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.
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  and transistors  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 , , , , , , , , , ,  and electronic ,  thermal diodes or rectifiers were developed, which later led to the proposition of thermal transistors , . Later, these concepts were extended to the case of thermal radiation in both the near field , ,  and the far field , , , , . The most interesting results were found through the use of phase change thermochrome materials , such as VO2 , . Recently, thermal transistors have been designed using similar properties , .
In the last few years, individual quantum systems, such as classical atoms ,  or artificial ones, as is the case of quantum dots , , have been proposed to develop photon rectifiers , , , transistors , , or even electrically controlled phonon transistors . 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 , ,  and led to several works on thermal rectification , , , .
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. , 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.
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.
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. . Each of the TLS is characterised by an angular frequency ωL or ωR. The coupling between the two TLS has the typical angular frequency ωLR. The Hamiltonian of the system is (in 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. HS 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  is based on the spin-boson model in the x component The two reservoirs P have their Hamiltonians equal to This modelling implies that baths can only flip one spin at a time. There are therefore four authorised transitions. The transitions 1↔3 and 2↔4 are induced by the thermal bath 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
of , 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 JL+JR=0, in accordance with the energy conservation.
The master equation is a system of four equations on the diagonal elements ρii. 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 (in kb=1 units): the master equation yields
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 ωL=1, ωR=0, and ωLR=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, 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 TL>TR, 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. 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 TL<TR, the energy transfers are reversed. Now imagine that TL is of the order of the transition energies, whereas TR is much lower. Then, energy will easily flow from reservoir L to reservoir R according to the process described above. On the contrary, if TR is much lower than the transition energies and TL<TR, then the energy transfer is poor since excitation by reservoir R through the transitions 4↔3 and 2↔1 is low. Hence, if we study the flux JL(TL, TR) with TR fixed at a value lower than the transition energies (e.g. TR=0.1, Fig. 3), we see that the flux is close to 0 when TL<TR. When TL is increased to values larger than TR, 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[ρ]=1). The exact expression of Γ can be found in . In the case studied here, this expression can be simplified and the current reads
where the transition from low current values, at low TL, to high current values, at higher TL, can be seen.
Let us note that the system proposed here constitutes a passive thermal switch at low temperature. As long as TL is larger than TR, the current in the structure is important and the thermal contact is good between the reservoirs L and R. However, when the temperature TL reduces to values below TR, 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.
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(TL, TR) variations with TL for different TR are represented in Figure 4. When TR is small enough (TR<1), rectification is strong except for values of TL very close to those of TR. When TR is larger, rectification is smaller, even for TL values that are greatly different from TR. We note, in particular, that rectification is low for high TR 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 TR is fixed, and TL goes to 0, then JL(TR, TL) 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.
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. . 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)
where ωP denotes the energy difference between the two spin states, whereas ωPQ stands for the interaction between the spin 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 JL+JM+JR=0, in accordance with the energy conservation.
The master equation is a system of eight equations on the diagonal elements ρii. 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 ρii whose resolution provides all state occupation probabilities as well as the currents JP.
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 JM it is possible to control JL or JR. Let us consider the following situation: the left and right TLS are both connected to thermal baths at respective temperatures TL and TR. The third bath at temperature TM changes the fluxes JL and JR by means of the current JM injected into the system. The dynamical amplification factor 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 JM makes a large one in JL or JR, that is, |αL,R |>1, then the thermal transistor effect can be identified. The system presented here exhibits many parameters: the frequencies ωP and ωPQ and the temperatures TL and TR. The last temperature TM, which is taken here between TL and TR, controls the transistor properties and is related to the current JM. 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 ωLM=ωMR=Δ, whereas ωRL 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 . The operating temperature TL is taken so that (TL/Δ≪0.25), whereas (TR/Δ≪0.0625).
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 ρI=ρ11+ρ88, ρII=ρ22+ρ77, ρIII=ρ33+ρ66, and ρIV=ρ44+ρ55. Using the net decaying rates between the states, the three currents read
Transitions between the different states are plotted in Figure 6, for TL/Δ=0.1, TR/Δ=0.01, and TM/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 III–II and IV–III transitions. JM is expected to be larger than JR and JL. This is shown in Figure 7, in which JL, JM, and JR are plotted versus TM, for TL/Δ=0.1 and TR/Δ=0.01. JL and JR increase linearly with TM, at low temperature, and behave sublinearly as TM gets close to TL. Note that over the whole range, JM remains lower than JL and JR, as expected. Thus, TM will be controlled by changing a little bit the current JM: a tiny change of JM can modify JL and JR in a larger proportion. Moreover, JL and JR are switched off when JM approaches 0, for small temperatures TM: the three TLS exhibit the transistor switching property. One also remarks that the JM slope is larger than the ones of JL and JR over a large part of the temperature range. Given the definition of the amplification factor α, the thermal current slopes are essential to figure out amplification.
In Figure 8, the two amplification coefficients αL and αR are plotted versus temperature TM. We see that α remains much larger than 1 (around 2.2×104) at low TM. One also notes that α diverges for a certain value of the temperature for which JM has a minimum. This occurs for TM0.07444Δ. In these conditions, an infinitely small change in JM makes a change in JL and JR. As TM approaches TL, 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 JM=0. This temperature is the one at which the bath M is at thermal equilibrium with the system since it does not put any thermal current in it. At this temperature (TM0.08581Δ), JL=−JR=3.325×10−6. Amplification still occurs since αL=831 and αR=−832.
Populations and current expressions explain these observations well. In the present case, if we limit the calculation to first order of approximations on and populations can be estimated by
ρIII remains very close to 1 and ρII to 10−2. ρI and ρIV change by one to two orders of magnitude with temperature and are much smaller than the two preceding ones.
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 TM. Note also that JL and JR seem to be driven by ρIV, the state population at the intermediate energy (EIV=0) since their expressions (21) and (20) are very similar. Examining the authorised transitions, one expects JM to be driven by the population of the most energetic state, that is, ρI. The main difference between ρIV and ρI is the temperature dependence, which is linear in one case and exponential (e−2Δ/T ) in the other one. The result is that even when TM is close to TL, ρI remains low. Therefore, JM keeps low values in the whole temperature range due to the low values of ρI. A careful look at JM shows that it is the sum of two terms. The first one is roughly linear on TM and is similar to the one that appears in ρIV. JM depends on the population of the state IV, which also changes the population of the state I with the transition IV–I. Increasing ρIV with TM facilitates the IV–I transition, and raises ρI. This increases the state I decay through the I–III transition. This term is negative and decreases as TM increases. This can be seen as a negative differential resistance since a decrease in JM (cooling in M) corresponds to an increase in the temperature TM. In this temperature range, the amplification factor (e10=22026.5). The second term in JM is the classical Boltzmann factor, which makes the population of the state I increase with TM. JM is a compromise between these two terms. At low temperature, the linear term is leading. As TM increases, the term dominates. Consequently, there exists a value where the increase in ρI reverses the 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 JM reaches a minimum and a second temperature where JM=0, as described above.
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 (JL and JR). Note that the amplification factor depends on and that the currents depend on Let us also recall that we have assumed up to now that Therefore, the best choice to have a transistor with a sufficiently collector or emitter current is to take the lowest Δ/TL with the condition and the criteria chosen to fulfil this last condition (here Δ/TL≈5).
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 JM much smaller than JL and JR, 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 JM on TM slow enough to obtain a high amplification.
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  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.
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.
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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