Determining the validity of cumulant expansions for central spin models

Open Access

Determining the validity of cumulant expansions for central spin models

Piper Fowler-Wright, Kristín B. Arnardóttir, Peter Kirton, Brendon W. Lovett, and Jonathan Keeling

Phys. Rev. Research 5, 033148 – Published 1 September 2023

Abstract

For a model with many-to-one connectivity it is widely expected that mean-field theory captures the exact many-particle $N \to \infty$ limit, and that higher-order cumulant expansions of the Heisenberg equations converge to this same limit whilst providing improved approximations at finite $N$ . Here we show that this is in fact not always the case. Instead, whether mean-field theory correctly describes the large- $N$ limit depends on how the model parameters scale with $N$ , and the convergence of cumulant expansions may be nonuniform across even and odd orders. Further, even when a higher-order cumulant expansion does recover the correct limit, the error is not monotonic with $N$ and may exceed that of mean-field theory.

Received 8 March 2023
Revised 12 June 2023
Accepted 1 August 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.033148

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Open quantum systems

Nonequilibrium systems Quantum many-body systems Quantum spin models

Mean field theory Mean-field & cluster methods

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Piper Fowler-Wright ¹, Kristín B. Arnardóttir ¹, Peter Kirton ², Brendon W. Lovett ¹, and Jonathan Keeling ¹

¹SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
²Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom

Article Text

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Vol. 5, Iss. 3 — September - November 2023

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Images

Figure 1
(a) Network of the model: a central site (index 0) couples to $N$ identical satellites ( $n = 1, ..., N$ ). (b) Each site is a two-level system (spin 1/2) subject to decay ( $κ$ or $Γ_{↓}$ ) and, in the case of the satellites, pump $Γ_{↑}$ .
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Figure 2
(a) Mean-field reduction to a two-body problem where expectations of a satellite evolve according to expectations of the central site ( $⇀$ ) which in turn evolve according to $N$ copies of the satellite expectations ( $↽$ ). (b) In the second-order cumulant expansion central-satellite and satellite-satellite expectations couple into the system [Eqs. (10) and (11)].
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Figure 3
(a) Central-site population, $p_{0}^{↑} = [1 + 〈 σ_{0}^{z} 〉] / 2$ , in mean-field (MF) and second-order cumulant (C2) approximations when $g \sqrt{N} = 3$ is fixed and $κ = 1$ in units of $ω$ . Exact data (blue dots) is included up to $N = 150$ . The horizontal scale $1 / N$ is such that $N \to \infty$ to the left. (b) MF, C2, and exact results when $κ / N = 1 / 16$ is fixed and $g = 3 / 4$ . The gray vertical line at $N = 50$ indicates points equivalent to those along the corresponding line in (c). (c) $p_{0}^{↑}$ vs $Γ_{↑} / Γ_{T}$ ( $Γ_{T} = Γ_{↑} + Γ_{↓}$ ) at fixed $κ / N = 1 / 16$ , $g = 3 / 4$ , and $N = 50$ (the low cost of the exact calculation allowed a continuous line to be plotted). The mean-field transition at $Γ_{↑} / Γ_{T} = R_{c} \approx 0.53$ is analogous to that in a driven-dissipative Tavis-Cummings model [15] and other models of lasing [37, 55]. (d) Error in MF and C2 results from (b). Other parameters used in these panels were $ε = ω = 1$ , $Γ_{T} = 2$ , and [except (c)] $Γ_{↑} = 3 / 2$ .
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Figure 4
(a) Central-site population $p_{0}^{↑}$ in mean-field (MF) and cumulant approximations up to fifth order (C2-5) at fixed $κ / N = 1 / 16$ [parameters and exact data as in Fig. 3]. Results at fourth and fifth order were derived using QuantumCumulants.jl [35]. Inset: the fifth-order solution has numerical noise beyond $N ≳ 2, 500$ , but is approaching a value distinct from the third-order limit. (b) Mean-field and cumulant results for the scaled photon number $〈 a^{†} a 〉 / N$ in the driven-dissipative Tavis-Cummings model using QuantumCumulants.jl. Exact results following a Fock-space truncation are included up to $N = 20$ ( $N_{phot.} = 20$ levels were sufficient to achieve convergence). Here the parameters were $g \sqrt{N} = 9 / 10$ , $ε = ω = κ = 1$ , $Γ_{T} = 1 / 2$ , and $Γ_{↑} = 3 Γ_{T} / 4$ . The fifth-order solution became unstable for $N ≳ 100$ .
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Figure 5
(a) Satellite-satellite and (b) central-satellite correlations in the steady state of the model with scaling $g \sqrt{N}$ fixed and parameters as in Fig. 3 ( $g \sqrt{N} = 3$ , $κ = 1$ ). Exact data (blue dots) up to $N = 150$ and second-order (C2) results for the correlations are included, as well as the mean-field (MF) approximations $〈 σ_{n}^{+} σ_{m}^{-} 〉 \approx {| 〈 σ_{n}^{+} 〉 |}^{2}$ and $〈 σ_{0}^{+} σ_{n}^{-} 〉 \approx 〈 σ_{0}^{+} 〉 〈 σ_{n}^{-} 〉$ . Note the use of $1 / \sqrt{N}$ on the horizontal axis: for this scaling $〈 σ_{n}^{+} σ_{m}^{-} 〉 = o (1 / \sqrt{N})$ and $Im [〈 σ_{0}^{+} σ_{n}^{-} 〉] = O (1 / \sqrt{N})$ as $N \to \infty$ (the real part of $〈 σ_{0}^{+} σ_{n}^{-} 〉$ vanishes at resonance). (c), (d) Same correlations when instead $κ / N$ is fixed with parameters as in Fig. 3 ( $κ / N = 1 / 16$ , $g = 3 / 4$ ). In this case both pairs of correlations remain finite for all $N$ .
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Figure 6
(a) Exact, mean-field (MF), and third-order (C3) results for the steady-state central-site population $p_{0}^{↑} = [1 + 〈 σ_{0}^{z} 〉] / 2$ of the central spin model at fixed $κ / N$ [parameters as in Fig. 3: $κ / N = 1 / 16$ , $g = 3 / 4]$ . Third-order results retaining symmetry-breaking terms in the equations are indicated with a dotted black line. (b) At $N = 26$ symmetry breaking $〈 σ_{0}^{+} 〉 \neq 0$ occurs in the steady state of these equations corresponding to the turning point of the dotted line in (a). (c) Central population versus $Γ_{↑} / Γ_{T}$ ( $Γ_{T} = Γ_{↑} + Γ_{↓}$ ) at $N = 50$ as in Fig. 3 ( $κ / N = 1 / 16$ , $g = 3 / 4$ ), now including third-order results with and without symmetry-breaking terms. The gray vertical line indicates data from (a). (d) Exact, mean-field (MF), and third-order (C3) results for the scaled photon number in the Tavis-Cummings model with parameters as in Fig. 4 ( $g \sqrt{N} = 9 / 10$ , $Γ_{↑} = 3 Γ_{T} / 4$ ).
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