Hysteretic Flux Response and Nondegenerate Gain of Flux-Driven Josephson Parametric Amplifiers

Stefan Pogorzalek, Kirill G. Fedorov, Ling Zhong, Jan Goetz, Friedrich Wulschner, Michael Fischer, Peter Eder, Edwar Xie, Kunihiro Inomata, Tsuyoshi Yamamoto, Yasunobu Nakamura, Achim Marx, Frank Deppe, and Rudolf Gross

Phys. Rev. Applied 8, 024012 – Published 17 August 2017

Abstract

Josephson parametric amplifiers (JPAs) have become key devices in quantum science and technology with superconducting circuits. In particular, they can be utilized as quantum-limited amplifiers or as a source of squeezed microwave fields. Here, we report on the detailed measurements of five flux-driven JPAs exhibiting a hysteretic dependence of the resonant frequency on the applied magnetic flux. We model the measured characteristics by numerical simulations based on the two-dimensional potential landscape of the dc superconducting quantum interference devices, which provide the JPA nonlinearity for a nonzero screening parameter $β_{L} > 0$ and demonstrate excellent agreement between the numerical results and the experimental data. Furthermore, we study the nondegenerate response of different JPAs and accurately describe the experimental results with our theory.

Received 30 September 2016

DOI:https://doi.org/10.1103/PhysRevApplied.8.024012

Physics Subject Headings (PhySH)

Josephson effect Optical parametric oscillators & amplifiers Quantum information processing with continuous variables Quantum information with solid state qubits Superconductivity

Microwave techniques SQUID Superconducting devices

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsNonlinear Dynamics

Authors & Affiliations

Stefan Pogorzalek^1,2,*, Kirill G. Fedorov^1,2, Ling Zhong^1,2,3, Jan Goetz^1,2, Friedrich Wulschner^1,2, Michael Fischer^1,2,3, Peter Eder^1,2,3, Edwar Xie^1,2,3, Kunihiro Inomata⁴, Tsuyoshi Yamamoto^4,5, Yasunobu Nakamura^4,6, Achim Marx¹, Frank Deppe^1,2,3, and Rudolf Gross^1,2,3,†

¹Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany
²Physik-Department, Technische Universität München, 85748 Garching, Germany
³Nanosystems Initiative Munich (NIM), Schellingstraße 4, 80799 München, Germany
⁴RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
⁵NEC IoT Devices Research Laboratories, Tsukuba, Ibaraki 305-8501, Japan
⁶Research Center for Advanced Science and Technology (RCAST), The University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan

^*stefan.pogorzalek@wmi.badw.de
^†rudolf.gross@wmi.badw.de

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Vol. 8, Iss. 2 — August 2017

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Figure 1
(a) Circuit diagram of a JPA consisting of a coplanar waveguide resonator with inductance $ℓ_{r}$ and capacitance $c_{r}$ per unit length which add up to the total inductance $L_{r}$ and capacitance $C_{r}$ . The dc-SQUID loop has a nonzero loop inductance $L_{loop}$ and, thus, inductively couples to a magnetic flux $Φ_{dc} + Φ_{rf}$ generated by a coil and a pump line, respectively. Crosses indicate Josephson junctions. (b) Setup for the characterization of JPAs with a VNA. The reflected signal from the JPA is separated from the input signal by a measurement circulator. The current through a superconducting coil determines the flux $Φ_{dc}$ through the dc-SQUID loop. $S$ and $P$ mark the signal and pump port, respectively.
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Figure 2
(a) Optical micrograph of a JPA sample chip. Red and green rectangles mark the coupling capacitor and the dc SQUID with the pump line, respectively. $S$ and $P$ mark the signal and pump ports, respectively. (b) Enlargements of coupling capacitors with a designed external quality factor of $Q_{ext} = 200$ (left panel), $Q_{ext} = 600$ (right panel). (c) Enlargement of the dc SQUID with the adjacent pump line. The size of the dc-SQUID loop is $3.4 \times 2.4 {μ m}^{2}$ .
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Figure 3
(a)–(f) Two-dimensional dc-SQUID potential $U$ for different external flux values $Φ_{ext}$ with $β_{L} = 0.6$ and $j_{tr} = 0$ . White dots denote the present position of the phase particle, whereas red dots indicate the positions of all other local minima. (a)–(d) and (e),(f) correspond to increasing and decreasing values of $Φ_{ext}$ , respectively. Arrows indicate a jump of the phase particle to an adjacent minimum, when the present local minimum disappears. (g) JPA resonant frequency $ω_{0}$ when sweeping $Φ_{ext}$ towards larger (blue) and smaller (orange) values, calculated from Eq. (1) with $L_{loop} = 0$ , $L_{r} = 10 L_{s, \min}$ , and $L_{s, \min} = Φ_{0} / (4 π I_{c})$ . Furthermore, the $ω_{0}$ values of the corresponding position of the phase particle in panels (a)–(f) are denoted by white dots in (g).
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Figure 4
JPA resonant frequency $ω_{0}$ of (a),(b) JPA 1 and (c) JPA 3 as a function of the applied flux $Φ_{ext}$ , as well as numerical fits (the red lines). Blue circles and green triangles mark the data taken for increasing and decreasing values of $Φ_{ext}$ , respectively, with arrows further indicating the sweep direction. The JPAs are stabilized at temperatures between 17 and 50 mK. Error bars of $ω_{0}$ from fits of Eq. (7) are smaller than the symbol size. The fit for JPA 1 in (a),(b) is shown for $γ = 5.8 \times 10^{- 3} {Wb}^{- 2}$ . The fitting results are summarized in Table 1.
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Figure 5
(a),(b) Experimental spectra of the nondegenerate signal and idler gains as a function of $ε$ and signal detuning $δ ω$ from half the pump frequency of $ω_{pump} / 4 π = 5.4 GHz$ for JPA 1. The smallest detuning is $| δ ω | / 2 π = 0.15 MHz$ . (c),(d) Theoretical calculations with $g ≃ 3.49 V^{- 1}$ of the signal and idler gains computed in Eqs. (8) and (9), respectively. (e) Maximal signal and idler gains extracted from Lorentzian fits along vertical cuts in (a),(b). (f) Signal and idler gains as a function of the signal frequency along the dashed lines. The symbols mark the experimental data, and the solid lines are fits of the data by Eqs. (8) and (9). The JPA temperature is stabilized at 50 mK.
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Figure 6
(a),(b) Experimental spectra of the nondegenerate signal and idler gains as a function of $ε$ and signal detuning $δ ω$ from half the pump frequency of $ω_{pump} / 4 π = 6.125 GHz$ for JPA 3. The smallest detuning is $| δ ω | / 2 π = 25 kHz$ . (c),(d) Theoretical calculations with $g ≃ 0.17 V^{- 1}$ of the signal and idler gains computed in Eqs. (8) and (9), respectively. (e) Maximal signal and idler gains extracted from Lorentzian fits along vertical cuts in (a),(b). (f) Signal and idler gains as a function of the signal frequency along the dashed lines. The symbols mark the experimental data, and the solid lines are fits of the data with Eqs. (8) and (9). The JPA temperature is stabilized at 30 mK.
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