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arXiv:cond-mat/0302383 · Physical review. B, Condensed matter

Measurement of coherent charge transfer in an adiabatic Cooper-pair pump

Rosario Fazio, F. W. J. Hekking, Jukka P Pekola

Published Aug 7, 2003• 24 citations
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Abstract

We study adiabatic charge transfer in a superconducting Cooper-pair pump, focusing on the influence of current measurement on coherence. We investigate the limit where the Josephson coupling energy ${E}_{J}$ between the various parts of the system is small compared to the Coulomb charging energy ${E}_{C}.$ In this case, the charge transferred in a pumping cycle ${Q}_{P}\ensuremath{\sim}2e,$ the charge of one Cooper pair: The main contribution is due to incoherent Cooper-pair tunneling. We are particularly interested in the quantum correction to ${Q}_{P},$ which is due to coherent tunneling of pairs across the pump and which depends on the superconducting phase difference ${\ensuremath{\varphi}}_{0}$ between the electrodes; $1\ensuremath{-}{Q}_{P}/(2e)\ensuremath{\sim}{(E}_{J}{/E}_{C})\mathrm{cos}{\ensuremath{\varphi}}_{0}.$ A measurement of ${Q}_{P}$ tends to destroy the phase coherence. We first study an arbitrary measuring circuit and then specific examples and show that coherent Cooper-pair transfer can, in principle, be detected using an inductively shunted ammeter.

DOI
10.1103/physrevb.68.054510

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We study adiabatic charge transfer in a superconducting Cooper-pair pump, focusing on the influence of current measurement on coherence. We investigate the limit where the Josephson coupling energy EJ{E}_{J}EJ​ between the various parts of the system is small compared to the Coulomb charging energy EC.{E}_{C}.EC​. In this case, the charge transferred in a pumping cycle QP\ensuremath∼2e,{Q}_{P}\ensuremath{\sim}2e,QP​\ensuremath∼2e, the charge of one Cooper pair: The main contribution is due to incoherent Cooper-pair tunneling. We are particularly interested in the quantum correction to QP,{Q}_{P},QP​, which is due to coherent tunneling of pairs across the pump and which depends on the superconducting phase difference \ensuremathφ0{\ensuremath{\varphi}}_{0}\ensuremathφ0​ between the electrodes; 1\ensuremath−QP/(2e)\ensuremath∼(EJ/EC)cos\ensuremathφ0.1\ensuremath{-}{Q}_{P}/(2e)\ensuremath{\sim}{(E}_{J}{/E}_{C})\mathrm{cos}{\ensuremath{\varphi}}_{0}.1\ensuremath−QP​/(2e)\ensuremath∼(EJ​/EC​)cos\ensuremathφ0​. A measurement of QP{Q}_{P}QP​ tends to destroy the phase coherence. We first study an arbitrary measuring circuit and then specific examples and show that coherent Cooper-pair transfer can, in principle, be detected using an inductively shunted ammeter.
We introduce and discuss a scheme for Cooper-pair pumping. The scheme relies on the coherent transfer of a superposition of charge states across a superconducting island and is realized by adiabatic manipulation of magnetic fluxes. Differently from previous implementations, it does not require any modulation of electrostatic potentials. We find a peculiar dependence of the pumped charge on the superconducting phase bias across the pump and that an arbitrarily large amount of charge can be pumped in a single cycle when the phase bias is $\ensuremath{\pi}$. We explain these features and their relation to the adiabatic theorem.
We investigate how coherent Cooper-pair tunneling depends on the competition between Josephson coupling energy E J and charging energy E C. Measurements were performed on a circuit consisting of two dc-SQUIDs in series, with a gate capacitively coupled to the central island. As the E J/E C ratio decreased, coherent Cooper pairs tended to tunnel incoherently. Measured data show good agreement with theoretical calculations in strong and weak coupling limits. This experiment implies the manifestation of Heisenberg's uncertainty principle in a superconductor.
We present the first measurements of the Berry phase in a superconducting Cooper pair pump. A fixed amount of Berry phase is accumulated to the quantum-mechanical ground state in each adiabatic pumping cycle, which is determined by measuring the charge passing through the device. The dynamic and geometric phases are identified and measured quantitatively from their different response when pumping in opposite directions. Our observations, in particular, the dependencies of the dynamic and geometric effects on the superconducting phase bias across the pump, agree with the basic theoretical model of coherent Cooper pair pumping.
We have analyzed and measured the quantum coherent dynamics of a circuit containing two coupled superconducting charge qubits. Each qubit is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Two qubits are coupled electrostatically by a small island overlapping both Cooper pair boxes. Quantum state manipulation ofthe qubit circuit is done by applying non-adiabatic voltage pulses to the common gate. We read out each qubit by means of probe electrodes connected to Cooper pair boxes through high-Ohmic tunnel junctions. With such a setup the measured pulse-induced probe currents are proportional to the probability for each qubit to have an extra Cooper pai1r after the manipulation. As expected from theory and observed experimentally the measured pulse-induced current in each probe has two frequency components whose position on the frequency axis can be externally controlled. This is a result ofthe inter-qubit coupling which is also responsible for the avoided level crossing that we observed in the qubits' spectra. Our simulations show that in the absence of decoherence and with a rectangular pulse shape the system remains entangled most ofthe time reaching maximally entangled states at certain instances.
Quantum mechanics constrains conjugate observables from being simultaneously measurable with arbitrary precision. In a Josephson junction, these are the transferred electric charge and the quantum-mechanical phase difference between the superconducting domains. Which of them fluctuates determines the supercurrent: a dissipative trickle of single Cooper pairs in one limit, a coherent dissipationless flow in the other. Bridging both regimes in one device remained elusive because the Josephson and charging energies are fixed at fabrication. Here, we use the tunable tunnel junction of a scanning tunneling microscope at millikelvin temperature to vary their ratio continuously over many orders of magnitude. In this way, we monitor the smooth transition between incoherent and coherent Cooper pair flow in a single junction, revealing the quantum-to-classical transition in a controlled way.
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.