Entanglement generation of arbitrary squeezed Fock states

Abstract: We propose an efficient and robust protocol for the generation of entanglement between a superconducting qubit and a squeezed cavity. By applying a parametric drive to the cavity coupled to the qubit, the dynamical evolution of the system is precisely described by an anisotropic Rabi model within a squeezed reference frame. Utilizing high-order time-averaging methods, we analytically derive the resonance conditions and the effective Rabi frequency for the high-order three-photon process. By implementing an adiabatic passage, slowly tuning the cavity frequency across the resonance, the system is steered into a maximally entangled state, e.g., between the three-photon state $\ket{g,3}$ and the qubit excited state $\ket{e,0}$ in the squeezed picture. Numerical simulation results confirm the high fidelity and robustness of the proposed protocol. Our method provides a practical pathway for generating complex non-Gaussian entangled states, which are of significant value for fault-tolerant quantum computation and quantum metrology beyond the standard quantum limit.

• The paper introduces a squeezed-frame protocol that maps a parametrically driven Jaynes–Cummings system to an anisotropic Rabi model, enabling three-photon coupling between |e,0⟩ and |g,3⟩ without ultrastrong bare coupling.
• The method reaches 99.63% population transfer to the squeezed three-photon state under ideal conditions, while analytical level splittings agree with numerical results within 3% for g < 0.047 at r = 0.9.
• The protocol produces Bell-like qubit–squeezed-Fock entanglement using about 5.6 dB of squeezing, achieving 0.77 fidelity and 0.71 concurrence under modeled realistic dissipation.φ

Overview

This paper proposes a protocol for generating hybrid entanglement between a superconducting qubit and a squeezed multi-photon Fock state, specifically the squeezed three-photon state $S(r)|3\rangle$ of a cavity mode. The scheme operates within the Jaynes-Cummings (JC) coupling regime, avoiding ultrastrong-coupling requirements, by combining two ingredients: a parametric drive on the cavity that engineers counter-rotating terms, and a transformation into a squeezed reference frame in which the dynamics map onto an anisotropic Rabi model. Within this frame, a three-photon resonance between $\ket{e,0}$ and $\ket{g,3}$ is established analytically via high-order time-averaging, and an adiabatic sweep of the cavity frequency through the avoided crossing deterministically prepares a maximally entangled Bell-like state. Back-transformation to the laboratory frame yields entanglement between the qubit and a non-Gaussian squeezed Fock state — a resource class previously underexplored relative to qubit–coherent-state or qubit–Fock-state hybrid entanglement.

Squeezed-frame effective Hamiltonian

The starting point is a parametrically driven JC Hamiltonian in the frame rotating at half the pump frequency:

$$H = \delta_c a^\dagger a + \frac{\delta_q}{2}\sigma_z - \frac{\lambda}{2}(a^{\dagger 2}+a^2) + g(a^\dagger\sigma_- + a\sigma_+),$$

with stability requiring $|\delta_c| > \lambda$. Applying the squeezing unitary $U_s(r) = \exp[\frac{r}{2}(a^2 - a^{\dagger 2})]$ with $\tanh 2r = \lambda/\delta_c$ removes the parametric term and yields an anisotropic Rabi model with effective cavity frequency $\omega_c = \delta_c\,\mathrm{sech}\,2r$ and couplings $\lambda_1 = g\sinh r$, $\lambda_2 = g\cosh r$. The transformation provides two advantages: exponential enhancement of the light-matter interaction ($\ket{e,0}$0) without increasing the bare coupling $\ket{e,0}$1, and the presence of counter-rotating terms proportional to $\ket{e,0}$2 that enable excitation-number-non-conserving multi-photon transitions forbidden under the rotating-wave approximation. Because the cavity vacuum in the squeezed frame corresponds to $\ket{e,0}$3 in the laboratory frame, any preparation in this frame automatically produces squeezed-state entanglement upon back-transformation.

Using a generalized James effective-Hamiltonian method [PhysRevA.95.032124], the authors derive second- and third-order effective terms. The third-order contribution,

$\ket{e,0}$4

directly couples $\ket{e,0}$5 and $\ket{e,0}$6. In the subspace $\ket{e,0}$7, the effective Rabi frequency is

$\ket{e,0}$8

and equating diagonal elements gives the shifted resonance condition

$\ket{e,0}$9

The analytical level splitting agrees with numerical diagonalization of the full anisotropic Rabi Hamiltonian to within 3% for $\ket{g,3}$0 (at $\ket{g,3}$1), validating the perturbative treatment. Numerical simulations with $\ket{g,3}$2, $\ket{g,3}$3 show clean three-photon Rabi oscillations, with the population of $\ket{g,3}$4 peaking at 99.63%; distinct oscillations persist for $\ket{g,3}$5, indicating robustness across a wide parameter range.

Role of the squeezing parameter

Decomposing the Hamiltonian into isotropic and anisotropic parts reveals the dual role of $\ket{g,3}$6. The isotropic component carries a coupling enhanced by $\ket{g,3}$7, so increasing $\ket{g,3}$8 shortens the oscillation period $\ket{g,3}$9 for fixed $$H = \delta_c a^\dagger a + \frac{\delta_q}{2}\sigma_z - \frac{\lambda}{2}(a^{\dagger 2}+a^2) + g(a^\dagger\sigma_- + a\sigma_+),$$0; simultaneously, the anisotropic correction scales as $$H = \delta_c a^\dagger a + \frac{\delta_q}{2}\sigma_z - \frac{\lambda}{2}(a^{\dagger 2}+a^2) + g(a^\dagger\sigma_- + a\sigma_+),$$1 and becomes negligible, driving the system from the anisotropic toward the isotropic Rabi regime. The period curves of the two models coincide beyond a threshold value of $$H = \delta_c a^\dagger a + \frac{\delta_q}{2}\sigma_z - \frac{\lambda}{2}(a^{\dagger 2}+a^2) + g(a^\dagger\sigma_- + a\sigma_+),$$2, confirming this crossover.

For fidelity, the maximum population of $$H = \delta_c a^\dagger a + \frac{\delta_q}{2}\sigma_z - \frac{\lambda}{2}(a^{\dagger 2}+a^2) + g(a^\dagger\sigma_- + a\sigma_+),$$3 peaks near unity at $$H = \delta_c a^\dagger a + \frac{\delta_q}{2}\sigma_z - \frac{\lambda}{2}(a^{\dagger 2}+a^2) + g(a^\dagger\sigma_- + a\sigma_+),$$4, corresponding to roughly 5.6 dB of squeezing. This is a practically significant result: while record squeezing levels exist in specialized setups, typical operational squeezing in qubit-coupled Josephson parametric devices is moderate due to pump depletion, higher-order nonlinearities, and decoherence under strong driving. The protocol's requirement falls well within the accessible range of current devices, which strengthens its experimental feasibility claim.

Adiabatic preparation of hybrid entanglement

The protocol sweeps the effective cavity frequency linearly, $$H = \delta_c a^\dagger a + \frac{\delta_q}{2}\sigma_z - \frac{\lambda}{2}(a^{\dagger 2}+a^2) + g(a^\dagger\sigma_- + a\sigma_+),$$5, starting from large positive detuning where the system occupies the instantaneous eigenstate adiabatically connected to $$H = \delta_c a^\dagger a + \frac{\delta_q}{2}\sigma_z - \frac{\lambda}{2}(a^{\dagger 2}+a^2) + g(a^\dagger\sigma_- + a\sigma_+),$$6. By the Landau-Zener argument, deterministic transfer requires the adiabatic parameter $$H = \delta_c a^\dagger a + \frac{\delta_q}{2}\sigma_z - \frac{\lambda}{2}(a^{\dagger 2}+a^2) + g(a^\dagger\sigma_- + a\sigma_+),$$7. At resonance, the squeezed-frame state becomes the Bell-like superposition $$H = \delta_c a^\dagger a + \frac{\delta_q}{2}\sigma_z - \frac{\lambda}{2}(a^{\dagger 2}+a^2) + g(a^\dagger\sigma_- + a\sigma_+),$$8, which maps in the laboratory frame to

$$H = \delta_c a^\dagger a + \frac{\delta_q}{2}\sigma_z - \frac{\lambda}{2}(a^{\dagger 2}+a^2) + g(a^\dagger\sigma_- + a\sigma_+),$$9

Numerical simulations with $|\delta_c| > \lambda$0, $|\delta_c| > \lambda$1, and $|\delta_c| > \lambda$2 confirm: smooth population transfer to equal superposition, mean photon number rising to $|\delta_c| > \lambda$3 consistent with the $|\delta_c| > \lambda$4 component, projection fidelity on the target instantaneous eigenstate remaining near unity throughout the sweep, concurrence reaching its maximum at the end of the passage, and Wigner-function evolution from squeezed vacuum to a squeezed three-photon Fock state. Notably, since larger $|\delta_c| > \lambda$5 exponentially increases $|\delta_c| > \lambda$6, it permits faster sweeps at fixed adiabaticity — reducing the total time exposed to dissipation.

Dissipative performance

Dissipation is modeled by a Lindblad master equation with cavity decay rate $|\delta_c| > \lambda$7 and qubit relaxation rate $|\delta_c| > \lambda$8, justified because the bare coupling remains deep in the JC regime ($|\delta_c| > \lambda$9). Under realistic parameters $U_s(r) = \exp[\frac{r}{2}(a^2 - a^{\dagger 2})]$0 and $U_s(r) = \exp[\frac{r}{2}(a^2 - a^{\dagger 2})]$1 (with $U_s(r) = \exp[\frac{r}{2}(a^2 - a^{\dagger 2})]$2), the target-state fidelity reaches 0.77 and the concurrence 0.71. With weaker dissipation ($U_s(r) = \exp[\frac{r}{2}(a^2 - a^{\dagger 2})]$3, $U_s(r) = \exp[\frac{r}{2}(a^2 - a^{\dagger 2})]$4, $U_s(r) = \exp[\frac{r}{2}(a^2 - a^{\dagger 2})]$5), both metrics approach unity. These results support experimental feasibility, though the fidelity degradation at realistic cavity loss rates is nontrivial and stems largely from photon loss acting directly on the three-photon component.

Circuit QED implementation

The authors detail a concrete implementation comprising an $U_s(r) = \exp[\frac{r}{2}(a^2 - a^{\dagger 2})]$6 resonator, a Transmon qubit capacitively coupled to it, and a SQUID loop biased by a time-dependent external flux $U_s(r) = \exp[\frac{r}{2}(a^2 - a^{\dagger 2})]$7 serving as the parametric pump. Expanding the SQUID potential to second order in resonator flux and modulating at $U_s(r) = \exp[\frac{r}{2}(a^2 - a^{\dagger 2})]$8 generates the term $U_s(r) = \exp[\frac{r}{2}(a^2 - a^{\dagger 2})]$9, with $\tanh 2r = \lambda/\delta_c$0 set by the flux modulation amplitude, $\tanh 2r = \lambda/\delta_c$1 fixed by the mutual capacitance $\tanh 2r = \lambda/\delta_c$2, and $\tanh 2r = \lambda/\delta_c$3 determined by the resonator frequency relative to the pump. In the frame rotating at $\tanh 2r = \lambda/\delta_c$4, the circuit realizes exactly the model Hamiltonian of Eq. (1), closing the loop between theory and hardware.

Limitations and open questions

Several caveats qualify the results. First, the analytical resonance condition and Rabi frequency are perturbative in $\tanh 2r = \lambda/\delta_c$5; their accuracy is demonstrated only up to $\tanh 2r = \lambda/\delta_c$6, and the protocol's quantitative predictions outside this dispersive window remain unverified. Second, the Markovian Lindblad treatment assumes weak system-bath coupling and neglects non-Markovian effects, pump-induced heating, and dephasing of the squeezing parameter itself — all relevant in strongly pumped parametric cavities. Third, the dissipative simulation at realistic loss rates yields a fidelity of only 0.77, leaving open how much improvement is achievable through optimized sweep rates, dynamical decoupling, or error mitigation. Finally, the paper demonstrates the three-photon case explicitly; extension to arbitrary squeezed Fock states $\tanh 2r = \lambda/\delta_c$7 with $\tanh 2r = \lambda/\delta_c$8 would require higher-order processes whose effective couplings scale as higher powers of $\tanh 2r = \lambda/\delta_c$9 and may fall below practical thresholds.

Conclusion

The paper establishes an analytically grounded route to qubit–squeezed-Fock-state hybrid entanglement using parametric driving, a squeezed-frame mapping to the anisotropic Rabi model, and adiabatic passage through a three-photon avoided crossing. Key quantitative claims include the 99.63% peak population of the squeezed three-photon state under ideal conditions, agreement between analytical and numerical level splittings within 3%, operation at a modest 5.6 dB of squeezing, and residual fidelity/concurrence of 0.77/0.71 under realistic circuit QED dissipation. The work supplies a concrete circuit-level implementation and identifies the squeezing parameter as the central control knob governing both speed and fidelity, while leaving open the extension to higher-order Fock states and performance under non-Markovian noise.