Large Deviations, Central Limit and dynamical phase transitions in the atom maser

Abstract

The theory of quantum jump trajectories provides a new framework for understanding dynamical phase transitions in open systems. A candidate for such transitions is the atom maser, which for certain parameters exhibits strong intermittency in the atom detection counts, and has a bistable stationary state. Although previous numerical results suggested that the "free energy" may not be a smooth function, we show that the atom detection counts satisfy a large deviations principle, and therefore we deal with a phase cross-over rather than a genuine phase transition. We argue however that the latter occurs in the limit of infinite pumping rate. As a corollary, we obtain the Central Limit Theorem for the counting process. The proof relies on the analysis of a certain deformed generator whose spectral bound is the limiting cumulant generating function. The latter is shown to be smooth, so that a large deviations principle holds by the Gartner-Ellis Theorem. One of the main ingredients is the Krein-Rutman theory which extends the Perron-Frobenius theorem to a general class of positive compact semigroups.

Figures (7)

• Figure 1: Mean photon number (black line) and photon number distribution (background) in the stationary state $\rho_{ss}$ as function of $\alpha = \sqrt{N_{\hbox{ex}}} \phi$
• Figure 2: The birth (blue) and death rates as functions of $\vartheta$ for different values of $\alpha$. The intersection points correspond to minima and maxima of the stationary distribution.
• Figure 3: Rescaled potentials $U(n)/N_{\text{ex}}$ as function of $n/N_{\text{ex}}$, for various finite $N_{\text{ex}}$ converge to a limit potential for $N_{\text{ex}} \rightarrow \infty$. For $\alpha < 1$ the potential is minimum at zero; for $1<\alpha <4.6$ it has a unique minimum away from $n=0$; for $4.6<\alpha< 7.8$ there are two local minima which become equal at $\alpha \approx 6.66$.
• Figure 4: Sample trajectories for the birth-death process describing the cavity state jumping on the ladder of Fock states $|k\rangle\langle k|$ (top left and right) and total measurement counts $\Lambda_{t}$ (bottom left and right) for $\alpha \approx 1$ (left) and $\alpha \approx 6.66$ (right) at $N_{\text{ex}}=50$. The corresponding to stationary state distributions (center) showing large variance at $\alpha \approx 1$ (red) and bistability at $\alpha \approx 6.66$ (green).
• Figure 5: Derivative $\lambda'(s)$ (panel (a)) and the spectral gap $g(s)$ of $L_{s}$ (panel (b)) as functions of $s$ and $\alpha = \phi /\sqrt{N_{\text{ex}}}$ (after Fig. 3 in Garrahan2011).

Theorems & Definitions (13)

• Definition 1: Fagnola1999 and Section 3.1.2 in Bratteli1979
• Proposition 2
• Proposition 3
• Theorem 4: Gärtner-Ellis theoremDembo2010, pp. 44-55
• Theorem 5
• Corollary 6
• Lemma 7
• Proposition 8
• Lemma 9
• proof