Damping of quasiparticles in a Bose-Einstein condensate coupled to an optical cavity

We present a general theory for calculating the damping rate of elementary density-wave excitations in a Bose-Einstein condensate strongly coupled to a single radiation field mode of an optical cavity. Thereby we give a detailed derivation of the huge resonant enhancement in the Beliaev damping of a density-wave mode, predicted recently by Kónya et al. [Phys. Rev. A 89, 051601(R) (2014)]. The given density-wave mode constitutes the polaritonlike soft mode of the self-organization phase transition. The resonant enhancement takes place, in both the normal and the ordered phases, outside the critical region. We show that the large damping rate is accompanied by a significant frequency shift of this polariton mode. Going beyond the Born-Markov approximation and determining the poles of the retarded Green's function of the polariton, we reveal a strong coupling between the polariton and a collective mode in the phonon bath formed by the other density-wave modes.

Figure 1

Schematic representation of the self-organization phase transition. (Left) Below a threshold value of the transverse laser pump power, the BEC fills the cavity homogeneously on the wavelength scale, and there is no light scattering from the pump laser into the cavity. (Right) Above threshold, the condensate self-organizes into a wavelength periodic pattern and Bragg scatters into the cavity. The field building up in the cavity traps the atoms in the patterned spatial structure, thereby stabilizing the ordered phase.

Figure 2

Phonon spectrum. The laser pump picks the polariton excitation whose frequency is denoted by a (red) dot with zero quasimomentum. The scattering processes underlying the Landau and Beliaev damping processes are schematically represented on the left and right sides, respectively.

Figure 3

Real part of the polariton quasiparticle frequency (only the positive frequency part is shown) as a function of the external laser drive strength. This polariton is the soft mode of the self-organization phase transition; hence, the frequency vanishes at a critical point. Without external driving $\left(y=0\right)$, the polariton frequency has just the value where the middle and the upper branches touch for $q=0$ in the phonon spectrum in Fig. 2.

Figure 4

Beliaev damping rate as a function of the control parameter that is the normalized laser pump strength at zero temperature, $T=0$. There appear resonant peaks both below and above the critical point. For the explanation of their origin, see text below. The quite overlapping curves correspond to various values of the phenomenological phonon damping parameter, $\epsilon =0.03,0.01,0.003,0.001$ in units of ${\omega }_R$, in order of increasingly sharp peaks.

Figure 5

Dependence of the Landau- and Beliaev-damping rates on the temperature. For temperatures below the recoil frequency, the Landau damping does not suppress the Beliaev damping peak. Parameters are the same as for Fig. 4, and $\epsilon =0.01$.

Figure 6

Beliaev frequency shift as a function of the normalized control parameter, indicating significant modification of the bare polariton frequency in conjunction with the enhanced damping rate. Parameters are the same as for Fig. 4.

Figure 7

Spectral function of the polariton mode for various driving strengths $y$. The vertical range is truncated thus the high peaks, which are the dominantly polariton ones, are cut. These peaks lie quite precisely on the thick line drawn in the bottom plane, which is the polariton eigenfrequency in the Bogoliubov approximation (cf. the curve in Fig. 3), except for the range around $y/y_{\mathrm{crit}}\sim 0.8$. As the increasing control parameter approaches this range, another peak grows up, which indicates that a significant phonon component mixes to the polariton, and an avoided crossing can be observed.

Figure 8

Complex analytic continuation of the retarded Green's function of the polariton mode for driving strength $y=0.629y_{\mathrm{crit}}$. The largest peak close to the real axis is dominantly the polariton mode. The phonon bath is represented by the multiple smaller peaks. The one closest to the large peak yields a strong phonon-polariton coupling influencing significantly the dependence of the polariton frequency as a function of the control parameter $y$ shown in Fig. 7.

Figure 9

The real (left) and imaginary (right) parts of the two most relevant poles of the Green's function. A well-resolved avoided crossing can be seen in the real part, indicating a considerable mixing of the polariton with a collective phonon mode. The smaller, imaginary part can be associated with the polariton damping rate which is then reduced compared with the Born-Markov prediction.