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arXiv:cond-mat/0511026 · Physical Review Letters

Poisson Transition Rates from Time-Domain Measurements with a Finite Bandwidth

Ofer Naaman, José Aumentado

Published Mar 16, 2006• 72 citations
arXivHTMLPDF

Abstract

In time-domain measurements of a Poisson two-level system, the observed transition rates are always smaller than those of the actual system, a general consequence of a finite measurement bandwidth in an experiment. This underestimation of the rates is significant even when the measurement and detection apparatus are 10 times faster than the process under study. We derive here a quantitative form for this correction, using a straightforward state transition model that includes the detection apparatus, and provide a method for determining a system's actual transition rates from bandwidth-limited measurements. We support our results with computer simulations and experimental data from time-domain measurements of quasiparticle tunneling in a single-Cooper-pair transistor.

DOI
10.1103/physrevlett.96.100201

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In time-domain measurements of a Poisson two-level system, the observed transition rates are always smaller than those of the actual system, a general consequence of a finite measurement bandwidth in an experiment. This underestimation of the rates is significant even when the measurement and detection apparatus are 10 times faster than the process under study. We derive here a quantitative form for this correction, using a straightforward state transition model that includes the detection apparatus, and provide a method for determining a system's actual transition rates from bandwidth-limited measurements. We support our results with computer simulations and experimental data from time-domain measurements of quasiparticle tunneling in a single-Cooper-pair transistor.
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The time of Zener tunneling, ${\ensuremath{\tau}}_{z}$, is calculated for a general two-level quantum-mechanical system by two methods. In the first we determine the width of the transition profile in time. In the second we apply an oscillating perturbation and examine how the final transition probability depends upon the perturbation frequency and phase. Both methods show, given that the coupling energy between the diabatic levels is \ensuremath{\Delta} and the time rate of change of the energy is \ensuremath{\alpha}, that in the adiabatic limit ${\ensuremath{\tau}}_{z}$ scales as \ensuremath{\Delta}/\ensuremath{\alpha} and in the sudden limit ${\ensuremath{\tau}}_{z}$ scales as (\ensuremath{\Elzxh}/\ensuremath{\alpha})1/2{)}^{1/2})1/2. We calculate the approximate transition probability at finite times for the two limits. The results can be applied to specific mesoscopic electronic systems.
This thesis presents measurements of quantum coherence in a single Cooper pair box, where quantum oscillations of charge between a reservoir and a small superconducting island have been observed. The single Cooper pair box (SCB) consists of a small metallic island which is connected to a reservoir via a tunnel junction. In the superconducting state, Cooper pairs are free to tunnel to and from the island, whose potential can be controlled by a gate voltage. A graded gap method was used to prevent quasiparticles from tunneling onto the island. For this purpose, a magnetic field suppressed the superconducting gap in the SCB leads more than in the island. This created an additional energy barrier for quasiparticles to tunnel to the island, making the gate voltage modulation periodic in Cooper pair charge. For a limited range of gate voltage, the SCB was found to behave as a model two-level quantum mechanical system. A non-adiabatic change in the induced island charge was used to bring two charge states into resonance. The resulting time evolution showed clear charge oscillations between the ground and excited state, with a frequency given by the energy level separation divided by Planck's constant. These oscillations had a longest coherence time of T2=9 ns at a point where the pure charge states are degenerate. The coherence time at this point was found to be limited by the relaxation rate, ?1. Away from this point, the coherence time was limited by the dephasing rate, ??. Away from the charge degeneracy, the dependence of T2 on gate charge suggested that low frequency fluctuators were the main source of dephasing. The charge of the Cooper pair box was measured by coupling the SCB capacitively to a radio frequency single electron transistor (RF-SET). The RF-SET was used as an electrometer by measuring the amount of reflection from a resonant microwave circuit in which a single electron transistor (SET) was embedded. A best charge sensitivity of 2.3·10-6 e/sqrt(Hz) and a 7 MHz bandwidth was reached in an improved setup. Another part of this thesis concerns an investigation of electrical filters for use in single electronics experiments. The characteristics of seven different types of cryogenic filters were evaluated.
A formalism of time-smoothed total transition probabilities and their rates is developed which employs Laplace averages of these quantities. Under conditions pertinent to scattering processes, the Laplace-average formalism is shown to yield results equivalent to those obtained from a stationary-state formalism. Rigorous lower and upper bounds are obtained for the Laplace-averaged quantities which reduce to equalities for two-level systems. The lower bounds appear to be potentially useful estimating lower bounds for total cross sections of various processes.