Quantum interference between independent reservoirs in open quantum systems

Ching-Kit Chan, Guin-Dar Lin, Susanne F. Yelin, and Mikhail D. Lukin

Phys. Rev. A 89, 042117 – Published 25 April 2014

Abstract

When a quantum system interacts with multiple reservoirs, the environmental effects are usually treated in an additive manner. We show that this assumption breaks down for non-Markovian environments that have finite memory times. Specifically, we demonstrate that quantum interferences between independent environments can qualitatively modify the dynamics of the physical system. We illustrate this effect with a two-level system coupled to two structured photonic reservoirs, discuss its origin using a nonequilibrium diagrammatic technique, and show an example when the application of this interference can result in an improved dark state preparation in a $Λ$ system.

Received 14 February 2014

DOI:https://doi.org/10.1103/PhysRevA.89.042117

Authors & Affiliations

Ching-Kit Chan^1,2, Guin-Dar Lin^1,2,3, Susanne F. Yelin^1,2,4, and Mikhail D. Lukin²

¹ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA
²Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
³Department of Physics, National Taiwan University, Taipei 10617, Taiwan
⁴Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA

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Vol. 89, Iss. 4 — April 2014

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Images

Figure 1
A demonstrative example showing the interference effect between two structured environments. (a) The model of a TLS interacting with two independent photonic vacua is shown. Each photonic bath is characterized by a non-Markovian spectral density $J_{i} (ω)$ . (b, c) The population dynamics of an initially excited TLS based on the additive ME and the exact solution, respectively, is presented. The ME solution produces a much suppressed oscillation due to the missing of the bath interference. The parameters used are $λ_{1} / γ_{1} = 0.01$ , $λ_{2} / γ_{1} = 0.02$ . Similar discrepancies are observed for other parameters. (d) The error $ε (t) = ρ_{s}^{exact} (t) - ρ_{s}^{add} (t)$ of the additive ME using $λ_{1} = λ_{2} = λ$ and $γ_{1} = γ_{2}$ is plotted. As $λ$ increases, memory times are reduced and the additive ME result starts to approach the exact solution in the Markovian limit.
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Figure 2
(a) The TLS oscillates between the excited and ground states with a non-Markovian decay rate $Γ_{12} (t)$ . In the additive ME, the overall decay rate is approximated by an incoherent sum $Γ_{12}^{exact} (t) \approx Γ_{12}^{additive} (t) = Γ_{1} (t) + Γ_{2} (t)$ . (b) The difference between $Γ_{12}^{exact} (t)$ (solid blue line) and $Γ_{12}^{additive} (t)$ (dashed red line) is shown. Here $γ_{2} = γ_{1}$ , $λ_{1} / γ_{1} = 0.01$ , $λ_{2} / γ_{1} = 0.02$ .
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Figure 3
Diagrammatic representations of the dynamics of a system interacting with two general baths from time 0 to $t$ . (a–c) Lowest-order diagrams that describe the interference effect between baths 1 (dashed lines) and 2 (dotted lines). Left: exact diagrams; right: diagrammatic correspondence of the additive ME. Similar diagrams obtained by exchanging the bath lines and flipping the diagrams are not shown. All the diagrams have the same order. The additive ME solution does not allow the baths to have time overlaps in (b), nor to cross in (c). (d) The additive ME corresponds to a ladder approximation in the diagrammatic structure and thus omits the interference between baths.
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Figure 4
Dephasing comparison of a driven $Λ$ system. (a) The model system in the presence of Markovian relaxations and non-Markovian dephasing. (b) The virtual dephasing evolution, in which the initial dark state $| D 〉 = (| 1 〉 - | 2 〉) / \sqrt{2}$ is flipped to the bright state and then pumped back. (c) The corresponding evolution of the additive ME solution. The missing of the bath interference prohibits the simultaneous action of the pumping (double line) and dephasing processes (dashed line). (d) The dark state infidelity $δ F (t) = 1 - P_{D} (t)$ calculated by various approaches using $Ω = Γ_{1} = Γ_{2} = 100 γ_{⊥}$ . (e) The dark state population reached by increasing the driving field when $γ_{⊥} t = 1 / 2$ and $Γ_{1} = Γ_{2} = 100 γ_{⊥}$ . The additive ME predicts a lower dark state fidelity.
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Figure 5
A comparison of the dark state populations of the driven $Λ$ system using various approaches. (a) The quantum solution, (b) the additive ME that neglects the bath interference, and (c) the alternative additive ME that includes the dressing of the optical drive, but still omits the bath interference. Parameters used: $Ω = Γ_{1} = Γ_{2} = 100 γ_{⊥}$ .
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Figure 6
Diagrammatic constructions for the dephasing problem based on different theoretical techniques. Each diagram, being second order in the dephasing interaction, represents the virtual evolution of the system from time 0 (left) to $t$ (right) with the dressing of the optical pumping (double purple-green line) and the influence of the dephasing (dashed brown line). The energy levels next to the diagram gives the virtual processes of the dark state evolution. (a) The quantum solution. (b) The additive ME approach that forbids both the relaxation and optical driving during the virtual dephasing process. (c) The alternative additive ME result that permits the drive, but not the relaxation during the virtual process. (d) The limit of Markovian dephasing. Similar diagrams obtained by flipping the bubbles are not shown.
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