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arXiv:1201.4951 · Physical Review B

Interference between magnetic field and cavity modes in an extended Josephson junction

Vincent F. C. Humbert, M. Aprili, Jan Hammer

Published Jul 20, 2012• 1 citation
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Abstract

An extended Josephson junction consists of two superconducting electrodes separated by an insulator and is therefore also a microwave cavity. The superconducting phase difference across the junction determines the amplitude as well as the spatial distribution of the supercurrent. Both external magnetic fields and resonant intracavity fields produce a spatial modification of the superconducting phase along the junction. The interplay between these two effects leads to interference in the critical current of the junction and allows us to continuously tune the coupling strength between the first cavity mode and the Josephson phase from 1 to $\ensuremath{-}0.68$. This enables static and dynamic control of the junction in the ultrastrong-coupling regime.

DOI
10.1103/physrevb.86.024520

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An extended Josephson junction consists of two superconducting electrodes separated by an insulator and is therefore also a microwave cavity. The superconducting phase difference across the junction determines the amplitude as well as the spatial distribution of the supercurrent. Both external magnetic fields and resonant intracavity fields produce a spatial modification of the superconducting phase along the junction. The interplay between these two effects leads to interference in the critical current of the junction and allows us to continuously tune the coupling strength between the first cavity mode and the Josephson phase from 1 to \ensuremath−0.68\ensuremath{-}0.68\ensuremath−0.68. This enables static and dynamic control of the junction in the ultrastrong-coupling regime.
We report a theoretical study of the macroscopic quantum dynamics in spatially extended Josephson systems. We focus on a Josephson tunnel junction of finite length placed in an externally applied magnetic field. In such a system, electromagnetic waves in the junction are excited in the form of cavity modes manifested by Fiske resonances, which are easily observed experimentally. We show that in the quantum regime, various characteristics of the junction such as its critical current Ic{I}_{c}Ic​, width of the critical current distribution $\ensuremath{\sigma}$, escape rate $\ensuremath{\Gamma}$ from the superconducting state to a resistive one, and the time-dependent probability P(t)P(t)P(t) of the escape are influenced by the number of photons excited in the junction cavity. Therefore, these characteristics can be used as a tool to measure the quantum states of photons in the junction, e.g., quantum fluctuations, coherent and squeezed states, entangled Fock states, etc.
An extended Josephson junction intrinsically couples the superconducting current to the microwave cavity in the insulating barrier. We demonstrate that this coupling produces sidebands in the microwave cavity resonances of the junction. By measuring the switching current distribution, we show that microwave radiation at sidebands brings the Josephson phase out of equilibrium. In particular, the effective phase temperature is reduced or enhanced through anti-Stokes and Stokes scattering, respectively. Phase cooling and heating both increase with microwave power.
Recently, we have proposed the half-Josephson laser (HJL): a device that combines lasing with superconducting leads, providing a locking between the optical phase and the superconducting phase difference between the leads. In this work, we propose and investigate two setups derived from a superconducting quantum interference device (SQUID), where two conventional Josephson junctions are replaced by two HJLs. In the first setup, the HJLs share the same resonant mode, while in the second setup two separate resonant modes of the two lasers are coupled optically. We dub the setup ``light-superconducting interference device'' (LSID). In both setups, we find the operating regimes similar to those of a single HJL. Importantly, the steady lasing field is significantly affected by the magnetic flux penetrating the SQUID loop, with respect to both amplitude and phase. This provides opportunities to tune or even quench the lasing by varying a small magnetic field. For the second setup, we find a parameter range where the evolution equation for the laser fields supports periodic cycles. The fields are thus modulated with the frequency of the cycle resulting in an emission spectrum consisting of a set of discrete modes. From this spectrum, two modes dominate in the limit of strong optical coupling. Therefore, the LSID can be also used to generate such modulated light.
de Broglie wave interferometers have been constructed, successfully utilizing extended superconducting links of tin to couple coherently the quantum phases of two Josephson junctions. The current transmitted by these devices was periodic in the enclosed flux (periodicity (.9 ± .3) h/2e) demonstrating unambiguously coherence of the superconducting order parameter over 1.33 meters. We may define a Josephson junction as one for which the quantum mechanical coupling energy is insufficient to overcome the disruptive effects of noise. Consequently, the relative phase of the coupled superconductors is not stabilized and the zero Josephson current fluctuates, (bandwidth ~ 2eVnoise/h) averaging to zero. Two such junctions have been coupled to an electromagnetic cavity formed by the superconducting arms of a quantum interferometer (junction separation 1. 33 meters). With an applied steady voltage such that the Josephson current excites a normal mode of the cavity, a coherent radiation field is built up. This coherent radiation feeds back to the junctions forcing time coherence on the phase precession, resulting in a dynamic stabilization of the junctions. Under these conditions a series of constant voltage has been observed in the I-V characteristic of one interferometer. The Josephson frequency associated with these steps is shown to be characteristic of the electromagnetic resonant modes of the cavity. Frequency modulation analysis, combined with a detailed analysis of the cavity-junction combination, predicts such maxima in the tunneling of pairs when a selection rule, (m + n) even, is obeyed. Here m/2 is the number of flux quanta (h/2e) linking the interferometer and n is the order of the cavity resonance excited. Experimental confirmation of the detailed predictions of the analysis is presented.
The possibility of using very-high-Q superconducting microwave cavities to obtain a strong coupling between the electromagnetic field and a Josephson junction has prompted a study of such systems. Expressions have been obtained describing the dc I-V characteristic of a junction biased so that the Josephson radiation frequency is near a resonance of the cavity in which it is located. The rf power coupled into an X-band (10 GHz) TE011 mode cavity from a weakly coupled junction is approximately 3×10−13 I02 Q W, where I0 is the critical current of the junction and Q is the cavity Q. When the coupling between the junction and cavity is large enough, the I-V characteristic becomes multiple valued and the power coupled to the cavity depends on the state of the system. The maximum power which can be coupled to the cavity is approximately 0.58I0Vdc, where Vdc is the voltage required to bias the junction to the resonant frequency of the cavity.
A dc superconducting quantum interference device (SQUID) consisting of a symmetric ring circuit with four Josephson junctions and two independent external currents I1{I}_{1}I1​ and I2{I}_{2}I2​, which enter the circuit at different points, has a phase boundary between the fully superconducting and the normal states that is different from that of a conventional SQUID. The control current I2{I}_{2}I2​ modifies the magnetic flux (\ensuremath{\varphi}) periodicity and the value of the maximum measuring current I1m{I}_{1m}I1m​(\ensuremath{\varphi}). For values of I2{I}_{2}I2​>Ic{I}_{c}Ic​ (critical current of Josephson junction) the fully resistive state exists over flux intervals centered around (n+(1/2)${\ensuremath{\varphi}}_{0}$, where n is an integer and ${\ensuremath{\varphi}}_{0}$ is the fluxoid quantum. A superconducting circuit without Josephson junctions is suggested which should have a phase boundary similar to that found here. Minimization procedures which are usually employed in analyzing SQUID's are discussed.