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Korkeamman kertaluvun approksimaatiot dekoherenssille ohjatussa kvanttikehityksessä

Salmilehto, Juha

Published Jan 1, 2010• 0 citations

Abstract

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.

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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.

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.

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.

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.

We derive a Markovian master equation that models the evolution of systems subject to driving and control fields. Our approach combines time rescaling and weak-coupling limits for the system-environment interaction with a secular approximation. The derivation makes use of the adiabatic time-evolution operator in a manner that allows for the efficient description of strong driving, while recovering the well-known adiabatic master equation in the appropriate limit. To illustrate the effectiveness of our approach, firstly we apply it to the paradigmatic case of a two-level (qubit) system subject to a form of periodic driving that remains unsolvable using a Floquet representation and lastly we extend this scenario to the situation of two interacting qubits, the first driven while the second one directly in contact with the environment. We demonstrate the reliability and broad scope of our approach by benchmarking the solutions of the derived reduced time evolution against numerically exact simulations using tensor networks. Our results provide rigorous conditions that must be satisfied by phenomenological master equations for driven systems that do not rely on first-principles derivations.

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.