Thermal transistor effect in quantum systems

TL;DR

This paper demonstrates how a three-qubit quantum system can function as a thermal transistor, with the interaction among qubits and system-bath characteristics being crucial for the effect, offering a more robust and realistic model.

Contribution

It introduces a new quantum thermal transistor model using three interacting qubits, extending parameter regimes and improving robustness over previous proposals.

Findings

The interaction among qubits is essential for the transistor effect.

The model is more robust to parameter variations.

Potential for implementation in quantum spin chains and nanomagnets.

Abstract

We study a quantum system composed of three interacting qubits, each coupled to a different thermal reservoir. We show how to engineer it in order to build a quantum device that is analogous to an electronic bipolar transistor. We outline how the interaction among the qubits plays a crucial role for the appearance of the effect, also linking it to the characteristics of system-bath interactions that govern the decoherence and dissipation mechanism of the system. By comparing with previous proposals, the model considered here extends the regime of parameters where the transistor effect shows up and its robustness with respect to small variations of the coupling parameters. Moreover, our model appears to be more realistic and directly connected in terms of potential implementations to feasible setups in the domain of quantum spin chains and molecular nanomagnets.

Figures

Tables

Table 1. Table 1: The possible transitions of positive frequency mediated by any of the three thermal baths by means of the interaction operator σ p x superscript subscript 𝜎 𝑝 𝑥 \sigma_{p}^{x} for the case of Sec.  III.2 . The first column gives the initial states, starting from the one with highest energy, whilst the other columns specify the qubit operator coupled to the corresponding thermal reservoir as prescribed by Eq. ​ ( 3 ) Eq. 3 \text{Eq.}\left(\ref{eq:hamiltonians}\right) , and the final states connected to the initial ones.

	${\sigma }_L^x$	${\sigma }_M^x$	${\sigma }_R^x$
$|{ϵ}_1⟩$	$|{ϵ}_2⟩$, $|{ϵ}_3⟩$, $|{ϵ}_5⟩$	$|{ϵ}_2⟩$, $|{ϵ}_3⟩$, $|{ϵ}_5⟩,|{ϵ}_8⟩$	$|{ϵ}_2⟩$, $|{ϵ}_3⟩$, $|{ϵ}_5⟩$
$|{ϵ}_2⟩$	$|{ϵ}_4⟩$, $|{ϵ}_6⟩$	$|{ϵ}_4⟩$, $|{ϵ}_6⟩,|{ϵ}_7⟩$	$|{ϵ}_4⟩$, $|{ϵ}_6⟩$
$|{ϵ}_3⟩$	$|{ϵ}_4⟩$, $|{ϵ}_7⟩$	$|{ϵ}_4⟩$, $|{ϵ}_6⟩,|{ϵ}_7⟩$	$|{ϵ}_4⟩$, $|{ϵ}_7⟩$
$|{ϵ}_4⟩$	$|{ϵ}_8⟩$	$|{ϵ}_5⟩,|{ϵ}_8⟩$	$|{ϵ}_8⟩$
$|{ϵ}_5⟩$	$|{ϵ}_6⟩$, $|{ϵ}_7⟩$	$|{ϵ}_6⟩,|{ϵ}_7⟩$	$|{ϵ}_6⟩,|{ϵ}_7⟩$
$|{ϵ}_6⟩$	$|{ϵ}_8⟩$	$|{ϵ}_8⟩$	$|{ϵ}_8⟩$
$|{ϵ}_7⟩$	$|{ϵ}_8⟩$	$|{ϵ}_8⟩$	$|{ϵ}_8⟩$

Equations

$$H_S=\frac{1}{2}\sum _{p=L,M,R}{\omega }_p{\sigma }_p^z+\sum _{p\ne q}{\hat{\mathbf{S}}}_p^T\lambda {\hat{\mathbf{S}}}_q.$$

$$H_S=\frac{1}{2}\sum _{p=L,M,R}{\omega }_p{\sigma }_p^z+\frac{1}{2}\sum _{p\ne q}{\eta }_{pq}\sum _{i=x,y,z}{\lambda }^i{\sigma }_p^i{\sigma }_q^i,$$

$$\begin{array}{rl}{H}_B& =\sum _{p=L,M,R}\sum _k{\omega }_k{a}_k^{p†}{a}_k^p\\ H_I& =\sum _{p=L,M,R}{\sigma }_p^x\otimes \sum _k{g}_k^p\left(a_k^{p†}+a_k^p\right),\end{array}$$

$$\dot{\rho }=-i[H_S+H_{LS},\rho ]+\sum _{p=1}^3{\mathcal{L}}_p[\rho ],$$

$$\mathcal{L}_{p}[\rho]=\sum_{\omega}\gamma(\omega)\Bigl{[}A_{p}(\omega)\rho A_{p}^{\dagger}(\omega)-\frac{1}{2}\{\rho,A_{p}^{\dagger}(\omega)A_{p}(\omega)\}\Bigr{]},$$

$$\sum _p{J}_p=\frac{\partial ⟨H_S⟩}{\partial t}.$$

$$J_p=\text{Tr}\left({\mathcal{L}}_p[\rho ]H_S\right).$$

$${\chi }_s={\left(\frac{\partial J_s}{\partial T_M}\right)}_{T_s=\text{const}}^{-1},s=L,R,$$

$${\alpha }_s=\frac{\partial J_s}{\partial J_M}=-\frac{{\chi }_s}{{\chi }_L+{\chi }_R}.$$

$$T_p^{\text{loc}}=\frac{{\omega }_p}{\ln ({\rho }_p^0/{\rho }_p^1)}.$$

$${\overline{\alpha }}_s^{i,j}=\int d{\lambda }^{i}d{\lambda }^j{\alpha }_s({\lambda }^i,{\lambda }^j)\mathcal{N}({\overline{\lambda }}^{i,j},{\Sigma }^{i,j}),$$

$${\zeta }^{i,j}=\frac{{\overline{\alpha }}_L^{i,j}}{{\alpha }_L^{\text{unp}}},$$

$$\frac{1}{2}\left(\begin{array}[]{cccccccc}\Omega_{LM}+\omega_{R}&0&0&\Lambda_{MR}^{x}&0&0&\Lambda_{LM}^{x}&0\\ 0&\Omega_{LM}-\omega_{R}&\Lambda_{MR}^{x}&0&0&0&0&\Lambda_{LM}^{x}\\ 0&\Lambda_{MR}^{x}&\Omega_{LR}-\omega_{M}&0&\Lambda_{LM}^{x}&0&0&0\\ \Lambda_{MR}^{x}&0&0&\omega_{L}-\Omega_{MR}&0&\Lambda_{LM}^{x}&0&0\\ 0&0&\Lambda_{LM}^{x}&0&-\omega_{L}+\Omega_{MR}&0&0&\Lambda_{MR}^{x}\\ 0&0&0&\Lambda_{LM}^{x}&0&\omega_{M}-\Omega_{LR}&\Lambda_{MR}^{x}&0\\ \Lambda_{LM}^{x}&0&0&0&0&\Lambda_{MR}^{x}&-\Omega_{LM}+\omega_{R}&0\\ 0&\Lambda_{LM}^{x}&0&0&\Lambda_{MR}^{x}&0&0&-\Omega_{LM}-\omega_{R}\\ \end{array}\right),$$

$$\begin{array}{rl}|{ϵ}_1⟩& =-a_1|111⟩-a_2|100⟩-a_3|010⟩-a_4|001⟩\\ |{ϵ}_8⟩& =+a_1|000⟩-a_2|011⟩+a_3|101⟩-a_4|110⟩\\ |{ϵ}_2⟩& =+b_1|000⟩+b_2|011⟩+b_3|101⟩+b_4|110⟩\\ |{ϵ}_7⟩& =+b_1|111⟩-b_2|100⟩+b_3|010⟩-b_4|001⟩\\ |{ϵ}_3⟩& =+c_1|000⟩-c_2|011⟩-c_3|101⟩+c_4|110⟩\\ |{ϵ}_6⟩& =-c_1|111⟩-c_2|100⟩+c_3|010⟩+c_4|001⟩\\ |{ϵ}_4⟩& =+d_1|111⟩-d_2|100⟩-d_3|010⟩+d_4|001⟩\\ |{ϵ}_5⟩& =-d_1|000⟩-d_2|011⟩+d_3|101⟩+d_4|110⟩\end{array},$$

Full text

I Introduction

The exploitation of classical thermodynamics has conducted to the technological revolution that shaped our world since the Nineteenth century Carnot . In recent years the exploration of the thermodynamics of quantum systems has given birth to quantum thermodynamics: a high impact research field either on the fundamental level and the applicative one QthermoBook . In analogy to classical models, quantum heat engines have been proposed and realized eng1 ; eng2 ; eng3 . In particular, the properties of non-equilibrium open quantum systems has been employed to study how to obtain many-body entanglement BA15 or efficient flux management LBA15 . Moreover, the implementation of non-equilibrium quantum heat machines has been discussed in solid state set-ups CPF15 or in more complex scenarios Ali15 .

The ability to manipulate quantum resources is a demanding but fruitful task that has to be pursued in order to build novel devices. A potential high impact apparatus will be the one aiming at the control of thermal energy transport in quantum systems, and at the amplification of heat fluxes among the different parts constituting a composite system. A promising approach in the aforementioned task is to pursue the flourishing path followed in electronics after the realization of rectifiers and transistors with semi-conducting materials BB48 that paved the way to build logic gates and to the information and computational technology millman .

Typically, thermal and electric currents phenomena are empirically well described by the Fourier’s and Ohm’s laws, respectively. In fact, the functional dependence upon the two respective control variables, temperature and voltage, is of the same type. Recently, evidences of the emergence of the Fourier’s law has been pointed out also in the quantum realm Qfourier1 ; Qfourier2 .

The proposal of such thermal analogues of the electronic rectifier and transistor is based on the assumption that in a thermal set-up the role played by batteries could be played by thermostats. This analogy allowed to come forward with various proposals of mesoscopic thermal devices such as rectifiers and diodes TPC02 ; LWC04 and transistors LLC06 , the latter operating either in near field NearField or in far field FarField regime of thermal radiation, and also thermal logic gates gate ; RevPhononics .

The theoretical efforts are bolstered by the experimental ingenuity involving a wide range of experimental platforms ranging from carbon and boron nitride nanotube structures carbnano and bulk oxide materials oxmat to semiconductors quantum dots qdots , magnonic systems DiodeMag and phase changing materials, such as vanadium dioxide VO2 vanad1 ; vanad2 , and ceramics materials that below a critical temperature behave like high temperature superconductors and above it as dielectrics ceram1 ; ceram2 .

In the same time, the quest of atomic-scale devices for quantum computational purposes is increasingly pushing the necessity to control and manipulate single or few atoms systems. In particular, the isolation and coherent manipulation of single spins, that are one of the best candidates for a non-optical implementation of a single qubit, is achieved using optical traps and electrical techniques manip . Recently, several novel quantum technology devices have been proposed, including isolators based on photonic transitions qtech1 , rectifiers qtech2 ; qtech3 , transistors in an electromagnetic controlled environment qtech4 ; qtech5 , and also phonon-thermoelectric transistors qtech6 .

Moreover, quantum systems suffer of an unavoidable coupling to their environment, typically modeled as a thermal reservoir of much bigger dimensions of those of the system BP02 ; Schaller . Despite some detrimental phenomena related to decoherence and dissipation, control and engineering of interacting quantum systems coupled with different thermal baths have led to several studies pointing out how to build the smallest thermal refrigerator refr1 ; refr2 , how to rectify a thermal current at the very quantum level QTD1 ; QTD2 ; QTD3 and recently the building blocks of a quantum thermal transistor have been discussed considering an integrable spin-chain model QTT and in a qubit-qutrit system QTT2 .

In this paper we analyze the ability to design a quantum thermal transistor employing a spin ring that can, in principle, be implemented on a molecular nanomagnet NanoMag . In comparison to a previous model QTT , we consider a more complex scenario where the collective quantum behavior of few spins is expected to play a crucial role in the functioning of the device.

The paper is structured as follows. In Section II, we examine the spin-ring model outlining the frameworks where the theoretical model finds a direct implementability. This section is also devoted to the quantum dissipative process that allows the system to exchange heat fluxes. In Section III, we discuss how the proposed systems can be adopted to build a quantum thermal transistor and some limits of their applicability. Having in mind a direct experimental implementation, in Section IV the robustness of the transistor effect against unavoidable fluctuations and perturbations is addressed. Section V concludes the paper with some final remarks and prospects. Some details can be found in the Appendix.

II The spin-ring model

Quantum critical spin chains have longly been considered as one of the best substrates to test and implement the future quantum computing and the novel quantum devices. Their main characteristic is to show wide versatility RevQC . In fact, considering the most general model for quantum magnetism, described by the Heisenberg Hamiltonian, one can explore a great variety of universality class by an appropriate tuning of the interaction strengths. Moreover, recent modern developments allow to realize spin chains of few atoms having net spin $s=1/2$, opening the road to the exploitation of the physics of some well known models, viz., Ising, XY and XXZ ones chain1 ; chain2 ; ExpIsing1 ; ExpIsing2 ; ExpIsing3 .

In parallel, molecular nanomagnets NanoMag are also attracting more interest as a feasible platform for quantum technological purposes TZ13 . Specifically, molecular nanomagnets, composed by $N=3$ main units, have been proved to be useful for quantum thermometry LMG14 and for coherent-manipulation of three-qubit states 3QMag .

Spin chains and nanomagnets are theoretically well described by the same effective Hamiltonian upon fixing the configuration geometry. Here, we consider a system of three two-level systems (qubits) of frequency $\omega_{p}$ $(p=L,M,R)$, embedded in a magnetic field, and we introduce the vector of the effective spin operators of any qubit $\hat{\mathbf{S}}_{p}=(\sigma^{x}_{p},\sigma^{y}_{p},\sigma^{z}_{p})^{T}$ and the 3-by-3 matrix $\boldsymbol{\mathsf{\lambda}}$ governing the coupling between spins along different polarization axis, where $\sigma^{i}_{p}$ ($i=x,y,z$) is the $i$-th Pauli matrix of the $p$-th qubit. A realistic description of a triangular spin chain with a qubit in each vertex labeled $L$, $M$ and $R$, as depicted in Fig. 1, is given by the following Hamiltonian (hereafter the reduced Plank constant and the Boltzmann constant are set equal to one, $\hbar=k_{\textup{B}}=1$):

[TABLE]

The first term is the sum of single particle free Hamiltonians of three qubits immersed in a magnetic field pointing along the direction of the $z-$axis and it is responsible of the non-zero field splitting of each qubit, while the second term is the intra-chain spin-spin exchange coupling. Such an Hamiltonian takes into account symmetric and antisymmetric exchange interactions. A proper tuning of $\boldsymbol{\mathsf{\lambda}}$ can give a situation where only the former type of interactions occurs, like in the standard Heisenberg Hamiltonian, or can introduce a coupling between non-homologous components of the spin operators, as in the Dzyaloshinskii-Moriya interaction Dz58 ; Mor60 . We omit here the overcomplexity introduced by an antisymmetric exchange, hence we assume that the tensor $\boldsymbol{\mathsf{\lambda}}$ contains only symmetric terms.

This allows us to consider spins interacting among them through an Heisenberg type Hamiltonian. Therefore, specifying their couplings $\eta_{pq}$ with $p,q=L,M,R$ and $p\neq q$, and the strengths along different polarization axis $\lambda^{i}$, the Hamiltonian of the system reads:

[TABLE]

where we have chosen equal $\lambda^{i}$ for the three qubits.

This Hamiltonian describes the simplest non-trivial example of spin ring whose properties can be observed with good approximation in a $\{\text{Cu}_{3}\}$-Type nanomagnet, a complex having an almost equilateral triangular shape with a $\text{Cu}^{2+}$ ion of spin$-1/2$ at each vertex Cu3 .

II.1 Comparison between two Ising-type rings

Joulain et al. in their seminal paper QTT have addressed how a thermal transistor can be realized with a quantum system of three interacting qubits, coupled to a thermal reservoir each, and showed how it is analogous to an electronic bipolar one. Their discussion is based on a peculiar choice of a very simple model, that implies a lot of strong assumptions on the underlying physics.

Let us refer to the model adopted in QTT as the $z-$axis. In fact, there the magnetic field applied on the three qubits is longitudinal to the direction of the spin-spin interactions. Being more specific, the exchange interaction in $H_{S}$ of $\text{Eq.}\left(\ref{eq:hsys}\right)$ is non-zero only along the same direction of the magnetic field, i.e. only $\lambda^{z}\neq 0$. Among the other assumptions, this is the one that, in our opinion, requires now a detailed discussion. In fact, this premise, along with some other minor approximations, has the advantage to let the model and the dissipative dynamics be solvable with analytical techniques, since the operators $\sigma_{p}^{z}\sigma_{q}^{z}$ commute with the free Hamiltonian. Essentially, the only effect of the zero field splitting terms $\sigma_{p}^{z}\sigma_{q}^{z}$ is to modify the eigenenergies of the system, but the collective eigenfunctions remain the same and can be still expressed as a tensor product of single-particle eigenstates. We note that this kind of spin ring, even if considered in a quantum context, is a semi-classical model. In fact, the spin$-1/2$ operators $\sigma^{z}_{p}$ can be alternatively viewed as a classical magnetic moment pointing only along two opposite directions. The $z-$axis Ising model, hence, takes correctly into account a quantization of the magnetic moment, but it does not consider quantum fluctuations due to other spatial components of the same spin operator $\hat{\mathbf{S}}_{p}$ and any possible critical behavior is disregarded.

It is worth noting that in order to propose a truly quantum model a more complex interaction should be taken into account. The reason why we discuss the class of Hamiltonians defined in $\text{Eq.}\left(\ref{eq:hsys}\right)$, where, in general, the sum of the single particle contributions does not commute with the interaction term is twofold: (i) from a purely theoretical point of view, these systems may exhibit strongly correlated and nonlocal eigenstates, opening the door to phenomena with an enhancement due to quantum correlations or criticality RevQC ; (ii) from an experimentally inspired perspective, nowadays, the control of nanomagnets, spin chains or even quantum dot molecules, has been well established for quantum critical systems ExpIsing1 ; ExpIsing2 ; ExpIsing3 .

For the aforementioned reasons our aim is then to propose a well established quantum setting, richer than the one considered in QTT , where the thermal transistor effect displays and to study its robustness with respect to the operating temperature and against spurious fluctuations of the exchange interactions.

In particular, we will mainly focus on the case of magnetic systems that belong to the universality class of the quantum critical Ising model in a transverse field and we will refer shorthand to it as the $x-$axis Ising model. It can be realized considering in $\text{Eq.}\left(\ref{eq:hsys}\right)$ that the coupling among the qubits acts only along a direction (that we name $x$) transverse to the applied magnetic field, i.e. only $\lambda^{x}\neq 0$. We remark that once the coupling with the environment is added (see next sub-section), the different directions transverse to the applied magnetic field are not equivalent anymore. We address in Section IV the presence of perturbations on both the $x$ and $z$ configurations.

The $x-$axis Ising model introduced as the exemplary model to study quantum phase transitions in magnetic systems and their critical behaviors has been employed in a wide range of protocols to characterize and fulfill quantum computational tasks RevQC . The main feature of such a model resides in the fact that its collective states are no longer a tensor product of the single particles states, but are non-separable. Its pivotal role for the comprehension of magnetism at the quantum scale has fostered the research of a model magnet where its properties can be experimentally measured with different techniques. Nowadays, the most germane realization has been achieved in the low-energy magnetic excitation of the insulator LiHoF4 ExpIsing1 or with crystals of the ferromagnetic CoNb2O6 where the spin resides on the Co2+ ion ExpIsing2 . A detailed presentation of the spectrum of the $x-$axis Ising model for the purposes of this paper is shown in Appendix A, while a complete discussion of the spectrum of $z-$axis one can be found in QTT .

II.2 Dissipative dynamics

Each qubit composing the spin ring described above suffers of an unavoidable coupling with its surrounding environment, that is modeled as a thermal bosonic reservoir. The temperatures of the three reservoirs are, in general, different, giving rise to an out of thermal equilibrium scenario where the temperatures are meant to be tunable at will. The total Hamiltonian is then $H=H_{S}+H_{B}+H_{I}$, where the bath and the system-bath Hamiltonians, respectively, read:

[TABLE]

where $a_{k}^{p}$ and $a_{k}^{p\,\dagger}$ are the bosonic operators of bath $p$ and $g_{k}^{p}$ are the coupling strengths. A schematic representation of the global system is sketched in Fig. 1. As spin ring Hamiltonian $H_{S}$, we consider the one of $\text{Eq.}\left(\ref{eq:hsys}\right)$.

The dynamics of the 3-qubit system is dissipative and, under the Born-Markov and secular approximations BP02 , the evolution of the density matrix is described by a master equation of the following form:

[TABLE]

where $H_{LS}=\sum_{p,\,\omega}s_{p}(\omega)A_{p}^{\dagger}(\omega)A_{p}(\omega)$ is the Lamb shift Hamiltonian, $A_{p}(\omega)=\sum_{\omega=\epsilon_{i}-\epsilon_{j}}|\epsilon_{j}\backslash\!\!\!/\epsilon_{j}|\sigma_{p}^{x}|\epsilon_{i}\backslash\!\!\!/\epsilon_{i}|$, $|\epsilon_{i}\rangle$ with $i=1,...,8$ are the dressed states of the Hamiltonian $H_{S}$, as described in the Appendix A, and the Lindblad operators are given by BP02

[TABLE]

where $\gamma_{p}(\omega)=\mathcal{J}_{p}(\omega)[1+n_{p}(\omega)]$ for $\omega>0$ and $\gamma_{p}(\omega)~{}=~{}\mathcal{J}_{p}(|\omega|)n_{p}(|\omega|)$ for $\omega<0$. The average number of excitations and the spectral density of the $p$-th reservoir are respectively $n_{p}(\omega)=[\mathrm{exp}\frac{\omega}{T_{p}}-1]^{-1}$ and $\mathcal{J}_{p}(\omega)$. In all the numerical computations, we will choose Ohmic reservoirs characterized by a linear spectral density $\mathcal{J}_{p}(\omega)=\kappa_{p}\omega$ (at least in the range of working frequencies), using values of $\kappa_{p}$ different for the three baths in order to suitably engineer the effect of the environments on the three qubits. It follows that $\gamma_{p}(0)=\mathrm{lim}_{\omega\rightarrow 0^{+}}\mathcal{J}_{p}(\omega)n_{p}(\omega)=\kappa_{p}T_{p}$. In the Appendix A, a scheme of the transitions induced by the three thermal baths is also reported.

We remark that as we will consider a strong coupling between the three qubits (comparable or higher than the bare frequencies $\omega_{p}$), we need the above microscopic approach to derive the Lindblad operators responsible of the non-unitary evolution of the density matrix, avoiding any extra simplification that can be somehow justified when the spin-spin interaction is much smaller than the bare frequencies of the qubits BP02 ; QTD1 . In this limit, we cannot consider phenomenological master equations where the operators $A(\omega)$ are built starting from the eigenstates of the bare system Hamiltonian. For the $z-$axis Ising model, these coincide with the dressed eigenstates, while for the $x-$type this is not the case.

We also note that since $H_{LS}$ commutes with $H_{S}$, it only leads to a renormalization of the unperturbed energy levels of $H_{S}$ induced by the coupling with the reservoirs and that the steady state is independent on it.

The three temperatures of the environment will be in general chosen to be different, leading to out of thermal equilibrium steady states, which are not expected to be three-qubit thermal states. At the same time, we expect that each qubit will not thermalize (its own reduced state) to the temperature of its own reservoir because of the collective nature of the quantum dynamics.

III Quantum thermal transistor

The particular operating principle of an electronic transistor makes possible to regulate the currents at two of its terminals, that could take also very high values, regulating a much smaller current injected through a third terminal. This peculiarity made the transistor particularly suitable to build logic gates commonly used in nowadays electronic. We propose here a thermal analogue of this device able to amplify some of the heat currents circulating in a system as in Fig. 1. The control parameter that will play the role of the gate potential in an electronic transistor is the temperature $T_{M}$ of the reservoir coupled to the middle qubit of frequency $\omega_{M}$.

III.1 Heat currents in a thermal transistor

A straightforward way to take into account the heat flow in and out a quantum system kosloff , is to link the variation of its mean energy $\langle H_{s}\rangle$ to the sum of the heat currents $J_{p}$ exchanged between the quantum system and each bath (no power comes from other external sources in our model):

[TABLE]

By substituting $\text{Eq.}\left(\ref{eq:master}\right)$ in the previous expression, one can compute the current $J_{p}$ that each thermostat exchanges with the system as

[TABLE]

We note that at the steady state $\rho_{ss}$, it holds $\dot{\rho}_{ss}=0$, so that the total energy is conserved in time. This implies that $\sum_{p}J_{p}=0$. A minor remark concerns the fact that the Lamb shift term appearing in $\text{Eq.}\left(\ref{eq:master}\right)$ does not contribute neither to the final expression of $J_{p}$ neither to the steady state, so that the steady currents are independent on it. In the following, we always address steady configurations and we use $J_{p}$ to indicate steady currents.

Following the geometry of the configuration of Fig. 1, we will refer to left, middle and right currents ($J_{L}$, $J_{M}$ and $J_{R}$) to those exchanged, collectively, between the spin ring and, respectively, the left, the middle and the right thermostat. From now on, we assume that the hot reservoir, providing the energy to the system, is the left one, whereas the cold one, that adsorbs the heat flux, is the one placed to the right, and finally the bath in the middle acts as a control. Borrowing the familiar terminology of the bipolar transistor, the right qubit is playing the role of the emitter, while the left one of the collector and the middle one is indeed the analogue of the base.

The thermal transistor effect happens when a small change in the the control temperature $T_{M}$ produce a significant variation of the two lateral currents in contrast with a tiny variation of $J_{M}$. To study the occurrence of this effect we define the differential thermal resistances

[TABLE]

and, as in the spirit of the electronic transistor BB48 , we introduce a dynamical amplification factor $\alpha_{s}$, function of the control temperature $T_{M}$, defined as

[TABLE]

The adimensional parameters in $\text{Eq.}\left(\ref{alpha}\right)$ are the figures of merit used to have a quantitative benchmark of the presence of the thermal transistor effect and they satisfy the relation $\alpha_{L}+\alpha_{R}=-1$. In particular, for regions where $|\alpha_{s}|\gg 1$ one can infer that we are in presence of a strong amplification of the lateral currents in comparison to that controlled by the heat bath in central position. Therefore, it is evident that to amplify the thermal currents one of the two differential thermal resistances should be negative. We underline that this condition was already discussed in the context of non-linear lattices LLC06 , three-terminal graphene devices graphene and band structured engineered silicene superlattices silicene .

III.2 Thermal transistor in an $x-$axis Ising model

As said in Sec. II.1 the main case we want to discuss is when the three qubits are described by an $x-$axis Ising model. In particular, we choose open boundary condition, i.e. $\eta_{LR}=0$.

In Fig. 2 we show the behavior of each current. In order to deal with adimensional quantities we plot the currents $J_{p}$ in units of $\kappa\Delta^{2}$, where $\kappa=\kappa_{L}$ is the parameter characterizing the strength of the dissipation of the left qubit and $\Delta=\omega_{L}$, $\omega_{L}$ being assumed as the reference frequency. The currents are function of the temperature $T_{M}$, given in units of $\Delta$. We observe that once expressed also $\kappa_{M}$ and $\kappa_{R}$ in units of $\kappa$, the steady state of $\text{Eq.}\left(\ref{eq:master}\right)$ is invariant with respect to $\kappa$, while the currents are just proportional to it. Its value must just be small enough to guarantee the validity of the approximations used to derive the master equation of $\text{Eq.}\left(\ref{eq:master}\right)$. In the following, all the values of temperatures are given in units of $\Delta$, while for currents they are in units of $\kappa\ \Delta^{2}$. It is possible to appreciate in Fig. 2 how the thermal transistor effect manifests itself and how the amplification of $J_{L}$ and $J_{R}$ is continuously achieved.

We stress here the crucial role played by the non-equilibrium steady state. In fact, the presence of three environments at different temperatures plugged to distant sites of a magnetic system is the responsible of the three heat flows circulating into it. In particular, it is not surprising observing that even at $T_{M}=0$ each of the currents has a non zero value.

In order to better describe the emergence of the thermal transistor effect, the amplification factors $\alpha_{s}$ are plotted in Fig. 3, where it is possible to see how the two lateral currents are amplified. Notably, around $T_{M}\simeq 0.600$, the amplification factor diverges, due to a local minimum of $J_{M}\simeq-2.90\times 10^{-3}$. To this extent, we can circumscribe the best working region of this quantum thermal transistor to an interval of the control quantity centered around this value. However, a plateau is present where the amplification factors have values still significantly high. In particular, at $T_{M}\simeq 4.17$, where $J_{M}=0$, $\alpha_{L}\simeq 18.3$. As a commentary, we notice that for this value of $T_{M}$, the reservoir $M$ does not need to inject or absorb energy to maintain the system in the steady state, or in other words we can say that the central reservoir is in a situation of dynamical thermal equilibrium between the hot and the cold heat baths.

We also remark that it is easy to see that the right qubit is the one showing $\chi_{R}<0$. We have thus shown how a purely quantum scenario can be used to implement a negative differential resistance device. Other choices of the values of the parameters entering either in $H_{S}$ and in the dissipator of $\text{Eq.}\left(\ref{eq:master}\right)$ give a behavior of the currents and of the amplifications analogous to the ones in Figs. 2 and 3. This has been observed in the case of asymmetric configurations.

III.3 Thermal transistor in a $z-$axis Ising model

In order to better assess the value of the model here proposed, we compare it with the model previously studied in QTT . There, the thermal transistor effect had been observed in a range of temperatures much narrower than in our case. We put ourself in the same zone of parameters identified in Fig. 2 and we look if it is possible to extend the working region to larger zones of temperatures as in the $x-$axis case.

In Fig. 4, we show that in the case of the $z-$axis Ising model the thermal transistor effect works only in a narrow interval of the control temperature, namely $T_{M}\in[0,0.2]$. When moving to higher values of temperature, the effect completely vanishes, no matter how large the thermal gradient between the hot and cold heat baths is. In fact, in our analysis we have also increased the difference $T_{R}-T_{L}$ in comparison to what has been shown in QTT and observed that the current amplification is always achieved in the same narrow working region.

Recently, the appearance of current amplification has been discussed in a model based on a qubit-qutrit system. Also in this case, the thermal transistor works in a narrow interval of the control temperature in comparison to the $x$-type Ising model analyzed in Sec. III.2, QTT2 .

III.4 Local behavior

Here, we discuss the possibility to interpret the results of Fig. 2 on the basis of considerations involving only the local behavior of the qubits composing the spin ring. Firstly, we can have access to some local features introducing the reduced density matrix of any qubit, $\rho_{p}=\text{Tr}_{q,r\neq p}(\rho_{ss})$, where $\rho_{ss}$ is the global steady state. We indicate with $\rho_{p}^{1}$ and $\rho_{p}^{0}$, respectively, the populations of the excited and ground state of the $p$-th qubit. Each $\rho_{p}$ is a mixed state in the local energy basis and such that we can define a local temperature for each qubit $(\rho_{p}^{0}\geq\rho_{p}^{1}),$ with respect to their free Hamiltonianan, as:

[TABLE]

We plot in Fig. 5 the local temperatures of the qubits as a function of the control temperature $T_{M}$. It is clear that by increasing $T_{M}$, the temperature gradient between the $L$ and $M$ qubits decreases, in contrast to what happens between the pair of qubits $M$ and $R$. In addition, the external qubits are never in thermal equilibrium with their respective environments, only the middle one reaches the equilibrium with the control thermostat $M$ at $T_{M}\simeq 4.17$. For this value it holds $J_{M}=0$ and $J_{L}=-J_{R}\simeq 0.179$. This means that all the heat injected by the hot reservoir into the system is transferred directly to the cold bath without any participation of the control bath.

Once analyzed the behavior of the local temperatures, we examine if we can qualitatively reproduce the behavior of the heat currents with the following local model. The three qubits are in contact with their local thermostats exactly as in our model, but the qubits do not interact between themselves. Instead, each qubit is also strongly coupled to another local thermal environment whose temperature is equal to (for any value of $T_{M}$) the temperature $T_{p}^{\text{loc}}(T_{M})$ depicted in Fig. 5. The steady state of each qubit can be made close at will to the thermal state of temperature $T_{p}^{\text{loc}}(T_{M})$ by suitably tuning the coupling between qubits and their own two local reservoirs. In this local model, the reduced state of each qubit is thus equal to the one obtained in our model, while the way each qubit dissipates is strongly different. In the local model the dissipation is governed by local Lindblad operators while in our model by collective ones, computed using the dressed states of $H_{S}$.

In Fig. 6 we compare the local currents with the exact ones of Fig. 2. We notice that in some cases, local currents well approximate the exact ones, like $J_{L}$, while in other they do not, like $J_{R}$. This is a clear indication of the relevant role played by the collective behavior of the spin ring when it exchanges heat with the local thermostats. The local temperatures induced by the collective dynamics are thus, in general, not enough by themselves to catch all the physics of this collective exchange and of the occurrence of the thermal transistor effect. In particular, we stress the absence of a local minimum for $J_{M}$, around which we observed the best performance of the device.

IV Robustness against perturbations

The results reported in the previous sections show how to build a quantum thermal transistor operating on a wide range of temperatures and starting from a microscopic derivation, with a potential implementation over some feasible physical settings. However, in any real physical implementation of an Ising Hamiltonian, spurious and undesired couplings pointing along different directions of the desired axis for the interaction could occur. This could entail a lost of the thermal transistor effect. Obviously, in any controlled setup, those interactions can be considered as small fluctuations and treated by means of perturbative corrections to the system Hamiltonian $H_{S}$. Nevertheless, in order to pursue our microscopical description of the effect and be precise, we use the full model for the three qubits as in $\text{Eq.}\left(\ref{eq:hsys}\right)$, introducing time by time more complex interactions and studying to which extent they alter the amplification performances.

In order to substantiate the process illustrated so far, we consider a physical implementation where the real axis of the interaction term of the model can differ from the expected one (indicated with $k$ in the following) ExpIsing1 , and we compute the amplification factors, $\alpha_{s}(\lambda^{i},\lambda^{j})$, that are now functions of the couplings, $\lambda^{i}$ and $\lambda^{j}$, along the other two possible polarization directions $i$ and $j$, and they may assume a different value from the unperturbed one. Only the coupling between the qubits $L$ and $M$ and between the qubits $R$ and $M$ are perturbed, while the coupling between the qubits $L$ and $R$ is maintained equal to zero as before.

To obtain a statistical detailed analysis we assume that the values of the perturbations are normally distributed and we evaluate the averaged amplification factors

[TABLE]

where $s=L,R$, and $\boldsymbol{\mathcal{N}}(\overline{\lambda}^{i,j},\Sigma^{i,j})$ is the bivariate Gaussian distribution of the perturbations. It is centered around their mean values $\overline{\lambda}^{i,j}=(\overline{\lambda}^{i},\overline{\lambda}^{j})$, with covariance matrix given by $\Sigma^{i,j}=\text{diag}(\Sigma^{i},\Sigma^{j})$. In particular, we choose that the spurious interactions can deviate of around $10\%$ from their mean value, i.e. $\Sigma^{i}=0.1\overline{\lambda}^{i}.$

As a tool to investigate the effect of small perturbations, we define the ratio

[TABLE]

where $\alpha^{\text{unp}}_{L}$ represents the value of the left amplification factor obtained in the absence of perturbation. If only one additional coupling of the type $\sigma_{p}^{i}\sigma_{q}^{i}$ perturbs the main interaction term $\sigma_{p}^{k}\sigma^{k}_{q}$, the above ratio is indicated with $\zeta^{i}$. For the sake of clarity, we point out that another quantity analogue to $\zeta^{i,j}$ can be in principle defined using the amplification factor of the right current $\alpha_{R}$. Nevertheless, its value depends on $\alpha_{L}$ in a simple way, such that for any $T_{M}$ this choice would lead to analogues dampings of $\zeta^{i,j}$ as function of the strength of the perturbation, $\overline{\lambda}^{i,j}$.

In Fig. 7 we show the quantity $\zeta$ as a function of the average strength of 3 types of typical perturbations that can occur in both settings discussed in Section III. Since $\alpha^{\text{unp}}_{L}$ can diverge for a certain value of the control temperature $T_{M}$, we fix it equal to $T_{M}=0.650$, close enough the one where the divergence appears in order to have $\alpha^{\text{unp}}_{L}\gg 1$ but where it is still finite, being $\alpha_{L}^{\text{unp}}\simeq 1.42\times 10^{3}$. We stress here that the value of $T_{M}$ in the presence of perturbations is kept the same of the unperturbed case. The quantity $\zeta$ is close to unity when the system still operates as a quantum transistor close to its unperturbed value $\alpha^{\text{unp}}_{L}$ but it vanishes as soon as the perpendicular couplings become more intense.

As it is apparent from the plot, the $x-$axis and $z-$axis Ising models show a completely opposite behavior in the presence of perturbations. Indeed, despite the fact that both the models have been perturbed around a quite optimal configuration, the $x-$axis model still shows the thermal transistor effect for extra couplings small enough. In contrast, the possibility to achieve a current amplification with the $z-$axis Ising model is completely lost in the presence of infinitesimal couplings among the spin in any direction perpendicular to the wanted interaction axis. Our numerical evidences only allow us to see that $\zeta$ is extremely low for values of the extra couplings larger than ten millionths of the one along the $z-$axis, but we cannot surmise how the unperturbed value is restored for smaller values of the spurious coupling $\overline{\lambda}^{i,j}$.

We also provide a qualitative mathematical explanation of this phenomenon which resides on the structure of the couplings between the qubits. In fact, as pointed out when discussing the two possible configurations, a coupling commuting with the bare Hamiltonian affects only the eigenfrequencies of the systems and it leaves unchanged the eigenstates entering in the definition of the operators $A(\omega)$, that are essential to derive the dissipative dynamics of $\text{Eq.}\left(\ref{eq:master}\right)$. Adding a perpendicular coupling, not commuting with the bare part of $H_{S}$ completely alters their structure and it imposes to consider the dressed-state basis to write the $A(\omega)$, that typically differs from the bare-state basis one.

V Conclusions

In this paper, we have addressed the design and the possible physical implementation of a quantum thermal transistor. This novel device acts on thermal currents as a bipolar electronic transistor does with electric currents, providing an amplification of the collector’s and emitter’s currents by a proper tuning of the gate potential, while the corresponding gate current is order of magnitude lower than the other two and practically constant.

Our model proposes a system of three qubits strongly interacting among them and each of them in contact with a different thermal reservoir. This setting imposes non-equilibrium dissipative dynamics that entail the presence of non-vanishing steady state currents flowing into the system via bath induced transitions. We have focused our attentions on a particular system that can lead to a feasible realization on several experimental platforms, such as molecular nanomagnets and few-body spin chains, and we have referred to this model as the spin-ring model.

In particular, we have devoted our efforts to describe how a quantum thermal transistor works in a scenario where a more complex spin-spin coupling is present, in comparison to a previous model recently presented in QTT . Throughout the paper, we have referred to these two models as the $x$-type Ising model and the $z$-type Ising model. The former is characterized by an interaction Hamiltonian not commuting with the bare one, or in other words the magnetic field acting on the three spins is perpendicular to the direction along which the spins interact; the latter, instead, is a model where magnetic fields and interaction are longitudinal, so that the two contributions of the system Hamiltonian commute.

Choosing as control parameter the temperature of the heat bath directly coupled to the middle qubit, we have shown that the $x$-type spin ring operates as a thermal transistor over a wider range of temperatures in comparison to the $z$-type one. The significant amplification of the lateral currents has been justified in the context of devices characterized by negative differential thermal resistance LLC06 ; graphene ; silicene and by means of the collective dynamics. In fact, considering a naive local model for the system, that partially takes into account collective phenomena, cannot explain the whole behavior exhibited by our model. Finally, we have discussed the robustness of both the possible implementations against spurious and uncontrollable couplings stemming from the magnetic interaction between the different pairs of the effective spins of the magnetic ring. The results of exact numerical computations suggest that, for reasonably small perturbations, the $x$-type molecule can be steered acting on the control temperature to behave still as a thermal transistor around its unperturbed configuration. In contrast, our computations show that the $z$-type molecule is really fragile in presence of those kinds of perturbations.

As final remarks, we would like to point out that even if we have performed a microscopic derivation of master equation describing the dynamics of the system, overriding any phenomenological simplification, a deeper comprehension of the very quantum effects conducting to such current amplification is still lacking. It could be matter of future investigation the role of system criticality and of the dissipation mechanism and how they contribute and compete in order to state a general scheme to build a quantum thermal transistor.

Acknowledgements.

A.M. acknowledges the “Région Bourgogne-Franche-Comté” for financial support throughout the mobility grant N. 2017Y-06400. B.B. acknowledges the support by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03). A.M. thanks Heinz-Peter Breuer for useful comments and Elena Servida for discussions about semiconductor electronics. B.B. thanks Benedetto Militello and David Viennot for helpful comments.

Appendix A Spectrum of the system Hamiltonian in the $x$-type Ising model.

In this appendix we give a sketch of the diagonalization of the system Hamiltonian introduced in $\text{Eq.}\left(\ref{eq:hsys}\right)$ and used to obtain the thermal transistor effect discussed throughout the paper. In particular, we focus on the case of the $x$-type Ising model with the left and right qubits not interacting, i.e. $\eta_{LR}=0$. The matrix of a system composed by three qubits with different splitting in a magnetic field with a transverse Ising interaction reads:

[TABLE]

where we have assumed $\eta_{qp}=\eta_{pq}$ and for shorthand we have introduced $\Omega_{pq}=\omega_{p}+\omega_{q}$ and $\Lambda_{pq}^{x}=\eta_{pq}\lambda^{x}$ with $p,q=L,M,R$.

The secular equation of $\text{Eq.}\left(\ref{eq:matrix}\right)$ can be reduced to a quartic equation, leading to four pairs of energy eigenvalues equal in modulus. We can, therefore, decompose $H_{S}$ in the base where it is diagonal, $H_{s}=\sum_{i=1}^{8}\epsilon_{i}|\epsilon_{i}\backslash\!\!\!/\epsilon_{i}|$, where we have ordered the energy eigenvalues using the following notation $\epsilon_{8}\leq\epsilon_{7}\leq...\leq\epsilon_{1}$ and $\epsilon_{9-i}=-\epsilon_{i},\,\forall i$.

Introducing the computational basis, defined as the tensor product $|l_{L}l_{M}l_{R}\rangle$, of all the dispositions of the eigenvectors of the three $\sigma_{p}^{z}$, i.e. $\sigma_{p}^{z}|l_{p}\rangle=(-1)^{(l_{p}+1)}~{}|l_{p}\rangle$, with $l_{p}=0,1$ and $p=L,M,R$, the eigenvectors of $H_{S}$, found for the main case discussed in the paper (Sec. III.2), are:

[TABLE]

where the 16 real coefficients $a_{i},b_{i},c_{i},d_{i}$ with $i=1,2,3,4$ in the above expressions are functions of the parameters entering in the Hamiltonian. In all the numerical trials, we have found the same above structure for the eigenvectors, all the coefficients being real.

It is worth noting that the above states have well-defined parity and that the states with opposite energy have different parity. In fact, it is easy to check it by measuring the parity operator $P=\sigma_{L}^{z}\otimes\sigma_{M}^{z}\otimes\sigma_{R}^{z}$ on any term appearing in $|\epsilon_{i}\rangle$.

Finally, we report in Table 1 the transitions mediated by any thermal reservoir. It is worth noting that we write here only the transitions that can be interpreted as the emission of a photon in the usual context of quantum optics, since they involve states with a positive energy difference between the initial and the final value.