Single-photon transistor based on superconducting systems

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Single-photon transistor based on superconducting systems

Marco T. Manzoni, Florentin Reiter, Jacob M. Taylor, and Anders S. Sørensen

Phys. Rev. B 89, 180502(R) – Published 12 May 2014

Abstract

We present a scheme for a single-photon transistor which can be implemented with only minor modifications of existing superconducting circuits. The proposal employs a three-level anharmonic ladder atom, e.g., a transmon qubit, placed in a cavity to mimic a $Λ$ -type atom with two long-lived states. This configuration may enable a wide range of effects originally studied in quantum optical systems to be realized in superconducting systems, and in particular allow for single-photon transistors. We study analytically and numerically the efficiency and the gain of the proposed transistor as a function of the experimental parameters, in particular of the level anharmonicity and of the various decay and decoherence rates. State-of-the-art values for these parameters indicate that error probabilities of $\sim 1 %$ and gains of the order of hundreds can be obtained.

Received 12 December 2013
Revised 24 April 2014

DOI:https://doi.org/10.1103/PhysRevB.89.180502

Authors & Affiliations

Marco T. Manzoni^1,2,3, Florentin Reiter¹, Jacob M. Taylor⁴, and Anders S. Sørensen^1,*

¹QUANTOP–Danish Quantum Optics Center, The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
²Dipartimento di Fisica, Università degli Studi di Milano, I-20133 Milano, Italy
³ICFO–Institut de Ciencies Fotoniques, 08860 Castelldefels (Barcelona), Spain
⁴Joint Quantum Institute/National Institute of Standards and Technology, 100 Bureau Dr. MS 8410, Gaithersburg, Maryland 20899, USA

^*anders.sorensen@nbi.dk

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Issue

Vol. 89, Iss. 18 — 1 May 2014

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Images

Figure 1
Setup. (a) Three-level ladder ( $Ξ$ ) system coupled to a cavity field. The cavity field resonantly couples the upper states $| g 〉$ and $| e 〉$ of a $Ξ$ system with a coupling constant $g$ (single line), while it is far off resonance by a detuning $Δ$ from the lower transition between $| f 〉$ and $| g 〉$ , where it interacts with a coupling constant $g^{'}$ . The lower transition is driven by a classical field (double line). (b) Numerical and analytical results for the error probability for the first step of the protocol as a function of deviation from $κ = g$ , parametrized by $α = (κ - g) / g$ . We use the parameters of [20] [coherence time $T_{2}^{*} = 2 / (γ_{g} + γ_{f}) = 92 μ$ s, and anharmonicity $Δ = (2 π) 206$ MHz], but a reduced coupling constant $g = (2 π) 5.3$ MHz. Full line: numerical results. Dashed line: analytical results. Dashed-dotted line: approximate optimum discussed in the text. The pulse length is chosen according to an approximate optimization [26] with $n = 8$ .
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Figure 2
Steps of the single-photon transistor operation. (a) First step. A gate field is sent in symmetrically from the left and right. This makes it indistinguishable whether the photon was reflected or transmitted, but still a phase difference is imparted on the atom (see text). This phase difference can be used to flip the atom depending on the presence or absence of a photon in the field. (b) Second step. A strong signal field is sent in from the left. The field is reflected if the atom is in state $| g 〉$ (full arrow), and transmitted if the atom is in state $| f 〉$ (dashed arrow).
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Figure 3
(a) Reflected intensity $I_{R}$ for an atom in state $| g 〉$ as a function of incoming intensity $I_{IN}$ obtained from a numerical solution of the master equation for various values of $g / κ$ . For weak incoming intensities the reflected intensity is equal to the total incoming intensity, but saturates in the range $g^{2} / 4 κ$ (dotted line) to $g^{2} / 2 κ$ as the intensity is increased. We have assumed that we only have population decay of the upper level at a rate $γ_{e} = 1.3 \times 10^{- 3} g$ corresponding to the values in Fig. 1 with $γ_{e} = 2 γ_{g}$ . The saturation of the reflected intensity provides an upper limit to how well one can distinguish an atom being in $| g 〉$ from an atom being in $| f 〉$ , which has a vanishing reflection. This saturation thus limits the gain of the transistor. (b) Alternative implementation based on a two-level system coupled to a cavity. Taking the subsystem to have a vacuum Rabi coupling $\tilde{g}$ , the nonlinearity of the Jaynes-Cummings Hamiltonian can mimic our example three-level system. Denoting the atomic levels by $\tilde{g}, \tilde{e}$ and the subsystem cavity Fock states with $\tilde{0}$ , etc., we identify $| f 〉 = | \tilde{g}, \tilde{0} 〉, | g 〉 = (| \tilde{g}, \tilde{1} 〉 - | \tilde{e}, \tilde{0} 〉) / \sqrt{2}, | e 〉 = (| \tilde{g}, \tilde{2} 〉 - | \tilde{e}, \tilde{1} 〉) / \sqrt{2}$ . This yields an anharmonicity parameter $Δ = 2 \tilde{g} (1 - \sqrt{2})$ , whereas the coupling for the effective three-level system $g$ is determined by the coupling of the anharmonic cavity subsystem to another cavity and can be as large as desired by appropriate cavity design.
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