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The Berry phase that stands alone: non-adiabaticity suppressed by an environment

Robert S. Whitney

Published Jun 30, 2008• 0 citations
OpenAlex

Abstract

We consider a two-level system coupled to a highly non-Markovian environment when the coupling axis rotates with time. The environment may be quantum (for example a bosonic bath or a spin bath) or classical (such as classical noise). We show that an Anderson orthogonality catastrophe suppresses transitions, so that the system's instantaneous eigenstates (parallel and anti-parallel to the coupling axis) can adiabatically follow the rotation. These states thereby acquire Berry phases; geometric phases given by the area enclosed by the coupling axis. Unlike in earlier proposals for environment-induced Berry phases, here there is little decoherence, so one does not need a decoherence-free subspace. Indeed we show that this Berry phase should be much easier to observe than a conventional one, because it is not masked by either the dynamic phase or the leading non-adiabatic phase. The effects that we discuss should be observable in any qubit device where one can drive three parameters in the Hamiltonian with strong man-made noise.

Related papers

Environmental suppression of non-adiabatic phases: easier observation of Berry phases

Robert S. Whitney

We consider a two-level system coupled to a highly non-Markovian environment when the coupling axis rotates with time. The environment may be quantum (for example a bosonic bath or a spin bath) or classical (such as classical noise). We show that an Anderson orthogonality catastrophe suppresses transitions, so that the system's instantaneous eigenstates (parallel and anti-parallel to the coupling axis) can adiabatically follow the rotation. These states thereby acquire Berry phases; geometric phases given by the area enclosed by the coupling axis. Unlike in earlier proposals for environment-induced Berry phases, here there is little decoherence, so one does not need a decoherence-free subspace. Indeed we show that this Berry phase should be much easier to observe than a conventional one, because it is not masked by either the dynamic phase or the leading non-adiabatic phase. The effects that we discuss should be observable in any qubit device where one can drive three parameters in the Hamiltonian with strong man-made noise.

•Jun 30, 2008•0 citations

Suppression of nonadiabatic phases by a non-Markovian environment: Easier observation of Berry phases

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We consider a two-level system coupled to a highly non-Markovian environment when the coupling axis rotates with time. The environment may be quantum (for example a bosonic bath or a spin bath) or classical (such as classical noise). We show that an Anderson orthogonality catastrophe suppresses transitions, so that the system's instantaneous eigenstates (parallel and antiparallel to the coupling axis) can adiabatically follow the rotation. These states thereby acquire Berry phases; geometric phases given by the area enclosed by the coupling axis. Unlike in earlier proposals for environment-induced Berry phases, here there is little decoherence, so one does not need a decoherence-free subspace. Indeed we show that this Berry phase should be much easier to observe than a conventional one because it is not masked by either the dynamic phase or the leading nonadiabatic phase. The effects that we discuss should be observable in any qubit device where one can drive three parameters in the Hamiltonian with strong man-made noise.

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We investigate the decoherence effect of a bosonic bath on the Berry phase of a spin- in a time-dependent magnetic field, without making the Markovian approximation. A two-cycle process resulting in a pure Berry phase is considered. The low-frequency quantum noise significantly affects the Berry phase. In the adiabatic limit, the high-frequency quantum noise only has a small effect. The result is also valid in some more general situations.

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Does Berry phase exist for a system coupled to its environment?

Robert S. Whitney, Yuval Gefen

Berry phase was originally defined for systems whose states are separated by finite energy gaps. One might naively expect that a system without a gap cannot have a Berry phase. Despite this we ask whether a Berry phase can be observed in a system which has a continuous spectrum because its coupling to the environment has broadened its energy levels. We find that, contrary to the above naive expectation, there are conditions under which the Berry phase is observable. However it is modified by the presence of the environment and no longer has a simple geometric interpretation. The model system we consider is a spin-half in a slowly rotating magnetic field, with the spin also coupled to a Ohmic environment of harmonic oscillators (spin-boson model). Here we discuss the high-temperature limit of this model. We then interpret our results in terms of a spin under the influence of a classical stochastic field.

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Berry's phase in the two-level model

A.C. Aguiar Pinto, M. Moutinho, M. T. Thomaz

We study the adiabatic evolution of a two-level model in the presence of an external classical electric field. The coupling between the quantum model and the classical field is taken in the electric dipole approximation. In this regime, we show the absence of geometric phases in the interacting two-level model in the presence of any periodic real time-dependent classical electric field. We obtain a conservative scalar potential in the calculation of Berry's phases of the instantaneous eigenstates of the model. For complex electric fields, we recover the existence of geometric phases. In particular, the geometric phases of the instantaneous eigenstates of the model in the presence of a positive or of a negative frequency component of the monochromatic electric field differ by an overall sign. As a check on our results, we map this interacting two-level model onto a spin-1/2 model under the action of a classical magnetic field. We confirm that the first one acquires Berry's phase only in the rotating wave approximation [RWA].

arXiv:0904.1774•Brazilian Journal of Physics•Jun 1, 2009•5 citations
We consider a two-level system coupled to a highly non-Markovian environment when the coupling axis rotates with time. The environment may be quantum (for example a bosonic bath or a spin bath) or classical (such as classical noise). We show that an Anderson orthogonality catastrophe suppresses transitions, so that the system's instantaneous eigenstates (parallel and anti-parallel to the coupling axis) can adiabatically follow the rotation. These states thereby acquire Berry phases; geometric phases given by the area enclosed by the coupling axis. Unlike in earlier proposals for environment-induced Berry phases, here there is little decoherence, so one does not need a decoherence-free subspace. Indeed we show that this Berry phase should be much easier to observe than a conventional one, because it is not masked by either the dynamic phase or the leading non-adiabatic phase. The effects that we discuss should be observable in any qubit device where one can drive three parameters in the Hamiltonian with strong man-made noise.
We consider a two-level system coupled to a highly non-Markovian environment when the coupling axis rotates with time. The environment may be quantum (for example a bosonic bath or a spin bath) or classical (such as classical noise). We show that an Anderson orthogonality catastrophe suppresses transitions, so that the system's instantaneous eigenstates (parallel and anti-parallel to the coupling axis) can adiabatically follow the rotation. These states thereby acquire Berry phases; geometric phases given by the area enclosed by the coupling axis. Unlike in earlier proposals for environment-induced Berry phases, here there is little decoherence, so one does not need a decoherence-free subspace. Indeed we show that this Berry phase should be much easier to observe than a conventional one, because it is not masked by either the dynamic phase or the leading non-adiabatic phase. The effects that we discuss should be observable in any qubit device where one can drive three parameters in the Hamiltonian with strong man-made noise.
We consider a two-level system coupled to a highly non-Markovian environment when the coupling axis rotates with time. The environment may be quantum (for example a bosonic bath or a spin bath) or classical (such as classical noise). We show that an Anderson orthogonality catastrophe suppresses transitions, so that the system's instantaneous eigenstates (parallel and antiparallel to the coupling axis) can adiabatically follow the rotation. These states thereby acquire Berry phases; geometric phases given by the area enclosed by the coupling axis. Unlike in earlier proposals for environment-induced Berry phases, here there is little decoherence, so one does not need a decoherence-free subspace. Indeed we show that this Berry phase should be much easier to observe than a conventional one because it is not masked by either the dynamic phase or the leading nonadiabatic phase. The effects that we discuss should be observable in any qubit device where one can drive three parameters in the Hamiltonian with strong man-made noise.
Berry phase was originally defined for systems whose states are separated by finite energy gaps. One might naively expect that a system without a gap cannot have a Berry phase. Despite this we ask whether a Berry phase can be observed in a system which has a continuous spectrum because its coupling to the environment has broadened its energy levels. We find that, contrary to the above naive expectation, there are conditions under which the Berry phase is observable. However it is modified by the presence of the environment and no longer has a simple geometric interpretation. The model system we consider is a spin-half in a slowly rotating magnetic field, with the spin also coupled to a Ohmic environment of harmonic oscillators (spin-boson model). Here we discuss the high-temperature limit of this model. We then interpret our results in terms of a spin under the influence of a classical stochastic field.
We study the adiabatic evolution of a two-level model in the presence of an external classical electric field. The coupling between the quantum model and the classical field is taken in the electric dipole approximation. In this regime, we show the absence of geometric phases in the interacting two-level model in the presence of any periodic real time-dependent classical electric field. We obtain a conservative scalar potential in the calculation of Berry's phases of the instantaneous eigenstates of the model. For complex electric fields, we recover the existence of geometric phases. In particular, the geometric phases of the instantaneous eigenstates of the model in the presence of a positive or of a negative frequency component of the monochromatic electric field differ by an overall sign. As a check on our results, we map this interacting two-level model onto a spin-1/2 model under the action of a classical magnetic field. We confirm that the first one acquires Berry's phase only in the rotating wave approximation [RWA].