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arXiv:1012.5184 · Physical Review B

Floquet theory of Cooper pair pumping

Angelo Russomanno, Stefano Pugnetti, Valentina Brosco, Rosario Fazio

Published Jun 6, 2011• 43 citations
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Abstract

We derive a general formula for the charge pumped in a superconducting nanocircuit. Our expression generalizes previous results in several ways; it is applicable in both the adiabatic and in the nonadiabatic regimes and it takes into account also the effect of an external environment. More specifically, by applying Floquet theory to Cooper pair pumping, we show that under a cyclic evolution the total charge transferred through the circuit is proportional to the derivative of the associated Floquet quasi-energy with respect to the superconducting phase difference. In the presence of an external environment the expression for the transferred charge acquires a transparent form in the Floquet representation. It is given by the weighted sum of the charge transferred in each Floquet state, the weights being the diagonal components of the stationary density matrix of the system expressed in the Floquet basis. To test the power of this formulation we apply it to the study of pumping in a Cooper pair sluice. We reproduce the known results in the adiabatic regime and we show new data in the nonadiabatic case.

DOI
10.1103/physrevb.83.214508

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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 ${Q}_{P}\ensuremath{\sim}2e,$ 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 study the effect of flux noise on the Cooper pair current of a superconducting charge pump. We generalize the definition of the current in order to take into account the contribution induced by the environment. It turns out that this dissipative current vanishes for charge noise but it is finite in general for noise operators that do not commute with the charge operator. We discuss in a generic framework the effect of flux noise and present a way to engineer it by coupling the system to an additional external circuit. We calculate numerically the pumped charge through the device by solving the master equation for the reduced density matrix of the system and show how it depends on the coupling to the artificial environment.
For adiabatically and periodically manipulated dissipative quantum systems we derive, using Floquet theory, a simple Markovian master equation. Contrary to some previous works we explicitly take into account the time dependence of the Hamiltonian and, therefore, obtain a master equation with a time-dependent dissipative part. We illustrate our theory with two examples and compare our results with the previously proposed master equations. In particular, we consider the problem of Cooper pair pumping and demonstrate the inadequacy of the secular (rotating wave) approximation when calculating the pumped charge. The secular approximation producing a master equation of the Lindblad type approximates well the quantum state (density matrix) of the system, while to determine the pumped charge a non-Lindblad master equation beyond the rotating wave approximation is necessary.
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.
We introduce a model for a flux-assisted Cooper-pair pump, the sluice, which is used to study geometric charge transport. Our model allows for a nonvanishing loop inductance going beyond the usual treatment with an exact phase bias. We derive the device Hamiltonian and current operators for different elements of the system and calculate the pumped charge carried by the ground state in the adiabatic limit. We show that extending the model beyond the exact phase bias has a weak but potentially non-negligible effect on the charge transport. This effect is observed to depend on the external flux bias. The adiabatic energy level and eigenstate structures are studied and the unusual features are explained. Finally, we discuss the requirements of adiabaticity.