content_image/401717.jpg content_image/401716.jpg content_image/401708.jpg content_image/401718.jpg content_image/401714.jpg content_image/401715.jpg
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 

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 

<img src='content_image/401717.jpg'>

<img src='content_image/401716.jpg'>

<img src='content_image/401708.jpg'>

with α$_{ss}$ given by Eq. (5), but now with 

The steady-state components of the condensate ampli- tudes β$_{n}$ ( t ) evolve as 

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. 

The equation of motion for the fluctuations, in this representation, is given by 

<img src='content_image/401718.jpg'>

<img src='content_image/401714.jpg'>

<img src='content_image/401715.jpg'>

