Critical relaxation with overdamped quasiparticles in open quantum systems

We study the late-time relaxation following a quench in an open quantum many-body system. We consider the open Dicke model, describing the infinite-range interactions between $N$ atoms and a single, lossy electromagnetic mode. We show that the dynamical phase transition at a critical atom-light coupling is characterized by the interplay between reservoir-driven and intrinsic relaxation processes in the absence of number conservation. Above the critical coupling, small fluctuations in the occupation of the dominant quasiparticle mode start to grow in time, while the quasiparticle lifetime remains finite due to losses. Near the critical interaction strength, we observe a crossover between exponential and power-law $1/\tau$ relaxation, the latter driven by collisions between quasiparticles. For a quench exactly to the critical coupling, the power-law relaxation extends to infinite times, but the finite lifetime of quasiparticles prevents aging from appearing in two-times response and correlation functions. We predict our results to be accessible to quench experiments with ultracold bosons in optical resonators.

Figure 1

Qualitative behavior of the quasiparticle characteristic frequency ${\omega }_{\mathrm{qp}}$ (gray) and inverse lifetime ${\kappa }_{\mathrm{qp}}$ (black), together with the system's damping rate ${\kappa }_{\mathrm{kin}}$ (blue), as a function of the final value $g$ of the light-matter coupling after a sudden quench from $g_i<g$. The dashed line corresponds to the prediction of the noninteracting theory ${\widehat{H}}^{\prime }=0$, where ${\kappa }_{\mathrm{kin}}={\kappa }_{\mathrm{qp}}$. For ${\omega }_{\mathrm{qp}}$, there is no difference between interacting and noninteracting predictions at large enough $N$.

Figure 2

Numerical values of the kinetic parameters ${\kappa }_{\mathrm{kin}},{\lambda }_{\mathrm{kin}}$ together with the inverse quasiparticle lifetime ${\kappa }_{\mathrm{qp}}$. ${\kappa }_{\text{kin}}$ is fitted with the scaling law (9) (blue line). The parameters used are $\kappa =2,{\omega }_0=2,{\omega }_z=2.1$, and $N=1000$, resulting in $g_c\approx 1.4491$. In Appendix pp3, we present results for $\kappa =0.2,1.0$.

Figure 3

Sketch of a log plot of the time evolution of the particle number near $g_c\left(N\right)$, separated into three regions. Starting from the vacuum, the system is for short-time scales described by the evolution according to the bare Green's function (region A), which will then cross over into an algebraic decay (region B), which continues to infinite times for $g=g_c\left(N\right)$. For $g<g_c\left(N\right)$, the final relaxation is exponential, as depicted in region C, whereas for $g>g_c\left(N\right)$, the population instead evolves linearly through that of the unstable steady state.

Figure 4

Self-energy diagrams used in the SCHF calculations. (a) The Hartree contribution, which is only a real shift of the retarded component. (b),(c) The Fock contribution, which is instead complex and frequency dependent. Here the solid (dashed) line correspond to a “classical”(“quantum”) field attached to the vertex.

Figure 5

Numerical values of the kinetic parameters ${\kappa }_{\text{kin}},{\lambda }_{\text{kin}}$ as well as the inverse quasiparticle lifetime ${\kappa }_{\text{qp}}$. The parameters used are the same as in the main text, apart from $\kappa =1$ (upper panel) and $\kappa =0.2$ (lower panel), which result in $g_c\approx 1.1420$ and $g_c\approx 1.0298$, respectively.

Figure 6

Behavior of the two eigenvalues of the photonic distribution function. The red dotted line corresponds to the integrable theory: $N=\infty$. The blue solid line is the result obtained from full SCHF theory, where qualitatively different features appear. These can be well described by an effective linearized theory given by Eq. (D2) (black dashed line) that shifts the atomic resonance in the complex plane and therefore contains only a single (complex) fit parameter ${\kappa }_b$. The parameters used are the same as in the main text, resulting in ${\kappa }_b\approx 0.0335+0.0009i$.