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Modern Physics Letters B

GEOMETRIC PHASE OF COMPOSITE SYSTEMS IN DEPHASING

Xin Li

Published Apr 20, 2010• 0 citations
Publisher

Abstract

Based on an exactly-solvable spin-boson model, the geometric phase (GP) of a system of two qubits collectively coupled to a quantum bath is investigated. In this scenario, the decomposition of GP into the sum of GP of the subsystem should be forbidden. We demonstrate that the environment-induced modifications of the GP usually include not only the pure decoherence effect, but also the information about the environment evolution. We also show the dependence of the GP upon the entangled initial state and extend it to the case of multi-qubits system.

DOI
10.1142/s0217984910023050

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Based on an exactly-solvable spin-boson model, the geometric phase (GP) of a system of two qubits collectively coupled to a quantum bath is investigated. In this scenario, the decomposition of GP into the sum of GP of the subsystem should be forbidden. We demonstrate that the environment-induced modifications of the GP usually include not only the pure decoherence effect, but also the information about the environment evolution. We also show the dependence of the GP upon the entangled initial state and extend it to the case of multi-qubits system.
Decoherence of qubits is one main limitation in quantum information processing, where the qubits themselves can be mixed because of decoherence and additional decoherence may generated through interactions with a bath. Here, we present a complete study of qubit-environment entanglement in the case of interactions that lead to pure dephasing of the qubit by allowing the initial state of the qubit to be mixed, exceeding previous restrictions. We find that the criteria for separability that hold for pure initial states of the qubit are reduced to witness status. Yet, the evolution of the witnesses reflects the array of features that entanglement can exhibit in the mixed case but are forbidden in the pure-qubit case. Most notably, this includes sudden birth of entanglement as the initial phase of the evolution is typically separable. Moreover, there is a stark numerical advantage to the study of entanglement for large environments when using the witness rather than the traditional measures available. We include the study of entanglement generated between a spin qubit and a sparse spin environment to illustrate the peculiarities of the evolution of entanglement in this case, as well as to emphasize the level of reliability of the entanglement witness.
We compute the geometric phase for a spin-1/2 particle under the presence of a composite environment, composed of an external bath (modeled by an infinite set of harmonic oscillators) and another spin-1/2 particle. We consider both cases: an initial entanglement between the spin-1/2 particles and an initial product state in order to see if the initial entanglement has an enhancement effect on the geometric phase of one of the spins. We follow the nonunitary evolution of the reduced density matrix and evaluate the geometric phase for a single two-level system. We also show that the initial entanglement enhances the sturdiness of the geometric phase under the presence of an external composite environment.
Geometric phases, arising from cyclic evolutions in a curved parameter space, appear in a wealth of physical settings. Recently, and largely motivated by the need of an experimentally realistic definition for quantum computing applications, the quantum geometric phase was generalized to open systems. The definition takes a kinematical approach, with an initial state that is evolved cyclically but coupled to an environment--leading to a correction of the geometric phase with respect to the uncoupled case. We obtain this correction by measuring the nonunitary evolution of the reduced density matrix of a spin one-half coupled to an environment. In particular we are interested in baths near a quantum phase transition, which are known to induce strong decoherence. The experiments are done with a NMR quantum simulator, where we emulate qualitatively the influence of a critical environment using a simple one-qubit model.
We explore the dynamics of spin-boson model for the Ohmic bath by employing the master equation approach and obtain an explicit expression of reduced density matrix. We also calculate the geometric phases of the spin-boson model by making use of the analytical results and discuss how the dissipative bosonic environment affects geometric phases. Furthermore, we investigate the dynamics of quantum discord and entanglement of two qubits each locally interacting with its own independent bosonic environments. It is found that the decay properties of quantum discord and entanglement are sensitive to the choice of initial state's parameter and coupling strength between system and bath.
The geometric phase of a qubit coupled dephasingly to ohmic or non-ohmic bosonic environment exhibits a strong dependence on both an initial state of the system and spectral properties of the environment. Analysing the exact reduced dynamics we show that the geometric phase can be more easily maintained for the subohmic and ohmic environments than for the superohmic environment. We show the existence of the initial states for which the geometric phase is robust against coupling to the environment.