Breakdown of Photon Blockade: A Dissipative Quantum Phase Transition in Zero Dimensions

The Jaynes-Cummings model with coherent drive is considered as an example of a nonlinear oscillator exhibiting photon blockade, where blockade by one, two, three, etc., photons occurs at a sequence of multiphoton absorption resonances. It is shown that with increasing drive strength, the blockade breaks down by way of a dissipative quantum phase transition. The transition is first order, except at a critical point in the space of drive amplitude and detuning, where a continuous transition is observed. Numerical solutions to the quantum master equation in the steady state are presented and compared with mean-field treatments based on Jaynes and Cummings’ semiclassical equations (strong coupling with conserved pseudospin) and the Maxwell-Bloch equations (spontaneous emission included). The concept of a “thermodynamic limit” in the absence of conserved particle number is explored. Contrasting the identity of large photon number with weak coupling (large volume) in other dissipative quantum phase transitions for photons (e.g., in the phase-transition analogy of laser threshold), the limit of large photon numbers is a strong-coupling limit.

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Bosons and fermions generally behave in different ways because only fermions obey the Pauli exclusion principle, which states that no two fermions may occupy the same quantum state (i.e., have the same spin and momentum); bosons are actually attracted to the same state via the process of stimulated emission. Under some conditions, however, analogies between bosonic and fermionic behavior arise. One such condition is “photon blockade,” an analog of Coulomb blockade for quantum-well electrons whereby an electric current is carried through a tunnel junction one electron at a time. The analog is realized when a photon interacts with an atom (or quantum dot or superconducting qubit) inside a small optical resonator forming a cavity polariton whose energy is shifted from the photon energy; the energy shift creates a pseudo-two-state system and prevents two photons from entering the resonator: The first photon “blocks” the entry of the second photon. Photon blockade has been observed experimentally, but its breakdown has not.

Using computer calculations aided by mean-field analysis, we show here how photon blockade breaks down as the photon flux illuminating the cavity is increased. Photon blockade is not absolute, but it breaks because of a quantum phase transition in zero dimensions far away from thermal equilibrium. Our analysis unifies the reported transition under blockade conditions with optical instabilities observed with nonlinear dielectrics in the 1980s, where the light quantum plays no essential role. We investigate how the mean photon number in the resonator changes with input photon flux and find that mean-field theory and a quantum analysis agree more closely for higher input flux.

Our results provide important background for ongoing studies of quantum phase transitions in photonic cavity arrays and may motivate experimental verification of the breakdown of photon blockade.

Figure 1

Mean photon number in the steady state as a function of drive detuning $\mathrm{\Delta }\omega /\kappa$ and amplitude $\mathcal{E}/\kappa$ for a dipole-coupling strength $g/\kappa =50$. The white dot in the uppermost panel locates the critical point at $\mathrm{\Delta }\omega /\kappa =0$ and $\mathcal{E}/\kappa =25$. Each panel on the left shows contours of the surface plot on the right.

Figure 2

(a) Region of drive-detuning-amplitude plane within which the steady-state $Q$ function corresponding to Fig. 1 is bimodal. Contours of $r=1-|h_1-h_2|/(h_1+h_2)$ are shown, with $h_1$ and $h_2$ the peak heights. Red lines outline the region of bistability according to Eqs. (10)–(12), e.g., as in Fig. 4. (b)–(e) Sample $Q$ functions plotted, respectively, for $(\mathcal{E}/\kappa ,\mathrm{\Delta }\omega /\kappa )=(10,16)$; (18,1.1); (25,0); (30,0).

Figure 3

(a) Mean photon number in steady state versus drive detuning $\mathrm{\Delta }\omega /\kappa$ for drive amplitude $\mathcal{E}/\kappa =18$ (red squares). Additional curves show the time evolution to the steady state for $\mathrm{\Delta }\omega /\kappa =\pm 1.1$ (blue triangles and blue circles) and $\mathrm{\Delta }\omega /\kappa =\pm 2.95$ (green triangles and green circles); the time axis appears at the top of the graph. (b) Surface plot and contours of the steady-state $Q$ function for $\mathrm{\Delta }\omega /\kappa =+1.1$; peak heights are $h_1=0.12$ (left peak) and $h_2=0.14$ (right peak).

Figure 4

Mean-field photon number $|\alpha {|}^2$ (red and blue curves), compared with the photon-number mean, including quantum fluctuations (gray), for a dipole-coupling strength $g/\kappa =50$ and drive amplitudes (a) $\mathcal{E}/\kappa =1$ (red and blue) and 1,5,8 (gray, lower to upper), (b) $\mathcal{E}/\kappa =12$, and (c) $\mathcal{E}/\kappa =30$. Steady states plotted as solid red (blue) curves are stable (unstable); those plotted as dashed red curves—running close to $|\alpha {|}^2=0$ in (a) and (b)—have $\zeta >0$ and are unstable.

Figure 5

Sample quantum trajectories show dynamical symmetry breaking as the drive detuning $\mathrm{\Delta }\omega /\kappa$ is scanned. Time-dependent photon-number expectations are superposed on the plot of Fig. 4, for scan times (one direction) of (a) $\kappa T=6\times {10}^2$ and (b) $\kappa T=6\times {10}^4$. Yellow (green) curves plot the scan from left to right (right to left).

Figure 6

(a) Mean photon number in the steady state versus detuning $\mathrm{\Delta }\omega /\kappa$ for drive amplitude $\mathcal{E}/\kappa =18$ in Fig. 1 and spontaneous emission rate $\gamma /\kappa =0$ (orange), 0.1 (cyan), and 1 (green). Additional curves (blue triangles) show the time evolution to steady state (blue circles) for $\mathrm{\Delta }\omega /\kappa =\pm 2.0$; the time axis appears at the top of the graph. (b) Surface plot of the steady-state $Q$ function for $\gamma /\kappa =0$ (upper left), 0.1 (upper right), and 1.0 (lower); left peak heights $h_l=0.006,0.14,0.19$; right peak heights $h_r=0.28,0.12,0.02$.

Figure 7

As in Fig. 1 with spontaneous emission at the rate (a) $\gamma /\kappa =0.1$, (b) $\gamma /\kappa =1$, (c) $\gamma /\kappa =10$, (d) $\gamma /\kappa =100$.

Figure 8

As in Fig. 2 with spontaneous emission at the rate (a) $\gamma /\kappa =0.1$, (b) $\gamma /\kappa =1$, (c) $\gamma /\kappa =10$, (d) $\gamma /\kappa =100$. The red lines in (d) outline the bistable region according to the state equation [Eq. (28)].