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arXiv:cond-mat/0501280 · Physical Review B

Running-phase state in a Josephson washboard potential

Gheorghe Sorin Paraoanu

Published Oct 31, 2005• 19 citations
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Abstract

We investigate the dynamics of the phase variable of an ideal underdamped Josephson junction in switching current experiments. These experiments have provided the first evidence for macroscopic quantum tunneling in large Josephson junctions and are currently used for state readout of superconducting qubits. We calculate the shape of the resulting macroscopic wave packet and predict that the propagation of the wave packet long enough after a switching event leads to an average voltage increasing linearly with time.

DOI
10.1103/physrevb.72.134528

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We investigate the dynamics of the phase variable of an ideal underdamped Josephson junction in switching current experiments. These experiments have provided the first evidence for macroscopic quantum tunneling in large Josephson junctions and are currently used for state readout of superconducting qubits. We calculate the shape of the resulting macroscopic wave packet and predict that the propagation of the wave packet long enough after a switching event leads to an average voltage increasing linearly with time.
This thesis examines Josephson tunnel junctions as candidate qubits for quantum computation. A large area current-biased junction, known as a phase qubit, uses the two lowest energy levels in a tilted washboard potential as the qubit states |0> and |1>. I performed experiments with 10 x 10 um^2 Nb/AlOx/Nb qubit junctions, with critical currents of roughly 30 uA. The state of a device was initialized by cooling below 50 mK in a dilution refrigerator. In order for quantum mechanical superpositions to be long-lived, it is necessary to isolate the junction from noisy bias leads that originate at room temperature. I studied two types of isolation: an LC filter, and a broadband scheme that used an auxiliary junction, resulting in a dc SQUID. One of the main goals of this work was to determine how well a simple Hamiltonian, derived assuming just a few lumped elements, describes the observed behavior of a macroscopic Josephson device, including coherent dynamics such as Rabi oscillations. I did this by comparing results to the expected behavior of ideal two-level systems and with more detailed master equation and density matrix simulations. I performed state manipulation by applying dc bias currents and resonant microwave currents, and through temperature control. The tunneling escape rate of the junction from the states |0> and |1> (zero voltage) to the running state (finite voltage) depends on the occupation probability of the energy levels and served as state readout. Experiments to measure the relaxation time T1 between |1> and |0> were performed by examining the dependence of the escape rate with temperature, yielding a maximum T1 = 15 ns. Measuring the decay to the ground state after applying a microwave pulse revealed at least two time constants, one of about 10 ns and another as long as 50 ns. The spectroscopic coherence time T2* was estimated to be roughly 5 ns by measuring resonance widths and the decay envelope of coherent Rabi oscillations was found to have a time constant T' = 10 ns over a wide range of conditions.
We propose to use a time-dependent imaginary potential to describe quantum mechanical tunneling through time-varying potential barriers. We use Gamow solutions for stationary tunneling problems to justify our choice of potential, and we apply our method to describe tunneling of a mesoscopic quantum variable: the phase change across a Josephson junction. The Josephson junction phase variable behaves as the position coordinate of a particle moving in a tilted washboard potential, and our general solution to the motion in such a potential with a time-dependent tilt reproduces a number of features associated with voltage switching in Josephson junctions. Apart from applications as artificial atoms in quantum information studies, the Josephson junction may serve as an electric field sensitive detector, and our studies provide a detailed understanding of how the voltage switching dynamics couples to the electric field amplitude.
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 investigated the dynamics of underdamped Josephson junctions with a relatively small Josephson coupling EJ{E}_{J}EJ​. In addition to the usual crossover between macroscopic quantum tunneling and thermally activated (TA) behavior we observe, for the first time, the transition from TA behavior to underdamped phase diffusion. Above the crossover temperature, the threshold for switching into the finite-voltage state becomes extremely sharp. We propose a (TTT, EJ{E}_{J}EJ​) phase diagram with various regimes and show that for a proper description of it, dissipation and level quantization in a metastable well are crucial.
Experiments to investigate the quantum behavior of a macroscopic degree of freedom, namely the phase difference across a Josephson tunnel junction, are described. The experiments involve measurements of the escape rate of the junction from its zero voltage state. Low temperature measurements of the escape rate for junctions that are either nearly undamped or moderately damped agree very closely with predictions for macroscopic quantum tunneling, with no adjustable parameters. Microwave spectroscopy reveals quantized energy levels in the potential well of the junction in excellent agreement with quantum-mechanical calculations. The system can be regarded as a "macroscopic nucleus with wires."
A macroscopic variable, the phase difference across a Josephson junction, is shown to obey quantum mechanics by observing the phenomena of energy-level quantization and quantum tunneling.The relevant parameters of the junction were determined .!.!! .ill!:!, in the thermal regime from the dependence of the escape rate on bias current and from resonant activation in the presence of microwaves.The first observation of quantized energy levels for this system is reported.The positions of these energy levels are in quantitative agreement with the quantum mechanical predictions based on the measured junction parameters.A model is presented that predicts correctly the observed resonance shapes.The measured value of the escape rate out of the zero-voltage state for an underdamped (Q -30) junction was found to be independent of temperature at low temperatures.This low temperature escape rate was in excellent agreement with the prediction for macroscopic quantum tunneling, with no adjustable parameters.Jolw rJt..