Fluctuations and photon statistics in a quantum metamaterial near a superradiant transition

The analysis of single-mode photon fluctuations and their counting statistics at the superradiant phase transition is presented. The study concerns the equilibrium Dicke model in a regime where the Rabi frequency, related to a coupling of the photon mode with a finite-number qubit environment, plays the role of the transition's control parameter. We use the effective Matsubara action formalism based on the representation of Pauli operators as bilinear forms with complex and Majorana fermions. Then, we address fluctuations of superradiant order parameter and quasiparticles. The average photon number, the fluctuational Ginzburg-Levanyuk region of the phase transition, and Fano factor are evaluated. We determine the cumulant-generating function which describes a full counting statistics of equilibrium photon number. Exact numerical simulation of the superradiant transition demonstrates quantitative agreement with analytical calculations.

Figure 1

[(a), (c)] Average photon number $\langle N_{\mathrm{ph}}\rangle$, fluctuations $\sqrt{\langle \langle {N_{\mathrm{ph}}^2\rangle \rangle }^{\phantom{0}}}$, and [(b), (d)] Fano factor $F$ as functions of collective Rabi frequency $\mathrm{\Omega }=g\sqrt{N}$ (in units of resonator mode frequency $\omega$). All curves are calculated for the full resonance limit between qubits and cavity mode frequencies, $\overline{\epsilon }=\omega$. The temperature is low, $T=0.1\omega$, and qubit number is $N=100$ in panels (a) and (b) and $N=1000$ in panels (c) and (d). White, light blue, and light green areas correspond to normal (N) phase, fluctuational region, and superradiant (SR) phase, respectively. The critical point is ${\mathrm{\Omega }}_{\mathrm{c}}=\omega$ and the width of the fluctuational region is $2{\mathrm{\Omega }}_{\mathrm{GL}}\approx 0.063\omega$ in panels (a) and (b) and $2{\mathrm{\Omega }}_{\mathrm{GL}}=0.02\omega$ in panels (c) and (d). Red and green curves stand for calculations based on $S_{\mathrm{eff},0}$ and $S_{\mathrm{eff},1}$ actions, respectively. The Fano factor [(b), (d)] in the normal phase demonstrates that $F_{\min }<F<1$. It means negative correlation between photons (antibunching effect). The horizontal dotted line $F=1$ separates the regions of negative ($F<1$) and positive ($F>1$) photon correlations. The fluctuational region in panels (b) and (d) demonstrates a growth of the Fano factor with a peak at $F_{\mathrm{c}}>1$ which means positive correlations between photons. The superradiant phase shows reentrance to the negative correlations with the decay of $F$.

Figure 2

Comparison of results obtained in effective action techniques and in exact numerical calculations based on equilibrium density matrix. The data for the average photon number $\langle N_{\mathrm{ph}}\rangle$ are presented. The temperature is low as $T/\omega =0.3$ for the panel (a) and $T/\omega =0.1$ for the panel (b) and qubit number is $N=10$. The range of cubit-cavity coupling covers the domain of the normal phase, fluctuational region, and the superradiant phase. The data obtained from numerical simulations are shown as blue curves. Results of field theoretical approaches based on $S_{\mathrm{eff},0}$ and $S_{\mathrm{eff},1}$ are shown as red dashed and green dotted curves, respectively. We note a surprisingly small deviation of solid blue lines from the green dotted line. Although the parameters are near the edge of applicability range of the theory, a good agreement between numerical results and theoretical calculations is clearly observed.

Figure 3

Average photon number $\langle N_{\mathrm{ph}}\rangle$ obtained by means of $S_{\mathrm{eff},1}$ as the function of the Rabi frequency $\mathrm{\Omega }$ and qubit energies ${\epsilon }_j=\overline{\epsilon }$ in nonresonant regimes of $\omega \ne \overline{\epsilon }$. The dark regions in the maps correspond to normal phase; bright regions correspond to superradiant phase. Qubit number $N=100$; $\mathrm{\Omega }$ and $\overline{\epsilon }$ are measured in units of resonator frequency $\omega$. (a) Data calculated for low-temperature regime $T=0.1\omega$. (b) Data calculated for intermediate temperature $T=\omega$. Red solid curve corresponds to the critical ${\mathrm{\Omega }}_{\mathrm{c}}$ as a function of $\overline{\epsilon }$ given by the relation (39). Insets in panels (a) and (b) demonstrate $\langle N_{\mathrm{ph}}\rangle$ as functions of $\mathrm{\Omega }/\omega$ for cuts marked by green dashed lines; red points mark the critical Rabi frequency for a given $T$ and $\overline{\epsilon }$ in the cut.