Diagrammatic approach for analytical non-Markovian time evolution: Fermi's two-atom problem and causality in waveguide quantum electrodynamics

Fatih Dinc

Phys. Rev. A 102, 013727 – Published 31 July 2020

Abstract

Non-Markovian time evolution of quantum systems is a challenging problem, often mitigated by employing numerical methods or making simplifying assumptions. In this work, we address this problem in waveguide quantum electrodynamics (QED) by developing a diagrammatic approach, which performs fully analytical non-Markovian time evolution of single-photon states. By revisiting Fermi's two-atom problem, we tackle the impeding question of whether the rotating-wave approximation violates causality in single-photon waveguide QED. Afterward, we introduce and prove the no upper half-plane poles (no-UHP) theorem, which connects the poles of scattering parameters to the causality principle. Finally, we visualize the time-delayed coherent quantum feedback mediated by the field and discuss the Markovian limit for microscopically separated qubits where short-distance causality violations occur and the emergence of collective decay rates in this limit. Our diagrammatic approach is a method to perform exact and analytical non-Markovian time evolution of multiemitter systems in waveguide QED.

Received 29 May 2020
Accepted 14 July 2020

DOI:https://doi.org/10.1103/PhysRevA.102.013727

Physics Subject Headings (PhySH)

Quantum channels Quantum communication

Waveguides

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Fatih Dinc ^*

Department of Applied Physics, Stanford University, Stanford, California 94305, USA

^*fdinc@stanford.edu

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Vol. 102, Iss. 1 — July 2020

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Images

Figure 1
(a) The time evolution of pulse scattering for a single qubit. The red dashed dotted arrows correspond to incident pulses, whereas solid black arrows correspond to transmitted pulses. (b) A pulse is generated by an initially excited qubit and travels to an adjacent qubit (top); a pulse is incident from the far left on a single qubit (bottom). Regardless of an incident pulse's source of origin, the two scenarios are equivalent in terms of the local equations of motion at the second qubit.
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Figure 2
A heuristic illustration of diagrams. (a) There are three physical components of the diagrams: qubits and right- and left-moving photons. (b) An example event whose amplitude can be calculated using the diagrams. A photon is emitted by the qubit A, propagates for some distance, and gets reabsorbed by the qubit B. (c) The illustration of the event in terms of diagrams. The diagrams break down the complete event into smaller unit cells; for each cell there is a special rule to calculate the input-output relation. Each unit cell is governed by the local equations of motion, whereas cascading defines the initial conditions for the next unit cell such that the output of one unit cell is the input of another. Diagrams are convenient ways of solving the Schrödinger equation for the single-photon subspace without any explicit computation.
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Figure 3
A visual summary for the set of diagrammatic rules. If followed, they provide exact and analytical non-Markovian time evolution for single-photon waveguide QED.
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Figure 4
Systematic approach for finding the event diagrams that contribute to the amplitude calculations. For illustration purposes, we consider $N = 3$ qubits with the initial condition that the middle qubit is excited. The qubits are separated by $L$ . To find the event diagrams, we start with a fractal tree and branch out at every scattering point. If the photon leaves the system, the branch is terminated, as is the case for the two out of four branches at the final step. Earlier branches describe earlier times, which are noted with horizontal dotted red lines. The event diagrams are illustrated on top of the branch arrows.
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Figure 5
Fermi's two-atom problem in waveguide QED (a) Two identical qubits are coupled to a one-dimensional waveguide. The first qubit is initially excited, the second one is initially in the ground state. The first qubit spontaneously decays with a rate $γ_{0}$ , whereas the emitted photon is picked up by the second qubit after some time delay due to photon propagation. (b) All diagrams contribute to the qubit excitation coefficient $〈 e_{1} | ψ (t) 〉$ , where $| ψ (t = 0) 〉 = | e_{- 1} 〉$ . Using the diagrams, one can answer Fermi's two-atom problem for waveguide QED.
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Figure 6
The time-delayed feedback leads to the periodic reexcitation of the second qubit with time intervals of $2 L$ . (a) The field mediates the interaction between the two qubits with some time delay. Here only the field ongoing from qubit 1 to the qubit 2 is shown, where the pale blue corresponds to the future state of the trapped photonic field moving to the right. The feedback waves arise from the internal reflections of the photon. The field amplitudes are illustrated using analytical expressions calculated in Appendix pp5. (b) The second qubit gets excited periodically due to the incoming feedback waves for a large separation of $L = 5 J_{0}^{- 1}$ . (c) The qubits are moderately separated with $L = 2 J_{0}^{- 1}$ , where interference between different feedback waves can be observed. (d) The qubits are closely separated with $L = 0.3 J_{0}^{- 1}$ such that from the interference between nearly instantaneous feedback waves emerges a collective behavior. For all plots, we pick $Ω = 200 J_{0}$ .
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