7 6 × 10 4 atoms, while the next state with n = ± 1, i.e. the atomic motional state with wavefunction cos k$_{0}$x con- taining only a few hundred atoms. Let us therefore re- strict the Hilbert space of the one-particle atomic mo- tion into this relevant 2-dimensional subspace, containing the homogeneous single particle wavefunction and that of cos k$_{0}$x . The atomic field operator becomes with the single-particle wavefunctions ϕ$_{0}$ ( x ) = L − 1 / $^{2}$, and ϕ$_{1}$ ( x ) = $^{√}$2 /L cos( k$_{0}$x ), and their corresponding an- nihilation operators b$_{0}$ and b$_{1}$ . The single-particle wave- function sin( k$_{0}$x ), corresponding to the antisymmetric combination of the n = ± 1 states, is omitted, since this wavefunction doesn’t couple to the photonic field due to symmetry reasons mentioned above. In this model the atomic motion is represented in a two-mode Fock space. The mean field expansion is with Ψ( x ) = ϕ$_{0}$ ( x ) b$_{0}$ + ϕ$_{1}$ ( x ) b$_{1}$, (26) i ℏ d d t δ ˜ b$_{n,q}$ = ( H$_{0}$ )$_{n,n}$ ′ δ ˜ b$_{n}$ $^{′}$,q + ℏ U$_{0}$δ$_{q,}$$_{0}$ ( α ∗ $_{ss}$δa + α$_{ss}$δa † ) ∑ n ′ L$_{n,n}$ ′ β$_{n}$ ′ + 2 g λp ∑ n$_{1}$,n$_{2}$,n ′ β$_{n}$$_{1}$β$_{n}$$_{2}$ $^{(}$δ ˜ b$_{n}$ $^{′}$,q + δ ˜ $_{b}$† − n $^{′}$, − q $^{)}$δ$_{n}$$_{+}$$_{n}$$_{1}$$_{,n}$ $^{′}$+ $_{n$_{2}$}$, (25b) ( H$_{0}$ )$_{n,n}$ ′ = ℏ $^{2}$( nk$_{0}$ + q ) 2 2 m δ$_{n,n}$ ′ + ℏ | α$_{ss}$ | $^{2}$U$_{0}$ L$_{n,n}$ ′ + gN Lp ∑ n$_{1}$,n$_{2}$ β$_{n}$$_{1}$β$_{n}$$_{2}$δ$_{n}$$_{+}$$_{n}$$_{1}$$_{,n}$ $^{′}$+ n$_{2}$ − µδ$_{n,n}$ ′ . (25c) b$_{n}$ = √$_{Nβ}$$_{n}$ + δb$_{n}$, n = 0 , 1; (27) δb$_{n}$ = β$_{n}$δb + γ$_{n}$δc, (28) where β$_{n}$ is again the representation of the condensate wavefunction in the 2-dimensional Hilbert space, normal- ized to unity. The presence of a Bose-Einstein condensate distinguishes a subspace of one single mode. Accordingly, the fluctuation operators can be expanded to a part par- allel to the condensate and to one orthogonal to it: For a qualitative understanding of Eqs. (25), let us con- sider first the weak photon-atom coupling case, when either the photon number inside the cavity is small or U$_{0}$ is small. In this situation the atomic distribution is essentially unmodified by the periodic potential and the condensate wavefunction is almost homogeneous, i.e. β$_{n}$ = δ$_{n,}$$_{0}$β$_{0}$ . Fluctuations in the atomic density are in- duced from the homogeneous condensate. In linear order, the atom-light interaction excites fluctuations to bands n ± 1 with q = 0, while s-wave interaction populates fluc- tuations with arbitrary n and q . However, as discussed earlier, in the limit ω$_{sw}$ ≪ ω$_{R}$ , s-wave scattering induces essentially only intra-band transitions, and therefore, for a homogeneous condensate, only states with n = 0 are populated by s-wave interaction. Figure 1 depicts graph- ically the population of fluctuations excited from the homogeneous condensate wavefunction in leading order. Moreover, in experiments [3, 4] the one atom light shift is a tunable parameter, it’s typical value is chosen to be on the order of the recoil frequency, i.e. U$_{0}$ ≈ 2 π × 4 kHz and is also much larger than the characteristic frequency of s-wave scattering. Therefore on the time scale in which the steady-state is reached, the effect of s-wave scattering is still not significant. When describing the steady state, one can safely neglect it as a first approximation and con- sider each band represented by a single state vector with q = 0. with γ$_{n}$ the unit vector orthogonal to the condensate: γ$_{n}$ = ( − β$_{1}$, β$_{0}$ ) $^{T}$. The part, δb , parallel to the conden- sate can be related to the arbitrariness of the phase of the condensate, and corresponds to a zero mode. The orthog- onal part, δc is the sole degree of freedom. Therefore the fluctuations of the atomic field, similarly to the photon field, has a single degree of freedom. The fluctuations of the combined atom-resonator field system has 2 degrees of freedom, and is analogous in many ways to cavities with a moving mirror, to the so called optomechanical systems. In order to obtain the mean-field equations and the fluctuation equations in linear order one can start from Eq. (2) and use the truncated field operator (26) and then the mean-field ansatz (27). The mean-field equations for the condensate now read Note that there is an interesting complementary regime in which the relevant states are localized, Wannier-type ones formed by the coherent superposition of many quasi- momentum states with q = 0. To describe many body effects of the atomic degree of freedom in this regime, one can resort to a Bose-Hubbard model with self-consistent parameters [32–37], in which collisions play a vital role but interband transitions are usually neglected. with P$_{1}$ = diag(0 , 1), the projection matrix to the sub- space of ϕ$_{1}$ ( x ); the matrix representing the light-shift is given by α$_{ss}$ given by Eq. (5), with 〈 U 〉 = β ∗ $_{k}$( t ) U$_{k,l}$β$_{l}$ ( t ). (Note the convention of automatic summation over repeated in- dices.) The condensate wavefunction also has the time In the experimental situation of Refs. [3, 4] the BEC wavefunction can be considered almost homogeneous, with a condensate fraction in the β$_{0}$ state containing i d d t β$_{k}$ ( t ) = $^{[}$4 ω$_{R}$ $^{(}$P$_{1}$) k,l + | α$_{ss}$ | $^{2}$U$_{k,l}$ $^{]}$β$_{l}$ ( t ) , (29) U$_{k,l}$ = 〈 ϕ$_{k}$ | U ( x ) | ϕ$_{l}$ 〉 = U$_{0}$ 2 [ 1 √$_{2}$$_{/}$$_{2}$ √$_{2}$$_{/}$$_{2}$ 1 ] , (30) F. Optomechanics