Steady-State Multiparticle Entanglement via Dissipative Engineering in Waveguide QED

Abstract

We propose a simple scheme for the dissipative generation of entangled states of multiple emitters coupled to a waveguide. Our approach exploits collective interactions arising from the formation of subradiant and superradiant excited states, combined with the quantum Zeno effect. We show that, starting from an arbitrary initial state, the system deterministically evolves toward a W-type entangled steady state, with an infidelity that scales inversely with the cooperativity. The protocol is scalable to an arbitrary number of emitters. We further analyze the impact of additional experimental imperfections and present a detailed implementation based on trapped $^{133}$Cs atoms.

Figures (9)

• Figure 1: (a) Schematic of the system. Two emitters are located in an optical waveguide. The inset shows the level structure of the emitters. The emitters decay to state $\mathinner{|{0}\rangle}$ through the waveguide at an enhanced rate $\Gamma_\mathrm{1D}$ and to the side with decay rates $\Gamma_0$ and $\Gamma_1$ to states $\mathinner{|{0}\rangle}$ and $\mathinner{|{1}\rangle}$ respectively. $\Omega_i$ and $\Delta_i$ represent the strength and the detuning respectively for the driving between the ground state $\mathinner{|{i}\rangle}$ and the excited state. The decay from $\mathinner{|{e}\rangle}$ to $\mathinner{|{0}\rangle}$ couples to the waveguide but not the one to $\mathinner{|{1}\rangle}$. We off-resonantly drive the ground states to the excited ones. Note that we add a phase difference between the drivings on $\mathinner{|{0}\rangle}_j$ to couple the state $\mathinner{|{00}\rangle}$ to the subradiant manifold. (b) Representation of singly-excited subradiant (purple) and superradiant (black) manifold, as well as the ground states, where $\mathinner{|{T}\rangle}$ is the desired state. $\mathinner{|{00}\rangle}$ and $\mathinner{|{S}\rangle}$ couple to a subradiant state, resulting in rapid pumping into $\mathinner{|{T}\rangle}$ at a rate $\sim\gamma^F$. $\mathinner{|{T}\rangle}$ is pumped back through superradiant states, suppressing the process to a slower rate $\sim\gamma^S$. On the other hand, $\mathinner{|{T}\rangle}$ effectively decays to $\mathinner{|{11}\rangle}$ through non-guided decay from a superradiant state, at an extremely slow rate $\gamma^{ES}$. The decay from $\mathinner{|{11}\rangle}$ to $\mathinner{|{T}\rangle}$ occurs through superradiant decay at the rate $\gamma^S$. (c) Effective ground states evolution separated into fast (green box) and slow (orange box) dynamics.
• Figure 2: (a) Comparison of the transient evolution of the system obtained from the total system dynamics (solid) and from the effective dynamics restricted to ground states (dashed). Starting from a completely mixed state, we observe excellent agreement between the two approaches, as well as how the population of the system converges into the entangled state $\mathinner{|{T}\rangle}$. We have used $\beta=0.99$. (b) Fidelity as a function of the cooperativity $C=\Gamma_{1D}/\Gamma'$, comparing the total system numerical result (solid) with the analytical prediction from Eq. (\ref{['eq:errstat']}). We observe good agreement between them, in particular for large $C$. For both figures, we use $\Gamma_1=\Gamma_2$, $\Omega_0=\Delta_0=-\Delta_1=\Gamma'/20$ and the optimal driving conditions $\Omega_1=\Omega_0\mathcal{R}$ (see Eqs. (\ref{['eq:optim_ratio']},\ref{['eq:opt_r']})).
• Figure 3: Comparison of the infidelity for the numerical simulation of the full system (solid) and the analytical results (dashed). (a) Infidelity of the steady state with respect to $\mathinner{|{T}\rangle}$ as a function of the difference in the detunings $\delta = \Delta_1 - \Delta_0$, with $\Delta_1 = -\Delta_0$, for $\Omega_0 / \Gamma' = 0.10$, $0.05$, and $0.02$. Different detuning regimes can be identified. For small $\delta$, the system experiences population trapping in dark states, leading to an infidelity given by $(1-F_{T})_{\mathrm{dark}}$ [Eq. (\ref{['eq:errdark']})]. At intermediate $\delta$, a plateau appears where the infidelity is limited by the cooperativity, $(1-F_{T})_{\mathrm{coop}}$ [Eq. (\ref{['eq:errstat']})]. For large $\delta$, detuning-induced errors dominate and the infidelity is described by $(1-F_{T})_{\mathrm{detu}}$ [Eq. (\ref{['eq:errdetu']})]. We observe a close match between the numerical simulation of the total system dynamics and the analytical results. (b) Infidelity including dephasing. In the presence of ground state dephasing, it is no longer desirable to have a very slow protocol, and $\Omega_0$ can be optimized based on the other parameters. The values used for these plots are $\Gamma_1=\Gamma_2$, $\beta=0.99$, $\Omega_1=\Omega_0\mathcal{R}$, and in (b) $\Delta_0=-\Delta_1=\Gamma'/20$ and $\Gamma_\mathrm{d}=10^{-5}\Gamma'$.
• Figure 4: (a) Dynamics of the ground states for $N$$\Lambda$-type emitters. $\mathinner{|{W_N}\rangle}$ is the state we want to prepare. The number of states to the left of $\mathinner{|{W_N}\rangle}$ grows with $N$, but not the number to its right. Increasing $\Omega_0/\Omega_1$ is equivalent to increasing the effective decay rate towards the further-right states. Thus, for optimal results, $\Omega_0/\Omega_1$ should increase with $N$. (b) Steady state infidelity for $\mathinner{|{W_N}\rangle}$ as a function of $C$ for different $N$. For $N=2$, we use the optimal $\Omega_1$, $\Delta_0$ and $\Delta_1$ derived in the text. For $N=3$, $4$ and $5$ we use $\Delta_0=0.4\cdot\Omega_0$ and $\Delta_1=-0.7\cdot\Omega_0$; and the ratio of driving strengths is $N=3$: $\Omega_1=0.88\cdot\Omega_0$; $N=4$: $\Omega_1=0.69\cdot\Omega_0$; $N=5$: $\Omega_1=0.59\cdot\Omega_0$. We have also assumed $\Gamma_0=\Gamma_1=\Gamma'/2$ and $\Omega_0=\Gamma'/100$.
• Figure 5: (a) Considered $^{133}$Cs levels scheme. The transition from $\mathinner{|{e}\rangle}$ to $\mathinner{|{0}\rangle}$ is well coupled to the waveguide, which we assume to be $\pi$-polarized. The entanglement is prepared between the states $\mathinner{|{0}\rangle}$ and $\mathinner{|{1}\rangle}$, and an additional source of error comes from $\mathinner{|{e}\rangle}$ also decaying into $\mathinner{|{2}\rangle}$. Therefore, we introduce a strong driving between $\mathinner{|{2}\rangle}$ and $\mathinner{|{e}\rangle}$ to pump the population out of that state. (b) Steady state fidelity of $\mathinner{|{T}\rangle}$ as a function of the ratios between the drivings for 2 emitters with $\beta=0.99$. For this simulation we use the decay rates from the $^{133}$Cs, $F'=4$, $m_{F'}=4$ level. The parameters used are $\Omega_0=\Gamma'/100$ and $\Delta_0=-\Delta_1=\Omega_2/2=\Gamma'/5$. (c) Steady state fidelity of $\mathinner{|{W_N}\rangle}$ including the additional ground state as a function of $\Gamma_\mathrm{1D}/\Gamma'$ for different $N$. The parameters we use are $N=2$: $\Omega_1= 1.14\cdot\Omega_0$; $N=3$: $\Omega_1=0.88\cdot\Omega_0$; $N=4$: $\Omega_1=0.69\cdot\Omega_0$; and for all of them $\Delta_0=-\Delta_1=\Omega_2/2=\Gamma'/5$ and $\Omega_0=\Gamma'/100$.