Rabi model beyond the rotating-wave approximation: Generation of photons from vacuum through decoherence

We study numerically the dynamics of the Rabi Hamiltonian, which describes the interaction of a single cavity mode and a two-level atom without the rotating wave approximation. We analyze this system subjected to damping and dephasing reservoirs, included via the usual Lindblad superoperators in the master equation. We show that the combination of the antirotating term and the atomic dephasing leads to linear asymptotic photon generation from the vacuum. We reveal the origins of the phenomenon and estimate its importance in realistic situations.

Figure 1

(Color online) Dynamics of the Rabi Hamiltonian for $\Delta =0$ and $g=0.1$ as a function of dimensionless time $\tau =\xi t$ $(\xi =0.1)$. (a) Mean photon number $⟨n⟩$ for initial state $\mid g,0⟩$ without dephasing (line 1) and with dephasing ${\gamma }_{\mathrm{ph}}=0.1$ (line 2). Line 2 shows photon generation from the vacuum due to atomic dephasing and the created field state is superpoissonian, since the Mandel factor $q>0$ (line 3). (b) $⟨n⟩$ for different initial states $\mid {\varphi }_i⟩$, $i=1,..,6$ (see the text), demonstrating that the asymptotic photon generation does not depend on the initial state. (c) Population inversion $⟨{\sigma }_z⟩$ for the states $\mid {\varphi }_i⟩$ shown in (b): as expected, $⟨{\sigma }_z⟩$ goes to zero asymptotically due to the atomic decoherence.

Figure 2

(Color online) (a)–(c) Mean photon number for RH $⟨n⟩$ vs $\tau$ for three particular trajectories, obtained using the quantum trajectories approach. The parameters are $\Delta =0$, $g=0.1$, ${\gamma }_{\mathrm{ph}}=0.1$, and the initial state is $\mid g,5⟩$. $⟨n⟩$ tends to increase with time. (d) $⟨n⟩$ for a single trajectory using JCH, showing that there is no increase in the photon number without the antirotating term. (e) Asymptotic increase of mean photon number for the RH (line 1) and the test Hamiltonian $H_E$ (line 2, see text), demonstrating that the phenomenon is due to the antirotating term and the limitation by vacuum, $a\mid 0⟩=0$.

Figure 3

Variation of asymptotic photon generation rate $\beta \equiv {\phantom{\mid }d⟨n\left(\tau \right)⟩∕d\tau \mid }_{as}$ with (a) $(\omega +{\omega }_0)$ for fixed $g$ and ${\gamma }_{\mathrm{ph}}$; (b) $g$ for fixed ${\gamma }_{\mathrm{ph}}$ and ${\omega }_0$; (c) ${\gamma }_{\mathrm{ph}}$ for fixed $g$ and ${\omega }_0$. We observe that $\beta \sim {\gamma }_{\mathrm{ph}}$ and is inversely proportional to ${({\omega }_0+\omega )}^2$. In the weak coupling regime $\beta \sim g^2$.

Figure 4

(Color online) Simulation of atomic dephasing via random atomic frequency fluctuations for parameters ${\Omega }_0=1$, $g=6\times {10}^{-2}$, $\epsilon =g$. (a) Spectrum of the frequency noise ${\omega }_0\left(t\right)-{\Omega }_0$. Line 1 (red) corresponds to the “low-” and line 3 (black) to the “high-” frequency noise. (b) $⟨n⟩$ for three individual simulation runs using high frequency noise. (c) ${⟨n⟩}_{\mathrm{av}}$ averaged out over many simulation runs, showing photon growth, dependent on the noise frequency. Here line 2 (blue) is the “middle-frequency” noise. (d) $P_e$ averaged out over many simulations. These curves agree qualitatively with the results obtained above using the master equation approach.

Figure 5

(Color online) (a) $⟨n⟩$ vs $\tau$ using the master equation (2) for initial state $\mid g,0⟩$ with parameters $\Delta =0$, $g=0.1$, $n_t=0$ and decay rates $({\gamma }_{\mathrm{ph}},\gamma ,\kappa )$ as follows. Line 1: $(0,0,1)\times {10}^{-1}$, line 2: $(0,1,0)\times {10}^{-1}$, line 3: $(1,1,1)\times {10}^{-1}$, line 4: $(1,0,1)\times {10}^{-1}$, line 5: $(1,1,0)\times {10}^{-1}$, line 6: (0,0,0). (b) $⟨n⟩$ vs $gt$ for initial state $\mid g,0⟩$ using circuit QED parameters $\Delta =0$, $g=2\times {10}^{-2}$, $n_t=6\times {10}^{-2}$ and distinct decay rates (line 1 denotes the thermal photon number $n_t$). Line 2: current parameters $(2,3,0.4)\times {10}^{-4}$. Line 3: expected future scenario $(2,3,0.4)\times {10}^{-5}$. Line 4: highly biased noise $(200,3,0.4)\times {10}^{-4}$. For current parameters there is no observable difference between $n_t$ and $⟨n⟩$, however, in the future or in a very noisy environment $⟨n⟩$ may become significantly larger than $n_t$ due to photon creation through decoherence. (c) $P_e$ corresponding to the parameters in (b); $P_e$ resembles the behavior of $⟨n⟩$.