Quantum transistors for heat flux in and out of working substance parts: harmonic vs transmon and Kerr environs

TL;DR

The paper develops a working-substance quantum thermal transistor (WTT) architecture in which three qubits exchange local heat currents with a non-Markovian collisional environment of sequential qutrits. Using CPTP dynamics from a global unitary on system+environment, it studies amplification of local currents as a function of modulating-bath temperature $oldsymbol{T_M}$, system–bath coupling $oldsymbol{g}$, and interaction time, including linear and nonlinear bath statistics (transmon/Kerr) and even environments of qubits. Key findings show robust transistor action across three, two, or symmetric/asymmetric couplings; nonlinearities enhance amplification in distinct temperature regimes; and non-Markovian memory (characterized by the BLP measure) correlates with, but is not fully captured by, the amplification dynamics. These results extend quantum thermal transistor design into non-Markovian, nonlinearly structured environments, offering guidance for local heat-flow control in nanoscale quantum devices.

Abstract

Quantum thermal transistors have been widely studied in the context of three-qubit systems, where each qubit interacts separately with a Markovian harmonic bath. Markovianity is an assumption that is imposed on a system if the environment loses its memory within short while, while non-Markovianity is a general feature, inherently present in a large fraction of realistic scenarios. Instead of Markovian environments, here we propose a transistor in which the interaction between the working substance and an environment comprising of an infinite chain of qutrits is based on periodic collisions. We refer to the device as a working-substance thermal transistor, since the model focuses on heat currents flowing in and out of each individual qubit of the working substance to and from different parts of the system and environment. We find that the transistor effect prevails in this apparatus and we depict how the amplification of heat currents depends on the temperature of the modulating environment, the system-environment coupling strength and the interaction time. We further show that there exists a non-zero amplification even if one of the environments, that is not the modulating one, is detached from the system. Additionally, the environment, being comprised of three-level systems, allows us to consider the effects of frail perturbations in the energy-spacings of the qutrit, leading to a non-linearity in the environment. We consider non-linearities that are either of transmon- or of Kerr-type. We find parameter ranges where there is a significant amplification for both transmon- and Kerr-type non-linearities in the environment. Finally, we detect the non-Markovianity induced in the system from a non-monotonic behavior of the amplification observed with respect to time, and quantify it using the distinguishability-based measure of non-Markovianity.

Figures (13)

• Figure 1: Schematic representation of the difference between the local and global approaches utilized to provide the definition of heat current used in previous models, referred to as BTTs versus the model studied in this paper, referred to as WTTs. The name, BTT essentially means that the transistor is conceptualized by looking at heat current flows in and out of each of the local baths to and from the entire system, as depicted by the red arrows in panel (a), and hence the transistor action is scrutinized by focusing on the "bath". The name, WTT suggests that, the heat is exchanged between each (local) qubit with the different parts of the system and the environment, as shown by the blue arrows in panel (b), and therefore the focus is on the "working substance". BTTs correspond to the usual QTT model previously studied in literature, while the class of transistor devices which we analyze in this paper are the WTTs. The bath interacts with the working substance via the collisional model where every qutrit sequentially interacts with the working substance. We show in the schematic that the $n^{\text{th}}$ qubit is interacting with the system while the $(n-1)^{\text{th}}$ qubit has already interacted. The interaction with the $(n+1)^{\text{th}}$ qubit is going to occur.
• Figure 2: (a) Variation of heat currents at time = 1 $\tilde{t}$ (b) Variation of $\partial J_X / \partial T_M$ with $T_M$, for $X\in\{L,M,R \}$, is depicted. Values of $\partial J_M / \partial T_M$ are much smaller than the other two, and becomes equal to zero at a particular value of $T_M$ which is equal to $6.65$ in units of $\Tilde{T}$ when $g = 4$. (c) Variation of $\partial{J_M}/\partial{T_M}$ with change in the coupling constant between the qubit and the environment is demonstrated. We see that, for every value of $g$, the quantity $\partial{J_M}/\partial{T_M}$ becomes zero at some value of $T_M = T_M^{critical}$, which makes the amplification factor undefined. This value of $T_M^{critical}$ increases as we increases the value of $g$. The black dotted line is for $\partial{J_M}/\partial{T_M} = 0.$ The horizontal axis has dimension $\tilde{T}$ while the vertical axis when multiplied by $\hbar/\Tilde{T}$ will have the dimension of ratio of rate of change of energy with temperature.
• Figure 3: Variation of the dynamical amplification factor, $\alpha_L$, along the vertical axis, versus the temperature of the middle bath, $T_M$, along the horizontal axis, for different values of system-bath interaction strength, $g$. We observe that there is a discontinuity in the value of $\alpha_L$ with respect to $T_M$, and an increase in the interaction strength increases and decreases the amplification to the right and left of the discontinuity respectively. The black dotted line signifies zero amplification. The quantity, $\alpha_L$, is dimensionless, while $T_M$ is in units of $\tilde{T}$.
• Figure 4: Variation of the dynamical amplification factor, $\alpha_X$, with time, $t$, at $T_M = 10$. Here $X\in L,R$ where $L$ and $R$ denote left and right qubits respectively. We find that the variation becomes periodic after certain number of interactions have happened between the working substance and the environment. The dotted black line signifies amplification equal to zero. The vertical axis is dimensionless and the horizontal axis is in units of $\tilde{t}$.
• Figure 5: Variation of $\alpha_L$ on the vertical axis as the coupling strength $g$ (on the horizontal axis) between the qubit and the environment is varied at $t = 1\tilde{t}$. We observe that when the coupling strength $g<4$, the amplification is small and positive, and when $g>4$, it is small and negative. A sharp peak in amplification occurs around $g= 4$, indicating a significant enhancement at this point.