Extension to the quantum Langevin equation in the incoherent hopping regime

Abstract

An extension to the quantum Langevin equation is derived, that is valid in the incoherent hopping regime, and which allows one to incorporate quantum tunneling events. This is achieved by the inclusion of additional stochastic variables in the Langevin equation representing the tunneling events. A systematic derivation of this extension and of its regime of validity is presented. The study is motivated by efforts to determine the error in reading the state of a super-conducting quantum bit. PACS numbers: 74.40.+k, 85.25.-j,05.40.-a, 03.65.DbThe quantum Langevin equation[1, 2] provides a physically appealing and numerically simple description of the dynamics of a system coupled to a heat bath. It describes the system's non-linear semiclasical dynamics and the effects of coupling to quantum and thermal fluctuations in the heat bath. It does not, however, describe quantum tunneling of the system. Instanton approaches [3,4,5] provide a powerful framework in which to describe these latter effects. Here, we synthesize these two methods within an extended quantum Langevin equation that includes quantum tunneling. We focus upon two effects; quantum tunneling through or reflection from an energy barrier. These may be incorporated into the Langevin equation by the inclusion of additional stochastic variables, representing the tunneling and reflection events. This is valid in the regime of incoherent hopping.

Anisotropic Landau-Lifshitz-Gilbert models of dissipation in qubits

“…In this appendix we derive the dynamical equation of a single spin coupled to a Drude bath and show that in an appropriate limit the long timescale θ dynamics are given by the Markovian equation ( 16) with φ remaining typically close to φ ≈ 0. Using the Drude density of states and evaluating eqs ( 9) and (10), in the…”

“…The dissipative dynamics of a two level system has been studied extensively 3 . In this work, we use a Keldysh path integral [8][9][10] to derive a Langevin 11,12 description of the dynamics of a flux qubit accounting for the anisotropic coupling to the environment. This extends the previous work on dissipative spin models [13][14][15][16][17][18][19][20][21][22] .…”