# Abstract

We study in this article how heat can be exchanged between two-level systems, each of them being coupled to a thermal reservoir. Calculations are performed solving a master equation for the density matrix using the Born–Markov approximation. We analyse the conditions for which a thermal diode and a thermal transistor can be obtained as well as their optimisation.

## 1 Introduction

Global warming and limited energy issues have increased the interest in the energy management and in particular heat losses. Indeed, heat wasted in energy production processes and thermal machines could in principle be better used in many applications if it could be guided or transport in a similar way as electricity. However, if heat pipes are proved to be good candidates for thermal guiding, there exist few devices at the moment that can switch or amplify heat as is the case for electricity.

In electricity, the development of diodes [1] and transistors [2] has led to its control at the scale of the electron, leading to the emergence of electronics. One can therefore wonder whether heat could be managed in the same way if the thermal equivalent of these two objects exists. In the last decade, several works focused on the development of thermal rectifiers, that is, devices for which the thermal fluxes flowing through them are different in magnitude when the temperatures are inverted at their ends. Thus, phononic [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] and electronic [12], [14] thermal diodes or rectifiers were developed, which later led to the proposition of thermal transistors [15], [16]. Later, these concepts were extended to the case of thermal radiation in both the near field [17], [18], [19] and the far field [20], [21], [22], [23], [24]. The most interesting results were found through the use of phase change thermochrome materials [25], such as VO_{2} [26], [27]. Recently, thermal transistors have been designed using similar properties [28], [29].

In the last few years, individual quantum systems, such as classical atoms [30], [31] or artificial ones, as is the case of quantum dots [32], [33], have been proposed to develop photon rectifiers [34], [35], [36], transistors [37], [38], or even electrically controlled phonon transistors [39]. Moreover, as quantum systems are always coupled to the environment, the question of how heat is transferred through a set of quantum systems in interaction naturally has arisen [40], [41], [42] and led to several works on thermal rectification [43], [44], [45], [46].

The goal of this article is to use elementary quantum objects, such as two-level systems (TLS) related to thermal baths, for developing thermal diodes and thermal transistors. To do so, we will use the classical quantum thermodynamics formalism proposed by Lindblad that is based on the resolution of a master equation for the density matrix. We show, following the work of Werlang et al. [45], that two TLS can easily make a thermal diode and that three TLS can make a thermal transistor. These three TLS related to thermal reservoirs are equivalent to the three entries of a bipolar electronic transistor. It is shown that a thermal current imposed at the base can drive the currents at the two other entries of the system.

## 2 Theory

We consider in the following that TLS are connected to a thermal bath and that they can be coupled to each other. Two configurations are studied in this article: two TLS coupled to each other make a thermal diode, whereas three coupled TLS make a thermal transistor.

### 2.1 Thermal Diode

The system under consideration consists of two coupled TLS, each of them related to a thermal bath, as depicted in Figure 1.

### Figure 1:

The two TLS are labelled with the letters *L* (left) and *R* (right), which is also the case of the temperature of the thermal baths related to TLS. We use the strong-coupling formalism developed by Werlang et al. [45]. Each of the TLS is characterised by an angular frequency *ω _{L}* or

*ω*. The coupling between the two TLS has the typical angular frequency

_{R}*ω*. The Hamiltonian of the system is (in

_{LR}*ћ*=1 units)

where
*P*=*L*, *R*) is the Pauli matrix *z*, whose eigenstates for the system *P* are the states ↑ and ↓. Therefore, angular frequencies *ω* have the dimension of energy. *H _{S}* eigenstates are given by the tensorial product of the individual TLS states, so that we have four eigenstates labelled as |1〉=|↑↑〉, |2〉=|↑↓〉, |3〉=|↓↑〉, |4〉=|↓↓〉. The coupling between the TLS and the thermal bath

*P*constituted of harmonic oscillators [47] is based on the spin-boson model in the

*x*component

*P*have their Hamiltonians equal to

*L*, whereas the transitions 1↔2 and 3↔4 are induced by the thermal bath

*R*. The transitions 1↔4 and 2↔3 are forbidden.

The system state is described by a density matrix *ρ*, which obeys a master equation. In the Born–Markov approximation, it reads

As in [45], [48], the Lindbladians ℒ_{P}[*ρ*] are written for an Ohmic bath according to classical textbooks [48], [49], so that we take the expression

of [45], where

and ℐ(*ω*)=*κω* in the case of a Ohmic bath with a coupling constant *κ*. We now consider a steady state situation. We define

the heat current injected by the bath *P* into the system. Averaging the master equation, we find *J _{L}*+

*J*=0, in accordance with the energy conservation.

_{R}The master equation is a system of four equations on the diagonal elements *ρ _{ii}*. If we introduce the net decaying rate from the state |

*i*〉 to the state |

*j*〉, due to the coupling with bath

*P*, with the help of the Bose–Einstein distribution

*k*=1 units):

_{b}from which it can be deduced that

The definition of the thermal currents (6) gives then the final expression of the thermal currents

As an example, let us consider the example of a system where *ω _{L}*=1,

*ω*=0, and

_{R}*ω*=0.1. The energy levels and the authorised transitions are depicted in Figure 2. Note that we took in this example, as well as in the rest of the paper,

_{LR}*κ*=1. This means that the currents have to be multiplied by

*κ*and divided by

*ћ*in order to be expressed in W. When

*T*>

_{L}*T*, the left reservoir populates level |1〉 from level |3〉 through the transition 1↔3. Level |1〉 de-excitates through level |2〉 by transferring energy to reservoir

_{R}*R*. Level |2〉 de-excitates through level |4〉 by transferring energy to reservoir

*L*and finally level |4〉 de-excitates through level |3〉 by transferring energy to reservoir

*R*. If

*T*<

_{L}*T*, the energy transfers are reversed. Now imagine that

_{R}*T*is of the order of the transition energies, whereas

_{L}*T*is much lower. Then, energy will easily flow from reservoir

_{R}*L*to reservoir

*R*according to the process described above. On the contrary, if

*T*is much lower than the transition energies and

_{R}*T*<

_{L}*T*, then the energy transfer is poor since excitation by reservoir

_{R}*R*through the transitions 4↔3 and 2↔1 is low. Hence, if we study the flux

*J*(

_{L}*T*,

_{L}*T*) with

_{R}*T*fixed at a value lower than the transition energies (e.g.

_{R}*T*=0.1, Fig. 3), we see that the flux is close to 0 when

_{R}*T*<

_{L}*T*. When

_{R}*T*is increased to values larger than

_{L}*T*, the current increases until it reaches saturation at high temperatures. The calculation of Γ, which gives the current, can be achieved by solving the system of equations on the populations (7). Note that the system of equations are not totally independent since the fourth equation is actually the sum of the three others. One has to use the fact that the trace of the density matrix is equal to 1 (Tr[

_{R}*ρ*]=1). The exact expression of Γ can be found in [45]. In the case studied here, this expression can be simplified and the current reads

### Figure 2:

### Figure 3:

where the transition from low current values, at low *T _{L}*, to high current values, at higher

*T*, can be seen.

_{L}Let us note that the system proposed here constitutes a passive thermal switch at low temperature. As long as *T _{L}* is larger than

*T*, the current in the structure is important and the thermal contact is good between the reservoirs

_{R}*L*and

*R*. However, when the temperature

*T*reduces to values below

_{L}*T*, the thermal current is drastically lowered, so that it can be seen as switched off. This system could therefore be used to isolate objects from a cold environment while it would be thermally linked to a hot environment. In a case of an environment with temperatures oscillating between high and low values, this simple quantum system can be seen as a passive heater and a thermal rectifier, that is, that heat flow through it depends on the direction of the heat flux.

_{R}There is actually another way to quantify the rectification of a system. This is the ratio between the sum of the fluxes through the system when the temperatures are reversed and the maximum of these two fluxes:

The rectification ratio *R*(*T _{L}*,

*T*) variations with

_{R}*T*for different

_{L}*T*are represented in Figure 4. When

_{R}*T*is small enough (

_{R}*T*<1), rectification is strong except for values of

_{R}*T*very close to those of

_{L}*T*. When

_{R}*T*is larger, rectification is smaller, even for

_{R}*T*values that are greatly different from

_{L}*T*. We note, in particular, that rectification is low for high

_{R}*T*temperature. In this latter case, there is no rectification, because heat transfer can occur with both reservoirs with the help of the energy transitions presented above. However, when

_{R}*T*is fixed, and

_{R}*T*goes to 0, then

_{L}*J*(

_{L}*T*,

_{R}*T*) tends to 0. Also, rectification rises to 1. This kind of device can thus be seen as a thermal diode, since the heat current through the system is nonzero when the heat flux is in a given direction and 0 when it is in the opposite one.

_{L}### Figure 4:

This type of system paves the way to develop more complicated ones. For example, it is well known that electronic transistors as the bipolar ones can be made up of NPN and PNP junctions whereas the PN junction constitutes a diode. One can therefore wonder if it is also possible to conceive a transistor with the elementary quantum system that constitutes the thermal diode that we have just studied in this section. This is the subject of the next section.

### 2.2 Thermal Transistor

We consider in this part a system constituted of three TLS coupled with each other. Each of the three TLS is also coupled to a thermal bath at thermal connection (Fig. 5). This system is therefore similar to the previous one with one supplementary TLS and reservoir. The three TLS are now indexed with the letters *L* (left), *M* (medium), and *R* (right). The thermal bath temperatures are labelled in the same way. As in the previous part, we use the strong-coupling formalism developed by Werlang et al. [45]. Similarly, TLS can be in the up state ↑ or in the down one ↓. Let us write the Hamiltonian of the system (in *ћ*=1 units)

### Figure 5:

where *ω _{P}* denotes the energy difference between the two spin states, whereas

*ω*stands for the interaction between the spin

_{PQ}*P*and the spin

*Q*. Following the preceding part on the quantum thermal diode, we have eight eigenstates labelled as |1〉=|↑↑↑〉, |2〉=|↑↑↓〉, |3〉=|↑↓↑〉, |4〉=|↑↓↓〉, |5〉=|↓↑↑〉, |6〉=|↓↑↓〉, |7〉=|↓↓↑〉, and |8〉=|↓↓↓〉. There are now 12 authorised transitions. The left bath (

*L*) induces the transitions 1↔5, 2↔6, 3↔7, and 4↔8, and the middle one (

*M*) drives the transitions 1↔3, 2↔4, 5↔7, and 6↔8. The right bath (

*R*) triggers the transitions 1↔2, 3↔4, 5↔6, and 7↔8. All other transitions flipping more than one spin are forbidden.

The master equation fulfilling the density matrix, in the Born–Markov approximation, reads

we now go to the steady state situation. Averaging the master equation, we find *J _{L}*+

*J*+

_{M}*J*=0, in accordance with the energy conservation.

_{R}The master equation is a system of eight equations on the diagonal elements *ρ _{ii}*. Introducing the net decaying rate from the state |

*i*〉 to the state |

*j*〉 due to the coupling with bath

*P*, the master equation becomes

The sum of these eight equations is 0 and therefore they are not independent. The condition Tr[*ρ*]=1 is added to the system in *ρ _{ii}* whose resolution provides all state occupation probabilities as well as the currents

*J*.

_{P}Let us now show that such a device makes a thermal transistor with a close analogy to an electronic one. In an electronic bipolar transistor, such as a PNP or an NPN transistor, a driven current at the base modulates, switches, or amplifies the collector and emitter currents. Therefore, switching, modulation, and amplification have to be exhibited in order to have a transistor. We are going to show here that by slightly changing *J _{M}* it is possible to control

*J*or

_{L}*J*. Let us consider the following situation: the left and right TLS are both connected to thermal baths at respective temperatures

_{R}*T*and

_{L}*T*. The third bath at temperature

_{R}*T*changes the fluxes

_{M}*J*and

_{L}*J*by means of the current

_{R}*J*injected into the system. The dynamical amplification factor

_{M}*α*, defined as

is a measure of the transistor ability to amplify a small heat flux variation at the base (*M*). If a small change in *J _{M}* makes a large one in

*J*or

_{L}*J*, that is, |

_{R}*α*

_{L,R}|>1, then the thermal transistor effect can be identified. The system presented here exhibits many parameters: the frequencies

*ω*and

_{P}*ω*and the temperatures

_{PQ}*T*and

_{L}*T*. The last temperature

_{R}*T*, which is taken here between

_{M}*T*and

_{L}*T*, controls the transistor properties and is related to the current

_{R}*J*. The number of parameters involved can be reduced by choosing a situation that will not change the physics of the system but will allow a good understanding of the physical phenomena involved. We therefore restrict our analysis to a case for which

_{M}*ω*=

_{LM}*ω*=Δ, whereas

_{MR}*ω*and the three TLS energies are equal to 0. As shown below, this configuration provides a good transistor effect, easy to handle with simple calculations. The transistor effect disappears when the three couplings are equal (symmetric configuration), but it still occurs and can even be optimised if the three TLS energies are nonzero [50]. The operating temperature

_{RL}*T*is taken so that

_{L}*T*/Δ≪0.25), whereas

_{L}*T*/Δ≪0.0625).

_{R}Under these conditions, the system states are degenerated 2 by 2. There are now only four states and three energy levels (see Fig. 6).The states |1〉 and |8〉 are now state |*I*〉, |2〉 and |7〉 state |*II*〉, |3〉 and |6〉 state |*III*〉, and |4〉 and |5〉 state |*IV*〉. One can define the new density matrix elements *ρ _{I}*=

*ρ*

_{11}+

*ρ*

_{88},

*ρ*=

_{II}*ρ*

_{22}+

*ρ*

_{77},

*ρ*=

_{III}*ρ*

_{33}+

*ρ*

_{66}, and

*ρ*=

_{IV}*ρ*

_{44}+

*ρ*

_{55}. Using the net decaying rates between the states, the three currents read

### Figure 6:

Transitions between the different states are plotted in Figure 6, for *T _{L}*/Δ=0.1,

*T*/Δ=0.01, and

_{R}*T*/Δ=0.05. The direction of the arrows shows the transition direction, whereas its width is proportional to the decay time. Energy exchanges are mainly dominated by the

_{M}*III*–

*II*and

*IV*–

*III*transitions.

*J*is expected to be larger than

_{M}*J*and

_{R}*J*. This is shown in Figure 7, in which

_{L}*J*,

_{L}*J*, and

_{M}*J*are plotted versus

_{R}*T*, for

_{M}*T*/Δ=0.1 and

_{L}*T*/Δ=0.01.

_{R}*J*and

_{L}*J*increase linearly with

_{R}*T*, at low temperature, and behave sublinearly as

_{M}*T*gets close to

_{M}*T*. Note that over the whole range,

_{L}*J*remains lower than

_{M}*J*and

_{L}*J*, as expected. Thus,

_{R}*T*will be controlled by changing a little bit the current

_{M}*J*: a tiny change of

_{M}*J*can modify

_{M}*J*and

_{L}*J*in a larger proportion. Moreover,

_{R}*J*and

_{L}*J*are switched off when

_{R}*J*approaches 0, for small temperatures

_{M}*T*: the three TLS exhibit the transistor switching property. One also remarks that the

_{M}*J*slope is larger than the ones of

_{M}*J*and

_{L}*J*over a large part of the temperature range. Given the definition of the amplification factor

_{R}*α*, the thermal current slopes are essential to figure out amplification.

### Figure 7:

In Figure 8, the two amplification coefficients *α _{L}* and

*α*are plotted versus temperature

_{R}*T*. We see that

_{M}*α*remains much larger than 1 (around 2.2×10

^{4}) at low

*T*. One also notes that

_{M}*α*diverges for a certain value of the temperature for which

*J*has a minimum. This occurs for

_{M}*T*0.07444Δ. In these conditions, an infinitely small change in

_{M}*J*makes a change in

_{M}*J*and

_{L}*J*. As

_{R}*T*approaches

_{M}*T*, the amplification factor drastically decreases to reach values below 1: the transistor effect does not exist anymore. Note also that, in between, there exists a temperature for which

_{L}*J*=0. This temperature is the one at which the bath

_{M}*M*is at thermal equilibrium with the system since it does not put any thermal current in it. At this temperature (

*T*0.08581Δ),

_{M}*J*=−

_{L}*J*=3.325×10

_{R}^{−6}. Amplification still occurs since

*α*=831 and

_{L}*α*=−832.

_{R}### Figure 8:

Populations and current expressions explain these observations well. In the present case, if we limit the calculation to first order of approximations on

*ρ _{III}* remains very close to 1 and

*ρ*to 10

_{II}^{−2}.

*ρ*and

_{I}*ρ*change by one to two orders of magnitude with temperature and are much smaller than the two preceding ones.

_{IV}We now explicitly present the three thermal current expressions and their dependence on temperature which is the core of our study.

These formulas are in accordance with the linear dependence of the thermal currents for small values of *T _{M}*. Note also that

*J*and

_{L}*J*seem to be driven by

_{R}*ρ*, the state population at the intermediate energy (

_{IV}*E*=0) since their expressions (21) and (20) are very similar. Examining the authorised transitions, one expects

_{IV}*J*to be driven by the population of the most energetic state, that is,

_{M}*ρ*. The main difference between

_{I}*ρ*and

_{IV}*ρ*is the temperature dependence, which is linear in one case and exponential (e

_{I}^{−2Δ/T}) in the other one. The result is that even when

*T*is close to

_{M}*T*,

_{L}*ρ*remains low. Therefore,

_{I}*J*keeps low values in the whole temperature range due to the low values of

_{M}*ρ*. A careful look at

_{I}*J*shows that it is the sum of two terms. The first one is roughly linear on

_{M}*T*and is similar to the one that appears in

_{M}*ρ*.

_{IV}*J*depends on the population of the state

_{M}*IV*, which also changes the population of the state

*I*with the transition

*IV*–

*I*. Increasing

*ρ*with

_{IV}*T*facilitates the

_{M}*IV*–

*I*transition, and raises

*ρ*. This increases the state

_{I}*I*decay through the

*I*–

*III*transition. This term is negative and decreases as

*T*increases. This can be seen as a negative differential resistance since a decrease in

_{M}*J*(cooling in

_{M}*M*) corresponds to an increase in the temperature

*T*. In this temperature range, the amplification factor

_{M}^{10}=22026.5). The second term in

*J*is the classical

_{M}*I*increase with

*T*.

_{M}*J*is a compromise between these two terms. At low temperature, the linear term is leading. As

_{M}*T*increases, the term

_{M}*ρ*reverses the

_{I}*I*–

*IV*transition, so that the

*I*–

*III*transition bids with both the

*I*–

*V*and

*I*–

*II*transitions.

*I*–

*III*is then reversed. With these two terms competing, there is a temperature for which

*J*reaches a minimum and a second temperature where

_{M}*J*=0, as described above.

_{M}One can wonder what are the conditions on the parameters to obtain the best transistor effect in the conditions studied here. There are two criteria that will quantify a good transistor. One is the amplification factor and the other one is the intensity of the heat currents at the emitter and the collector (*J _{L}* and

*J*). Note that the amplification factor depends on

_{R}*T*with the condition

_{L}*T*≈5).

_{L}One can summarise the conditions needed for the system to undergo a thermal transistor effect. Two baths (here *L* and *R*) induce transitions between two highly separated states with an intermediate energy level, whereas the third one (*M*) makes only a transition between the two extremes. This will first make *J _{M}* much smaller than

*J*and

_{L}*J*, and second, it will set a competition between a direct decay of the highest level to the ground level and a decay via the intermediate one. This competition between the two terms makes the thermal dependence of

_{R}*J*on

_{M}*T*slow enough to obtain a high amplification.

_{M}Finally, one can wonder what could be the type of system that could be used in order to make a thermal diode or a thermal transistor. Electrons in quantum dots that are used to form qubits [51] and in particular electron qubits are good candidates if they can be coupled to a thermal reservoir that can be controlled in temperature. The idea would be to place these qubits on nano objects the temperature of which would be controlled by electrical currents. The thermal currents calculated in this article would correspond to the heat deposited by the electrical currents applied. Using spin qubits, one could naturally study the influence of the qubit quantum coherence on the thermal objects presented here as well as the time scale at which decoherence would kill coherence effects and retrieve the thermal behaviour presented in this article.

## 3 Conclusions

We have shown that coupled TLS linked to thermal reservoirs can make systems exhibiting thermal rectification. In the case of two TLS, a thermal diode can be made where one of the entries is set at a certain temperature of the order of the system transition. When the other end of the diode is set at a lower temperature, the system is blocked, whereas it is opened when the temperature is higher. This kind of device can isolate a system from cold sources. In the case of three TLS, we have shown that it is possible to make a thermal transistor. We found a temperature regime where a thermal current variation imposed at the base generates an amplified variation at the emitter and the collector. This regime is typically such that the temperature corresponds to an energy one order of magnitude smaller than the coupling energy between the TLS. With this kind of thermal transistor one can expect to modulate or amplify thermal fluxes in nanostructures made up of elementary quantum objects.

# Acknowledgement

This work pertains to the French Government Program “Investissement d’avenir” (LABEX INTERACTIFS, Agence Nationale de la Recherche, ‘ANR-11-LABX-0017-01’).

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**Received:**2016-9-14

**Accepted:**2016-11-29

**Published Online:**2017-1-11

**Published in Print:**2017-2-1

©2017 Walter de Gruyter GmbH, Berlin/Boston