Fock-state stabilization and emission in superconducting circuits using dc-biased Josephson junctions

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Fock-state stabilization and emission in superconducting circuits using dc-biased Josephson junctions

J.-R. Souquet and A. A. Clerk

Phys. Rev. A 93, 060301(R) – Published 21 June 2016

Abstract

We present and analyze a reservoir-engineering approach to stabilizing Fock states in a superconducting microwave cavity which does not require any microwave-frequency control drives. Instead, our system makes use of a Josephson junction biased by a dc voltage which is coupled to both a principal storage cavity and a second auxiliary cavity. Our analysis shows that Fock states can be stabilized with an extremely high fidelity. We also show how the same system can be used to prepare on-demand propagating Fock states, again without the use of microwave pulses.

Received 1 February 2016

DOI:https://doi.org/10.1103/PhysRevA.93.060301

Physics Subject Headings (PhySH)

Quantum error correction

Quantum cavities

Josephson junctions

Quantum Information, Science & Technology

Authors & Affiliations

J.-R. Souquet and A. A. Clerk

Department of Physics, McGill University, 3600 rue University, Montreal, Quebec, Canada H3A 2T8

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Vol. 93, Iss. 6 — June 2016

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Images

Figure 1
Schematic of the proposed setup. A dc-biased Josephson junction of energy $E_{J}$ is in series with two cavities. Each cavity has an impedance close to $R_{K} = h / e^{2}$ and is damped via a coupling to a transmission line.
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Figure 2
Matrix elements $w_{n + l, n} [λ]$ for junction-driven cavity Fock-state transitions as a function of the zero-point voltage fluctuation amplitude $λ$ for different values of $n$ and $l$ . These functions are highly nonlinear with respect to $λ$ and can cancel for specific values. Roots of $w_{n + l, n} [λ]$ , denoted ${\tilde{λ}}_{n + l, n}$ , are indicated.
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Figure 3
Schematic depictions of biased-junction cavity pumping processes. The boxed digit indicates the number of Cooper pairs that have tunneled, whereas dots in the parabolas indicate the intracavity photon number. (a) Single-cavity setup for an impedance yielding $λ = {\tilde{λ}}_{2, 1}$ . Cooper-pair tunneling can take the cavity between the zero- and one-photon states, but transitions to the two-photon state are blocked. (b) Two-cavity setup for Fock-state stabilization, where $λ_{s} = {\tilde{λ}}_{2, 1}$ . Starting from vacuum, Cooper-pair tunneling can cause oscillations between the $| 0, 0 〉$ and $| 1, 1 〉$ photon Fock states. Photon decay from the aux cavity, however, freezes the system into the desired $| 1, 0 〉$ state. When the storage-cavity photon decays (due to $κ_{s}$ ), the cycle repeats.
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Figure 4
Steady-state probability $p_{s} [n]$ for the storage cavity to be in a Fock state ${| n 〉}_{s}$ , as obtained from the RWA master equation in Eq. (10). We take $E_{J} = κ_{aux} ≪ ω_{s}, ω_{aux}$ , and $λ_{aux} = 1 / 2$ ; $λ_{s}$ and $κ_{s}$ are indicated in the plots. For each $n$ , the rightmost bar represents $λ$ closest to perfect tuning. (a) Probabilities when $λ_{s}$ is tuned close to ${\tilde{λ}}_{2, 1}$ (for stabilizing $| 1 〉$ ). (b) The same for $λ_{s}$ tuned close to ${\tilde{λ}}_{3, 2}$ (for stabilizing $| 2 〉$ ). Even with imperfect tuning, the target Fock state is stabilized with a high fidelity. (c) and (d) Corresponding Wigner function $W [α]$ for the storage-cavity state, showing that negative values are obtained; note that $W [α]$ is rotationally invariant in all cases.
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Figure 5
Time dependence of storage-cavity photon-number occupancies $p_{s} [n, t]$ , where the bias $V_{dc} = (ω_{aux} + ω_{s}) / 2 e$ is turned on at $t = 0$ and the cavities start from vacuum. Dashed line: $κ_{s}$ chosen to optimize the steady-state value of $p_{s} [1, t]$ (thick dashed line) with the Fock-state stabilization protocol. Solid line: alternate choice of $κ_{s}$ which optimize $p_{s} [1, t]$ (thick solid line) at intermediate times with the pulsed protocol. In this latter protocol, the voltage can be turned off near $t_{off}$ , resulting with high probability in the production of a propagating Fock state in the transmission line coupled to the storage cavity. In both cases, we assume the realistic situation where the cavity impedance has not been tuned perfectly (here, $λ_{s} = 0.95 {\tilde{λ}}_{21}$ and $λ_{aux} = 0.13 {\tilde{λ}}_{21}$ ). The inset shows $p_{s} [n, t \to \infty]$ for the pulsed protocol.
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