Some implications of superconducting quantum interference to the application of master equations in engineering quantum technologies

Abstract

In this paper we consider the modelling and simulation of open quantum systems from a device engineering perspective. We derive master equations at different levels of approximation for a Superconducting Quantum Interference Device (SQUID) ring coupled to an ohmic bath. We demonstrate that the master equations we consider produce decoherences that are qualitatively and quantitativly dependent on both the level of approximation and the ring's external flux bias. We discuss the issues raised when seeking to obtain Lindbladian dissipation and show, in this case, that the external flux (which may be considered to be a control variable in some applications) is not confined to the Hamiltonian, as often assumed in quantum control, but also appears in the Lindblad terms.

Open quantum systems, effective Hamiltonians, and device characterization

“…Here, the heuristic adding/canceling of terms becomes much more involved [30], which necessarily raises difficult issues as high-precision control of such a system inevitably requires precise device characterization [31]. The purpose of this paper is to highlight such problems, using the SQUID as an example system, and to investigate whether a measurement of its magnetic susceptibility might be capable of providing some resolution of these difficulties.…”

“…We consider the following standard model of a SQUID, with chargeQ and fluxˆ as canonically conjugate variables (so that [ˆ ,Q] = ih), coupled both inductively and capacitively to an Ohmic bath environment, modeled as an infinite set of harmonic oscillators, mode index n, with chargeQ n and fluxˆ n ([ˆ n ,Q m ] = ihδ nm ) [30]. The dimensionless total Hamiltonian H/hω 0 may be written as a sum of the Hamiltonians of the SQUID (Ĥ S ), the bath (Ĥ B ), and the coupling between them (Ĥ I ):…”

“…(1) into the dissipator integral, and evaluating the bath correlation functions (i.e., B XBX (−τ ) , etc. ), yields [9,30] …”

“…(6) survive, so that the external flux control parameter x disappears from the dissipator at first order, appearing only in the second order BCH model in that case [30]. As Eq.…”

“…H δ is routinely canceled by introducing an identical counterterm, i.e., by settingĤ LS =Ĥ δ , while the impact ofĤ XP has previously been discussed in terms of frequency shifts and squeezing [30,32], and in other contexts too [19,21,[21][22][23][24][25][26]29]; however, it has also been the source of some debate as discussed in the Introduction: arranging for a counterterm inĤ LS allows the master equation to display additional properties [21], while such cancellation does come at the cost of sacrificing translational invariance [19] and Ehrenfest [23].…”