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arXiv:1007.2347 · Physical Review B

Decoherence of adiabatically steered quantum systems

Paolo Solinas, Mikko Möttönen, Juha Salmilehto, Jukka P Pekola

Published Oct 12, 2010• 41 citations
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Abstract

We study the effect of Markovian environmental noise on the dynamics of a two-level quantum system which is steered adiabatically by an external driving field. We express the master equation taking consistently into account all the contributions to the lowest nonvanishing order in the coupling to the Markovian environment. We study the master equation numerically and analytically and we find that, in the adiabatic limit, a zero-temperature environment does not affect the ground-state evolution. As a physical application, we discuss extensively how the environment affects Cooper-pair pumping. The adiabatic ground-state pumping appears to be robust against environmental noise. In fact, the relaxation due to the environment is required to avoid the accumulation of small errors from each pumping cycle. We show that neglecting the nonsecular terms in the master equation leads to unphysical results, such as charge nonconservation. We discuss also a possible way to control the environmental noise in a realistic physical setup and its influence on the pumping process.

DOI
10.1103/physrevb.82.134517

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We study the effect of Markovian environmental noise on the dynamics of a two-level quantum system which is steered adiabatically by an external driving field. We express the master equation taking consistently into account all the contributions to the lowest nonvanishing order in the coupling to the Markovian environment. We study the master equation numerically and analytically and we find that, in the adiabatic limit, a zero-temperature environment does not affect the ground-state evolution. As a physical application, we discuss extensively how the environment affects Cooper-pair pumping. The adiabatic ground-state pumping appears to be robust against environmental noise. In fact, the relaxation due to the environment is required to avoid the accumulation of small errors from each pumping cycle. We show that neglecting the nonsecular terms in the master equation leads to unphysical results, such as charge nonconservation. We discuss also a possible way to control the environmental noise in a realistic physical setup and its influence on the pumping process.
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.
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.
An irreversible time evolution of a quantum system interacting with a surrounding environment is described by a quantum master equation. When an external field is applied to a quantum system, a non-Markovian master equation is derived in a rigorous way, where the relaxation terms in the quantum master equation include the effect of the external field that is ignored in the conventional treatment. It is shown that when the external field is a sequence of phase-modulation π-pulses, the decoherence of the quantum system can be suppressed. In particular, when the pulse separation is sufficiently short in comparison with the correlation time of the environmental system, dynamical decoupling takes place. To see the effect of the phase-modulation π-pulses, the time evolution of qubit entanglement is investigated in detail.
In the utilization of quantum information, it is essential to control the quantum register accurately. A prominent solution is to utilize adiabatic steering which is typically considered robust against certain types of error. A recent study introduced a way, in which the steering and noise can be accounted for in a consistent manner using the Markovian approximation. This thesis introduces a new way to derive the full first order master equation for adiabatic steering in the ground state. The method can potentially he used to derive analytically higher-order master equations in a simpler manner than the originally proposed method. We introduce two means of deriving higher-order bases to he used in conjunction with our master equation. We show how the phases of the basis states can he selected in an optimal manner. A Cooper pair sluice is introduced as a model system for the utilization of our master equation. It can he used for Cooper pair pumping intimately related to the Berry phase accumulated during the adiabatic evolution. We perform numerical simulations on the sluice using both the already utilized adiabatic basis and the novel first super-adiabatic basis. The previous analytical result of zero-temperature Markovian environment not affecting the system dynamics in the adiabatic limit is confirmed. We show the effect of optimal phase selection in the adiabatic basis and compare our results with previous literature. The break-down of the adiabatic condition is examined. Using high-order bases in the simulation is shown to reduce significantly non-physical phenomena stemming from the lack of complete positivity of our master equation. Unlike in the case of the adiabatic basis, increasing the environmental coupling strength using the super-adiabatic basis does not restore ground state pumping far from the adiabatic limit, as expected.
We introduce an alternative way to derive the generalized form of the master equation recently presented by J. P. Pekola et al. [Phys. Rev. Lett. 105, 030401 (2010)] for an adiabatically steered two-level quantum system interacting with a Markovian environment. The original derivation employed the effective Hamiltonian in the adiabatic basis with the standard interaction picture approach but without the usual secular approximation. Our approach is based on utilizing a master equation for a nonsteered system in the first superadiabatic basis. It is potentially efficient in obtaining higher-order equations. Furthermore, we show how to select the phases of the adiabatic eigenstates to minimize the local adiabatic parameter and how this selection leads to states which are invariant under a local gauge change. We also discuss the effects of the adiabatic noncyclic geometric phase on the master equation.