Enhancement and state tomography of a squeezed vacuum with circuit quantum electrodynamics

Matthew Elliott and Eran Ginossar

Phys. Rev. A 92, 013826 – Published 15 July 2015

Abstract

We study the dynamics of a general quartic interaction Hamiltonian under the influence of dissipation and nonclassical driving. We show that this scenario could be realized with a cascaded superconducting cavity-qubit system in the strong dispersive regime in a setup similar to recent experiments. In the presence of dissipation, we find that an effective Hartree-type decoupling with a Fokker-Planck equation yields a good approximation. We find that the stationary state is approximately a squeezed vacuum, which is enhanced by the $Q$ factor of the cavity but conserved by the interaction. The qubit nonlinearity, therefore, does not significantly influence the highly squeezed intracavity microwave field but, for a range of realistic parameters, enables characterization of itinerant squeezed fields.

Received 19 January 2015

DOI:https://doi.org/10.1103/PhysRevA.92.013826

Authors & Affiliations

Matthew Elliott and Eran Ginossar

Advanced Technology Institute and Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom

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Vol. 92, Iss. 1 — July 2015

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Images

Figure 1
Plot of the $P$ -quadrature uncertainty in the second cavity obtained in our theoretical model as a function of the effective cavity-cavity detuning ${\tilde{Δ}}_{12}$ and the pump strength $ε_{1}$ . The other system parameters are fixed at $κ_{1} / κ_{2} = 50, g / κ_{2} = 56, Δ_{q} / κ_{2} = 600$ . The total squeezing increases with pump strength, reaching the same uncertainty as the no-qubit case for any value of $ε_{1}$ . The optimum detuning shifts as the total squeezing in the cavity is increased.
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Figure 2
Plot of the minimum quadrature uncertainty that can be obtained in our system with $κ_{1} / κ_{2} = 50$ and $g / κ_{2} = 56$ by choosing the optimum value of ${\tilde{Δ}}_{12}$ (i.e., the base of the valley in Fig. 1). Mean-field values in the Hartree approximation, which are identical to the no-qubit case, represent ideal squeezed states and fall on the boundary of the shaded region, which is inaccessible. Yellow circles show simulation data for no-qubit case, with deviations from theory caused by Fock basis truncation. Red diamonds and green squares show simulation data with $Δ_{q} / κ_{2} = 1200$ and $Δ_{q} / κ_{2} = 600$ , respectively. With the qubit detuned further from the cavity, higher-order interaction terms are smaller, the qubit behaves more like a spectator and greater squeezing can be observed. With $Δ_{q} / κ_{2} = 1200$ and $ε_{1} / κ_{2} = 12$ we find 7 dB of squeezing can be achieved, much greater than the factor of two [28, 29] that can be achieved in the internal field of a degenerate parametric amplifier, here marked with a dashed line.
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Figure 3
Wigner functions of the squeezed-cavity state in the second cavity with parameters fixed at $κ_{1} / κ_{2} = 50, g / κ_{2} = 56, Δ_{q} / κ_{2} = 600$ and using the optimum value of $Δ_{12}$ (a) in the no-qubit case with $ε_{1} / κ_{1} = 10$ the state produced is purely Gaussian and significant squeezed, with artifacts caused by basis truncation. (b) If the full qubit interaction is included rippling can be seen in the Wigner function, which damages the squeezing. (c) If only terms up to fourth order are considered, almost identical rippling is seen, suggesting that the breakdown of our mean-field approximation is responsible for such distortions. (d) If the pump is increased to $ε_{1} / κ_{2} = 12$ the distortions become larger and damage the squeezing further.
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Figure 4
Plot of percentage error between the factorized fourth moment ${〈 a^{†} a a^{†} a 〉}_{R}$ and the exact moment $〈 a^{†} a a^{†} a 〉$ . Green squares correspond to $Δ_{q} / κ_{2} = 600$ and red diamonds correspond to $Δ_{q} / κ_{2} = 1200$ , matching the curves in Fig. 2, with other parameters set to $g / κ_{2} = 56, κ_{1} / κ_{2} = 50$ . For the closer qubit, the error increases at relatively low pump strengths and is more than 5% for the largest values of $ε$ . This corresponding to the destruction of squeezing and non-Gaussian steady-state Wigner functions we see for these pumps in full simulations. For the further detuned qubit, however, the error is very low up to $ε_{1} / κ_{2} = 13$ , leading to much better agreement between the model and simulations when $Δ_{q} / κ_{2} = 1200$ .
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Figure 5
Probability $P (N)$ of observing $N$ photons in the second cavity in the steady state. Solid bars show simulated number distribution in the steady state with $κ_{1} / κ_{2} = 50, ε_{1} / κ_{2} = 10, g / κ_{2} = 56, Δ_{q} / κ_{2} = 600$ , and $Δ_{12} / κ_{2} = 4.95$ . Empty outlines show a comparison with an ideal squeezed cavity state of the same average photon number. Almost all additional probability is in the $N = 0$ state, which is not shown to make these differences clearer. The simulated distribution is in good agreement with the ideal even-odd behavior, which has not been observed experimentally to date.
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