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Dynamics of a qubit in a high-impedance transmission line from a bath perspective

Soumya Bera, Harold U. Baranger, and Serge Florens
Phys. Rev. A 93, 033847 – Published 28 March 2016
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  • INTRODUCTION
  • SINGLE COHERENT STATE DYNAMICS
  • POPULATION DECAY AND COHERENCE BUILDUP
  • SCATTERING PROPERTIES: NUMERICAL…
  • SCATTERING PROPERTIES: COHERENT STATE…
  • CONCLUSION
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We investigate the quantum dynamics of a generic model of light-matter interaction in the context of high-impedance waveguides, focusing on the behavior of the photonic states generated in the waveguide. The model treated consists simply of a two-level system coupled to a bosonic bath (the Ohmic spin-boson model). Quantum quenches as well as scattering of an incident coherent pulse are studied using two complementary methods. First, we develop an approximate ansatz for the electromagnetic waves based on a single multimode coherent state wave function; formally, this approach combines in a single framework ideas from adiabatic renormalization, the Born-Markov approximation, and input-output theory. Second, we present numerically exact results for scattering of a weak intensity pulse by using numerical renormalization group (NRG) calculations. NRG provides a benchmark for any linear response property throughout the ultrastrong-coupling regime. We find that in a sudden quantum quench, the coherent state approach produces physical artifacts, such as improper relaxation to the steady state. These previously unnoticed problems are related to the simplified form of the ansatz that generates spurious correlations within the bath. In the scattering problem, NRG is used to find the transmission and reflection of a single photon, as well as the inelastic scattering of that single photon. Simple analytical formulas are established and tested against the NRG data that predict quantitatively the transport coefficients for up to moderate environmental impedance. These formulas resolve pending issues regarding the presence of inelastic losses in the spin-boson model near absorption resonances, and could be used for comparison to experiments in Josephson waveguide quantum electrodynamics. Finally, the scattering results using the coherent state wave-function approach are compared favorably to the NRG results for very weak incident intensity. We end our study by presenting results at higher power where the response of the system is nonlinear.

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    • Received 4 January 2016

    DOI:https://doi.org/10.1103/PhysRevA.93.033847

    ©2016 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Numerical Renormalization GroupQuantum description of light-matter interactionQuantum electrodynamicsScattering theory
    1. Physical Systems
    Waveguides
    Atomic, Molecular & Optical

    Authors & Affiliations

    Soumya Bera1,2, Harold U. Baranger3, and Serge Florens2

    • 1Max-Planck-Institut fuer Physik komplexer Systeme, 01187 Dresden, Germany
    • 2Institut Néel, CNRS and Université Grenoble Alpes, F-38042 Grenoble, France
    • 3Department of Physics, Duke University, P. O. Box 90305, Durham, North Carolina 27708, USA

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    Issue

    Vol. 93, Iss. 3 — March 2016

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    Images

    • Figure 1
      Figure 1

      Population decay σz(t) (upper panel) and coherence buildup σx(t) (lower panel) of the two-level system for several values of dissipation α=0.1,0.3,0.5 at Δ/ωc=0.1. The dashed lines on the vertical axis of the lower panel denote the values for the ground-state coherence that are expected from the static single coherent state approximation. The discrepancy with the long time limit of σx originates in spurious correlations between the bath and the two-level system.

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    • Figure 2
      Figure 2

      Real-space displacements f(x) and h(x) associated with the ↑ and ↓ components of the wave function, respectively, shown for a long time after the instantaneous quench process discussed in the text (parameters are Δ=0.1 and α=0.2). At short distances, the Silbey-Harris ground state is stabilized, while at large distances a wave packet is propagating away from the qubit located at the origin. The lack of factorization f(x)≠g(x) for x>220 is associated with spurious correlations between the emitted wave packet and the two-level system, that are responsible for the improper relaxation seen in Fig. 1.

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    • Figure 3
      Figure 3

      Top panel: coherence σx(t) as a function of time for three different switching processes (from sudden to adiabatic), that are given by the temporal profile of the local field shown in the rightmost middle panel. The saturation to the equilibrium value is obtained for an adiabatic switching, while the sudden quench shows improper relaxation. The leftmost middle panels shows the associated population decay σz(t). Bottom panel: real-space displacement f(x) and h(x), associated to the ↑ and ↓ components of the wave function, respectively, shown for a long time after the adiabatic quench. Now, factorization is correctly recovered, in agreement with the proper relaxation value of σx(t) at long times.

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    • Figure 4
      Figure 4

      Waveguide setup considered here, where a qubit is side-coupled in a two-terminal geometry to a photonic waveguide. Transport properties are described by scattering processes depicted by a set of ingoing and outgoing wave packets.

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    • Figure 5
      Figure 5

      Reflection coefficient (top panel), transmission coefficient (middle panel), and total inelastic losses (bottom panel) from the NRG calculations at three values of dissipation α=0.2,0.4,0.6 for Δ=0.1. The curves with largest α show a resonant peak at the smallest value of the renormalized qubit splitting ΔR.

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    • Figure 6
      Figure 6

      Reflection coefficient comparing the RWA Lorentzian line shape (24) (dashed lines, upper panel) and the NIBA line shape (27) of Ref. [12] (dashed lines, lower panel) to the exact NRG calculations (full lines) at two values of dissipation α=0.2,0.6 for Δ=0.1. An effective renormalized qubit splitting ΔR and a renormalized linewidth Γ were used as fitting parameters within the RWA and NIBA formulas, for better comparison of the actual line shapes.

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    • Figure 7
      Figure 7

      Reflection coefficient (top panel) and inelastic deficit (bottom panel) comparing the phenomenological line shape (31) (dashed lines) to the exact NRG calculations (full lines) at two values of dissipation α=0.2,0.6 (rightmost and leftmost curves, respectively). An effective renormalized qubit splitting ΔR and a renormalized linewidth Γ were used as fitting parameters within the phenomenological formula, for better comparison of the actual line shapes.

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    • Figure 8
      Figure 8

      In the quantum optics regime α=0.05, the reflection coefficient obtained from the variational dynamics (symbols) matches the RWA line shape (dashed line), which was however corrected with the proper renormalized qubit splitting. Parameters here are Δ/ωc=0.1 and the variational dynamics is done at weak power (n¯=0.2) in order to be in the linear response regime. The wave-packet width is taken very small σ=0.0005 to allow good spectral resolution.

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    • Figure 9
      Figure 9

      In the ultrastrong-coupling regime α=0.2, the reflection coefficient obtained from the variational dynamics (symbols) compares favorably to the exact NRG line shape (solid line), while the RWA (dashed line) presents clear deviations. The same parameters as in Fig. 8 were used.

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    • Figure 10
      Figure 10

      Reflection coefficient at α=0.2 and Δ/ωc=0.2 for increasing irradiation power. The incident power is characterized by the average number of photons n¯ in the incoming coherent state wave packets; results for up to n¯=5 are shown.

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