves can have indices n differing only by Δ n = ± 1. On the other hand, the s-wave scattering mixes operators with different n and different q momenta. The total momentum is conserved by the δ function imposing ( n + n$_{1}$ − n$_{2}$ − n$_{3}$ ) k$_{0}$ + ( q + q$_{1}$ − q$_{2}$ − q$_{3}$ ) = 0, which allows for normal scattering, when q + q$_{1}$ = q$_{2}$ + q$_{3}$ , and n + n$_{1}$ = n$_{2}$ + n$_{3}$ , and also for umklapp scattering processes, where the differ- ence of the quasi momentum in the scattering is equal to a reciprocal lattice vector: q + q$_{1}$ = q$_{2}$ + q$_{3}$ − Δ n · k$_{0}$ , and n + n$_{1}$ = n$_{2}$ + n$_{3}$ + Δ n . It is useful at this point to estimate the characteristic frequency corresponding to the atom-atom s-wave scattering based on the physical parameters relevant to the experimental situation in eg. Refs. [3, 4]. The s-wave scattering length for the | 1 , − 1 〉 states of the $^{87}$Rb atoms is about 5 . 3 nm. Assuming a particle number of N = 6 × 10 4 atoms distributed in the cavity lattice of period L = λ/ 2 = 390 nm and the cavity egg-crate potential containing p = 460 val- leys, corresponding to a cavity length of 180 µ m, and the waist of the optical potential as w = 25 µ m, the char- acteristic frequency of s-wave interaction can be around ω$_{sw}$ = 4 π ℏ aN/ ( Lpw $^{2}$m ) ≈ 2 π × 4 Hz. (Here we have neglected any external trapping potential.) This means that the characteristic energy of s-wave scattering is 3 orders of magnitude lower than that of the recoil en- ergy separating the bands from each other. As a re- sult s-wave scattering dominantly occurs only with atoms in the same band and umklapp processes can be ne- glected. When incorporating the effects of the confin- ing parabolic optical potential in Ref. [3], the condensate density grows roughly by a factor of 100, and results in the ratio ω$_{sw}$/ω$_{R}$ ∼ 0 . 1. In this situation the contribution of umklapp processes is still small enough to expect that it does not significantly change the forthcoming results. The mean-field equation for the condensate is obtained by the substitution The steady-state components of the condensate ampli- tudes β$_{n}$ ( t ) evolve as with α$_{ss}$ given by Eq. (5), but now with 〈 U 〉 = U$_{0}$ ∑ n,n ′ β ∗ $_{n}$( t ) L$_{n,n}$ ′ β$_{n}$ ′ ( t ) . (23) β$_{n}$ ( t ) = β$_{n}$ e − iµt/ $^{ℏ}$, (24) b$_{n,q}$ = √$_{Nδ}$$_{q,}$$_{0}$$_{β}$$_{n}$$_{(}$$_{t}$$_{) +}$ δb$_{n,q}$ ( t ) , (21) i ℏ d d t β$_{n}$ ( t ) = ℏ $^{2}$n$^{2}$k2 0 2 m β$_{n}$ ( t ) + ℏ | α$_{ss}$ | $^{2}$U$_{0}$ ∑ n ′ L$_{n,n}$ ′ β$_{n}$ ′ ( t ) + 2 g λp ∑ n$_{1}$n$_{2}$n$_{3}$ β ∗ $_{n$_{1}$}$( t ) β$_{n}$$_{2}$ ( t ) β$_{n}$$_{3}$ ( t ) δ$_{n}$$_{+}$$_{n}$$_{1}$$_{,n}$$_{2}$$_{+}$$_{n}$$_{3}$, (22) i d d t δa ( t ) = $^{[}$− Δ$_{C}$ + N 〈 U 〉 − iκ $^{]}$δa ( t ) + NU$_{0}$α$_{ss}$ ∑ n,n ′ β$_{n}$L$_{n,n}$ ′ $^{(}$δ ˜ b$_{n}$ $^{′}$, 0 + δ ˜ $_{b}$† − n $^{′}$, $_{0}$) + iξ ( t ) , (25a) with µ being the chemical potential. It can be seen from Eq. (22) that β$_{n}$ can also be chosen real for all n . Since L$_{n,n}$ ′ = $^{L}$− n, − n ′ , and due to the symmetry of Eq. (22) it follows that, β$_{n}$ = $^{β}$− $_{n}$, which means, that the Fourier expansion of the condensate wavefunction contains only cosine terms. with β$_{n}$ the mean value of the annihilation operators. The δ$_{q,}$$_{0}$ condition for the mean-field comes from the fact that the ground state of the translationally invariant sys- tem is also invariant under discrete translation, i.e. the condensate wavefunction is periodic also with L = λ/ 2. (None of the terms in Eq. (21) violates the discrete trans- lation of the system, and having q = 0 would cost in kinetic energy.) The mean-field equation reads then The equation of motion for the fluctuations, in this representation, is given by