Quantum Heat Transformers

We begin by considering three qubits, labeled as $q_{1}$, $q_{2}$, and $q_{3}$, each locally connected to three thermal reservoirs at temperatures $\tilde{\tau}_{1}$, $\tilde{\tau}_{2}$, and $\tilde{\tau}_{3}$, respectively, and each qubit is in thermal equilibrium with its associated reservoir. Let $q_{2}$ denote the qubit serving as the common qubit for both the primary and secondary thermal junctions. We take the free Hamiltonian of the three-qubit composite system as $\mathcal{H}_{0}=\frac{\mathcal{K}}{2}\sum_{j=1}^{3}\mathcal{E}_{j}\sigma_{j}^{z}$, where $\sigma^{z}_{j}$ is the $z$ component of the Pauli matrix and $\mathcal{K}\mathcal{E}_{j}$ is the energy gap between the two levels of the $j^{\text{th}}$ qubit. Here, $\mathcal{K}$ is a constant with the unit of energy and $\mathcal{E}_{j}$s are dimensionless energy parameters. Now, the crucial task is to select the interaction Hamiltonian for this three-qubit system that will effectively demonstrate the properties of QHT while ensuring the protocol remains self-contained. To meet the self-contained condition, we find that, for the free system Hamiltonian $\mathcal{H}_{0}$, various interaction Hamiltonians can be crafted. Without assuming any of the $\mathcal{E}_{j}$ to be zero, the three-qubit free Hamiltonian is subject to some specific self-contained conditions, as follows:

Here, $\ket{1}$ and $\ket{0}$ are the eigenvectors of Pauli-$z$ matrix corresponding to the eigenvalues $-1$ and $+1$, respectively. $g$ is the dimensionless coupling strength of the interaction. As each qubit resides in its respective thermal state initially, the state of each qubit at the initial time, $\tilde{t}=0$, can be expressed as

for $j=1$, $2$, and $3$. Here, $p_{j}^{\text{in}}$ is the initial probability to find the qubit in the state $\ket{0}$. Now, considering the coupling between the systems and their respective reservoirs as weak, we are able to make use of the Born-Markov and secular approximations [29, 30, 31, 32, 33, 34, 35] to govern the evolution of the composite system under the influence of the local thermal baths. Consequently, the dynamics of the reduced system, $\rho(t)$, can be described using the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, as

Here, we introduce the dimensionless time as $t=\mathcal{K}\tilde{t}/\hbar$. We define $\mathscr{H}_{0}=\mathcal{H}_{0}/\mathcal{K}$, and $\mathscr{H}_{\text{int}}=\mathcal{H}_{\text{int}}^{x}/\mathcal{K}$, where $x$ can be any of $1$, $2$, $3$, and $4$. The dissipative term, $\mathcal{L}(\rho(t))=\sum_{j,\omega}\gamma_{j}(\omega)\left[A_{j}(\omega)\rho A_{j}^{\dagger}(\omega)-\frac{1}{2}\{A_{j}^{\dagger}(\omega)A_{j}(\omega),\rho\}\right]$, arises due to the presence of thermal environments. Here $j$ runs from $1$ to $3$. Additionally, $\omega$ represents the transition frequencies, $\gamma_{j}(\omega)$ denotes the decay constants, and $A_{j}(\omega)$ are the Lindblad operators. Suppose $\lambda_{1}$ and $\lambda_{2}$ represent two eigenvalues of $\mathcal{H}_{0}+\mathcal{H}^{x}_{\text{int}}$. Then, the transition frequency linked to the transition between these two energy eigenstates is given by $\omega=\frac{1}{\hbar}(\lambda_{2}-\lambda_{1})$. See [29, 30, 31, 32, 33, 35] for detailed discussions on transition energies, decay constants, and the construction of Lindblad operators. For the validation of Born-Markov approximations, the condition, $\max\{\frac{\hbar}{\mathcal{K}}\gamma_{j}\}<<\min\{\mathcal{E}_{j},g\}$, has to be satisfied. Under the dynamical evolution given by Eq. (3), the reduced density matrices of the individual subsystems, denoted as $\rho_{j}(t)$ at any given time $t$, retain their diagonal form as given in Eq. (2). However, in this evolving scenario, the equilibrium probabilities $p_{j}^{\text{in}}$ are replaced by time-dependent probabilities $p_{j}(t)$ and it is connected to the local temperature of each qubit at time $t$ as $p_{j}(t)=\frac{1}{1+\exp(\mathcal{K}\mathcal{E}_{j}/k_{B}\tilde{T}_{j}(t))}$, where $k_{B}$ is the Boltzmann constant and the index $j$ goes to $1$, $2$, and $3$. From now on, we define the dimensionless temperatures of the qubits as $T_{j}(t)=\frac{k_{B}}{\mathcal{K}}\tilde{T}_{j}(t)$ and those of the baths as $\tau_{j}=\frac{k_{B}}{\mathcal{K}}\tilde{\tau}_{j}$ for $j=1$, $2$, and $3$. Thus, at every instant of time, we have the temperature gradients, $\Delta T_{p}(t)=|T_{1}(t)-T_{2}(t)|$ for the primary junction, and $\Delta T_{s}(t)=|T_{3}(t)-T_{2}(t)|$ for the secondary junction. A schematic diagram of this three-qubit QHT is depicted in FIG. 1. To design a step-down quantum transformer, it is essential to ensure that, at any time $t$, $\Delta T_{p}(t)>\Delta T_{s}(t)$.

We now analyze the four self-contained conditions for designing the three-qubit QHTs, given in Eq. (II.1). Let us consider the initial temperature of the three qubits, i.e., the temperature of the three reservoirs follows the condition $\tau_{1}>\tau_{2}>\tau_{3}$, and also $\mathcal{E}_{2}>\mathcal{E}_{1}>0$. In this scenario, one approach to designing an effective QHT involves an increase in the temperature of $q_{1}$ over time, accompanied by a corresponding decrease in the temperature of $q_{2}$. Consequently, the temperature difference $\Delta T_{P}(t)=|T_{1}(t)-T_{2}(t)|$ at the primary junction at time $t$ is expected to increase with time. Conversely, as the temperature of $q_{2}$ decreases, the temperature of $q_{3}$ should increase with time. This adjustment ensures that the temperature difference at the secondary junction $\Delta T_{S}(t)=|T_{3}(t)-T_{2}(t)|$ decreases over time. Therefore, under these circumstances, the condition demands $\mathcal{H}_{\text{int}}^{2}$ as the interaction Hamiltonian between the qubits, with the choices of parameters facilitating the system transitioning from an initial state of $|101\rangle$ to the state $|010\rangle$ as time progresses with more probability than the opposite process. Note that the interaction Hamiltonian $\mathcal{H}^{2}_{\text{int}}$ is commonly utilized in the design of quantum refrigerators, as discussed in the literature [8]. However, in this QHT protocol involving the interaction Hamiltonian $\mathcal{H}_{\text{int}}^{2}$, does not exhibit the characteristics of a quantum refrigerator. The objective of refrigeration is typically achieved when the qubit with the lowest initial temperature among the three qubits undergoes cooling. Yet, in our protocol, the qubit having the lowest temperature, undergoes a transition to a higher excited state, indicating an increase in its temperature over time. Now, if we alter the considerations and designate $q_{3}$ as the common qubit, with the initial temperatures of the three reservoirs following the sequence $\tau_{1}>\tau_{3}>\tau_{2}$, and $\mathcal{E}_{1}>0$, $\mathcal{E}_{2}>0$, a parallel methodology reveals $\mathcal{H}_{\text{int}}^{3}$ as the compatible interaction Hamiltonian. By appropriately selecting parameters with this interaction Hamiltonian, we can ensure that the system initially resides in the $|110\rangle$ state with a higher probability and transitions to the $|001\rangle$ state as time progresses. This transition ensures that over time, the temperature of $q_{1}$ increases while correspondingly, the temperature of $q_{3}$ decreases, thereby causing the primary junction temperature difference $\Delta T_{P}(t)=|T_{1}(t)-T_{3}(t)|$ to increase. Also, the temperature of $q_{2}$ rises over time to decrease the secondary junction temperature difference $\Delta T_{S}(t)=|T_{2}(t)-T_{3}(t)|$. Similarly, if we designate $q_{1}$ as the common qubit, $\mathcal{H}_{\text{int}}^{4}$ emerges as the most appropriate Hamiltonian for constructing a QHT for some fixed set of initial conditions.

For a more deeper insight into the operational characteristics of QHTs, we will now delve into the operational aspects of QHTs utilizing the interaction Hamiltonian $\mathcal{H}_{\text{int}}^{1}$. Let us consider a scenario where the three qubits are interacting through $\mathcal{H}_{\text{int}}^{1}$, and the three thermal baths are bosonic baths, each consisting of an infinite number of harmonic oscillators. Here we set $\mathcal{E}_{1}>0$ and $\mathcal{E}_{2}>0$. The Hamiltonian of each bath can be expressed as

for $j=1$ to $3$. Here the operators $b^{j}_{\omega^{\prime}}$ ($b^{j\dagger}_{\omega^{\prime}}$) represent the bosonic annihilation (creation) operators corresponding to the energy mode $\omega^{\prime}$. These operators, having the unit of $1/\sqrt{\omega^{\prime}}$, satisfy the commutation relation $[b_{\omega^{\prime}}^{j},b_{\omega^{\prime\prime}}^{j\dagger}]=\delta(\omega^{\prime}-\omega^{\prime\prime})$. Here, $\tilde{\omega}$ is an arbitrary constant with units of frequency, and $\Omega_{j}$ denotes the cutoff frequency for the $j^{\text{th}}$ bath. We take the coupling between the $j^{\text{th}}$ qubit and $j^{\text{th}}$ bath to be

where $\chi_{j}(\omega^{\prime})$, a dimensionless function of $\omega^{\prime}$, modulates the coupling strength of the local interaction between the $j^{\text{th}}$ qubit and $j^{\text{th}}$ bath. Specifically, we have $\tilde{\omega}\chi_{j}^{2}(\omega^{\prime})=J_{j}(\omega^{\prime})$, where $J_{j}(\omega^{\prime})$ denotes the spectral density function of the $j^{\text{th}}$ bosonic bath. For our purposes, we adopt the Ohmic spectral density function, which takes the explicit form $J_{j}(\omega^{\prime})=\alpha_{j}\omega^{\prime}\exp(-\omega^{\prime}/\Omega_{j})$. Here $\alpha_{j}$s are dimensionless parameters. For the choice of the system-bath intection Hamiltonian given in Eq. (5), we obtain the decay constants as

where $f(\omega,T_{j})=[\exp(\hbar\omega/k_{B}T_{j})-1]^{-1}$ being the Bose-Einstein distribution function. Suppose we have two eigenvectors, $\ket{k}$ and $\ket{l}$, of the system Hamiltonian $\mathcal{H}_{0}+\mathcal{H}^{1}_{\text{int}}$, corresponding to eigenvalues $\lambda_{k}$ and $\lambda_{l}$ respectively. The Lindblad operators for the interaction Hamiltonian $H_{SB_{j}}$ [Eq. (5)] can then be written as

Let us now choose the parameters of the system and the interaction Hamiltonians to ensure that the initial probability of finding the qubits in the state $\ket{111}$ exceeds that of $\ket{000}$. To achieve a higher probability of being in the state $\ket{111}$ initially compared to $\ket{000}$, the initial composite probability must satisfy the condition

where $\mathcal{P^{\text{in}}}_{111}=(1-p_{1}^{\text{in}})(1-p_{2}^{\text{in}})(1-p_{3}^{\text{in}})$ and $\mathcal{P}^{\text{in}}_{000}=p_{1}^{\text{in}}p_{2}^{\text{in}}p_{3}^{\text{in}}$. With this initial condition, upon activating the interaction $\mathcal{H}_{\text{int}}^{1}$, the transition of the system from the state $\ket{111}$ to the state $\ket{000}$ becomes more probable. In this scenario, qubits $q_{1}$ and $q_{2}$ are initially in their ground state, while $q_{3}$ is in the excited state. With the progress of time, qubits $q_{1}$ and $q_{2}$ will transition to their excited state, while $q_{3}$ will transition to the ground state. Note that the reverse condition of Eq. (8) is also feasible, but we have opted for this particular choice arbitrarily.

In Fig. 2, we illustrate the time evolution of the temperature gradients of the primary ($\Delta T_{p}(t)$) and secondary ($\Delta T_{s}(t)$) thermal junctions, by changing the initial temperature of one of the qubits, say the third qubit $q_{3}$. Here we set the initial temperature of $q_{1}$ as $T_{1}(0)=\tau_{1}=1.0$ and the initial temperature of $q_{2}$ as $T_{2}(0)=\tau_{2}=2.0$, and vary the value of $T_{3}(0)$. For the initial investigation, we set the initial temperature of $q_{3}$ as $T_{3}(0)=\tau_{3}=3.0$. This ensures that at time $t=0$, we have $\Delta T_{P}^{i}=\Delta T_{P}(0)=|T_{1}(0)-T_{2}(0)|=1.0$ and $\Delta T_{S}^{i}=\Delta T_{S}(0)=|T_{3}(0)-T_{2}(0)|=1.0$. Hence, the difference between the primary and secondary junctions starts at zero. If we now allow the system to evolve under the influence of the thermal baths, we can observe that the difference between $\Delta T_{p}(t)$ and $\Delta T_{s}(t)$ increases over time until it saturates at a value of $1.768$. See the first panel of Fig. 2. Subsequently, as the initial temperature of $q_{3}$ decreases incrementally compared to its previous value in different scenarios, as studied in the subsequent panels of Fig. 2, the differences between the saturated values of $\Delta T_{p}(t)$ and $\Delta T_{s}(t)$ reduce, and their differences also decrease. Thus, by adjusting the initial temperature of $q_{3}$, we can effectively regulate the performance of the transformer.

Now, the performance of the QHTs can be well understood by defining a parameter, capacity of thermal control ($\mathcal{C}_{H}$), which quantifies the ability of a QHT to either decrease or increase the temperature of the secondary thermal junction compared to the primary one, for step-down or step-up operation modes respectively. It delineates the degree of temperature alteration achievable in both step-down and step-up operational modes. Let us consider, at any given time $t=t_{f}$, the temperature difference at the primary junction be $\Delta T_{p}^{f}$, and at the secondary junction be $\Delta T_{s}^{f}$. Hence, the capacity of thermal control of a QHT can be defined as

For a step-down quantum transformer, $\mathcal{C}_{H}>0$. Conversely, for a step-up QHT, $\mathcal{C}_{H}<0$. Based on the analysis of Fig. 2, it is evident that, over a sufficiently long period, the difference $\Delta T=\Delta T_{p}(t)-\Delta T_{s}(t)$ varies with alterations in the initial temperature of $q_{3}$, represented by $\tau_{3}$. Moreover, we can control $\Delta T$ by adjusting the initial temperature of $q_{2}$ or $q_{1}$ as well. Therefore, $\mathcal{C}_{H}$ can be regulated by the initial temperature of qubits. The relationship between $\mathcal{C}_{H}$ and $\tau_{3}$ is depicted in Fig. 3. Here, we have evaluated $\mathcal{C}_{H}$ at time $t_{f}=0.5\times 10^{5}$. We observe that $\mathcal{C}_{H}$ monotonically increases with the increase of $\tau_{3}$.

If we examine the interaction Hamiltonian $\mathcal{H}_{\text{int}}^{1}$ and its self-contained condition, it becomes evident that it yields a negative $\mathcal{E}_{3}$ value, regardless of the chosen values for $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$. Consequently, for qubit $q_{3}$, the state $\ket{0}$ corresponds to the ground state, while the state $\ket{1}$ corresponds to the excited state. When we examine each qubit individually, we observe that the temperatures of $q_{1}$ and $q_{2}$ increase over time $t$. However, the temperature of $q_{3}$ initially decreases for a short period before stabilizing back near to its initial value. So, this QHT scheme is not as straightforward as the previous two cases discussed for $\mathcal{H}_{\text{int}}^{2}$ and $\mathcal{H}_{\text{int}}^{3}$. The most effective approach to comprehend this protocol is by analyzing the rates of temperature change relative to the evolution time, rather than focusing solely on the individual temperature changes of each qubit. The rationale behind constructing the QHT framework using $\mathcal{H}_{\text{int}}^{1}$ lies in the fact that the rates of temperature change with respect to evolution time $\left(\frac{\partial T_{j}(t)}{\partial t}\right)$ differ among the qubits. From Fig. 2, we can infer that the profile of the temperatures is oscillatory, indicating that the quantities $\frac{\partial T_{j}(t)}{\partial t}$ will also oscillate around zero, having positive and negative values. Therefore, the rate at which the temperature increases will depend on the absolute values of $\frac{\partial T_{j}(t)}{\partial t}$. Now, to increase the temperature difference between two qubits associated with the primary junction $(\Delta T_{p}(t))$ over time through the interaction Hamiltonian $\mathcal{H}_{\text{int}}^{1}$, it is crucial for the absolute rate of temperature increase of $q_{1}$ $\left(\Big{|}\frac{\partial T_{1}(t)}{\partial t}\Big{|}\right)$ to be less than the absolute rate of temperature increase of $q_{2}$ $\left(\Big{|}\frac{\partial T_{2}(t)}{\partial t}\Big{|}\right)$. Consequently, to reduce the temperature difference between two qubits associated with the secondary junction $(\Delta T_{s}(t))$ over time, it is necessary for the rate of temperature increase of $q_{3}$ $\left(\Big{|}\frac{\partial T_{3}(t)}{\partial t}\Big{|}\right)$ to be less than the rate of temperature increase of $q_{2}$ $\left(\Big{|}\frac{\partial T_{2}(t)}{\partial t}\Big{|}\right)$. Additionally, between $q_{1}$ and $q_{3}$, the rate of temperature increase of $q_{1}$ should be less than the rate of temperature increase of $q_{3}$. So, we can write the order of rate of change of temperature of qubits in QHT model involving the interaction Hamiltonian $\mathcal{H}_{\text{int}}^{1}$ as follows,

For a deeper understanding of these observations, let us consider the scenario depicted in the first panel of Fig. 2. By examining the rate of temperature change for each qubit for this case, as shown in Fig. 4, we find that the absolute values of the rates $\Big{|}\frac{\partial T_{j}(t)}{\partial t}\Big{|}$ for $j=1$, $2$, and $3$, satisfy Eq. (10) in the transient regime. This fulfillment leads to the realization of an effective QHT (see the first panel of Fig. 2). Hence, it is evident that for various sets of initial conditions, we can design three-qubit self-contained QHTs with different interaction Hamiltonians. The operation of these QHTs is contingent upon different conditions related to the local temperature or the change in temperature of the qubits over time. This highlights the strength versatility of the three-qubit QHT model, as it can be tailored to construct a perfectly self-contained QHT model for various initial conditions. We have the flexibility to choose from various configurations for the quantum heat transformer and subsequently select its self-contained interaction Hamiltonian, thus enabling the creation of an intended QHT model.

Now, for completeness, we proceed to discuss another example of a QHT model involving the interaction Hamiltonian $\mathcal{H}_{\text{int}}^{2}$, which has already been briefly introduced earlier. In this QHT model, we take the initial temperatures of the three qubits in the order $\tau_{1}>\tau_{2}>\tau_{3}$. As previously mentioned, we configure the parameters of the systems and the interaction Hamiltonians such that, $\mathcal{P}^{\text{in}}_{101}>\mathcal{P}^{\text{in}}_{010}$, with $\mathcal{P}^{\text{in}}_{101}=(1-p_{1}^{\text{in}})p_{2}^{\text{in}}(1-p_{3}^{\text{in}})$ and $\mathcal{P}^{\text{in}}_{010}=p_{1}^{\text{in}}(1-p_{2}^{\text{in}})p_{3}^{\text{in}}$. In Fig. 5, we present the operational characteristics of the QHT for this configuration. The temperature gradients of the primary and secondary junctions, $\Delta T_{p}(t)$ and $\Delta T_{s}(t)$, exhibit a different behavior compared to the case described in Fig. 2. Initially, both $\Delta T_{p}(t)$ and $\Delta T_{s}(t)$ oscillate,and then gradually decrease to reach a steady-state value over time. Furthermore, as the initial temperature of $q_{3}$ decreases, the steady-state difference between $\Delta T_{p}(t)$ and $\Delta T_{s}(t)$ increases. This indicates that for steady-state regime, $\mathcal{C}_{H}$ increases with the decrease of $\tau_{3}$. This behavior contrasts with the scenario depicted in Fig. 2.

So far, our investigation has focused on cases where the QHT models function as step-down transformers in both steady-state and transient regimes. These models are undeniably crucial for designing effective step-down quantum transformers. However, in certain scenarios, the initial conditions may impose constraints such that they yield a steady-state step-up transformer, whereas the requirement is for a step-down operation. In such cases, a necessarily transient step-down transformer model can prove to be beneficial. The term “necessarily transient step-down transformer” implies that the desired step-down mode can be achieved within the transient domain in an steady-state step-up QHT model. These models hold significance, particularly for specific initial conditions. In the next section, we will present an example of a necessarily transient step-down quantum heat transformer and delve into its operational characteristics.