Collision model for non-Markovian quantum trajectories

S. J. Whalen

Phys. Rev. A 100, 052113 – Published 19 November 2019

Abstract

We present an algorithm to simulate genuine, measurement-conditioned quantum trajectories for a class of non-Markovian systems, using a collision model for the environment. We derive two versions of the algorithm: the first corresponding to photodetection and the second to homodyne detection with a finite local oscillator amplitude. We use the algorithm to simulate trajectories for a system with delayed coherent feedback, as well as a system with a continuous memory.

Received 23 July 2019

DOI:https://doi.org/10.1103/PhysRevA.100.052113

Physics Subject Headings (PhySH)

Open quantum systems

Statistical Physics & Thermodynamics

Authors & Affiliations

S. J. Whalen ^*

The Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand

^*simon.whalen@gmail.com

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 100, Iss. 5 — November 2019

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Two time steps of the unitary collision model described in Sec. 2. In the first step (top) the system interacts with the $n th$ environment subsystem for a time $Δ t$ . Between the first and second steps, the unitary $Δ U_{E}$ is applied, which as indicated by Eq. (10) can be thought of as translating the environment to the right. As such, in the second step (bottom) the system interacts with the $(n + 1) st$ subsystem. The open-system dynamics depicted in this diagram are Markovian, because the system $S$ couples to the environment at a single location.
Reuse & Permissions
Figure 2
Schematic illustration of the trajectory algorithm described in Sec. 3, for a Markovian open system. From left to right: first, the system interacts with the environment, represented by a collision model, for a time $Δ t$ ; second, a simulated measurement is performed on the zeroth environment oscillator, disentangling it from the rest of the combined system; third, the environment oscillators are shifted along by one step, and interaction with the system resumes.
Reuse & Permissions
Figure 3
Comparison of “counting” distributions for the homodyne detection of squeezed light. The system is a harmonic oscillator with Hamiltonian $H_{S} = i ζ ({a^{†}}^{2} - a^{2})$ , and the local oscillator field has amplitude $α e^{i θ}$ . Orange lines (front) show results from the conventional trajectory algorithm with jump operators given by Eq. (20); with this phase convention, $θ = 0$ corresponds to detection of the squeezed $Y$ quadrature and $θ = π / 2$ corresponds to the unsqueezed $X$ quadrature. Blue lines (back) correspond to an analogous collision model simulation, with $γ Δ t = 0.01$ and up to 249 excitations in the local oscillator. The horizontal axis is the sum of measurements in an interval of length $10 / γ$ , after a “burn-in” time of $4 / γ$ that reduces the influence of transient dynamics. The squeezing parameter is given by $ζ = 0.1 γ$ , and the system Hilbert space is truncated at nine excitations. Each histogram was prepared from 50 000 trajectories.
Reuse & Permissions
Figure 4
Collision model of an open quantum system with delayed coherent feedback. The system (yellow, below) couples to two different environment subsystems (blue, above), creating a feedback loop with delay $τ = M Δ t$ .
Reuse & Permissions
Figure 5
Trajectory simulations of a qubit with delayed coherent feedback. The upper four plots show the occupation number: (a), (b) for photodetection, and (c), (d) for homodyne detection with $α^{2} = 100 γ$ . Panels (a) and (c) correspond to emission from an initially excited system without driving, while panels (b) and (d) correspond to a driven qubit with $Ω / γ = 1$ . Twenty sample trajectories are plotted as solid blue lines for each configuration. The opacity of the blue curves provides an indication of how common a given trajectory is. Note in particular the trajectories without jumps that appear in darker blue in panels (a) and (b). The corresponding ensemble averages (25 000 trajectories) are shown as dotted orange lines. Trajectories of the Pauli $Y$ operator are shown in panels (e) and (f), corresponding exactly to the driven cases (b) and (d); the onset of the feedback is visible in the ensemble average at $t = τ$ . The other parameters are $ϕ = π, γ τ = 0.5, γ Δ t = 0.01$ , and local oscillator dimension 250.
Reuse & Permissions
Figure 6
Comparison of simulations with the exponential coupling (23), with $L = 5$ . Fifty sample photodetection trajectories are plotted as thin blue lines, with the ensemble average (10 000 trajectories) in orange. The broken lines show equivalent simulations using the Markovian master equation for the Jaynes–Cummings model: for the dashed green curve, the cavity Hilbert space was truncated at the single-excitation level, while up to 20 excitations were allowed for the dotted red curve. The trajectory simulations used $K_{max} = 1$ , so the ensemble average agrees well with the single-excitation master equation. The multiple-excitation master equation begins to diverge at $γ t \approx 1$ , as the environment excitation level increases. The other parameters are $Ω / γ = 1, λ / γ = 1$ , and $γ Δ t = 0.005$ . The size of the environment has been truncated at $L = 5 γ^{- 1}$ .
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information