Continuous generation and stabilization of mesoscopic field superposition states in a quantum circuit

Ananda Roy, Zaki Leghtas, A. Douglas Stone, Michel Devoret, and Mazyar Mirrahimi

Phys. Rev. A 91, 013810 – Published 7 January 2015

Abstract

While dissipation is widely considered to be harmful for quantum coherence, it can, when properly engineered, lead to the stabilization of nontrivial pure quantum states. We propose a scheme for continuous generation and stabilization of Schrödinger cat states in a cavity using dissipation engineering. We first generate nonclassical photon states with definite parity by means of a two-photon drive and dissipation, and then stabilize these transient states against single-photon decay. The single-photon stabilization is autonomous, and is implemented through a second engineered bath, which exploits the photon-number-dependent frequency splitting due to Kerr interactions in the strongly dispersive regime of circuit QED. Starting with the Hamiltonian of the baths plus cavity, we derive an effective model of only the cavity photon states along with analytic expressions for relevant physical quantities, such as the stabilization rate. The deterministic generation of such cat states is one of the key ingredients in performing universal quantum computation.

Received 7 November 2014

DOI:https://doi.org/10.1103/PhysRevA.91.013810

Authors & Affiliations

Ananda Roy^1,2,3,*, Zaki Leghtas¹, A. Douglas Stone¹, Michel Devoret¹, and Mazyar Mirrahimi^3,1

¹Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06511, USA
²Laboratoire Pierre Aigrain, Ecole Normale Suprieure, CNRS (UMR 8551), Universit Pierre et Marie Curie, Universit D. Diderot 24, rue Lhomond, 75231 Paris Cedex 05, France
³INRIA Paris-Rocquencourt, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France

^*ananda.roy@yale.edu

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Vol. 91, Iss. 1 — January 2015

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Figure 1
Schematic for the stabilization of the “cat state,” $| C_{α = 2}^{+} 〉$ . The two-photon drive and dissipation [denoted by $H_{2 ph}, κ_{2 ph} D (a_{s}^{2})$ ] act on the even and odd manifolds, shown in blue and orange, respectively. In the absence of single-photon loss and starting from vacuum, the odd manifold remains unpopulated, while the even manifold population is distributed to realize an even cat state. However, single-photon loss (shown in red) denoted by $κ_{1 ph} D (a_{s})$ transfers some of the population to the odd manifold, where it is distributed as in an odd cat state due to the two-photon drive and/or dissipation. We propose to engineer a dissipation interaction from $| 5 〉$ to $| 4 〉$ (in green) denoted by $κ_{ps} D (J_{4})$ . This dissipation, together with the two-photon process, transfers excitations from the odd to even manifold and stabilizes the desired cat state.
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Figure 2
Fidelity with respect to the target state $| C_{α = 2}^{+} 〉$ (left panel) and Wigner function of the steady state (right panel). The parity-selecting dissipation and the two-photon dissipation and/or drive, in the presence of single-photon loss, stabilizes the even cat state. The dissipation rates are $κ_{2 ph} = 250 κ_{1 ph}, κ_{ps} = 760 κ_{1 ph}$ . The evolution of fidelity is shown in the absence of single-photon loss (blue dashed line), in the presence of single-photon loss and the absence of parity selection (red double-dashed line), and, lastly, in the presence of single-photon loss and parity selection (green solid line). The steady-state Wigner function of the stabilized cat state is shown in the presence of single- and two-photon loss and parity selection.
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Figure 3
Schematic of experimental setup realizing the stabilization scheme. Josephson junction ${JJ}_{1}$ bridges the storage and readout cavity $r_{1}$ . This, together with the stiff off-resonant pump at $ω_{p} = 2 ω_{s} - ω_{r_{1}}$ , and the weak resonant drive at $ω_{r_{1}}$ incident on $r_{1}$ , gives rise to the two-photon drive and dissipation. Josephson junction ${JJ}_{2}$ bridges the storage and readout cavity $r_{2}$ providing a nonlinear coupling between the modes $a_{s}$ and low-Q mode $a_{r_{2}}$ . An off-resonant pump incident on $r_{2}$ at frequency $ω_{p^{'}} = (ω_{r_{2}} - ω_{s} - 2 \tilde{n} χ_{s r_{2}}) / 2$ gives rise to beam-splitter-like interaction between $a_{s}$ and $a_{r_{2}}$ : $g_{ps} e^{2 i ω_{p^{'}} t} a_{s}^{†} a_{r_{2}} + c . c .$ This beam-splitter-like interaction acts conditioned on the mode $a_{s}$ having $2 \tilde{n} + 1$ photons in the storage cavity. When the condition is realized, this interaction transfers one quantum of excitation from the $a_{s}$ mode to the $a_{r_{2}}$ mode, which is then lost irreversibly to the environment.
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Figure 4
Scattering processes taking place through the nonlinear elements. (a) One photon in readout mode $a_{r_{1}}$ , together with one photon of pump at $ω_{p}$ , gets converted to two photons in mode $a_{s}$ , giving rise to the two-photon drive. (b) Two photons of the mode $a_{s}$ are converted into one photon in pump mode at frequency $ω_{p}$ and one photon in mode $a_{r_{1}}$ , which then irreversibly decays to the environment, giving rise to two-photon dissipation. (c) One photon in the mode $a_{s}$ , along with two photons in the pump with the adequate frequency $ω_{p^{'}}$ , are converted conditionally into a photon in mode $a_{r_{2}}$ , which then irreversibly decays to the environment. This process occurs only when the number of photons in the storage cavity is $2 \tilde{n} + 1$ , giving rise to the parity-selection mechanism.
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Figure 5
Fidelity with respect to the target cat state, obtained by solving Eq. (11), vs parameters $g_{2 ph} / κ_{1 ph}, g_{ps} / κ_{1 ph}$ . We choose $κ_{r_{1}} = κ_{r_{2}} = 10^{3} κ_{1 ph}, χ_{s r_{2}} = 2.5 \times 10^{4} κ_{1 ph}$ . The ratio of $ε_{r_{1}}$ and $g_{2 ph}$ is chosen to be 4 so that the target cat state is $| C_{α = 2}^{+} 〉$ . The white square denotes the point of optimal fidelity, $≃ 0.94$ for this choice of parameters ( $g_{2 ph} = 250 κ_{1 ph}, g_{ps} = 400 κ_{1 ph}$ ; cf. Fig. 2). The black square is the point in the shown range of parameters where the adiabatic elimination of Sec. 3c works best.
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Figure 6
Comparison between the evolution of fidelities for the full three-mode master equation [Eq. (11)] in solid lines, and that obtained from the reduced dynamics [Eqs. (3) and (26)] in dashed lines. The two sets of parameters are chosen from Fig. 5: the white square (purple curves) corresponding to $g_{ps} = 400 κ_{1 ph}, g_{2 ph} = 250 κ_{1 ph}$ , and $ε_{r_{1}} = 1000 κ_{1 ph}$ , and the black square (orange curves) corresponding to $g_{ps} = 120 κ_{1 ph}, g_{2 ph} = 50 κ_{1 ph}$ , and $ε_{r_{1}} = 200 κ_{1 ph}$ . For both sets of curves, $χ_{s r_{2}} = 2.5 \times 10^{4} κ_{1 ph}, κ_{r_{2}} = κ_{r_{1}} = 1000 κ_{1 ph}$ and the target state is $| C_{α = 2}^{+} 〉$ . The model reduction [Eq. (3)] approaches the full three-mode master equation [Eq. (11)] as the adiabatic approximation ( $g_{2 ph} / κ_{r_{1}} ≪ 1, g_{ps} / κ_{r_{2}} ≪ 1$ ) and the rotating wave approximation ( $g_{ps} / χ_{s r_{2}} ≪ 1, κ_{r_{2}} / χ_{s r_{2}} ≪ 1$ ) become more and more accurate.
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