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arXiv:cond-mat/0206507

Tunnelling-charging Hamiltonian of a Cooper pair pump at large E_J\lesssim E_C: Modified Hamiltonians and renormalisability

Matias Aunola

Published Jun 26, 2002• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0206507

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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.
The Hamiltonian operator for an unbiased array of Josephson junctions with gate voltages is constructed when only Cooper pair tunneling and charging effects are taken into account. The supercurrent through the system and the pumped current induced by changing the gate voltages periodically are discussed with an emphasis on the inaccuracies in the Cooper pair pumping. Renormalization of the Hamiltonian operator is used in order to reliably parametrize the effects due to inhomogeneity in the array and nonideal gating sequences. The relatively simple model yields an explicit, testable prediction based on three experimentally motivated and determinable parameters.
The standard phenomenological Hamiltonian of a small superconducting Josephson junction in the charge regime (Cooper Pair Box) produces a model of the effective charge qubit with possible applications to quantum information processing. In this note a new model based on the BCS Hamiltonian with individual tunneling yields an effective multi-level picture with a highly degenerated level placed between the ground state and the excited state. Unlike in the standard approach, the excited Cooper pairs play here an important role. For such a system coupled to a zero temperature bath the additional levels act as a probability sink. In contrast to the standard large-spin model the coupling to phonons can be an effective source of dissipation. This model provides also alternative explanations of various effects observed in experiments and sheds new light on the issue of Josephson junctions as macroscopic quantum systems.
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.
In this thesis transport phenomena in the arrays consisting of two (SSET) or three (CPP) or more Josephson junctions are discussed. A quantitative theory of adiabatic Cooper pair transport in phase biased arrays is developed. The theory predicts that the quantum inaccuracy of the Cooper pair pumping in arrays with a small number of junctions is very large. The effects due to inhomogeneous arrays or nonideal gating sequences are also quantitatively treated. It is also shown explicitly how the pumped charge in the Cooper pair pump can be understood as a partial derivative of Berry’s phase with respect to the phase difference φ\varphiφ across the array. This makes possible to obtain information about Berry’s phase by measuring the pumped current in the CPP. Also the decoherence time τφ\tau_{\varphi}τφ​, which is one of the central quantities in quantum computing, is quantitatively estimated in a dissipative electromagnetic environment of the circuit. This method allows comparison of the suitability of a system as a quantum bit. Also a direct measurement of τφ\tau_{\varphi}τφ​ as a crossover between coherent and incoherent pumping in the single Cooper pair pump is suggested. In experiments with Cooper pair pump it is demonstrated in practise how the CPP can be used as a turnstile with help of dissipation. Also a regular pumping experiment is performed. To explain the twofold behavior of the CPP, sometimes yielding eee-periodicity while sometimes 2e2e2e-periodicity, the energy-minimisation model is developed. It is shown also experimentally that Nb\text{Nb}Nb based junctions with high critical temperature (Tc≈8.5 KT_c \approx 8.5\text{ K}Tc​≈8.5 K) and Josephson coupling can be fabricated using the regular self-alignment technique. The measured Al/Nb/Al\text{Al/Nb/Al}Al/Nb/Al SSETs show a clear signature of resonant tunnelling of Cooper pairs combined with elastic cotunnelling of quasiparticles, q-MQTq\text{-MQT}q-MQT, through the barrier of ΔNb\Delta_{\text{Nb}}ΔNb​.
We present a detailed theoretical study of adiabatic Cooper-pair pumping in a one-dimensional array of Cooper-pair boxes and discuss its experimental feasibility. The Josephson tunneling and the gate voltage are two adiabatic couplings of our system. Using the concept of Berry phase and Abelian bosonization, we show that the evolution of the ground state follows a closed loop in the flux-voltage plane. Its adiabatic motion yields the Cooper-pair pumping in the system. We find the condition for blocking the Cooper-pair pumping even in the perfect pumping condition.
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.