Improving photon blockade, entanglement and mechanical-cat-state generation in a generalized cross-Kerr optomechanical circuit

TL;DR

This work introduces a tripartite microwave optomechanical circuit that leverages a generalized cross-Kerr nonlinearity to simultaneously enhance photon blockade, photon-induced tunneling, and photon–phonon entanglement, while enabling mechanical Schrödinger cat-state generation even when the primary optomechanical coupling vanishes. It develops an effective Hamiltonian with radiation-pressure, cross-Kerr, and generalized cross-Kerr terms, and uses both analytical (polaron-derived energy shifts) and numerical (master-equation) methods to characterize PB, PIT, and entanglement across parameter regimes. A zero-g0 regime is shown to yield multi-component mechanical cat states via the generalized CK term, with Wigner-function evidence and robustness against dissipation. The paper also demonstrates that generalized CK nonlinearity can substantially boost steady-state microwave–mechanical entanglement in large-red-detuning scenarios, supported by linearized Langevin analysis and stability considerations. Experimental feasibility is discussed with realistic circuit parameters and precision requirements, highlighting potential applications in microwave quantum sensing, telecommunications, and information processing.

Abstract

We propose a feasible experimental scheme to improve the few-photon optomechanical effects, including photon blockade and mechanical-Schrodinger cat-state generation, as well as photon-phonon entanglement in a tripartite microwave optomechanical circuit. The system under consideration is formed by a single-Cooper-pair transistor, a microwave LC resonator, and a micromechanical resonator. Our scheme is based on an additional higher-order (generalized) nonlinear cross-Kerr type of coupling, linearly dependent on photon number while quadratically dependent on mechanical phonon one, which can be realized via adjusting the gate charge of the Cooper-pair transistor. We show, both analytically and numerically, that the presence of both cross-Kerr and generalized cross-Kerr nonlinearities not only may give rise to the enhancement of one- and two-photon blockades as well as photon induced tunneling but can also provide more controllability over them. Furthermore, it is shown that in the regime of zero optomechanical coupling, with the aid of generalized cross-Kerr nonlinearity, one can generate multi-components mechanical superposition states which exhibit robustness against system dissipations. We also study the steady-state entanglement between the microwave and mechanical modes, the results of which signify the role of generalized cross-Kerr nonlinearity in enhancing the entanglement in the regime of large-red detuning. The proposed generalized cross-Kerr optomechanical system can be found potential applications in microwave quantum sensing, quantum telecommunication, and quantum information protocols.

Figures (13)

• Figure 1: (a) Schematic circuit diagram of the considered tripartite microwave optomechanical system heikkila2014 composed of a single-Cooper-pair transistor with Josephson energies $E_{J1,J2}$ and Josephson capacitances $C_{1,2}$ (the brown box), a microwave LC resonator (the blue box), and a micromechanical resonator with gate capacitance $C_{g0}$ which couples via a time-dependent capacitance $C_g(t)\equiv C_{g}[x(t)] \equiv C_g$ (the green box). Here, we consider the gate capacitor can be vibrated through modulating the movable part of the gate capacitance. (b) Equivalent cavity optomechanical system where the cavity mode $\hat{a}$ with frequency $\omega_c$ is coupled to the mechanical mode $\hat{b}$ with frequency $\omega_M$ through the radiation-pressure, the CK, and an additional higher order CK- types of interaction with coupling strengths $g_0,\bar{g}_{\rm CK}$, and $g'_{\rm CK}$, respectively (see the text for details).
• Figure 2: (Color online). Variation of (a) radiation pressure coupling, (b) CK coupling, and (c) the generalized CK coupling for different values of $E_J/E_C$ as a function of $\delta n_{g0}$. Here, $E_J/\hbar=10$ GHz.
• Figure 3: The qualitative (unscaled) diagram of the eigenenergy of the Hamiltonian $\hat{H}$. The horizontal axis shows photon subspace associated with $n=0,1,2$.
• Figure 4: (Color online) Contour plots of the steady-state second-order correlation function $g^{(2)}(0)$ versus the normalized cavity-laser detuning $\Delta_c /\omega_M$ and gate charge deviation $\delta n_{g0}$ for different coupling regimes $(a)$$E_J/E_C=1/20$, and $(b)$$E_J/E_C=1/30$. The other system parameters are taken as $\omega_c/2\pi=5$ GHz, $\omega_M/2\pi=10$ MHz, $\kappa/\omega_M=0.01$, $\gamma/\omega_M=0.001$, $\Omega/\omega_M=0.001$, and $\bar{n}_{\rm th}=0$.
• Figure 5: (Color online) Steady-state second-order correlation function, $g^{(2)}(0)$, versus the normalized cavity-laser detuning $\Delta_c/\omega_M$ for (a) $E_J/E_C = 1/20,\delta n_{g0}=0.533$, and (b) $E_J/E_C = 1/30,\delta n_{g0}=0.527$. The corresponding values of the coupling strengths are also given. The blue-solid and red-dashed lines are, respectively, referred to the analytical and numerical solutions. Other system parameters are the same as in Fig. \ref{['fig4']}.