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arXiv:0710.5623 · Physical Review Letters

Experimental Determination of the Berry Phase in a Superconducting Charge Pump

Mikko Möttönen, Juha J. Vartiainen, Jukka P Pekola

Published Apr 28, 2008• 117 citations
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Abstract

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.

DOI
10.1103/physrevlett.100.177201

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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.
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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 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 ${Q}_{P}\ensuremath{\sim}2e,$ 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 QP,{Q}_{P},QP​, 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 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 have studied dynamic properties of a special Josephson charge qubit. The device consists of a single-Cooper-pair transistor (with capacitive gate) closed by a superconducting inductive loop. The phase bias is implemented by an external flux that threads the loop. Restricting ourselves to a two-band model, we have derived analytical expressions for the oppositely circulating ring currents in terms of gate charge and phase bias. They permit the determination of important device characteristics such as critical current, loop parameters, and Josephson inductance. For the purpose of measurements in connection with macroscopic quantum coherence, we investigate the qubit's microwave response within the frame of Bloch equations with phenomenological relaxation rates. We find the (nonlinear) qubit impedance in terms of quality, detuning, and the quantum resistance unit, which reflects the competition between irradiation and dephasing.
In adiabatic Cooper pair pumps, which are operated by means of gate voltage modulation only, the quantization of the pumped charge during a cycle is limited due to the quantum coherence of the macroscopic superconducting wave function. In this work, we show that it is possible to obtain very accurate pumps in the nonadiabatic regime by a suitable choice of the shape of the gate voltage pulses. We determine the shape of these pulses by applying quantum optimal control theory to this problem. In the optimal case, the error, with respect to the quantized value, can be of the order of ${10}^{\ensuremath{-}6}e$, i.e., reduced by up to 5 orders of magnitude with respect to the adiabatic pumping. In order to test the experimental feasibility of this approach, we consider the effect of charge noise and the deformations of the optimal pulse shapes on the accuracy of the pump. Charge noise is assumed to be induced by random background charges in the substrate, which is responsible for the observed 1/f1/f1/f noise. Inaccuracies in the pulse shaping are described by assuming a finite bandwidth for the pulse generator. In realistic cases, the error increases to 1 order of magnitude at most as compared to that of the optimal case. Our results are promising for the realization of accurate and fast superconducting pumps.
The properties of the tunnelling-charging Hamiltonian of a Cooper pair pump are well understood in the regime of weak and intermediate Josephson coupling, i.e. E_J\lesssim E_C. Instead of perturbative treatment of charging effects, the present work applies the charge state representation in the the strong coupling case. From the discrete Hamiltonian we construct effective, truncated PDE Hamiltonians which yield approximate, but analytical, wave functions and energies for the ground state. The validity of the expressions is confirmed by comparing the results against numerically obtained eigenstates and eigenenergies, whenever possible. For uniform arrays, the results converge rapidly and even ϕ-dependence of the wave function is rather well described. In the inhomogeneous case we find the Hamiltonian to be parametrically renormalisable in order to obtain an accurate ground state energy. A method for for finding an inhomogeneous trial wave function is explained. As a passing note, we briefly examine the intertwined connection between the pumped charge and the Berry's phase. As addendum, we have explicitly validated the ground state ansatz for ϕ=0 when N<43.