Photon-Mediated Interactions: A Scalable Tool to Create and Sustain Entangled States of $N$ Atoms

We propose and study the use of photon-mediated interactions for the generation of long-range steady-state entanglement between $N$ atoms. Through the judicious use of coherent drives and the placement of the atoms in a network of cavity QED systems, a balance between their unitary and dissipative dynamics can be precisely engineered to stabilize a long-range correlated state of qubits in the steady state. We discuss the general theory behind such a scheme and present an example of how it can be used to drive a register of $N$ atoms to a generalized $W$ state and how the entanglement can be sustained indefinitely. The achievable steady-state fidelities for entanglement and its scaling with the number of qubits are discussed for presently existing superconducting quantum circuits. While the protocol is primarily discussed for a superconducting circuit architecture, it is ideally realized in any cavity QED platform that permits controllable delivery of coherent electromagnetic radiation to specified locations.

Popular Summary

One of the holy grails of modern quantum information technology is the preparation and control of quantum states whose bits of information are delocalized between many constituents. Quantum entanglement has been successfully achieved in few-qubit systems, but the most significant challenge now is to devise robust schemes that can be scaled up to larger systems. A major hindrance comes from the destructive interferences rapidly caused by an uncontrolled environment. Instead of targeting perfect isolation from the environment, an alternative approach is to use dissipative environments as a resource rather than an obstacle. Here, we detail a simple scheme, readily achievable with current cavity quantum-electrodynamics technology, which consists of using a carefully crafted electromagnetic medium as a host for qubits.

We show, both analytically and numerically, that striking the precise balance between external microwave drives and the dissipative mechanisms can create large-scale entangled states and stabilize their peculiar quantum nature indefinitely. Furthermore, our work includes an analysis of the scaling of the achievable fidelities with the number of qubits.

Figure 1

(a) One-dimensional array of cavity-qubit systems coupled by photon exchange and subject to one or several ac microwave drives, cavity decay at rate $\kappa$, qubit relaxation $\gamma$, and pure dephasing ${\gamma }_{\varphi }$. (b) Implementation with superconducting transmon qubits embedded in interconnected microwave cavities and driven by external continuous-wave generators.

Figure 2

Engineered qubit many-body spectrum. The nonequilibrium drive and the resulting photon fluctuation bath are used to create a dominant transition from the trivial ground state to a target entangled $W$ state in the one-excitation manifold. The existence of a nonzero spontaneous decay $\gamma$ is critical to avoid populating higher excited-state manifolds.

Figure 3

Schematics of the pumping rate ${\mathrm{\Gamma }}_{0\to k}$ as a function of the drive frequency ${\omega }_{\mathrm{d}}$ for $N=5$. (a) Driving first cavity only: Any of the five photon-fluctuation modes (responsible for the five peaks) can be used for dissipative stabilization. Driving to $|k⟩$ while avoiding populating the nearest state $|k^{\prime }⟩$ necessitates $\kappa <\mathrm{\Delta }\tilde{E}\equiv |{\tilde{E}}_k-{\tilde{E}}_{k^{\prime }}|\sim 2\pi J(g/\mathrm{\Delta }{)}^2/N$. (b) Driving all cavities equally: Only one of the five modes can channel the mechanism, and driving to the nearest state $|k^{\prime }⟩$ is avoided if $\kappa <\mathrm{\Delta }\omega \equiv |{\omega }_k-{\omega }_{k^{\prime }}|\sim 2\pi J/N$.

Figure 4

Fidelities to all the possible $|k⟩$ states versus drive frequency ${\omega }_{\mathrm{d}}$ for open boundary conditions. (Top panel) Only the first of the $N=5$ cavities is driven; the energy difference between driving to two neighboring $|k⟩$ states is controlled by $\mathrm{\Delta }\tilde{E}\sim 2\pi J(g/\mathrm{\Delta }{)}^2/N$. (Bottom panel) However, when all cavities are driven equally, the energy differences are controlled by $\mathrm{\Delta }\omega \sim 2\pi J/N$, allowing a much better control over stabilizing the desired target state. Please note the widely different scales for the frequency ranges shown in the two plots. ${\epsilon }^{\mathrm{d}}=0.3$, ${\omega }_{\mathrm{c}}=6$, $\mathrm{\Delta }=1$, $g=J=10^{-1}$, $\kappa =10^{-4}$, $\gamma =10^{-5}$, ${\gamma }_{\varphi }=10^{-6}$ in units of $2\pi \mathrm{GHz}$.

Figure 5

Fidelities to the $|k_{+}⟩$ states defined in Eq. (25) versus drive frequency ${\omega }_{\mathrm{d}}$ in case only the first of $N=5$ cavities is driven and with periodic boundary conditions. Same parameters as in Fig. 4.

Figure 6

Light-mediated interactions offer a highly versatile platform to design and control networks of interacting qubits. (a) The qubits (blue arrows) are embedded in an electromagnetic environment, which is the solution of the Maxwell equations in a given scattering geometry. (b) Integrating out the EM degrees of freedom yields effective interactions between the qubits, forming a network that can sustain large-scale entangled many-body states.