Single-shot conditional displacement gate between a trapped atom and traveling light

TL;DR

The paper addresses enabling universal control of hybrid quantum systems by a single-shot atom–traveling-light gate. It proposes a reflection-based conditional displacement (RCD) gate implemented via a cavity-mediated interaction with synchronized atomic driving, described by an effective Hamiltonian $\hat{H}^{\text{eff}}_{\text{sys}}(t)=\hat{\sigma}_{x}[\lambda(t)\hat{c}^{\dagger}+\lambda^{*}(t)\hat{c}]$ and a unitary that factors into $\text{CD}_{\text{out}}(\alpha)$, $\text{CD}_{\text{loss}}(\sqrt{\eta_{\text{ex}}^{-1}-1}\,\alpha)$, a beamsplitter $\hat{B}(\phi)$, and a phase flip $\hat{R}_{\text{out}}(\pi)$ in the long-pulse limit. The authors derive concise models incorporating cavity loss and atomic decay, providing analytic expressions for gate imperfections such as $\epsilon_{\text{pulse}}$ and $p_{\text{sp}}$, and show performance scales with coupling efficiency $\eta_{\text{ex}}$ and internal cooperativity $C_{\text{in}}$. They also examine coherent-state input via the Mollow transformation, enabling reduced master-equation simulations for practical gate assessment. The work offers a hardware-efficient path to connecting stationary atoms with itinerant light, with potential applicability to circuit QED and broader hybrid quantum information tasks, by clarifying design regimes and optimization targets for high-fidelity, single-shot CV–DV gates.

Abstract

We propose a single-shot conditional displacement gate between a trapped atom as the control qubit and a traveling light pulse as the target oscillator, mediated by an optical cavity. Classical driving of the atom synchronized with the light reflection off the cavity realizes the single-shot implementation of the crucial gate for the universal control of hybrid systems. We further derive a concise gate model incorporating cavity loss and atomic decay, facilitating the evaluation and optimization of the gate performance. This proposal establishes a key practical tool for coherently linking stationary atoms with itinerant light, a capability essential for realizing hybrid quantum information processing.

Figures (3)

• Figure 1: Schematic of the reflection-based conditional displacement (RCD) gate, along with the conceptual representation in the phase space.
• Figure 2: System and proof of concept for the RCD gate. (a) Detail of the cavity-QED system. The two optical transitions couple with a single cavity mode $\hat{c}$ at strength $g$, and the other diagonal transitions are driven by $\sigma_+$- and $\sigma_-$- polarized laser fields of Rabi frequency $\Omega(t)$ at the cavity frequency $\omega_c$. The detuning $\Delta = \omega_e-\omega_c$ of the atomic transition frequency $\omega_e$ is sufficiently large to suppress the atomic excitation. The one-sided optical cavity is coupled to the external optical field at rate $\kappa_\text{ex}$ while this also has a nonzero internal loss at rate $\kappa_\text{in} (< \kappa_\text{ex})$; the total cavity decay rate is $\kappa = \kappa_\text{ex} + \kappa_\text{in}$. The atom can decay spontaneously from excited states at rate $\gamma$. (b) State-dependent output-field intensity $I_\text{out}(t)$ via numerical simulation of the full model shown in panel (a) (solid lines) and the effective model (dashed lines). For the input coherent state $\ket{\beta=1}$ in a Gaussian pulse \ref{['eq:gaussian_pulse']}, centered at $t=4\tau$, applying the RCD gate $\text{CD}(\alpha = 1)$ ideally yields the coherent state $\ket{\beta = 2}$ ($\ket{\beta=0}$, the vacuum state) if the initial qubit state is in $\ket{+}_\text{q}$ ($\ket{-}_\text{q}$). The system parameters are $(\Delta, \kappa, \kappa_\text{in}, \gamma) = (20, 1, 0.01, 0.1)g$ and $\kappa \tau = 50$, resulting in $\epsilon_\text{pulse} = 8.0e-4$ and the atomic decay probablity $p_\text{sp}\simeq 0.1$ [see Eqs. \ref{['eq:epsilon_pulse']}\ref{['eq:p_sp']}].
• Figure 3: Optimization of the coupling efficiency $\eta_\text{ex}$ to maximize the gate performance. (a) Numerical optimization of $\eta_\text{ex}$ based on our analytical results \ref{['eq:no-atomic_decay_channel_2']}\ref{['eq:p_sp']}, in the RCD gate $\text{CD}(1j)$ acting on $\ket{0}_\text{q}\ket{\beta}$. The dashed line represents $1-\eta_\text{ex} = 1/(1+\sqrt{1+2C_\text{in}})$, which captures well the result for $\beta=0$. (b) Infidelity $1-F$ for $\beta = 0.5$ with the optimized $\eta_\text{ex}$ [see panel (a)], where we numerically solve the master equation of an effective model, as explained in Sec. \ref{['ap:numerical_simulation_method']}. Given $C_\text{in}$, the system parameters are $(\kappa_\text{in}, \Delta) = (0.01, 30)g$, which gives $\gamma = g^2/(2\kappa_\text{in} C_\text{in})$, while $\kappa_\text{ex}$ is determined by the optimized $\eta_\text{ex}$. The pulse length satisfies $g\tau = 300$. The dashed and dashdot lines respectively represent $1-F_\text{LB}-p_\text{sp}$ and $1-F_\text{LB}$ calculated by the analytical results \ref{['eq:no-atomic_decay_channel_2']}\ref{['eq:p_sp']}. (c) Wigner functions of the output light with the postselection of the qubit states for $C_\text{in} = 1000$. Measuring the qubit state in $\ket{0(1)}_\text{q}$ projects the light state onto the non-classical state $\propto [\hat{D}(1j) \pm \hat{D}(-1j)]\ket{\beta=0.5}$ in the ideal case. Both of the projected states clearly show the Wigner negativity.