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arXiv:quant-ph/0010015 · Modern Physics Letters B

q-DEFORMED DYNAMICS AND JOSEPHSON JUNCTION

Ramandeep S. Johal

Published Dec 10, 2000• 1 citation
arXivHTMLPDF

Abstract

We define a generalized rate equation for an observable in quantum mechanics, that involves a parameter q and whose limit q→1 gives the standard Heisenberg equation. The generalized rate equation is used to study dynamics of current-biased Josephson junction. It is observed that this toy model incorporates diffraction-like effects in the critical current. Physical interpretation for q is provided which is also shown to be a q-deformation parameter.

DOI
10.1142/s0217984900001166

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We define a generalized rate equation for an observable in quantum mechanics, that involves a parameter q and whose limit q→1 gives the standard Heisenberg equation. The generalized rate equation is used to study dynamics of current-biased Josephson junction. It is observed that this toy model incorporates diffraction-like effects in the critical current. Physical interpretation for q is provided which is also shown to be a q-deformation parameter.
Deep theoretical understanding of the electrical response of Josephson junctions is indispensable regarding both recent discoveries of new kinds of superconductivity and technological advances such as superconducting quantum computers. Here, we study the microscopic theory of the dc current-biased I-V characteristics of Josephson tunnel junctions. We derive an analytical formula of the I-V characteristics of generic junctions. We identify subharmonics of the I-V characteristics and their underlying mechanism as the feedback effect of intrinsic ac currents generated by voltage pulses in the past. We apply our theory to analytically solve the Werthamer equation and describe various dc current-biased I-V characteristics as a function of softening of the superconducting gap. Strikingly, we identify voltage staircases of the I-V characteristics in a genuine Josephson junction without ac current bias or qubit dynamics. Our general analytical formalism opens new avenues for a microscopic understanding of I-V characteristics of Josephson junctions that have been limited to phenomenological models so far.
We consider quantum fluctuations in a current biased overdamped Josephson junction in the regime when the bias current is larger than the critical current. The fluctuations of the voltage and phase across the junction are assumed to be initiated by equilibrium current fluctuations in the shunting resistor. This corresponds to low enough temperatures, when fluctuations of the normal current in the junction itself can be neglected. Quantum effects are important when the resistor temperature is low compared to the Josephson frequency, even in the case when noise is measured at frequencies lower than the temperature. We used the quantum Langevin equation in terms of random variables related to the limit cycle of the nonlinear Josephson oscillator. This allows to go beyond the perturbation theory and calculate the widths of the Josephson radiation lines.
Quantum fluctuations in a current-biased Josephson junction, described in terms of the RCSJ model, are considered. The fluctuations of the voltage and phase across the junction are assumed to be initiated by equilibrium current fluctuations in the shunting resistor. This corresponds to low enough temperatures, when fluctuations of the normal current in the junction itself can be neglected. We used the quantum Langevin equation in terms of random variables related to the limit cycle of the nonlinear Josephson oscillator. This allows to go beyond the perturbation theory and to calculate the widths of the Josephson radiation lines.
We discuss correlated lattice models with a time-dependent potential across a barrier and show how to implement a Josephson-junction-like behavior. The pairing occurs by a correlation effect enhanced by the symmetry of the system. In order to produce the effect we need a mild distortion which causes avoided crossings in the many-body spectrum. The Josephson-like response involves a quasi-adiabatic evolution in the time-dependent field. Besides, we observe an inverse-Josephson (Shapiro) current by applying an AC bias; a supercurrent in the absence of electromotive force can also be excited. The qualitative arguments are supported by explicit exact solutions in prototype 5-atom clusters with on-site repulsion. These basic units are then combined in ring-shaped systems, where one of the units sits at a higher potential and works as a barrier. In this case the solution is found by mapping the low-energy Hamiltonian into an effective anisotropic Heisenberg chain. Once again, we present evidence for a superconducting flux quantization, i.e. a Josephson-junction-like behavior suggesting the build-up of an effective order parameter already in few-electron systems. Some general implications for the quantum theory of transport are also briefly discussed, stressing the nontrivial occurrence of asymptotic current oscillations for long times in the presence of bound states.
Josephson junction is one of the promising devices to be used as a quantum bit or known as qubit in a quantum computer. The dynamics of current-bias Josephson junction can be described by using the Resistively and Capacitively Shunted Junction (RCSJ) model which is analogous to the classical motion of a particle in a tilted washboard potential. In this work, quantum theory and approximation methods are used to calculate and explain quantum properties of the Josepson junction. We found that the energy levels in the washboard potential can be quantum mechanically quantized, and thus used as a qubit in quantum computation. The tunneling rate from the n[superscript th] energy level exponentially increases with bias current and the first excited state is three orders of magnitude more than the ground state. The wave function of the state in the potential well is relatively similar to the harmonic oscillator wave function because of the shape of the washboard potential. The transition rate between the energy levels due to an electromagnetic radiation depends on the amplitude and the frequency of microwave current. The period of the oscillation is in the order of nanoseconds. Finally the decoherence time in Josephson junction can be observed by considering the Rabi's oscillation in the presence of current noise. The decoherence time depends on the magnitude of the current noise. This result can be used to determine the feasibility of using Josephson Junction as a qubit
We have performed a quantitative electrodynamic experiment on a Josephson oxide tunneling junction. Using a current source at 30 Mc/sec, we have verified the time dependence of the Josephson equation, j=j0j={j}_{0}j=j0​ $sin [{\ensuremath{\gamma}}_{l}\ensuremath{-}{\ensuremath{\gamma}}_{r}\ensuremath{-}(\frac{2e}{\ensuremath{\hbar}})\ensuremath{\int}\mathrm{A}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{dl}]$, where $(\frac{\ensuremath{\partial}}{\ensuremath{\partial}t})[{\ensuremath{\gamma}}_{l}\ensuremath{-}{\ensuremath{\gamma}}_{r}\ensuremath{-}(\frac{2e}{\ensuremath{\hbar}})\ensuremath{\int}{l}^{r}\mathrm{A}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{dl}]=\ensuremath{-}(\frac{2eV}{\ensuremath{\hbar}})$. Thus the limiting voltage is shown to obey the quantum relationship relationship, $2eV=\ensuremath{\hbar}\ensuremath{\omega}$. Measurements were made completely in the coherent state showing the diffraction nature of the junction as a function of applied magnetic field. The Josephson (parametric) inductance of the junction was directly observed and is in good agreement with the theory.