Optimal rectification in the ultrastrong coupling regime

T. Werlang, M. A. Marchiori, M. F. Cornelio, and D. Valente

Phys. Rev. E 89, 062109 – Published 5 June 2014

Abstract

We study the effect of ultrastrong coupling on the transport of heat. In particular, we present a condition for optimal rectification, i.e., flow of heat in one direction and complete isolation in the opposite direction. We show that the strong-coupling formalism is necessary for correctly describing heat flow in a wide range of parameters, including moderate to low couplings. We present a situation in which the strong-coupling formalism predicts optimal rectification whereas the phenomenological approach predicts no heat flow in any direction, for the same parameter values.

Received 3 April 2014

DOI:https://doi.org/10.1103/PhysRevE.89.062109

Authors & Affiliations

T. Werlang^*, M. A. Marchiori, M. F. Cornelio, and D. Valente

Instituto de Física, Universidade Federal de Mato Grosso, Cuiabá MT, Brazil

^*thiago_werlang@fisica.ufmt.br

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Vol. 89, Iss. 6 — June 2014

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Figure 1
(a) Two spins coupled with strength $Δ$ . Two independent thermal reservoirs are defined at temperatures $T_{L}$ ( $T_{R}$ ) for the bath on the left (right). (b) and (c) Case $Δ < h$ . $| 1 〉$ to $| 4 〉$ are the eigenstates of $H_{S}$ in the corresponding case. (b) Energy transitions that allow for heat flow from the left hot bath to the right cold bath ( $J_{L} > 0$ , as defined in the text). $Γ_{i j}^{L (R)}$ is the net rate of transition from state $| i 〉$ to $| j 〉$ , driven by the left (right) bath. The colored thick arrows indicate the flow of heat involved in each step of the cycle. (c) The same as (b) with reverted temperatures, and correspondingly reverted cycle. Note that in both cases the transition between states $| 2 〉$ and $| 3 〉$ involve heat exchange in the opposite sense with respect to the net heat flow. This is the key ingredient for the establishment of rectification, as explained in the text.
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Figure 2
Left current $J_{L}$ as a function of left bath temperature $k_{B} T_{L}$ for $T_{R} = 0$ and $Δ = 0.01 h$ (black dotted line, weak regime), $0.1 h$ (red dashed line, intermediate regime), and $0.5 h$ (blue solid line, strong regime). (Inset) Left current $J_{L}$ from the microscopic approach (black dotted line) and the phenomenological approach (red solid line) as a function of $k_{B} T_{L}$ for $T_{R} = 0$ in the weak-coupling regime.
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Figure 3
(a) Left current $J_{L}$ as a function of $Δ / h$ for (a) $k_{B} T_{L} = 10 h$ and $k_{B} T_{R} = 0, 0.1 h, 0.3 h$ , and (b) $k_{B} T_{R} = 10 h$ and $k_{B} T_{L} = 0, 0.1 h, 0.3 h$ . We used a black dotted line for $k_{B} T_{R (L)} = 0$ , a red dashed line for $k_{B} T_{R (L)} = 0.1 h$ , and a blue solid line for $k_{B} T_{R (L)} = 0.3 h$ . When the temperature of the cold reservoir is set to absolute zero, we find optimal rectification for any value of the coupling parameter $Δ$ . On the other hand, if the temperature of the cold reservoir is finite, it is necessary to reduce the coupling parameter to achieve optimal rectification. (Inset) Left current $J_{L}$ as a function of the temperature gradient $δ T = k_{B} (T_{L} - T_{R})$ for $Δ = 0.5 h$ and $\bar{T} = 0.5 h$ (black solid line), and $5 h$ (red dashed line). Here $\bar{T} = k_{B} (T_{L} + T_{R}) / 2$ denotes the average temperature. As the average temperature decreases, the curve around $δ T = 0$ becomes asymmetric, indicating the presence of rectification.
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Figure 4
Left current $J_{L}$ as a function of left bath temperature $T_{L}$ for the XY model with $N = 2$ and $N = 4$ . We fixed $T_{R} = 0$ . While the left current $J_{L}$ obtained through microscopic ME (black solid line) saturates at a finite value, as the left temperature $T_{L}$ increases, the current obtained by the phenomenological ME (red dashed line) vanishes. The left current $J_{L}$ for $N = 2$ and $N = 4$ have the same behavior.
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