content_image/392080.jpg content_image/392092.jpg content_image/392091.jpg
ics of the two coupled harmonic oscillators of the res- onator field and the atomic collective motion is described by Eqs. (32). These are linearized equations of motion around the steady-state configurations. For very small fluctuations the semiclassical time evolution can be in- terpreted as orbitals around the steady-state fixed points with frequencies plotted in the left panels of Fig. 4. Due to the imaginary parts the trajectories spiral closer to (for fluctuations around the cooling solution) or farther from (for fluctuations around the heating solution) the corresponding fixed point. From Fig. 4 we can see, that the frequency of atomic motion dominated fluctuations is close to 4 ω$_{R}$ for parameters not very close to the bistable region. In the bistable regime two steady-state solutions exist, and correspondingly there are two fixed points in the harmonic oscillator phase space. According to Fig. 4 the oscillation frequency for the cooling fixed point re- mains around 4 ω$_{R}$ and softens close to the endpoint, while for the heating solution the oscillation frequency grows to around 10 ω$_{R}$ before softening at its endpoint. This kind of renormalization of the atomic dynamics by the interaction with radiation is often referred to as the ‘optical spring effect’. The approximately 10 ω$_{R}$ angular frequency quantitatively gives back the experimentally found 42 kHz oscillation frequency of the Fig. 3C inset of Ref. [4]. 

It is interesting to point out that in the bistable region even small atomic fluctuations can result in a phase-space trajectory that orbits around both of the fixed points [see Fig. 3 of Ref. [3]]. In this case such a lineariza- tion strategy simply can not work because the transition between the two fixed points is necessarily a nonlinear effect. However, if the trajectory would lie in the basin of the linear region of the corresponding fixed points ev- erywhere except the small region of the separatrix, one could expect the oscillation to be described by the above two frequencies: when the system is in a part of the tra- jectory inside the attraction basin of the cooling fixed point the angular frequency would be around 4 ω$_{R}$ , while after crossing the separatrix it would change to around 10 ω$_{R}$ . Such a trajectory of coherent oscillations would also cause a periodic output photon signal but both of the frequencies would appear, i.e. the count rate peaks would come in pairs. 

Figure 5 depicts the regions in the parameter space which represent the heating and (cooling) solutions, i.e. the solutions with dynamical instability (stability). The bistable region, where both a heating and a cooling so- lution can be found, is wedged between the two distinct regions. The tip of the wedge corresponds to the critical point (Δ$_{Cc}$ , η$_{c}$ ). 

Second order correlations of the fluctuations are calcu- lated with the help of the quasi-normal modes and the 

## C. Correlations and entanglement 

noise correlations, since by virtue of Eqs. (11) and (15) 

For a stable system, where all the eigenvalues have nega- tive imaginary parts, the first term on the right hand side, corresponding to the initial condition of the fluctuations, vanishes for times much longer than the characteristic decay times of the system. For the second term one can use the noise correlation function (2b), the definition of the vector noise Z and that of Q . For t → ∞ it follows straightforwardly, that 

where we index the components of the row vectors R and Z starting from 1, and also dropped the exponential term vanishing for large times in the case of a stable system. Note that the zero mode, i.e., δb in (28) representing the phase fluctuations of the condensate, do not contribute to this sum because its eigenfunction does not have photon component, i.e., l (zero mode) 1 = l (zero mode) 2 = 0. 

To be able to relate our results more explicitly to other works, let us introduce quadrature operators, according to δx = ( δa + δa † ) $_{/}$√$_{2,}$ δy = − i ( δa − δa † ) $_{/}$√$_{2,}$ δX = ( δc + δc † ) $_{/}$√$_{2,}$ δY = − i ( δc − δc † ) $_{/}$√$_{2. These quadratude}$ operators are hermitian operators, easily expressed with 

<img src='content_image/392080.jpg'>

<img src='content_image/392092.jpg'>

<img src='content_image/392091.jpg'>

