Landau-Zener extension of the Tavis-Cummings model: Structure of the solution

Chen Sun and Nikolai A. Sinitsyn

Phys. Rev. A 94, 033808 – Published 7 September 2016; Erratum Phys. Rev. A 95, 069907 (2017)

Abstract

We explore the recently discovered solution of the driven Tavis-Cummings model (DTCM). It describes interaction of an arbitrary number of two-level systems with a bosonic mode that has linearly time-dependent frequency. We derive compact and tractable expressions for transition probabilities in terms of the well-known special functions. In this form, our formulas are suitable for fast numerical calculations and analytical approximations. As an application, we obtain the semiclassical limit of the exact solution and compare it to prior approximations. We also reveal connection between DTCM and $q$ -deformed binomial statistics.

Received 23 June 2016

DOI:https://doi.org/10.1103/PhysRevA.94.033808

Physics Subject Headings (PhySH)

Feshbach resonance Nonperturbative methods Tavis-Cummings model

Atomic, Molecular & Optical

Erratum

Erratum: Landau-Zener extension of the Tavis-Cummings model: Structure of the solution [Phys. Rev. A 94, 033808 (2016)]

Chen Sun and Nikolai A. Sinitsyn

Phys. Rev. A 95, 069907 (2017)

Authors & Affiliations

Chen Sun^1,2,* and Nikolai A. Sinitsyn^2,†

¹Department of Physics, Texas A&M University, College Station, Texas 77843, USA
²Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

^*chen.sun.whu@gmail.com
^†nsinitsyn@lanl.gov

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Vol. 94, Iss. 3 — September 2016

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Images

Figure 1
Samples of transition probabilities $P_{I \to F}$ vs $g$ predicted by Eq. (7) for $N_{s} = 7$ at $N_{B} = 0$ . In all cases, $I = | ↓, ↓, ↑, ↓, ↑, ↓, ↓ 〉$ and (a) $F = | ↓, ↓, ↑, ↓, ↑, ↓, ↓ 〉$ (the same state); (b) $F = | ↓, ↑, ↑, ↓, ↓, ↑, ↓ 〉$ ; (c) $F = | ↑, ↑, ↓, ↑, ↑, ↑, ↑ 〉$ ; (d) $F = | ↑, ↑, ↓, ↑, ↓, ↑, ↑ 〉$ (the “opposite spin” state). For these transitions, Eq. (7) provides the following probabilities: (a) $p_{5}^{5} p_{6}^{2}$ ; (b) $p_{4} p_{5}^{2} p_{6} q_{5}^{3}$ ; (c) $p_{4} q_{2} q_{3}^{3} q_{4} q_{5}$ ; (d) $q_{3} q_{4}^{5} q_{5}$ . Among them, (a) represents the main diagonal elements of the transition probability matrix, and (d) represents the antidiagonal elements.
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Figure 2
Comparison of exact predictions (discrete dots) and continuous approximations (solid curves) for transition probabilities in (a), (b) the forward and (c), (d) the inverse processes at different values of $S$ 's and $g$ 's, and at $N_{B} = 0$ . (a), (b) For the forward process, the exact prediction is Eq. (23), and Eq. (33) is its continuous approximation. As $g$ increases, the curves shift toward larger $ν$ values. (c), (d) For the inverse process, the exact prediction is Eq. (24) and its continuous limit is well represented by the Gaussian approximation in Eq. (40). As $g$ increases, the curves shift to smaller $ν$ values.
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Figure 3
Average final number of bosons $n_{b}$ vs square of the coupling $g^{2}$ at $S = 200$ and $N_{B} = 0$ for (a) forward and (b) inverse processes. Exact predictions of Eqs. (23) and (24) are shown as discrete dots, and approximations are marked as solid curves. (a) For the forward process, the black curve shows prediction (49) [Eq. (32) in Ref. 8] that captures the behavior at small and moderate $g$ values $[g < 1 / (2 S)]$ . The green (light gray in the grayscale version) curve shows prediction (50) [Eq. (15) in Ref. 10] that was derived to describe the large- $g$ limit $[g > 1 / (2 S)]$ . Our own approximation in Eq. (47) is shown only in the inset (red) because it would almost coincide with the green curve in the main part of the same figure. The inset shows $n_{b}$ vs $1 / g^{2}$ for the extremely large- $g$ region. It demonstrates that Eq. (50) (green) deviates from the exact result, although this difference is suppressed by a factor $\sim 1 / S$ . In contrast, our approximation (47) shows no visible deviations from the exact prediction in this case at all. (b) For the inverse process, our prediction in Eq. (51) (red) fits the exact results very well at almost all values of $g$ . The inset shows that $n_{b}$ is growing linearly with $1 / g^{2}$ at large- $g$ values $[g > 1 / (2 S)]$ . Our result (51) produces quantitatively the same prediction for this slope as the prior semiclassical prediction of Ref. [9].
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