Single-Photon-Resolved Cross-Kerr Interaction for Autonomous Stabilization of Photon-Number States

E. T. Holland, B. Vlastakis, R. W. Heeres, M. J. Reagor, U. Vool, Z. Leghtas, L. Frunzio, G. Kirchmair, M. H. Devoret, M. Mirrahimi, and R. J. Schoelkopf

Phys. Rev. Lett. 115, 180501 – Published 26 October 2015

Abstract

Quantum states can be stabilized in the presence of intrinsic and environmental losses by either applying an active feedback condition on an ancillary system or through reservoir engineering. Reservoir engineering maintains a desired quantum state through a combination of drives and designed entropy evacuation. We propose and implement a quantum-reservoir engineering protocol that stabilizes Fock states in a microwave cavity. This protocol is realized with a circuit quantum electrodynamics platform where a Josephson junction provides direct, nonlinear coupling between two superconducting waveguide cavities. The nonlinear coupling results in a single-photon-resolved cross-Kerr effect between the two cavities enabling a photon-number-dependent coupling to a lossy environment. The quantum state of the microwave cavity is discussed in terms of a net polarization and is analyzed by a measurement of its steady state Wigner function.

Received 13 April 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.180501

Authors & Affiliations

E. T. Holland¹, B. Vlastakis¹, R. W. Heeres¹, M. J. Reagor¹, U. Vool¹, Z. Leghtas¹, L. Frunzio¹, G. Kirchmair^1,2,3, M. H. Devoret¹, M. Mirrahimi^1,4, and R. J. Schoelkopf¹

¹Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
²Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
³Institute for Experimental Physics, University of Innsbruck, A-6020 Innsbruck, Austria
⁴INRIA Paris-Rocquencourt, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France

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Vol. 115, Iss. 18 — 30 October 2015

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Figure 1
Storage and cooling cavity spectra. (a) Spectroscopy is performed on the storage cavity with a single cw drive. With a large amplitude drive, we observe the two-photon transition $f_{s, 0 \to 2} / 2$ . From this measurement we infer the location of the $f_{s, 1 \to 2}$ transition (solid black line) and determine its detuning from the $f_{s, 0 \to 1}$ transition as 4.0 MHz, which we define as the anharmonicity of the storage cavity. (b) A 5 ns square pulse, which displaces the storage cavity state and whose amplitude gives $\bar{n} \approx 1.5$ in the storage cavity, enables the observation of a single-photon-resolved cross-Kerr interaction between the two cavities, $χ_{sc} / 2 π = 2.59 \pm 0.06 MHz$ .
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Figure 2
Ideal cavity spectrum and Fock state stabilization protocol. (a) Left: sketch of idealized storage cavity spectrum. The storage cavity must have unequal energy levels spacing ( $ℏ A_{s}$ ), inherited from the coupled qubit, to selectively drive storage cavity transitions. On the right is the idealized cooling cavity spectrum. The frequency shift of the cooling cavity due to photons in the storage cavity, the cross-Kerr effect ( $χ_{sc}$ ), must be larger than either cavity linewidth to selectively drive this transition. (b) Energy level diagram for the coupled cavity-cavity system tracing over the qubit state. Excitations ascending vertically in the storage cavity while excitations ascend horizontally in the cooling cavity. A microwave drive $Ω_{S}$ is applied on the storage cavity so that population only oscillates between vacuum and the first Fock state of the storage cavity. Simultaneously, a drive, $Ω_{C}$ , is applied on the cooling cavity such that it is resonant provided there is exactly one excitation in the storage cavity. Once resonant, the cooling cavity is pumped to a mean photon-number set by the strength of the drive. Cavity decays, decaying arrows, close the autonomous loop of this protocol returning the population to $| 0, 0 ⟩$ allowing the preparation to be repeated.
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Figure 3
Storage cavity polarization. (a) The Fock state stabilization protocol described in Fig. 2 is applied for a duration $T_{s}$ , followed by a 300 ns wait to evacuate excitations from the cooling cavity, a photon selective $π$ pulse is then performed on the qubit determining the probability of each photon state of the storage cavity up to three photons. (b) Storage cavity state polarization as a function of drive amplitude and frequency. The frequency of the cooling cavity drive is plotted as $Δ = ω_{c}^{0} - ω_{dc}$ , and normalized by the cross-Kerr effect $χ_{sc}$ . As the frequency of the drive applied to the cooling cavity is brought in resonance with the first photon peak of the storage cavity $Δ / χ_{sc} \approx 1$ the protocol stabilizes the first Fock state of the storage cavity. In simulation, we determine maximum polarization inversion corresponds to roughly four photons on average in the cooling cavity which explains why maximal stabilization occurs further detuned. The inset is a simulation plot with the same axis and color scale as the experimental result. (c) Line cuts for a weak drive power and a drive power resulting in a polarization inversion. (d) As the duration of the stabilization protocol is varied the polarization of the storage cavity alters and for infinite time reaches its steady state solution.
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Figure 4
Wigner tomography of stabilized steady state of the storage cavity. (a) The previously described stabilization protocol is used to reach the desired steady state. Then Wigner tomography is performed on the state of the storage cavity. (b) Left: Measured Wigner function for the steady state of the storage cavity which is a statistical mixture of an $N = 1$ and $N = 0$ Fock state. Right: Simulated steady state of the protocol. (c) Line cuts along $Im (α)$ and $Re (α)$ for the measured Wigner function and the simulated steady state Wigner function. Although not a pure $N = 1$ Fock state of the storage cavity our steady state solution does have negativity in the Wigner function indicative of a quantum state.
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