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Decimation technique for open quantum systems: A case study with driven-dissipative bosonic chains

Álvaro Gómez-León, Tomás Ramos, Diego Porras, and Alejandro González-Tudela
Phys. Rev. A 105, 052223 – Published 26 May 2022
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  • INTRODUCTION
  • GREEN'S FUNCTIONS IN OPEN QUANTUM…
  • DECIMATION FOR LATTICE GREEN'S FUNCTIONS
  • EXAMPLES: DRIVEN-DISSIPATIVE BOSONIC…
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (nonunitary) dynamics, which can be radically different from closed-system scenarios. Such open quantum system dynamics is generally described by Lindblad master equations, whose dynamical and steady-state properties are challenging to obtain, especially in the many-particle regime. Here, we introduce a method to deal with these systems based on the calculation of a (dissipative) lattice Green's function with a real-space decimation technique. Compared to other methods, such a technique enables us to obtain compact analytical expressions for the dynamics and steady-state properties, such as asymptotic decays or correlation lengths. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity, including the Hatano-Nelson model. The latter is especially illustrative because its surface and bulk dissipative behavior are linked due to its nontrivial topology, which manifests in directional amplification.

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    • Received 3 March 2022
    • Accepted 6 May 2022

    DOI:https://doi.org/10.1103/PhysRevA.105.052223

    ©2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Dissipative dynamicsOpen quantum systemsOpen quantum systems & decoherencePhotonicsQuantum opticsTopological effects in photonic systems
    1. Techniques
    Green's function methods
    Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

    Authors & Affiliations

    Álvaro Gómez-León*, Tomás Ramos, Diego Porras, and Alejandro González-Tudela†

    • Institute of Fundamental Physics IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain

    • *a.gomez.leon@csic.es
    • †a.gonzalez.tudela@csic.es

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    Vol. 105, Iss. 5 — May 2022

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    Images

    • Figure 1
      Figure 1

      Schematic for the system: a bosonic chain with neighboring sites coupled by (coherent and incoherent) tunneling, t̃±. Sites are also subject to local gain P and loss γ processes. The dashed rectangles represent the two different decimation schemes: method 1, based on decimating site 1 until just site 0 remains; and method 2, based on adding sites at the edge until their Green's function remains constant with the number of sites. Method 1 is more appropriate to determine the Green's function for finite-size systems, while method 2 is more effective in the semi-infinite limit.

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    • Figure 2
      Figure 2

      Comparison between the exact value of ε̃1N for a finite chain with N sites and the one obtained in the semi-infinite limit (horizontal red lines) for γ/tc=0.1, P/tc=0.05, ε/tc=−0.2, and ω/tc=0. The tendency towards the semi-infinite limit is not monotonous, but is correct for large systems. The vertical dashed line indicates the corresponding correlation length, Re[ξ(ω)]−1, for these parameters [see Eq. (31) below for details]. This length is in one-to-one correspondence with the size required to reach convergence to the semi-infinite limit.

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    • Figure 3
      Figure 3

      Change in the local density of states (DOS), Dj(ω), obtained from the exact Green's function from decimation, Eq. (31), as one moves away from the edge, j=0 (blue), towards the bulk, j→∞ (red). Intermediate sites are labeled in the figure: 1, 2, 10, and 50. Notice the renormalization of the band-edge Van Hove singularities in the bulk as j≠0 approaches the edge j=0.

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    • Figure 4
      Figure 4

      Density of states (DOS), Dj(ω), as defined in the text, at the boundary (blue and green) and deep in the bulk (orange and red). Solid lines refer to the lossless case, characterized by a sharp disappearance of states beyond |ω|≥2tc and the Van Hove singularities for the bulk case. In contrast, the dissipative case softens and renormalizes the singularities, extending the surface/bulk DOS to frequencies beyond |ω|=2tc.

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    • Figure 5
      Figure 5

      |Gj,0(ω)| normalized by |G0,0(ω)| for ω=ωa (red), ω=ωa+2tc (blue), and ω=ωa with γ/tc=0 (black). In the lossless case, two-point correlations remain constant between distant sites. In contrast, adding dissipation introduces a decay of correlations, which is accentuated at the band edges. Vertical dashed lines indicate the corresponding coherence length 1/Re[ξ(ω)] from Eq. (31).

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    • Figure 6
      Figure 6

      Comparison between G5,4(ω) from exact diagonalization for a finite array with N=45 sites (dot and square markers) and the semi-infinite case (solid and dashed lines), computed using Eq. (51). Parameters: γ/tc=3, P/tc=3, ε/tc=0.1, and ϕ=0.9π/2.

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    • Figure 7
      Figure 7

      Top: ξ(ω) for the topological phase (ϕ=π/2). The blue region represents the region of amplification due to Re[ξ(ω)]>0. The vertical dashed lines indicate the critical points from the phase diagram. The inset shows the Im[ξ(ω)]. Bottom: Phase diagram obtained from the surface Green's function. It gives identical results to the calculation of W1(ω) for PBC using Eq. (53). Parameters: γ/tc=2, P/tc=4, and ε/tc=0.

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    • Figure 8
      Figure 8

      Comparison between the exact and the analytical solution for the real part of 〈aj(t)〉 for N=15, ϕ=0, P/tc=0, γ/tc=0.5, and ε/tc=0.1. The inset shows a zoom at intermediate time, where a difference between the two solutions arises around t∼12(unitsoftc−1) due to a revival. This is the Poincaré recurrence time for the finite system, where the excitation has bounced off the opposite edge. Enlarging the unit-cell size delays this effect.

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    • Figure 9
      Figure 9

      Regimes of amplification dynamics for the topological phase. The top plot shows the case with local dissipation dominating over a collective pump (γ/tc=2), where the excitation is amplified as it propagates, but rapidly damping after hopping to the next site. The bottom plot shows the regime dominated by a collective pump (γ/tc=1), where the excitation is delocalized between sites and constantly amplified over time. Parameters: ϕ=π/2, P/tc=1.4, and ε/tc=0.

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    • Figure 10
      Figure 10

      Stability diagram and its relation with the topological phase. The trivial phase is always stable (white region). In contrast, the topological phase is divided in two: a stable region (blue) and an unstable one (red). The crossing between dashed lines indicates the two points chosen in Fig. 9 to represent the different dynamical regimes of amplification. Parameters: ϕ=π/2 and ε/tc=0.

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    • Figure 11
      Figure 11

      Top: Gain vs ω at different sites. The input port is at the edge, and as the signal propagates, it is exponentially amplified with the distance, for a finite range of frequencies. Bottom: Normalized added noise during the amplification process. The frequencies within the topological phase show a noise-to-signal ratio nearly at the quantum limit njadd(ω)=1 (dashed line). Parameters: ε/tc=0, ϕ=π/2, γ/tc=4, and P/tc=3.6.

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