Exactly solvable model of calorimetric measurements

Brecht Donvil, Paolo Muratore-Ginanneschi, and Dmitry Golubev

Phys. Rev. B 102, 245401 – Published 1 December 2020

Abstract

Calorimetric measurements are experimentally realizable methods to assess thermodynamics relations in quantum devices. With this motivation in mind, we consider a resonant level coupled to a Fermion reservoir. We consider a transient process in which the interaction between the level and the reservoir is initially switched on and then switched off again. We find the time dependence of the energy of the reservoir, of the energy of the level, and of the interaction energy between them at weak, intermediate, strong, and ultrastrong coupling. We also determine the statistical distributions of these energies.

Received 31 July 2020
Revised 13 November 2020
Accepted 17 November 2020

DOI:https://doi.org/10.1103/PhysRevB.102.245401

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics Quantum thermodynamics

Quantum dots Two-dimensional electron gas

Full counting statistics

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Brecht Donvil and Paolo Muratore-Ginanneschi

Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014 Helsinki, Finland

Dmitry Golubev

Pico group, QTF Centre of Excellence, Department of Applied Physics, Aalto University, P.O. Box 15100, 00076 Aalto, Finland

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Issue

Vol. 102, Iss. 24 — 15 December 2020

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Images

Figure 1
The contour $C$ incorporating the Bromwich contour $B$ of the Laplace antitransform (6) for positive $t$ . $B, V_{I}^{(0)}$ , and $V_{I}^{(ε_{c})}$ (red) are the contour components lying on the first Riemann sheet. $V_{II}^{(0)}$ and $V_{II}^{(ε_{c})}$ (blue) are on the second Riemann sheets. The remaining parts of the contour are at infinity and give vanishing contributions to the occupation amplitude. Circled crosses stand for poles encompassed by the contour.
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Figure 2
Time dependence of the energies $E_{0} (t)$ and $Δ E_{R} (t)$ at (a) and (b) weak, (c) and (d) intermediate, (e) strong, and (f) ultrastrong coupling. The dots are obtained by numerical evaluation of the exact time evolution (5). The solid lines show the analytical weak-coupling predictions (31), and the dashed lines show the ultrastrong coupling predictions (40) and (41). We have assumed $T = 0, n_{0} = 1, ε_{0} = 1.1 μ$ , and $ε_{c} = 10 μ$ . For (a) the number of modes in the reservoir is $N = 7000$ , and the level spacing $Δ ω = ε_{c} / N = 1 / 700 μ$ ; for (b), $N = 500, Δ ω = 1 / 50 μ$ ; and for (c)–(f) $N = 200, Δ ω = 1 / 20 μ$ .
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Figure 3
Average values of the long-time asymptotic energies of the resonant level $〈 E_{0}^{as} 〉$ (34), of the reservoir $〈 Δ E_{R}^{as} 〉$ (35), and of the interaction energy $〈 E_{I}^{as} 〉$ (36) plotted versus the energy of the level $ε_{0}$ . The dots are obtained by numerical evaluation of the exact dynamics (5). The temperature is zero, $T = 0$ . In the left panel we assume $n_{0} = 1$ ; that is, the energy level is initially populated. In the right panel we put $n_{0} = 0$ . The coupling rate between the level and the reservoir is $Γ_{0} = 0.05 μ, ε_{c} = 10 μ, N = 200$ , and $Δ ω = 1 / 20 μ$ .
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Figure 4
Long-time nonexponential behavior of ${| φ (t) |}^{2}$ for $Γ_{0} = 0.0623 μ, ε_{c} = 10 μ$ , and $ε_{0} = 1.1 μ$ : there are three curves, with two overlapping. The dark red curve shows the numerical evaluation of the integral (6). The orange solid line and yellow dashed respectively show the numerical evaluation of the vertical contour contribution plus the residue of (27) and asymptotic evaluation (24) of the vertical contour integral plus the residue contribution. The small discrepancy between the three curves originates from the residues' contributions. The timescales in the plot require retaining all terms in the analytic asymptotic, as reported in the second line of (24).
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Figure 5
Probability distributions of the reservoir, interaction, and resonant level energies. The left column corresponds to the weak-coupling regime with $Γ_{0} = 0.02 μ$ , and the right one corresponds to the strong-coupling limit $Γ_{0} = 18$ . Other parameters have the values $T = 0, n_{0} = 1, ε = 1.1 μ, ε_{c} = 10 μ, N = 200$ , and $Δ ω = 1 / 20 μ$ . The inset in the bottom left panel shows a Lorentzian fit of the peak in the reservoir energy distribution, $P_{R} (Δ E, t) = a / [{(Δ E - {\tilde{ε}}_{0})}^{2} + Γ_{0}^{2} / 4]$ . We find from the fit $Γ_{0} = 0.0193 μ$ , which is close to the expected value $0.02 μ$ .
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Figure 6
Sketch of a possible experimental setup. The control potential $V_{b}$ allows one to tune the height of the barrier between the quantum dot hosting a spin-polarized energy level and the 2DEG reservoir, the gate potential $V_{g}$ tunes the position of the level relative to the Fermi energy of the reservoir, the electrometer allows one to monitor the number of electrons in the dot, and the thermometer in the bottom left corner measures the temperature of the reservoir.
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