Stabilization of nonclassical states of one- and two-mode radiation fields by reservoir engineering

We analyze a quantum reservoir engineering method, originally introduced by Sarlette et al. [Phys. Rev. Lett. 107, 010402 (2011)], for the stabilization of nonclassical field states in high-quality cavities. We generalize the method to the protection of mesoscopic entangled field states shared by two nondegenerate field modes. The reservoir consists of a stream of atoms consecutively interacting with the cavity. Each individual atom-cavity interaction follows the same time-varying Hamiltonian, combining resonant with nonresonant parts. We gain detailed insight into the competition between the engineered reservoir and decoherence. We show that the operation is quite insensitive to experimental imperfections and that it could thus be implemented in the near future in the context of microwave cavity or circuit quantum electrodynamics.

Figure 1

Wigner functions $W\left(\gamma \right)$, with the same vertical, horizontal axes and color scales for all frames. (a)–(d) Nonclassical field states $e^{-i{t}_K{\mathbf{H}}_K}|\alpha \rangle$ generated by propagation of an initial coherent state $|\alpha \rangle$ with a mean photon number ${\left|\alpha \right|}^2=2.7$ through a Kerr medium: (a) 2-component MFSS given by Eq. (2) for $t_K{\gamma }_K=\pi /2$; (b) 3-component MFSS for $t_K{\gamma }_K=\pi /3$; (c) “banana-state” for $t_K{\gamma }_K=0.28$, and (d) squeezed state for $t_K{\gamma }_K=0.08\ll \pi$. (e)–(h) Similar states stabilized, despite decoherence, by the atomic reservoir onto which we focus in this paper. Frame (e) corresponds to our reference two-component MFSS.

Figure 2

Scheme of current ENS CQED experiment.

Figure 3

Top frame shows a scheme of the propagators corresponding to the successive steps in the composite interaction. Bottom frame shows a schematic time profile of $\delta \left(t\right)$ [difference between the tunable frequency ${\omega }_0$ of the atomic transition and the fixed frequency ${\omega }_c$ of the single cavity mode (full line)] and $\Omega \left(vt\right)$ (dashed line) during cavity crossing by one atomic sample. We take $t=0$ when the atom is at cavity center.

Figure 4

Reservoir action with resonant atom-field interaction. (a) Mean photon number $\langle {\psi }_{\infty }\left|\mathbf{N}\right|{\psi }_{\infty }\rangle$ of the pointer state $|{\psi }_{\infty }\rangle$. Grayscale axis is linear in $\sqrt{\langle {\psi }_{\infty }\left|\mathbf{N}\right|{\psi }_{\infty }\rangle }$. The shaded zone is delimited such that the corresponding states have at least a $99\%$ fidelity $|\langle {\psi }_{\infty }|{\alpha }_{*}\rangle {|}^2$ to a coherent state $|{\alpha }_{*}\rangle$ of same mean photon number $|{\alpha }_{*}{|}^2=\langle {\psi }_{\infty }\left|\mathbf{N}\right|{\psi }_{\infty }\rangle$. On the top-left corner, pointer states have significant population outside the truncated Hilbert space. On the top-right part, $|\langle {\psi }_{\infty }|{\alpha }_{*}\rangle {|}^2$ drops to $\sim 98\%$ as $u$ approaches $\pi /2$. (b) Evolution of the fidelity indicator $\ln |\ln \langle {\psi }_{\infty }|{\rho }_k|{\psi }_{\infty }\rangle |$ as a function of the number of atom-field interactions (i.e., Kraus map iterations) $k$, starting from vacuum ${\rho }_0=|0\rangle \langle 0|$. For illustration we have chosen an arbitrary typical case with parameters $u=0.5$ and ${\theta }_r=0.4$, giving $\langle {\psi }_{\infty }\left|\mathbf{N}\right|{\psi }_{\infty }\rangle =6.21$. (c) Convergence rate ${\lambda }_{\mathrm{conv}}$ as a function of ${\theta }_r$ for $u=0.1$ (dotted curve) and $u=1$ (dashed curve). Dependency in $u$ is small. We also represent (full curve) the analytic result of the simplified model [Eq. (19)]. This model slightly overestimates the convergence speed.

Figure 5

Mean photon number $\langle {\psi }_{\infty }\left|\mathbf{N}\right|{\psi }_{\infty }\rangle$ of pointer state $|{\psi }_{\infty }\rangle$ stabilized by $\mathbf{Y}\left({\theta }_{\mathbf{N}}^c\right)$, with ${\delta }_0=2.2{\Omega }_r$ (a) and ${\delta }_0=10{\Omega }_r$ (b). The grayscale axis is linear in $\sqrt{\langle {\psi }_{\infty }\left|\mathbf{N}\right|{\psi }_{\infty }\rangle }$. The 2-component MFSS in Fig. 1 corresponds to $u=0.9\pi /2$ and ${\theta }_r=\pi /2$ with ${\delta }_0=2.2{\Omega }_r$ [black dot on (a)], for which $\langle {\psi }_{\infty }\left|\mathbf{N}\right|{\psi }_{\infty }\rangle =2.96$. The shaded zone is delimited such that all states have at least $99\%$ fidelity $|\langle {\psi }_{\infty }|{\alpha }_{*}\rangle {|}^2$ to a coherent state $|{\alpha }_{*}\rangle$ of the same mean photon number ($|{\alpha }_{*}{|}^2=\langle {\psi }_{\infty }\left|\mathbf{N}\right|{\psi }_{\infty }\rangle$).

Figure 6

(a) Kerr-effect-inducing $D{\phi }_n^c={\phi }_{n+1}^c-{\phi }_n^c$ as a function of photon number $n$ and ${\delta }_0/{\Omega }_r$. Since ${\phi }_0^c$ is undefined, we start with $D{\phi }_1^c$. (b) Corresponding velocities $v$: for each ${\delta }_0/{\Omega }_r$, we adjust $v$ to have $D{\phi }_n^c=\pi$ at $n=2.96$. This value is chosen to cover the parameter values $v=70$ m/s, ${\delta }_0/{\Omega }_r=1/2.2$ (black dots) used for the 2-component MFSS in Fig. 1. An ideal $h_n^c$, proportional to ${\mathbf{H}}_K$, requires $D{\phi }_n^c$ to be constant in $n$. Small ${\delta }_0/{\Omega }_r$ values are disadvantageous for this criterion, but makes it possible to use higher velocities and hence more frequent reservoir atoms for a same mean $D{\phi }^c$. We have selected a constant value ${\theta }_r=\pi /2$ for this figure.

Figure 7

(a) Convergence rate ${\lambda }_{\mathrm{conv}}/T$ giving slope, in inverse time units (${\mathrm{s}}^{-1}$), of the convergence toward the reservoir pointer state $|{\psi }_{\infty }^c\rangle$, according to $\ln |\ln \langle {\psi }_{\infty }^c|{\rho }_k|{\psi }_{\infty }^c\rangle |=\ln |\ln \langle {\psi }_{\infty }^c|{\rho }_0|{\psi }_{\infty }^c\rangle |-{\lambda }_{\mathrm{conv}}k$ (see Fig. 4). For each ${\theta }_r$ and ${\Omega }_r/{\delta }_0$, we adjust $v$ as in Fig. 6 to keep $D{\phi }_n^c\approx \pi$ and $u$ to keep $\langle {\psi }_{\infty }^c\left|\mathbf{N}\right|{\psi }_{\infty }^c\rangle =2.96$. The choice of these reference values corresponds to ${\theta }_r=\pi /2$, $u=0.45\pi$, ${\Omega }_r/{\delta }_0=2.2$, $v=70$ m/s (black dot) used for Fig. 1e. Time $T=3w/v$ between consecutive atoms changes as we adjust $v$. (b) Fidelity of the same $|{\psi }_{\infty }^c\rangle$ to a 2-component MFSS $|c_{{\alpha }_{\infty }}^{\prime }\rangle =\left(\right|{\alpha }_{\infty }\rangle +i{e}^{i\beta }|-{\alpha }_{\infty }\rangle )/\sqrt{2}$, where we tune ${\alpha }_{\infty }$ and $0\le \beta <\pi$ to optimize fidelity. It turns out that $\left|\beta \right|<0.005$ for most parameter values, while $|{\alpha }_{\infty }{|}^2$ decreases as fidelity decreases, below 2.7 for the lowest values of ${\Omega }_r/{\delta }_0$. The black dot marks the settings for the 2-component MFSS in Fig. 1. For ${\theta }_r$ values larger than those represented, there is no $u$ value stabilizing a mean photon number $2.96$; see also Fig. 5. The two plots together illustrate a tradeoff between fidelity in absence of decoherence and convergence speed.

Figure 8

Wigner functions $W\left(\gamma \right)$ illustrating a stabilized 2-component MFSS (colorbar as in Fig. 1). Left panel shows $\rho =|{\psi }_{\infty }^c\rangle \langle {\psi }_{\infty }^c|$ which is the pointer state of our reservoir with composite interaction. Parameter values are ${\theta }_r=\pi /2$, $u=0.45\pi$, ${\delta }_0=2.2{\Omega }_0$, $v=70$ m/s [i.e., those used for Fig. 1e, except $T_c$ set to infinity here]. Right panel shows target state $\rho =|c_{\alpha }\rangle \langle c_{\alpha }|$.

Figure 9

Evolution of the cavity field coupled to a reservoir stabilizing a 2-component MFSS, immediately after a relaxation-induced photon loss. We use the same parameter values as for Fig. 8. A photon loss out of the reservoir pointer state occurs between the frames labeled $0^{-}$ and $0^{+}$. The other frames are labeled by the number of atomic interactions, that have taken place after the photon loss, with no further relaxation. Left two columns: simplified model, described by Eq. (35). We show the Wigner functions of the cavity state, ${\rho }^{\prime }$, on column 1 and of ${\rho }^{\prime h}$ on column 2. Right two columns: same plots for the actual reservoir characterized by ${\mathbf{U}}_c$ ($\rho$ on column 3 and ${\rho }^h$ on column 4).

Figure 10

Steady state of cavity field coupled to atomic reservoir and to relaxation-inducing environment with $T_c=65$ ms. Top (bottom) panel shows marginal distribution of the Wigner function along the imaginary (real) quadrature for the simplified model ${\rho }_{\infty }^{\prime h}$ (dashed line) and for our reservoir ${\rho }_{\infty }^h$ (solid line). These states approximately correspond to the quantum Monte Carlo average of the sequence presented in Fig. 9.

Figure 11

Schematic timing of composite interaction of atom with two cavity modes $a$ and $b$ at frequencies ${\omega }_b>{\omega }_a$. Bottom frame, solid line is time profile of $\delta$ (difference between the atomic frequency ${\omega }_0$ and the mean frequency ${\omega }_m$ of the two cavity modes) during cavity crossing by one atomic sample. For $\delta$ values of 0, $+\Delta$, and $-\Delta$, ${\omega }_0$ coincides with ${\omega }_m=\left({\omega }_b+{\omega }_a\right)/2$, ${\omega }_b$, and ${\omega }_a$, respectively. The red dot represents a $\pi$ pulse acting on the atomic state. Bottom frame, dashed line shows coupling strength $\Omega \left(vt\right)$, taking $t=0$ when the atom is at cavity center. Top frame shows scheme of propagators corresponding to successive steps in the composite interaction.

Figure 12

Simulation of reservoir stabilizing a two-mode entangled state. Solid line shows fidelity of $\rho$, the cavity state starting at vacuum, with respect to an ideal optimized entangled state of the two modes $|{\overline{c}}_{\alpha }\rangle$, as a function of time in units of the sample interaction time $T$. The reservoir operates up to $t/T=200$ and is then switched off. Dashed line shows maximum Bell signal ${\mathcal{B}}^{\max }$ as a function of time. A ${\mathcal{B}}^{\max }$ value above the thin dash-dotted line (${\mathcal{B}}^{\max }=2$) proves entanglement of $\rho$.

Figure 13

Cut in the plane [$\mathrm{Re}\left({\gamma }_a\right)=\mathrm{Re}\left({\gamma }_b\right)=0$] of the two-mode Wigner function $\overline{W}({\gamma }_a,{\gamma }_b)$ of ${\rho }_{200}$. The fringes and negative values in $\overline{W}$ are a signature of the “quantumness” of the stabilized state. The white dots show the points that maximize the Bell signal (45).

Figure 14

Maximum Bell signal ${\mathcal{B}}^{\max }$ of ${\rho }_{200}$ as a function of cavity lifetime $T_c$, for $\Delta /\left(2\pi \right)=300$ kHz, $v=30$ m/s (solid blue line); $\Delta /\left(2\pi \right)=400$ kHz, $v=22$ m/s (dashed-dotted green line); $\Delta /\left(2\pi \right)=500$ kHz, $v=18$ m/s (dashed red line).