Entanglement and decoherence of $N$ atoms and a mesoscopic field in a cavity

We investigate the behavior of $N$ atoms resonantly coupled to a single electromagnetic field mode sustained by a high quality cavity, containing a mesoscopic coherent field. We show with a simple effective Hamiltonian model that the strong coupling between the cavity and the atoms produces an atom-field entangled state, involving $N+1$ nearly coherent components slowly rotating at different paces in the phase plane. The periodic overlap of these components results in a complex collapse and revival pattern for the Rabi oscillation. We study the influence of decoherence due to the finite cavity quality factor. We propose a simple analytical model, based on the Monte Carlo approach to relaxation. We compare its predictions with exact calculations and show that these interesting effects could realistically be observed on a two or three atoms sample in a 15 photon field with circular Rydberg atoms and superconducting cavities.

Figure 1

Schematic evolution of the entangled state for one atom and a mesoscopic coherent state in a cavity for real positive $\alpha$. The atomic dipole states are represented as arrows. The field coherent states are represented as an uncertainty disk at the tip of the classical amplitude. Each component $\mid D_{\pm }\left(t\right)⟩\otimes \mid {\psi }_{\pm }\left(t\right)⟩$ of the superposition involves an atomic polarization and a field state slowly rotating in the phase plane at velocities $\pm g∕4\sqrt{\overline{n}}$.

Figure 2

Spontaneous revivals of the Rabi oscillation signal for one atom with initial condition $(\mid e⟩=\mid m=1∕2⟩)\otimes \mid \alpha ⟩$ and a coherent state with mean photon number (a) $\overline{n}=15$ and (b) $\overline{n}=40$. The solid line shows the probability $P\left(\varphi \right)$ for finding the atom in $\mid e⟩$, as a function of the dimensionless time $\varphi =gt∕2\sqrt{\overline{n}}$, computed using numerical integration. The dashed curves show the upper and lower envelopes predicted by our mesoscopic approximation.

Figure 3

Schematic view of the Hilbert space for $N=3$ atoms. The oblique gray lines represent the stable subspaces. The action of ${\mathcal{J}}_{\pm }$ on ${\mathcal{H}}_3$ is depicted.

Figure 4

Position of the Gea-Banacloche states at the times of first occurrence of spontaneous Rabi oscillation revivals for $N=3$ $(J=3∕2)$. (a) Collapse of the Rabi oscillations when the various components split apart. (b) First spontaneous revival for $\varphi =2\pi ∕3$. (c) Second spontaneous revival for $\varphi =\pi$. (d) Complete spontaneous revival involving all atomic polarizations for $\varphi =2\pi$.

Figure 5

(Color online) Spontaneous revivals of the Rabi oscillation signal $P=P_{m=J}$ for $N=3$ atoms $(J=3∕2)$, as a function of the dimensionless time $\varphi =gt∕2\sqrt{\overline{n}}$. The initial state $\mid m=3∕2⟩\otimes \mid \alpha ⟩$ with an average photon number $\overline{n}=\mid \alpha {\mid }^2=15$. The signal is computed using the mesoscopic approximation (gray line), its upper and lower envelopes (plain lines), and the exact solution (plain line with rapid oscillations) in the dissipationless case. Letters (a), (b), (c), and (d) refer to the overlaps of cavity field states for each revival depicted in Fig. 4. As expected, spontaneous revivals (b) and (c) are symmetric with respect to the flat signal and involve only one rapid frequency, respectively, given by $3g\sqrt{\overline{n}}$ and $2g\sqrt{\overline{n}}$.

Figure 6

Spontaneous revivals of the Rabi oscillation signal for $\overline{n}=15$, without dissipation, computed from a numerical integration of the Schrödinger equation for (a) $N=1$ atom, (b) $N=2$ atoms, (c) $N=3$ atoms as a function of the dimensionless time $\varphi =gt∕2\sqrt{\overline{n}}$. The initial condition is $m=N∕2$ (all atoms excited).

Figure 7

Influence of dissipation on the spontaneous revivals of the Rabi oscillation signal $P=P_{m=J}$ (as a function of $\varphi =gt∕2\sqrt{\overline{n}}$) for one atom and $m=1∕2$ (atom excited) with $\overline{n}=15$ photons initially. The graph on the left depicts the analytical envelopes $P_{\pm }\left(t\right)$ for (a) no dissipation, (b) ${\gamma }^{-1}\simeq 14\mathrm{ms}$, (c) ${\gamma }^{-1}\simeq 5\mathrm{ms}$, and (d) ${\gamma }^{-1}\simeq 1\mathrm{ms}$. The right part of the figure presents the associated Rabi oscillation signals obtained from quantum Monte Carlo simulations (plain lines) as well as the associated analytical envelopes. The vertical dashed line corresponds to the largest reachable $\varphi$ (atoms at $100\mathrm{m}{\mathrm{s}}^{-1}$).

Figure 8

Influence of dissipation on the spontaneous revivals of the Rabi oscillation signal $P=P_{m=J}$ (as a function of $\varphi =gt∕2\sqrt{\overline{n}}$) for $N=2$ atoms. The initial condition is $m=1$ (all atoms excited) with $\overline{n}=15$ photons. The graph on the left depicts the analytical envelopes $P_{\pm }\left(t\right)$ for (a) no dissipation, (b) ${\gamma }^{-1}\simeq 14\mathrm{ms}$, (c) ${\gamma }^{-1}\simeq 5\mathrm{ms}$, and (d) ${\gamma }^{-1}\simeq 1\mathrm{ms}$. The right part of the figure presents the associated Rabi oscillation signals obtained from quantum Monte Carlo simulations (plain lines) as well as the associated analytical envelopes.

Figure 9

Influence of dissipation on the spontaneous revivals of the Rabi oscillation signal $P=P_{m=J}$ (as a function of $\varphi =gt∕2\sqrt{\overline{n}}$) for $N=3$ atoms. The initial condition is $m=3∕2$ (all atoms excited) with $\overline{n}=15$ photons. The graph on the left depicts the analytical envelopes $P_{\pm }\left(t\right)$ for (a) no dissipation, (b) ${\gamma }^{-1}\simeq 14\mathrm{ms}$, (c) ${\gamma }^{-1}\simeq 5\mathrm{ms}$, and (d) ${\gamma }^{-1}\simeq 1\mathrm{ms}$. The right part of the figure presents the associated Rabi oscillation signals obtained from quantum Monte Carlo simulations (plain lines) as well as the associated analytical envelopes.

Figure 10

Modulus of the decoherence coefficient ${\mathcal{F}}_q\left(t\right)$ for $q=1$ (plain lines), $q=2$ (dashed lines), and $q=3$ (dotted lines) as a function of $\varphi$ for dissipation times 14, 5, and $1\mathrm{ms}$ in the case of $\overline{n}=10$ photons (left graph) and $\overline{n}=20$ photons (right graph). The vertical dashed line corresponds to the largest reachable $\varphi$ (atoms at $100\mathrm{m}{\mathrm{s}}^{-1}$).

Figure 11

Echo signal (function of $\varphi =gt∕2\sqrt{\overline{n}}$) simulated for $\overline{n}=15$ at $t_{\pi }=30\mathrm{\mu }\mathrm{s}$ and $g∕2\pi =49\mathrm{kHz}$. All graphs on a row correspond to the same dissipation time (a) ${\gamma }^{-1}=14\mathrm{ms}$, (b) ${\gamma }^{-1}=5\mathrm{ms}$, and (c) $N=3$ and ${\gamma }^{-1}=1\mathrm{ms}$. All graphs in a column correspond to the same number of atoms. The arrow shows the time of the echo pulse. The vertical dashed line corresponds to the largest reachable $\varphi$ value (slowest atoms at $100\mathrm{m}{\mathrm{s}}^{-1}$). The initial condition is $m=3∕2$ (all atoms excited).

Figure 12

Rabi oscillation signals obtained from quantum Monte Carlo simulations for one atom using the thermalization protocol producing a thermal state with $0.15$ photons displaced in Fresnel plane by an amplitude corresponding to $\overline{n}=15$ photons, for (a) ${\gamma }^{-1}=14\mathrm{ms}$, (b) $5\mathrm{ms}$, and (c) $1\mathrm{ms}$. Graph (d): comparison between zero temperature signals (dashed line) and finite temperature signals (full line) at short times for ${\gamma }^{-1}=5\mathrm{ms}$. Graph (e): comparison between zero temperature signals (full line) and finite temperature signals (dotted line) for ${\gamma }^{-1}=5\mathrm{ms}$. Dashed lines represent the upper and lower envelopes $P_{\pm }$ obtained by taking temperature into account by rescaling $\gamma \mapsto (1+2n_{\mathrm{th}})\gamma$.

Figure 13

Rabi oscillation signals obtained from quantum Monte Carlo simulations for $N=3$ atoms using the thermalization protocol producing a thermal state with $0.15$ thermal photon number displaced in Fresnel plane with $\overline{n}=15$ for (a) ${\gamma }^{-1}=14\mathrm{ms}$, (b) $5\mathrm{ms}$, and (c) $1\mathrm{ms}$. Graph (d): comparison between zero temperature signals (dashed line) and finite temperature signals (full line) at short times for ${\gamma }^{-1}=5\mathrm{ms}$. Graph (e): comparison between zero temperature signals (full line) and finite temperature signals (dotted line) for ${\gamma }^{-1}=5\mathrm{ms}$. Dashed lines represent the upper and lower envelopes $P_{\pm }$ obtained by taking temperature into account by rescaling $\gamma \mapsto (1+2n_{\mathrm{th}})\gamma$.