Autonomous quantum refrigerator in a circuit QED architecture based on a Josephson junction

Patrick P. Hofer, Martí Perarnau-Llobet, Jonatan Bohr Brask, Ralph Silva, Marcus Huber, and Nicolas Brunner

Phys. Rev. B 94, 235420 – Published 16 December 2016

Abstract

An implementation of a small quantum absorption refrigerator in a circuit QED architecture is proposed. The setup consists of three harmonic oscillators coupled to a Josephson junction. The refrigerator is autonomous in the sense that it does not require any external control for cooling, but only thermal contact between the oscillators and heat baths at different temperatures. In addition, the setup features a built-in switch, which allows the cooling to be turned on and off. If timing control is available, this enables the possibility for coherence-enhanced cooling. Finally, we show that significant cooling can be achieved with experimentally realistic parameters and that our setup should be within reach of current technology.

Received 20 July 2016
Revised 29 November 2016

DOI:https://doi.org/10.1103/PhysRevB.94.235420

Physics Subject Headings (PhySH)

Quantum electrodynamics Quantum thermodynamics

Quantum heat engines & refrigerators

Josephson junctions

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Patrick P. Hofer^1,*, Martí Perarnau-Llobet^1,2, Jonatan Bohr Brask³, Ralph Silva¹, Marcus Huber^3,4, and Nicolas Brunner¹

¹Département de Physique Théorique, Université de Genève, 1211 Genève, Switzerland
²ICFO-Institut de Ciencies Fotoniques, Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain
³Group of Applied Physics, University of Geneva, Switzerland
⁴Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria

^*patrick.hofer@unige.ch

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Issue

Vol. 94, Iss. 23 — 15 December 2016

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Images

Figure 1
(a) Sketch of the refrigerator. A Josephson junction is coupled to three $L C$ circuits acting as harmonic oscillators which are themselves coupled to thermal baths. The phase across the junction is determined by the oscillators as well as a magnetic flux through the loop structure. The three-body interaction mediated by the Josephson junction allows for two photons with energies $Ω_{c}$ and $Ω_{h}$ to be converted into a photon with energy $Ω_{r} = Ω_{c} + Ω_{h}$ . A hot bath at temperature $T_{h}$ ensures an elevated occupation number in oscillator $h$ favoring the process illustrated with blue (solid) arrows over the process illustrated with red (dashed) arrows. (b) On/off refrigeration. Plot of the temperature in the $c$ oscillator $θ_{c}$ obtained from the energy of the reduced density matrix (see main text). At the dashed vertical lines, the refrigerator is switched on or off respectively allowing for on-demand cooling of a harmonic oscillator. As discussed in Appendix pp3, the reduced state of the $c$ oscillator is very close to a thermal state. The green dashed line is obtained with the simplified model in Eqs. (12a) and (12b) with $E_{J}^{'} / E_{J} = 0.059$ and $E_{J}^{''} / E_{J} = 0.00135$ .
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Figure 2
Performance of fridge as function of $E_{J}$ . The higher the coupling between the oscillators, the stronger the cooling. Here we are limited by the condition $E_{J} ≪ Ω_{α}$ which ensures the validity of our master equation. The inset shows the steady-state temperature as a function of the hot temperature. While increasing $T_{h}$ generally enhances cooling, the nonlinear operators given in Eq. (4) reduce cooling for high temperatures $T_{h}$ because they only couple weakly to Fock states with a high photon number. The crosses show values as given in Table 1. All other parameters are as in Table 1.
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Figure 3
Transient cooling. Blue (solid) line: The temperature in the $c$ oscillator oscillates before reaching the steady-state temperature $T_{c}^{S}$ . Green (dashed) line: Switching off the fridge when the temperature reaches its first minimum allows for cooling below the steady-state temperature. For low coupling to the cold bath, a temperature below $T_{c}^{S}$ can be maintained for a substantial amount of time (shaded area). Parameters are as in Table 1 except for $κ_{h} = κ_{c} = κ_{r} = 0.001 Ω_{c}$ and $k_{B} T_{h} = 8 Ω_{c}$ which corresponds to half of the value in Table 1.
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Figure 4
Temperature of the cold harmonic oscillator $θ_{c}$ as a function of time. At the dashed line the fridge is switched on or off respectively. The solid (blue) line is obtained using Eqs. (3b) and (3c), the dashed (green) line using the full Hamiltonian in Eq. (1). Apart from a small overestimation of the cooling power, the RWA approximation describes the system extremely well.
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Figure 5
Fock-state occupation probabilities. The light colors show the Fock-state occupation probabilities for the reduced state in the $c$ oscillator if cooling is on (blue) and off (red) respectively. The dark colors show Fock-state occupation probabilities for thermal states at the corresponding temperatures. The reduced states are very close to thermal states. Note that the reduced states are diagonal in the Fock-state basis. Parameters are given in Table 1.
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