7 intensity I propagating along z -axis passes on its way through an atomic cloud with density distribution ρ ( r ). The resulting intensity change as a function of transverse coordinates x, y is Δ I ( x, y ) = Iσ$_{L}$ ∫ dzρ ( r ), where inte- gration is done over the length of the cloud and σ$_{L}$ is the absorption cross-section. This cross-section depends in the standard way on the light intensity and the fre- quency detuning. Generally, because of the magnetic- field induced frequency shift and the Doppler shift σ$_{L}$ oscillates in time. The light intensity I also oscillates in time. However, for comparatively small modulation am- plitudes, we can disregard these oscillations. Then we get for typical experimental conditions σ$_{L}$ ≈ 5 . 6 × 10 − 15 m $^{2}$. The atomic density distribution ρ ( r ) can be assumed Gaussian with the same width w$_{t}$ ≈ 1 mm in all direc- tions, which was achieved in the experiment by tuning the transverse beam intensities. According to the op- timal path picture, during switching atoms most likely move along the MOT axis, and for such atoms the light intensity change is determined by ρ ( r ) on the axis. The extra force on the switching atom as it moves between the clouds is then directed toward the more populated cloud and is equal to f$_{sh}$ | N$_{1}$ − N$_{2}$ | , with Here, k is the photon wave number, Γ$_{p}$ is the recipro- cal lifetime of the excited state, and s = ( I/I$_{s}$ )[1 + (2 δ/ Γ$_{p}$) $^{2}$] − 1 is the resonant absorption strength ( I$_{s}$ is the saturation intensity and δ is the detuning of the radiation frequency from the atomic transition frequency). Equa- tion (29) gives f$_{sh}$ ≈ 2 . 5 × 10 − 32 N. Note that, since the clouds are oscillating, the force ∝ f$_{sh}$ is oscillating as well, its time dependence is determined by the sgn- function. The effect of the shadow force can be conveniently an- alyzed in the rotating frame by changing from ( z $^{i}$, ˙ z $^{i}$) to slowly varying two-component vectors q $^{i}$, Eq. (26). In the RWA the equation of motion for q i becomes ( i = 1 , . . . , N$_{tot}$ ). Here, f $^{i}$( τ ) is the random force on i th atom (random forces on different atoms are uncorre- lated), ˆ ϵ is the permutation tensor, and H$_{sh}$ is the inter- action Hamiltonian in the slow variables, The shift of the n th attractor ( n = 1 , 2) given by Eq. (33) and the corresponding change of the vibration amplitude of the n th cloud is proportional to the num- ber of atoms in the other cloud, $^{N$_{3}$}$− $_{n}$. Therefore the vibration amplitudes of the clouds become different in the broken-symmetry state. This prediction and the lin- ear dependence of the amplitude change on the number of atoms in the other cloud are in full agreement with the experiment; the numerical estimate for the simplified model of sinusoidal 1D vibrations is within a factor of 2 from the measured value. Here, ¯ V$_{sh}$ is given by Eq. (31) for V$_{sh}$ in which f$_{sh}$ is replaced with ¯ f$_{sh}$ . The quantity ¯ f$_{sh}$ characterizes the force from the shadow effect which is averaged over the cross-section of the cloud transverse to the MOT axis. Such averaging occurs as atoms are moving within the cloud. For Gaussian clouds of the same cross-section, ¯ f$_{sh}$ = f$_{sh}$/ 2. An important effect that allows one to independently compare the many-atomic theory with experiment is the change of the vibration amplitudes due to the interac- tion. This change is determined by the change of the equilibrium positions q A n in the rotating frame. For weak inter-atomic interaction it can be found by linearizing equations of motion (30). The average force on an atom from other atoms in the same cloud is equal to zero. The shift of the equilibrium position is due only to the force from the atoms in the other cloud. Disregarding small fluctuations about the equilibrium positions, for the shift of n th attractor in the rotating frame δ q A n we obtain Equations (28) and (32) were used to find the coef- ficients α$_{nm}$ and the critical value of the total number of atoms N$_{c}$ = 2 k$_{B}$T/ ( α$_{11}$ − α$_{12}$ ) where the symmetry breaking transition occurs. The obtained N$_{c}$ was within a factor of 2 from the value of N$_{c}$ observed in the exper- iment. This is reasonable given the uncertainty in the temperature, the renormalization of N$_{c}$ by the finite life- time of atoms in the MOT, cf. Eq. (23), and the fact that the light intensity modulation was not weak and, respectively, the atomic vibrations noticeably deviated from sinusoidal. where the prime indicates that i = j . The general expression for the interaction-induced cor- rection to the activation energy R$_{n}$ , Eq. (5), in the RWA has the form where λ$_{n}$ ( τ ) and q (opt) n ( τ ) are the solutions of the single- atom problem of minimizing the functional (28) for switching from the state n . α$_{nm}$ = ∫ ∞ −∞ dτ λ$_{n}$ ( τ )ˆ ϵ∂$_{q}$ A $_{m}$V$_{sh}$ ( q (opt) n ( τ ) − q A $_{m}$) , (32) ( δ q A $_{n}$∂$_{q}$ A $_{n}$) K ( q A n ) + $^{N$_{3}$}$− $_{n}$ˆ ϵ∂$_{q}$ A n ¯ V$_{sh}$ ( q A n − q A 3 − $_{n}$) = 0 . (33) f$_{sh}$ = ℏ k Γ$_{p}$ σ$_{L}$s/ 4 π ( s + 1) w 2 $_{t}$. (29) d q i dτ = K $^{i}$( q $^{i}$) + ˆ ϵ∂$_{q}$ i H$_{sh}$ + f $^{i}$( τ ) (30) H$_{sh}$ = 1 2 ∑ ′ ij V$_{sh}$ ( q i − q $^{j}$) , V$_{sh}$ ( q ) = 8 f$_{sh}$ πm$_{a}$ω 2 $_{0}$ϵ$_{F}$ C$_{RWA}$ | q | , (31) C. The shadow effect in the rotating frame