Theory of microwave spectroscopy of Andreev bound states with a Josephson junction

L. Bretheau, Ç. Ö. Girit, M. Houzet, H. Pothier, D. Esteve, and C. Urbina

Phys. Rev. B 90, 134506 – Published 8 October 2014

Abstract

We present a microscopic theory for the current through a tunnel Josephson junction coupled to a nonlinear environment, which consists of an Andreev two-level system coupled to a harmonic oscillator. It models a recent experiment [Bretheau, Girit, Pothier, Esteve, and Urbina, Nature (London) 499, 312 (2013)] on photon spectroscopy of Andreev bound states in a superconducting atomic-size contact. We find the eigenenergies and eigenstates of the environment and derive the current through the junction due to inelastic Cooper pair tunneling. The current-voltage characteristic reveals the transitions between the Andreev bound states, the excitation of the harmonic mode that hybridizes with the Andreev bound states, as well as multiphoton processes. The calculated spectra are in fair agreement with the experimental data.

Received 24 June 2014
Revised 22 September 2014

DOI:https://doi.org/10.1103/PhysRevB.90.134506

Authors & Affiliations

L. Bretheau^1,*, Ç. Ö. Girit¹, M. Houzet^2,3, H. Pothier^1,†, D. Esteve¹, and C. Urbina¹

¹Quantronics Group, Service de Physique de l'État Condensé (CNRS, URA 2464), IRAMIS, CEA-Saclay, 91191 Gif-sur-Yvette, France
²Université Grenoble Alpes, INAC-SPSMS, F-38000 Grenoble, France
³CEA, INAC-SPSMS, F-38000 Grenoble, France

^*Present address: Laboratoire Pierre Aigrain, Ecole Normale Supérieure, CNRS (UMR 8551), 75231 Paris Cedex 05, France.
^†hugues.pothier@cea.fr

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Vol. 90, Iss. 13 — 1 October 2014

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Images

Figure 1
Principle of the photon spectroscopy of Andreev bound states. (a) Phase $(δ)$ dependence of the Andreev levels with energies $\mp E_{A}$ in a short conduction channel of transmission $τ .$ $Δ$ is the superconducting gap. (b) A Cooper pair tunneling across the spectrometer junction releases energy $2 e V$ as a photon of frequency $ν,$ which is absorbed either in the atomic contact by exciting the Andreev transition at energy $2 E_{A}$ (a) or in a harmonic oscillator mode (c) present in the embedding circuit of the atomic contact (see Fig. 2).
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Figure 2
Simplified diagram of the experimental setup. A voltage-biased Josephson junction (orange cross) is used as a spectrometer. It emits microwaves in its environment, an atomic SQUID formed by an ancillary Josephson junction (green cross in parallel with a capacitor $C$ ) and an atomic point contact (blue triangles) of channel transmission probabilities $\{τ_{i}\}$ . The absorption of a photon by the environment is accompanied by the transfer of a Cooper pair through the spectrometer. The phases $γ$ and $δ$ across the SQUID junction and the atomic contact are linked by the external reduced flux $φ$ threading the loop: $δ = γ + φ$ . The phase across the spectrometer is given by $α = γ + 2 e V t$ .
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Figure 3
Phase dependence of the coupling energies $Ω_{x}$ (red) and $Ω_{z}$ (blue) in units of $\sqrt{z} Δ$ , for $τ = 0.8$ .
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Figure 4
(a) Energy spectrum diagrams of $H_{SQ}$ for a single channel: each state is labeled $|-, n〉$ or $|+, n〉$ for the Andreev pair in the ground ( $-$ ) or excited ( $+$ ) state and $n$ photons in the plasma mode. At degeneracy $2 E_{A} = h ν_{p}$ , the plasma and Andreev modes hybridize, which leads to an avoided crossing visible in (b). (b) and (c) Excitation energies $ε$ (in units of $Δ$ ) (b) and transition probabilities $P$ (c) of the first resonances, as a function of the reduced flux $φ$ , for a channel of transmission $τ = 0.98$ . These lines are obtained by numerical resolution of a $14 \times 14$ Hamiltonian matrix. Each color encodes a transition from the ground state towards a different excited state, as represented in (a).
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Figure 5
Comparison between calculated (left) and experimental (right) spectra $I (φ, V)$ for contact $A C 1$ (transmissions $0.942, 0.26$ ) (top) and $A C 2$ (transmissions $0.985, 0.37$ ) (bottom). The two insets show theoretical (black lines) and experimental (green lines) $I (V)$ cuts for $φ = π$ , for each contact. Right: The experimental spectra are extracted from Fig. 3 in Ref. [10]. The gray regions at $ν < 4$ GHz and around 25 GHz are not accessible because there the biasing of the spectrometer is unstable. Left: The calculated spectra are obtained using Eq. (7) with a phenomenological damping parameter $Γ / h = 2$ GHz. The calculated currents have been multiplied by 0.6 to match the measured ones. The transition probabilities and excitation energies are computed numerically, using seven photon levels in the plasma mode.
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Figure 6
Excitation energies $ε$ (in units of $Δ$ ) (top) and transition probabilities $P$ (bottom) of the first resonances, as a function of the reduced flux $φ$ , in the Jaynes-Cummings approximation for a channel of transmission $τ = 0.98$ . Each color encodes a transition towards a different state, as shown in Fig. 4.
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