Non-Markovian open quantum systems: Input-output fields, memory, and monitoring

Abstract

Principles of monitoring non-Markovian open quantum systems are analyzed. We use the field representation of the environment (Gardiner and Collet, 1985) for the separation of its memory and detector part, respectively. We claim the system-plus-memory compound becomes Markovian, the detector part is tractable by standard Markovian monitoring. Because of non-Markovianity, only the mixed state of the system can be predicted, the pure state of the system can be retrodicted. We present the corresponding non-Markovian stochastic Schrödinger equation.PACS numbers: 03.65. Yz, 42.50.Lc In a seminal paper [1] Gardiner and Collett used quantum white-noise and the related Markovian quantum field to represent the dynamics of a quantum oscillator bath in the Markovian (memory-less) limit. This allowed the construction of exact stochastic differential equations to describe the influence of the bath B on the embedded (i.e.: open) quantum system S, the reaction of S on B, and the time-continuous monitoring of S. The theory became standard in quantum optics [2] and in many fields where a quantum system is open to natural or designed environmental influence [3]. If the memory of B cannot be ignored for S then Markovian tools become jeopardized. In non-Markovian (NM) case, S is coherently interacting with a finite part of B over a finite time. From different theoretical efforts [4-10] we distill a central question: how can we divide the environment B into the memory M and detector D? Part M is continuously entangled with S but the compound S+M becomes a Markovian open system, as we shall argue. Part D contains information on S and can be continuously disentangled, i.e.: monitored, without changing the dynamics of S.As a matter of fact, the Markovian field representation [1] of B is capable to account for memory effects and leads to a natural separation between M and D. The local Markov field interacts with S in a finite range: this part makes the memory M. The output field carries away information on S, it makes the detector D. Markovian field, non-Markovian coupling. The composite S+B dynamics is based on the total HamiltonianwhereĤ S is the Hamiltonian of S, the bath Hamiltonian isĤ B = ωb † ωbω dω andĤ SB = iŝ κ ωb † ω dω+h.c. is their interaction, whereŝ is a S-operator that couples to the Bmodes. Hereb ω are boson annihilation operators for the ω-frequency modes of B, satisfying. B can be called Markovian because of the flat spectrum. Memory effects are fully encoded in the coupling κ ω . If the coupling is frequency-independent, κ ω = const, then S is Markovian open system, otherwise it has a memory. We are interested in the latter case, i.e., in NM open systems. We assume that S and B are initially uncorrelated. Let, for simplicity's, the initial B-state be the vacuum |0 defined byb ω |0 = 0 for all ω.We switch for an abstract field representation [1-3]. The bath fieldb(z) is defined bŷwhere z is a real 1-dimensional spatial coordinate. For convenience, we set the velocity of propagation to 1. The canonical commutation r...

Non-Markovian quantum input-output networks

“…It can be easily verified that Eq. (16) reduces to the quantum stochastic differential equation (8) and the output equation (17) reduces to Eq. (10).…”

“…Quantum input-output theory itself has also mainly been limited to the Markovian regime, although it was developed for quantum systems about twenty years ago [6]. It was extended to non-Markovian systems only quite recently [8].…”

“…In Ref. [8] the non-Markovian inputoutput relation was derived, but a quantum measurement was imposed on the output field so that the dynamics of the system is described by a stochastic Schrödinger equation. Such a formalism cannot be used to describe coherent cascade connections between systems because the quantum coherence in the output field is deteriorated by the measurement.…”

“…III, we use an alternative method to derive the non-Markovian inputoutput relations in Ref. [8], and obtain the dynamical equation for the system such that it can be easily used for networks. We derive these here as a natural extension of the original CollettGardiner quantum input-output theory.…”

“…There has been increasing interest in recent years in non-Markovian open quantum systems, and a number of analytical approaches have been devised to describe them. These include the projection-operator partitioning technique [25], the non-Markovian quantum trajectory approach [26], and very recently a non-Markovian input-output formalism [8].…”