Geometric phase in dephasing systems

Abstract

Beyond the quantum Markov approximation, we calculate the geometric phase of a two-level system driven by a quantized magnetic field subject to phase dephasing. The phase reduces to the standard geometric phase in the weak coupling limit and it involves the phase information of the environment in general. In contrast with the geometric phase in dissipative systems, the geometric phase acquired by the system can be observed on a long time scale. We also show that with the system decohering to its pointer states, the geometric phase factor tends to a sum over the phase factors pertaining to the pointer states.PACS numbers: 03.65. Vf, 03.65.Yz Quantum mechanics tell us that physical states are equivalent up to a global phase, which in general does not contain useful information about the described system and thus can be ignored. This is not the case, however, for a system transported round a circuit by varying the parameters s = (s 1 , s 2 , ...) in its Hamiltonian H( s). As Berry showed [1], the phase can have a component of geometric origin called geometric phase with important observable consequences, such as the Aharonov-Bohm effect [2] and the spin-1 2 particle driven by a rotating magnetic field [1]. The geometric phases that only depend on the path followed by the system during its evolution, have been investigated and tested in a variety of settings and have been generalized in several directions [3]. The geometric phases are attractive both from a theoretical perspective, and from the point of view of possible applications, among which geometric quantum computation [4,5,6,7] is one of the most importance.As realistic systems always interact with their environment, the study on the geometric phase in open systems become interesting. Garrison and Wright [8] were the first to touch on this issue by describing open system evolution in terms of a non-Hermitian Hamiltonian. This is a pure state analysis, so it did not address the problem of geometric phases for mixed states. Toward the geometric phase for mixed states in open systems, the approaches used involve solving the master equation of the system [9,10,11,12,13], employing a quantum trajectory analysis [14,15] or Krauss operators [16], and the perturbative expansions [17,18]. Some interesting results were achieved, briefly summarized as follows: nonhermitian Hamiltonian lead to a modification of Berry's phase [8,17], stochastically evolving magnetic fields produce both energy shift and broadening [18], phenomenological weakly dissipative Liouvillians alter Berry's phase by introducing an imaginary correction [11] or lead to damping and mixing of the density matrix elements [12]. However, almost all these studies are performed for dissipative systems, and thus the representations are applicable for systems whose energy is not conserved. For open systems with conserved energy (dephasing systems), the problem beyond the Markov approximation remains untouched to our best knowledge. Because the systemenvironment interaction H I and the free system Hamilton...

Geometric phase as a determinant of a qubit– environment coupling

“…for the ground state | − 1 ). This finding corroborates the results for the phase in the exactly solvable model of pure dephasing with arbitrary (not only weak) coupling [49][50][51][52]. For the presentation as in Fig.…”

Non-Markovian dynamics of the mixed-state geometric phase of dissipative qubits

“…Later it is proved to be merely a result of RWA and disappears in Rabi model [27], where RWA is not performed. JC model and Rabi model all suppose that the electromagnetic field inside the cavity is monochromatic (single mode), while in reality the imperfection of cavity mirrors will broaden the spectral line, which serves as a bosonic environment (bath), and there are already some works on how geometric phase is affected by dephasing and dissipative environment [28,29].…”

Geometric phase of a qubit interacting with a squeezed-thermal bath

“…In this paper we make use of the method of Tong et al [10] to study the GP of a qubit (a two-level quantum system) interacting with different kinds of system-bath (environment) interactions, one in which there is no energy exchange between the system and its environment, i.e., a quantum non-demolition (QND) interaction and one in which dissipation takes place [24,25]. Throughout, we assume the bath to start in a squeezed thermal initial state, i.e., we deal with a squeezed thermal bath.…”