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Physical review. B, Solid state

Theory of Josephson Radiation. I. General Theory

Patrick A. Lee, Marlan O. Scully

Published Feb 1, 1971• 47 citations
Publisher

Abstract

We consider the problem of a Josephson junction interacting with a quantized electromagnetic field contained in a microwave cavity. Inasmuch as the cavity has a quality factor $Q$, and an external current $J$ flows in and out of the junction, this is a problem in nonequilibrium statistical mechanics. We use a technique that eliminates adiabatically the radiation field directly from the total-density-matrix equation of motion. The resulting equation for the reduced density matrix is found to have a simple physical interpretation. The spectrum of the voltage fluctuations and the linewidth of the emitted radiation are then calculated for the case when the resonance cavity is at absolute zero.

DOI
10.1103/physrevb.3.769

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Physical review. B, Condensed matter•May 1, 1988•0 citations
We consider the problem of a Josephson junction interacting with a quantized electromagnetic field contained in a microwave cavity. Inasmuch as the cavity has a quality factor QQQ, and an external current JJJ flows in and out of the junction, this is a problem in nonequilibrium statistical mechanics. We use a technique that eliminates adiabatically the radiation field directly from the total-density-matrix equation of motion. The resulting equation for the reduced density matrix is found to have a simple physical interpretation. The spectrum of the voltage fluctuations and the linewidth of the emitted radiation are then calculated for the case when the resonance cavity is at absolute zero.
Recent experiments on Josephson junction arrays (JJAs) in microwave cavities have opened up a new avenue for investigating the properties of these devices while minimizing the amount of external noise coming from the measurement apparatus itself. These experiments have already shown promise for probing many-body quantum effects in JJAs. In this work, we develop a general theoretical description of such experiments by deriving a quantum phase model for planar JJAs containing quantized vortices. The dynamical susceptibility of this model is calculated for some simple circuits, and signatures of the injection of additional vortices are identified. The effects of decoherence are considered via a Lindblad master equation.
A major factor in the practical utilization of any Josephson device for microwave and higher frequency applications (such as the generation and detection of electromagnetic radiation) is the success with which the junction device can be coupled to an external transmission line. As part of an experimental investigation of the electrodynamics of small arrays of Josephson tunnel junctions, the properties of the coherent radiation emitted both by individual junctions and by coupled junction systems have been studied at frequencies between 2 and 12 GHz. Near 9 GHz, more than 10-9 W of coherent radiation has been detected from a single Pb-Pb oxide-Pb junction coupled via waveguide. The development of thin film devices in which the tunnel junctions form part of a microstrip line have made it feasible to observe directly (in a single device) the radiation emitted both at the fundamental geometrical frequency and at the low order harmonics. The observed temperature dependence of the junction cavity Q indicate the low temperature losses are primarily due to cavity loading of the external transmission line.
With the recent technical breakthroughs, physicists fully play with the beauty of quantum mechanics. In this theoretical work on the mesoscopic Josephson effect, we take advantage of the collaborations with experimentalists as well as the exchanges between the cold atoms and condensed matter communities. We consider various systems based on the Josephson junction, beginning with its quantum description in the underdamped regime. Using the Keldysh formalism, we obtain the current-voltage characteristics from the classical to the zero temperature regime and the quantum Smoluchowski equation in the quasi-classical limit. We then study the quantum dynamics of a phase qubit realized with a SQUID in a novel configuration where tunneling events occur through two quartic barriers. The escape rate in this camel-back potential, calculated with the instantons technique, allows us to describe the experiments. Circuit quantum electrodynamics predicts that a lasing effect appears when a qubit is coupled to a resonant cavity. We consider the case of a charge qubit and also that of a transmon which exploits the Purcell effect. With the Lindbladian we obtain the density matrix from which we derive the spectrum of the output field. Finally, we study a gas of cold atoms in a ring shape trap comprising a barrier, thus creating a Bose Josephson junction. The low energy physics is described through the correlation functions with the Luttinger liquid theory. We show that the quantum fluctuations in the ring induce a renormalization of the Josephson energy.
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.
We develop a quantum field-theoretic method to describe the effect of frequency mixing using Josephson junctions or using an electron beam traversing a succession of microwave cavities. One of our purposes is to discuss the basic mechanism of current modulation and the subsequent radiation in the complete quantum mechanical context. We use the Heisenberg picture, so that the field operators depend on time and state vectors are fixed. It is essential to take quantum mechanical expectation values only for the quantities that are actually observed. Thus, in the case of frequency mixing, we do not evaluate the current but rather the radiated power, which is observable. An evaluation of the current would destroy correlations between Fourier components of the current produced in the first cavity or in the first junction. Leaving the current as an operator permits the correlations produced in the first cavity (junction) to affect the radiation in the second cavity (junction). Another purpose of the approach is to find the possible dependence of the output radiation on the quantum state of the incident radiation preexisting in cavities or in Josephson junctions. In the case of frequency mixing using two microwave cavities where the fields are treated as independent, we show that it is possible to distinguish the case where both cavities are in definite phase-certain coherent states from the phase-uncertain number states by the value of the wave function renormalization constant. However, if the relative phase of the two cavities cannot be defined, this distinction disappears. This indicates that the radiation occurs at the beat frequency even when the cavity fields are taken to be in energy eigenstates, so that no phase whatever can be associated with the fields in those cavities and thus a classical velocity modulation picture does not apply.
A theory is presented for the nonequilibrium voltage states of an irradiated Josephson junction shunted by an external resistor but with no external current or voltage biasing. This device, referred to as a free-running Josephson junction, is modeled in a small--radiation-amplitude, deterministic regime extending the previous work of Shenoy and Agarwal. The time-averaged induced voltage is treated as a dynamical variable, the external radiation is modeled as a current source, and the induced junction-radiation vector potential, with and without a mode structure, is treated to first order in the driving currents. A dynamical equation for the time-averaged induced voltage yields a (nonequilibrium) steady-state relation between the time-averaged induced voltage and the incident radiation amplitude valid for a wide range of voltages, including zero. Regions of bistability occur in the voltage--versus--incident-amplitude curves, some of which are dependent on the external resistor. The zero-voltage state breaks down, as the external radiation amplitude is increased, at a critical value of the incident-radiation amplitude inversely proportional to the external resistance.