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arXiv:0911.3750 · Physical Review Letters

Decoherence in Adiabatic Quantum Evolution: Application to Cooper Pair Pumping

Jukka P Pekola, Valentina Brosco, Mikko Möttönen, Paolo Solinas, Alexander Shnirman

Published Jul 12, 2010• 68 citations
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Abstract

One of the challenges of adiabatic control theory is the proper inclusion of the effects of dissipation. Here we study the adiabatic dynamics of an open two-level quantum system deriving a generalized master equation to consistently account for the combined action of the driving and dissipation. We demonstrate that in the zero-temperature limit the ground state dynamics is not affected by environment. As an example, we apply our theory to Cooper pair pumping, which demonstrates the robustness of ground state adiabatic evolution.

DOI
10.1103/physrevlett.105.030401

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One of the challenges of adiabatic control theory is the proper inclusion of the effects of dissipation. Here we study the adiabatic dynamics of an open two-level quantum system deriving a generalized master equation to consistently account for the combined action of the driving and dissipation. We demonstrate that in the zero-temperature limit the ground state dynamics is not affected by environment. As an example, we apply our theory to Cooper pair pumping, which demonstrates the robustness of ground state adiabatic evolution.
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Accurate control of a quantum system is complicated to achieve partly due to the system being coupled to its surrounding environment. The coupling induces dissipation and decoherence not only destroying the coherent quantum state but making the application of control unpredictable. When the control results from manipulation of external fields, it is usually referred to as driving and its joint effect with decoherence constitutes an active field of study in reduced-density-operator theory. Recently, this field has been pushed forward by its necessity in simulating Cooper pair pumping where the geometric nature of quantum evolution allows for controlled transport of charge carriers in superconducting circuits. Such circuits themselves are under constant investigation as they grant access to fundamental quantum phenomena and facilitate promising applications such as those in quantum information processing.\nIn this dissertation, driven quantum systems under decoherence are investigated. Extensions and improvements to a recent master equation for nearly adiabatic driving are presented and analyzed. The emergent properties in quantum evolution are studied both analytically and numerically. Related to the findings, a general conservation law of operator current is derived and used to explain observed nonconservation in previous theoretical studies. The derived theory is applied to Cooper-pair pumping and shown to lead to important properties such as superadiabatic ground-state pumping. A recent pumping experiment is simulated and the breakdown point for ideal pumping is found with feasible physical parameters. A study of charge transport in the presence of flux noise is presented leading to detectable dissipative currents and the typical description of the device used for pumping is extended to include a nonvanishing loop inductance.\nA novel approach to implement control of quantum systems is proposed based on constructing a tunable coupling to an artificial environment using either a coplanar waveguide cavity or coupled quantum LC resonators. Tunability allowing for both efficient initialization and protected evolution is theoretically demonstrated. Finally, a general framework for quantum driving is constructed without the typical assumption of a classical driving force leading to peculiar results.\nThis dissertation presents original research on both modeling the control of open quantum systems as well as the realization of such control. The work simulates physical phenomena in superconducting circuits and makes predictions for future experiments. In addition, it introduces novel theoretical tools and approaches that advance the state of the art.
We introduce a method where successive coordinate transformations are applied to decrease the error in the adiabatic master equation resulting from truncation in the local adiabatic parameter. Our method reduces the nonphysical behavior stemming from the lack of complete positivity. The strong environment-induced relaxation at high Cooper pair pumping frequencies leads to adiabatic ground-state pumping only in the lowest-order approximation. We illustrate the robustness of the frequency where the adiabaticity breaks down using the high-order theory and show the emergence of an optimal environmental coupling strength, for which ideal pumping is preserved for the highest frequency. Finally, we study the effect of quantum interference on the pumped current and give an estimate for the relaxation rate of an experimentally measured system.
The adiabatic theorem and shortcuts to adiabaticity for time-dependent open quantum systems are explored in this paper. Starting from the definition of dynamical stable decoherence-free subspace, we show that, under a compact adiabatic condition, the quantum state remains in the time-dependent decoherence-free subspace with an extremely high purity, even though the dynamics of the open quantum system may not be adiabatic. The adiabatic condition mentioned here in the adiabatic theorem for open systems is very similar to that for closed quantum systems, except that the operators required to change slowly are the Lindblad operators. We also show that the adiabatic evolution of decoherence-free subspaces depends on the existence of instantaneous decoherence-free subspaces, which requires that the Hamiltonian of open quantum systems be engineered according to the incoherent control protocol. In addition, shortcuts to adiabaticity for adiabatic decoherence-free subspaces are also presented based on the transitionless quantum driving method. Finally, we provide an example that consists of a two-level system coupled to a broadband squeezed vacuum field to show our theory. Our approach employs Markovian master equations and the theory can apply to finite-dimensional quantum open systems.
We demonstrate controlled pumping of Cooper pairs down to the level of a single pair per cycle, using an rf-driven Cooper-pair sluice. We also investigate the breakdown of the adiabatic dynamics in two different ways. By transferring many Cooper pairs at a time, we observe a crossover between pure Cooper-pair and mixed Cooper-pair-quasiparticle transport. By tuning the Josephson coupling that governs Cooper-pair tunneling, we characterize Landau-Zener transitions in our device. Our data are quantitatively accounted for by a simple model including decoherence effects.
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.