Exponentially Enhanced Light-Matter Interaction, Cooperativities, and Steady-State Entanglement Using Parametric Amplification

We propose an experimentally feasible method for enhancing the atom-field coupling as well as the ratio between this coupling and dissipation (i.e., cooperativity) in an optical cavity. It exploits optical parametric amplification to exponentially enhance the atom-cavity interaction and, hence, the cooperativity of the system, with the squeezing-induced noise being completely eliminated. Consequently, the atom-cavity system can be driven from the weak-coupling regime to the strong-coupling regime for modest squeezing parameters, and even can achieve an effective cooperativity much larger than 100. Based on this, we further demonstrate the generation of steady-state nearly maximal quantum entanglement. The resulting entanglement infidelity (which quantifies the deviation of the actual state from a maximally entangled state) is exponentially smaller than the lower bound on the infidelities obtained in other dissipative entanglement preparations without applying squeezing. In principle, we can make an arbitrarily small infidelity. Our generic method for enhancing atom-cavity interaction and cooperativities can be implemented in a wide range of physical systems, and it can provide diverse applications for quantum information processing.

Figure 1

Schematics of the proposed method for enhancing cooperativity and maximizing steady-state entanglement. (a) Two driven atoms are trapped inside a single-mode cavity, which contains a ${\chi }^{(2)}$ nonlinear medium strongly pumped at amplitude ${\mathrm{\Omega }}_p$, frequency ${\omega }_p$, and phase ${\theta }_p$. The cavity couples to a squeezed-vacuum reservoir, which is generated by optical parametric amplification (OPA) with a squeezing parameter $r_e$ and a reference phase ${\theta }_e$. As depicted in (b), the three-level atoms (in the $\mathrm{\Lambda }$ configuration) are coupled to the cavity mode with a strength $g$. In addition, the transition with Rabi frequency $\mathrm{\Omega }$ (${\mathrm{\Omega }}_{\mathrm{MW}}$) is driven by a laser (microwave) field of frequency ${\omega }_L$ (${\omega }_{\mathrm{MW}}$). We also assume that, along with a cavity decay rate $\kappa$, the excited state $|e⟩$ of the atoms decays to the ground states $|g⟩$ and $|f⟩$ at rates ${\gamma }_g$ and ${\gamma }_f$, respectively.

Figure 2

Cooperativity enhancement $C_s/C$ versus the squeezing parameter $r_p$. For $C=0.2$, the gray and yellow shaded areas represent the WCR ($C_s<1$) and the SCR ($C_s>1$), respectively. For $C=20$, the two shaded areas represent the regions, respectively, with $C_s<100$ and $C_s>100$. The inset shows the exponentially-enhanced effective coupling, $g_s$, between atom and cavity.

Figure 3

(a) Evolution of the entanglement infidelity $\delta$ for different driving strengths $\mathrm{\Omega }=0.5\gamma$, $1.0\gamma$, and $1.5\gamma$, with the cooperativity $C=20$. We assumed ${\mathrm{\Delta }}_f=\mathrm{\Omega }/2^{7/4}$ and ${\mathrm{\Delta }}_f=\mathrm{\Omega }/2^{7/4}+|g_s^{\prime }{|}^2/(2{\mathrm{\Delta }}_e)$, ${\mathrm{\Delta }}_e=200g_s^{\prime }$ when using the effective (thick curves) and full (symbols) master equations, respectively. This yields an excellent agreement especially for time $t\in [0,500/\gamma ]$. The steady-state error decreases as $\mathrm{\Omega }$ and becomes closer to $6/(e^{2r_p}C)$ (thin solid line), far below both $1/\sqrt{C}$ (thin dashed line) and $1/C$ (thin dotted-dashed line). (b) Entanglement infidelity at $t=200/\gamma$ as a function of $C$ and $\mathrm{\Omega }$. Here, due to excellent agreement between our predictions based on the full and effective master equations in panel (a), only the latter equation was used in panel (b). The solid line represents the optimal drive resulting in the smallest error for a given cooperativity. In both plots, we have assumed that ${\gamma }_g=\gamma /2$, $\kappa =2\gamma /3$, ${\mathrm{\Omega }}_{\mathrm{MW}}=\sqrt{2}{\mathrm{\Delta }}_f$, $r_p=3$, ${\theta }_p=\pi$, while the initial state of the atoms is $(\mathcal{I}-|{\psi }_{-}⟩⟨{\psi }_{-}|)/3$ and the cavity is initially in the vacuum.