We present a quantum thermal diode model based on a coupled qutrit–qubit system designed to control heat flow between two thermal baths with unprecedented efficiency. This differs from previous models in terms of the asymmetry introduced by spin particles and their interaction. By exploiting the interactions between degenerate states within the coupled qutrit–qubit system, our model demonstrates diode-like behavior that is both robust and energy-efficient. Utilizing the frameworks of open quantum systems and the quantum Markovian master equation, with the Born and rotating wave approximations, we comprehensively analyze the system’s behavior. Numerical simulations reveal significant thermal rectification across a wide temperature range, positioning our model as a groundbreaking solution for nanoscale heat management. In addition, we employ state transition diagrams to elucidate the transition rate characteristics that underpin the diode behavior. Finally, we explore the potential for physical implementation using superconducting circuits, highlighting the practical relevance of our design.

Topics
Thermal diode, Open quantum systems

The emerging challenge of high thermal energy dissipation in electronic circuits is a key factor contributing to the failure of Moore’s Law.1,2 As transistor density3 grows, power consumption also increases, resulting in increased temperatures within the circuits. This excessive heat can lead to performance degradations, increased failure rates, and a reduced overall lifespan of electronic components.3–5 Hence, effective thermal management is crucial to maintain the reliability and efficiency of electronic devices.

Thermotronics, the study and development of devices that manipulate heat flux in a way analogous to how electronics manipulate electric energy, has become an emerging topic among researchers in recent decades.6–10 A thermal diode11–16 and a thermal transistor17,18 are two such implementations developed to control heat flows. A thermal diode exhibits an asymmetric thermal conduction in a way similar to the current flow in an electronic diode, and a thermal transistor controls the thermal flow between two of its terminals in response to the temperature of its third terminal. Practical implementation of such thermal devices is possible using phase-transition materials,19,20 asymmetric structures in nano-membranes,21 graded materials,22 etc.

With the advancement of quantum theories,23–25 researchers focused on heat control using quantum thermodynamics.26,27 Segal and Nitzan28 have proposed a spin-boson nanojunction model, which results in rectification behavior due to the system’s inherent anharmonicity combined with the asymmetry introduced by different couplings to thermal baths. Werlang et al.29 have proposed a quantum system that consists of a pair of spins coupled via Ising interaction. They have shown that this model can conduct heat in one direction and isolate in the opposite direction under strong coupling between the spins. Another approach developed by Ordonez-Miranda et al.30 has shown that two interacting spinlike systems with different excitation frequencies can act as a quantum thermal diode when connected between thermal baths, and the Kargi et al.31 quantum thermal diode model uses an optical field to control the thermal flow between two thermal baths through a pair of qubits interacting anisotropically with each other. Iorio et al.32 have shown that two interacting flux qubits coupled asymmetrically to two heat baths can be modeled as dissipative RLC resonators and result in the rectification of heat current. Bhandari et al.33 have shown the thermal rectification effect of a nonlinear harmonic system coupled asymmetrically between two thermal baths. The model proposed by Upadhyay et al.34 uses a Dzyaloshinskii–Moriya type anisotropic interaction to couple two off-resonant qubits, and they have studied the variations in heat rectification with different system parameters. Yang et al.35 have used three pairwise coupled qubits, two coupled to a common thermal bath and the other to an independent bath, to obtain diode behavior. Left–right asymmetry is a common feature in most of these diode models. Experimental realization of quantum thermal diodes can be achieved through the use of quantum dots,36 superconducting artificial atoms,37 etc. In addition, researchers have extended the theory of quantum thermal diodes to model quantum thermal transistors38–41 and multi-transistor systems.42,43 All these models effectively control the heat flow at the nanoscale.

As discussed, the most currently available quantum thermal diode models have been developed using the simplest spin particle, i.e., spin-12 or qubits, which can be represented using quantum systems with two energy levels. Moreover, the models consisting of either two-qubits29,31 or qubit-chains44,45 require external energy sources, such as magnetic fields, to align the spin particles in a desired direction or to tune interactions between their spin particles. However, the latest technologies, such as quantum thermal machines, quantum computing, and quantum communication, use multi-level spin particles, i.e., qudits, in their applications due to the significant advantages qudits offer in terms of increased computational space, enhanced efficiency, and flexibility.46–49 These benefits have made qudits a promising alternative for advancing the development of quantum technology. Even so, in thermotronics, only a few approaches have been taken to go beyond the use of qubits and instead use multi-level spin particles in the development of quantum thermal devices. For example, Poulsen and Zinner50 have modeled heat rectification using a qutrit, where the low-temperature bath forces the system to a dark state to block the thermal flow in the reverse direction. However, this model requires two harmonic oscillators between the qutrit and two thermal baths to induce only a set of allowed transitions between the qutrit levels. Malavazi et al.51 have introduced a coupled qubit–qutrit–qubit based quantum thermal device, where the qutrit acts as a control element to result in different thermal functionalities. This model also requires an external mechanism to control the interactions between the qubits and the qutrit, resulting in thermal diode characteristics. Our study is motivated by these gaps in the available quantum thermal diode models and uses a qutrit–qubit based system with an autonomous interaction between the qubit and qutrit to result in diode characteristics.

Autonomous or self-contained quantum thermal machines are of significant interest in quantum thermodynamics because of their capability to perform different thermodynamic operations without any external source of work.52,53 Most of the available autonomous quantum thermal machines use the resonance interaction between degenerate quantum states to create coupling between their spin particles.52,54–58 When a quantum system has degenerate states, the free energy from the connected thermal baths can develop an interaction between the system’s degenerate energy levels while suppressing the interactions between other states. We can use a similar strategy in modeling thermotronic devices. For instance, Guo et al.39 have used a resonance interaction between a qutrit and a qubit to model a quantum thermal transistor. The experimental realization of such a quantum thermal transistor has been introduced by Majland et al.59 using a superconducting circuit. Drawing inspiration from these quantum thermal machines and device models, in our research we develop a qutrit–qubit based quantum thermal diode model that does not require additional sources to create interactions between the qutrit and qubits.

This paper is structured as follows: In Sec. II, we formulate the proposed model considering Hamiltonian formation. Next, in Sec. III, we analyze the dynamic and steady-state behavior of the thermal properties and quantum states of the system using the quantum master equation. In Sec. IV, we discuss the parameter selection, simulated diode characteristics, and mechanism of operation of the diode model. The conditions that should be satisfied by the model to result in diode behavior, possible improvements, and physical implementation are discussed in Sec. V, and finally, in Sec. VI, we write our conclusion.

Figure 1(a) shows an electronic diode connected between two terminals A and B. If the potential at terminal A is greater than that at terminal B, there will be a current flow from A to B through the diode. In contrast, if the potential at B is greater than the potential at A, there would not be a current flow. This is the asymmetric electron conduction behavior in an electronic diode. Figure 1(b) is the proposed equivalent thermal diode model, where we connect the diode model S between two terminals A and B. Here, terminal A is a thermal bath (BL) at temperature TL and B is a thermal bath (BR) at temperature TR. When TL > TR, there will be a thermal flow from bath BL to BR through our diode model S, whereas there is no thermal flow from BR to BL, although we make TR > TL.

FIG. 1.
(a) Electronic diode and (b) quantum thermal diode model with a qutrit, qubit, and two thermal baths BL and BR with temperatures TL and TR, respectively. |01⟩, |11⟩, and |21⟩ with energies E0, E1, and E2 are the three energy states of the qutrit, respectively, while |02⟩ and |12⟩ with energies E0 and E3 are the two energy states of the qubit, respectively. Ground state energies, E0, are zero, and g is the interaction energy between the qutrit and qubit. JP is the thermal energy flow from bath BP to system S for P = L and R.
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(a) Electronic diode and (b) quantum thermal diode model with a qutrit, qubit, and two thermal baths BL and BR with temperatures TL and TR, respectively. |01⟩, |11⟩, and |21⟩ with energies E0, E1, and E2 are the three energy states of the qutrit, respectively, while |02⟩ and |12⟩ with energies E0 and E3 are the two energy states of the qubit, respectively. Ground state energies, E0, are zero, and g is the interaction energy between the qutrit and qubit. JP is the thermal energy flow from bath BP to system S for P = L and R.

FIG. 1.
(a) Electronic diode and (b) quantum thermal diode model with a qutrit, qubit, and two thermal baths BL and BR with temperatures TL and TR, respectively. |01⟩, |11⟩, and |21⟩ with energies E0, E1, and E2 are the three energy states of the qutrit, respectively, while |02⟩ and |12⟩ with energies E0 and E3 are the two energy states of the qubit, respectively. Ground state energies, E0, are zero, and g is the interaction energy between the qutrit and qubit. JP is the thermal energy flow from bath BP to system S for P = L and R.
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(a) Electronic diode and (b) quantum thermal diode model with a qutrit, qubit, and two thermal baths BL and BR with temperatures TL and TR, respectively. |01⟩, |11⟩, and |21⟩ with energies E0, E1, and E2 are the three energy states of the qutrit, respectively, while |02⟩ and |12⟩ with energies E0 and E3 are the two energy states of the qubit, respectively. Ground state energies, E0, are zero, and g is the interaction energy between the qutrit and qubit. JP is the thermal energy flow from bath BP to system S for P = L and R.

Close modal

The proposed diode model consists of a coupled qubit and a qutrit, which form system S, and the two thermal baths BL and BR interact individually with qutrit and qubit. These two thermal baths form an environment for system S. The quantum states in our model energize and de-energize by exchanging energy with thermal baths and contribute to a thermal energy flow from one bath to another. By properly selecting system parameters, here we achieve an asymmetric heat flow that is similar to the current flow in an electronic diode.

As described earlier, qutrit is a system consisting of three quantum states. We represent the energy states corresponding to the three levels as column vectors |01⟩ = [0,0,1]T, |11⟩ = [0,1,0]T, and |21⟩ = [1,0,0]T, where [ ]T is the transpose. These quantum states have energies E0, E1, and E2, respectively, and we assume the ground level energy E0 as zero. Then, according to Sakurai and Napolitano,60 we write the Hamiltonian for the qutrit as
(1)
Similarly, a qubit is a quantum system with two energy levels. Ground state |02⟩ = [0,1]T has an energy E0, which we consider zero in our case, and an excited state |12⟩ = [1,0]T has an energy E3. Consequently, the Hamiltonian of the qubit is
(2)
We get the Hilbert space of the combined system using the tensor product between individual systems.61 Accordingly, our free Hamiltonian of the combined system Ĥ0 becomes a 6 × 6 matrix,
(3)
where ⊗ denotes the tensor product, Î2 and Î3 denote the 2 × 2 and 3 × 3 identity matrices in Hilbert space, respectively. The combined system has six quantum states given by the tensor product between individual states of the qutrit and qubit as |mn⟩ = |m1⟩ ⊗ |n2⟩ for m = {0, 1, 2} and n = {0, 1}. We can obtain the same states from the eigenvectors of the free Hamiltonian Ĥ0⁠. Corresponding eigenvalues give the energy associated with the quantum state,
(4)
We make the energy level E2 = E1 + E3 and create two degenerate states |11⟩ and |20⟩ in the system. This enables transitions between |11⟩ and |20⟩ states to occur without input from external energy sources. Hence, we write the interaction Hamiltonian Ĥint between the qutrit and the qubit as
(5)
where g is the interaction energy between the qutrit and qubit. For the validity of the interaction between the degenerate state, it is important to ensure g ≪ Ei for i = {1, 2, 3} such that the eigenstates and eigenvalues of the system remain governed by Ĥ0⁠. This condition certainly implies weak internal couplings.
With Eqs. (3) and (5), we write the Hamiltonian ĤS for system S as
(6)
To find the eigenstates and eigenvalues of system S, we diagonalize the Hamiltonian ĤS⁠, using a transformation matrix denoted by A. Matrix A should be invertible and have the normalized eigenvectors of ĤS as its columns. Then, the diagonalized system Hamiltonian Ĥsys becomes
(7)
where |i⟩ are the quantum states in system S with ɛi energy. These can be listed as
(8)
Figure 2 shows the energy level diagram of system S.
FIG. 2.
Energy level diagram of system S. It has six energy states and eight possible energy jumps, as shown with the arrows. Red arrows show the allowed energy transitions induced by bath BL, and blue arrows show the allowed energy transitions induced by bath BR in the system.
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Energy level diagram of system S. It has six energy states and eight possible energy jumps, as shown with the arrows. Red arrows show the allowed energy transitions induced by bath BL, and blue arrows show the allowed energy transitions induced by bath BR in the system.

FIG. 2.
Energy level diagram of system S. It has six energy states and eight possible energy jumps, as shown with the arrows. Red arrows show the allowed energy transitions induced by bath BL, and blue arrows show the allowed energy transitions induced by bath BR in the system.
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Energy level diagram of system S. It has six energy states and eight possible energy jumps, as shown with the arrows. Red arrows show the allowed energy transitions induced by bath BL, and blue arrows show the allowed energy transitions induced by bath BR in the system.

Close modal
As the system consists of six energy levels, there are 15 possible energy transitions between S and thermal baths. To jump from a lower energy level to a higher energy level, system absorb energy from thermal baths. In contrast, when jumping from a higher energy level to a lower energy level, the system releases energy to the thermal baths. However, we model the system such that the qutrit can only interact with BL and the qubit can only interact with BR. Thus, an energy exchange between the system and thermal bath BL can only flip the quantum state of the qutrit, while an energy exchange between the system and thermal bath BR can only flip the quantum state of the qubit. In addition, it is assumed that in qutrit, transitions |01⟩ ↔ |11⟩ and |11⟩ ↔ |21⟩ are allowed while |01⟩ ↔ |21⟩ is forbidden, which conveys a cascade type qutrit.62,63 These restrictions limit the number of allowed thermal bath induced transitions, as listed below.

Transitions induced by bath BL:

|1⟩ ↔ |2⟩, |1⟩ ↔ |3⟩, |2⟩ ↔ |5⟩, |3⟩ ↔ |5⟩, |2⟩ ↔ |4⟩, |3⟩ ↔ |4⟩, |4⟩ ↔ |6⟩

Transitions induced by bath BR:

|1⟩ ↔ |2⟩, |1⟩ ↔ |3⟩, |2⟩ ↔ |4⟩, |3⟩ ↔ |4⟩, |5⟩ ↔ |6⟩

Suppose that initially the system is at its lowest energy state, i.e., |6⟩. When TL > TR, the qutrit absorbs energy from BL and changes the system state to a higher energy state. As higher energy states are unstable in nature, the system will release its energy to bath BR by changing the state of the qubit. This process makes an energy flow from bath BL to bath BR. When TR > TL, this process happens in the reverse direction, but with a much smaller rate, and therefore, the reverse thermal flow is negligible compared to the forward thermal flow.

Our total system consists of S and two thermal baths. Similar to Caldeira and Leggett,64 we think of the thermal baths as a collection of decoupled harmonic oscillators, each with a different characteristic frequency ωk. Then we write the Hamiltonian of the thermal bath BP as
(9)
where âkP and âkP† denote the annihilation and creation operators of the kth harmonic oscillator in bath BP, and ℏ is the reduced Planck constant. We can write the Hamiltonian for the interaction between system S and bath BP using the spin-boson model in the x-component,29,
(10)
where gk is the coupling strength between S and kth oscillation mode in bath BP. σ̂xP for P = {L, R} represent the Gell–Mann matrix65 in the x-direction and Pauli matrix65 in the x-direction, respectively. Equations (9) and (10) effectively capture environmental effects on the system.
Using Eqs. (7), (9), and (10), we get Hamiltonian Ĥ for the complete system,
(11)
The quantum thermal diode model S interacts and exchanges energy with thermal baths. These thermal baths create an external environment for our system. Thus, we need to use the theory of open quantum systems to describe the dynamic behavior of the proposed system. This interaction with the environment creates entanglements between the system and the environment’s degrees of freedom. As a result, the system may exist in a mixed state rather than a pure state, which is known as decoherence. A mixed state is a statistical ensemble of pure states, and its time evolution is described by the von Neumann or Liouville–von Neumann equation,61,
(12)
Equation (12) is the von Neumann equation in interaction picture, where ρ̂T(t) is the total density operator or total density matrix, HInt(t) is the interaction Hamiltonian between the system and baths in interaction picture, and [ , ] denotes the commutator operator.66 The density matrix’s diagonal elements are known as populations, and the off-diagonal elements represent the coherence between states.67 When system S interacts with thermal baths, this coherence behavior will be lost, and only diagonal elements will remain in the density matrix. If we only want to evaluate the dynamics of our system, we take partial trace over the environment on both sides of Eq. (12). This is known as reduced dynamics of the system.61,68 With Born approximation, Markov approximation, and rotating wave approximation, this results in the following Schrödinger picture quantum master equation in Lindblad form. See  Appendix A for the derivation of the following equation:
(13)
where
(14)
In Eq. (13), ρ̂(t) is the reduced density matrix, a 6 × 6 matrix in our case, and LP[ρ̂(t)] for P ∈ {L, R} is the Lindblad superoperator given in Eq. (14). Lindblad superoperator represents the dissipation that occurs due to coupled thermal baths.69 Here, we have dropped the explicit time dependence for clarity, and { , } denotes the anticommutator operator.66, ω is the list of released energies when a transition from state |i⟩ to |j⟩ occurs and is given by ωij = (ɛi − ɛj)/ℏ. Here, ɛi is the energy in state |i⟩ and ɛj is the energy in state |j⟩. In Eq. (14), nP(ω) is the Bose–Einstein distribution, symbolizing the average number of excitations in bath BP. When bath temperature is TP, excitation frequency is ω, and kB is the Boltzmann constant,
(15)
JP(ω) is the bath spectral density function, and it describes the coupling strength between the system and the bath P at different frequencies,64,70
(16)
In our case, we take the ohmic bath spectral case,71,72 which simplifies Eq. (16) to JP(ω)=κω⁠, where κ is a dimensionless constant.
The Lindblad operator ÂP(ω) is defined as
(17)
where Π̂(ε) is the projection operator40 given by
(18)
for eigenvalues (ɛj) of Ĥsys⁠. Here, |j⟩ is the corresponding eigenvector. It is important to note here that we can also define a new Lindblad operator to reduce the transitions corresponding to the same frequency and find the Lindblad superoperator.

Equations (17) and (18) are collectively known as the jump operator if the unperturbed Hamiltonian Ĥsys is diagonalized.73 Then, substituting unperturbed Hamiltonian eigenvectors to the jump operator will result in non-zero values only for the allowed state transitions in the system.

As we diagonalized the system Hamiltonian, the first term in Eq. (13) vanishes and only Lindblad terms will be left. By expanding the master equation, we get 36 first-order differential equations. We can solve this set of differential equations with given initial conditions to obtain the evolution of density matrices in time. Here, as we have shown in the supplementary material, all off-diagonal solutions decay to zero with time while diagonal density matrix element solutions remain non-zero. Differential equations corresponding to these diagonal elements are
(19)
where ΓjkP is the transition rate from |j⟩ to |k⟩ induced by bath BP and given by
(20)
Note that ΓjkP=−ΓkjP⁠.
We use the definitions used by Alicki74 to find the thermal current injected by the bath BP into system S. If JP is the thermal current flowing from bath to system,
(21)
The complete derivation of Eq. (21) is given in  Appendix B. With this definition, we obtain the following two equations for the thermal energy inflows to our system S:
(22)
Here, ɛjk = ɛj − ɛk. By comparing Eq. (22) and Fig. 2, we can see that every term in energy flows corresponds to a state transition in the system induced by the coupled thermal baths.
At non-equilibrium steady state, ρ̇ii for i = {1, 2, …, 6} becomes zero, and we can derive the following relationships from Eq. (19):
(23)
At this state, although the expectation values of the system operators are time-independent, there will be a steady heat flux from the hot thermal bath to the cold thermal bath through the systems. The reduced density matrix will converge to a fixed but arbitrary state, known as a “relaxation.” Substituting the corresponding energy values from Eq. (8) and transition rates from Eq. (23) along with ɛ2 ≈ ɛ3 in the weak internal interaction condition reduces energy inflows in Eq. (22) to
(24)
which conveys the conservation of energy, the first law of thermodynamics75 in our complete system.
The rectification factor is a measurement of the effectiveness of a diode. It refers to the ability of the diode to conduct in one direction and isolate in the opposite direction. According to the literature, one can define the rectification factor in different ways.15 We use the following definition:30,31
(25)
to evaluate the performance of our model, where JL(TL, TR) is the thermal inflow in the forward condition and JL(TR, TL) is the thermal inflow in the reverse condition. According to the definition, R(TL, TR) can take values between 0 and 1. An ideal diode will have a rectification factor of 1, which means it conducts heat effectively when TL > TR, i.e., in forward condition, and completely isolates when TR > TL, i.e., in reverse condition. A rectification factor of 0 means similar heat flows in both forward and reverse conditions.

We use Mathematica V14 to model and numerically evaluate the behavior of our system. We work with SI units and take ℏ = 1.0546 × 10−34 Js and kB = 1.3806 × 10−23 J/K.

To ensure a proper energy exchange between the thermal baths and the system, we need to make the overall scale of the system energy levels greater than the thermal bath energy levels.40 For computations, we suppose system energy levels are 5 times greater than the thermal bath energy levels. If ℏω is the average system energy and T is the maximum bath temperature, the above condition implies
(26)
Furthermore, recently developed quantum thermal systems work in the milli-Kelvin temperature range.76–78 Accordingly, if we choose to work with a maximum of 300 mK temperature, the above conditions suggest that our system energy scale should be Δ = ℏω = 2.0709 × 10−23.

In our quantum thermal diode model, we let qutrit and qubit ground level energies to zero and make E2 = E1 + E3 to create degenerate states. Thus, we select E1 = 0.95Δ and E3 = 0.05Δ to make E2 = Δ. In addition, to have a valid interaction Hamiltonian, energies of the states |2⟩ and |3⟩ should be resonant, i.e., g ≪ Ei for i = {1, 2, 3}. Therefore, we select g = 0.01Δ for our simulations. Furthermore, for simplicity, we make κ = 1.

With the above system parameters and Eq. (22), we calculate the steady-state thermal current inflows to the systems. Figure 3 shows the variation of steady-state thermal current flow JL from bath BL to system S.

FIG. 3.
Variation of thermal current flow JL from bath BL to system S at steady state. The blue curve shows the variation in JL when TR is fixed at 150 mK and TL is varying along the T axis. The orange curve shows the variation in JL when TL is fixed at 150 mK and TR is varying along the T axis.
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Variation of thermal current flow JL from bath BL to system S at steady state. The blue curve shows the variation in JL when TR is fixed at 150 mK and TL is varying along the T axis. The orange curve shows the variation in JL when TL is fixed at 150 mK and TR is varying along the T axis.

FIG. 3.
Variation of thermal current flow JL from bath BL to system S at steady state. The blue curve shows the variation in JL when TR is fixed at 150 mK and TL is varying along the T axis. The orange curve shows the variation in JL when TL is fixed at 150 mK and TR is varying along the T axis.
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Variation of thermal current flow JL from bath BL to system S at steady state. The blue curve shows the variation in JL when TR is fixed at 150 mK and TL is varying along the T axis. The orange curve shows the variation in JL when TL is fixed at 150 mK and TR is varying along the T axis.

Close modal

When TR is fixed at 150 mK temperature and TL is varying, we see an exponential increase in JL for TL > 150 mK (i.e., in forward condition). This is equivalent to the forward current in an electronic diode. In addition, when TR is fixed at 150 mK and TL < 150 mK (i.e., in reverse condition), thermal flow JL is close to zero. Conversely, when TL is fixed at 150 mK and TR is varying, JL is always close to zero.

Figure 4 shows the variation of the rectification factor calculated with Eq. (25) for the same set of system parameters.

FIG. 4.
Variation of rectification factor with TL when TR is fixed at 150 mK.
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Variation of rectification factor with TL when TR is fixed at 150 mK.

FIG. 4.
Variation of rectification factor with TL when TR is fixed at 150 mK.
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Variation of rectification factor with TL when TR is fixed at 150 mK.

Close modal

Our system behaves as a perfect diode within a considerably large temperature range except around 90–130 mK. As stated in Sec. III D, we cannot obtain an idea about the magnitudes of the thermal flows from the rectification factor. For example, the rectification factor is 1 for TL < 90 mK as per Fig. 4. However, in Fig. 3, when TL < 90 mK, thermal flows are negligible.

To understand the mechanism of operation of our model, we plot the state transition diagrams for TL > TR and TR > TL cases as shown in Fig. 5.

FIG. 5.
State transition diagram in steady state when (a) TL = 300 mK and TR = 150 mK, i.e., in diode forward condition; and (b) TR = 300 mK and TL = 150 mK, i.e., in diode reverse condition. Red arrows show the transitions induced by bath BL, and blue arrows show transitions induced by bath BR. Arrows are directed toward the transition direction, and arrow thickness is proportional to the transition rate.
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State transition diagram in steady state when (a) TL = 300 mK and TR = 150 mK, i.e., in diode forward condition; and (b) TR = 300 mK and TL = 150 mK, i.e., in diode reverse condition. Red arrows show the transitions induced by bath BL, and blue arrows show transitions induced by bath BR. Arrows are directed toward the transition direction, and arrow thickness is proportional to the transition rate.

FIG. 5.
State transition diagram in steady state when (a) TL = 300 mK and TR = 150 mK, i.e., in diode forward condition; and (b) TR = 300 mK and TL = 150 mK, i.e., in diode reverse condition. Red arrows show the transitions induced by bath BL, and blue arrows show transitions induced by bath BR. Arrows are directed toward the transition direction, and arrow thickness is proportional to the transition rate.
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State transition diagram in steady state when (a) TL = 300 mK and TR = 150 mK, i.e., in diode forward condition; and (b) TR = 300 mK and TL = 150 mK, i.e., in diode reverse condition. Red arrows show the transitions induced by bath BL, and blue arrows show transitions induced by bath BR. Arrows are directed toward the transition direction, and arrow thickness is proportional to the transition rate.

Close modal
Here, red(blue) arrows show the transitions induced by bath BL(BR), and arrow thickness is proportional to the transition rates. Arrows are directed toward the direction of the transition, and upward arrows absorb energy from coupled thermal baths to jump to higher energy states, whereas downward arrows release energy to thermal baths and come to lower energy states. By comparing the widths of the arrows, we see that the transition rates in the reverse condition are comparatively smaller than the rates in the forward condition. We can use the definition of transition rates given in Eq. (20) to explain this behavior. According to this definition, the transition rate is a complicated function of the bath temperature TP, the transition energy difference ℏωjk, and state population densities ρjj and ρkk. For a given TP and ωjk, the transition rate becomes zero only when the following condition is satisfied:
(27)
If the population densities deviate from this ratio, ΓjkP will generate an opposing flow between |j⟩ and |k⟩ to restore the population densities back to their original ratio. In addition, according to Eq. (20), bath spectral density, JP(ω) also affects the magnitude of the transition rates. Since we assumed ohmic spectral densities, large energy differences will enhance the transition rates and, therefore, the thermal flows.

From Fig. 5, we can identify the transition cycle that contributes to an energy flow in both forward and reverse conditions. It is important to note that although there are transitions happening toward and away from state |1⟩, the total contribution of those transitions induced from baths BL and BR cancels with each other. Therefore, we can neglect the |1⟩ ↔ |2⟩ and |1⟩ ↔ |3⟩ transitions in Fig. 5 for the simplicity of understanding the mechanism. In addition, when the transitions are happening in opposite directions (e.g., |5⟩ → |3⟩ and |2⟩ → |5⟩ in Fig. 5), we can use the resultant. Accordingly, the main transition cycle contributing to thermal flow in our model is shown in Fig. 6.

FIG. 6.
State transitions occur in (a) forward condition and (b) reverse condition.
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State transitions occur in (a) forward condition and (b) reverse condition.

FIG. 6.
State transitions occur in (a) forward condition and (b) reverse condition.
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State transitions occur in (a) forward condition and (b) reverse condition.

Close modal

Here, +L indicates an energy absorption from bath BL, and −L indicates an energy release to bath BL. Similarly, +R indicates an energy absorption from bath BR, and −R indicates an energy release to bath BR. Suppose the probability of the complete system to be in state |6⟩, i.e., in the state with the lowest energy, is 1 at time t = 0. Then, in the forward condition, state transitions start with the |6⟩ → |4⟩ transition by absorbing an energy E1 from bath BL. Afterward, the system again absorbs heat from bath BL to jump to the degenerate states |2⟩ and |3⟩. Thereafter, the system de-energizes to its lowest energy state |6⟩ through |5⟩ by releasing energy. In reverse conditions, this happens in the other direction, but with much smaller rates.

Our reduced system dynamic model utilized Born approximation, Markov approximation, and rotating wave approximation.61 To perform Markov approximation, it is important that the environmental correlation times are significantly shorter than the timescale of the system’s evolution. This requirement is essential to ensure that memory effects can be neglected, such that the system’s evolution can be described without considering the effects of its past interactions with the environment. In addition, the coupling between our system and environment should be weak to satisfy the Born approximation. Then, under secular approximation, we average the rapidly oscillating terms, and under RWA, we neglect the non-secular terms in the master equation. Secular approximation helps to decouple diagonal and off-diagonal elements of the density matrix by ensuring off-diagonal elements evolve on a much faster timescale compared to the diagonal elements when the system’s characteristic frequencies are well-separated.

When our system S is coupled to thermal baths, this interaction can lead to a shift in the system energy levels, which is known as the energy level renormalization.61,79,80 As defined in Eq. (7), if ɛi is the energy that corresponds to quantum state |i⟩, we take δɛi(t, t0) as the variation in energy that occurs in state |i⟩ due to the interaction with the thermal baths at a time t after the initial time t0. Then, the renormalized system Hamiltonian Ĥsysr(t,t0) is
(28)
where εir(t,t0) is the renormalized system energy corresponding to the quantum state |i⟩ at time t. This shift can also be written in terms of system eigenfrequency representation,
(29)
where ωir(t,t0) is the renormalized system eigenfrequency, ωi is the bare state eigenfrequency, and δωi(t, t0) is the variation in eigenfrequency corresponding to the quantum state |i⟩ at time t.
We can calculate the system eigenfrequency (energy) shift δω using the self-energy function,80,81 ϒ(ω) of the system. In simple terms, ϒ(ω) is a complex function of eigenfrequency ω, which describes the effects of environmental interactions on the energy levels of a quantum system and can be written as
(30)
when the bath spectral density function J(ω) is defined for positive frequencies and decays to zero when ω → ∞. The real part of the self-energy function, δ(ω), represents the shift in system energy levels, whereas the imaginary part represents the broadening (decay) in energy levels. Accordingly, the eigenfrequency (energy) shift as a function of ω is
(31)
where P is the Cauchy principal value.82 

This renormalization effect is significant when the system–environment coupling is strong, and for such a system we should use the exact quantum master equation with a renormalized system Hamiltonian. However, in this study, we assume a very weak system–bath coupling, where Δ(ω) → 0, such that the energy level shift can be ignored. Therefore, the system evolves in time according to the master equation derived under the weak coupling limit.

Our analysis assumes that the thermal baths are at fixed temperatures. To ensure this, the diode models are connected between large baths, keeping their terminals at fixed temperatures. However, the use of such large baths introduces indirect interactions with diode terminals that are not directly connected, leading to additional transitions. This effect is enhanced when the device is fabricated on a substrate, as the thermal baths can serve as a common environment.58,77,83 Figure 7 illustrates system S when thermal baths, BL and BR interact with both qutrit and qubit. In such a condition, the total interaction Hamiltonian between system S and baths is35,84
(32)
Here, we select g2j = μjg1j and g1k = μkg2k, where μj and μk are the two parameters that account for the asymmetry in the system–bath coupling.
FIG. 7.
System with common baths BL and BR. aj† and aj (bk† and bk) are the annihilation and creation operators of the jth (kth) harmonic oscillator in bath BL (BR) with characteristic frequency ωjL (ωkR). g1j, g2j, g1k, and g2k represent the coupling strength between the baths jth and kth oscillation modes with qutrit and qubit.
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System with common baths BL and BR. aj† and aj (⁠bk† and bk) are the annihilation and creation operators of the jth (kth) harmonic oscillator in bath BL (BR) with characteristic frequency ωjL (ωkR)⁠. g1j, g2j, g1k, and g2k represent the coupling strength between the baths jth and kth oscillation modes with qutrit and qubit.

FIG. 7.
System with common baths BL and BR. aj† and aj (bk† and bk) are the annihilation and creation operators of the jth (kth) harmonic oscillator in bath BL (BR) with characteristic frequency ωjL (ωkR). g1j, g2j, g1k, and g2k represent the coupling strength between the baths jth and kth oscillation modes with qutrit and qubit.
View largeDownload slide

System with common baths BL and BR. aj† and aj (⁠bk† and bk) are the annihilation and creation operators of the jth (kth) harmonic oscillator in bath BL (BR) with characteristic frequency ωjL (ωkR)⁠. g1j, g2j, g1k, and g2k represent the coupling strength between the baths jth and kth oscillation modes with qutrit and qubit.

Close modal
The system–bath interaction Hamiltonian given in Eq. (32) determines the non-vanishing jump operators ÂPα(ω)⁠, for all eigenfrequencies of Ĥsys⁠, ω=ωij=εi−εjℏ>0⁠, corresponding to the transitions between states |i⟩ and |j⟩. Then, the non-vanishing jump operators are
(33)
These unavoidable extra-state transitions change the performance of the proposed quantum diode model by reducing the symmetry of the system. Therefore, it is important to restrict the undesired transitions by performing reservoir engineering.77,83 One such technique is bath spectral filtering.84–87 When a thermal bath is connected to the system through a bandpass filter, only the frequencies near the resonance frequency of the filter will interact with the system. Only desired transitions will occur between the thermal baths and the system. These reservoir engineering techniques can also be used to enhance the performance of quantum thermal devices in common environments.58,77

In addition, our system consists of two thermal reservoirs, which can interact directly with each other. However, to obtain diode-like behavior, this direct interaction should be prevented, and thermal energy should only flow through our system. In addition, we only showed the thermal flow behavior of the model for system parameters; E1 = 0.95Δ and E3 = 0.05Δ. However, this model behaves as a diode for any set of E1 and E3 selections, given E1 ≫ E3. When we make E1 = E3, the diode behavior vanishes, resulting in similar forward and reverse thermal flows. If we make E1 ≪ E3, the behavior inverts and gives a diode in the reverse direction. Furthermore, E1 + E3 = E2 and g ≪ Ei for i = {1, 2, 3} conditions should be satisfied all the time to create degenerate states and maintain an effective interaction between the qutrit and qubit. Furthermore, in our model, we always connect the qutrit to the left side and the qubit to the right side. However, we can have the same characteristics by interchanging those.

In our study, we derived the average dynamics and thermal flows of the system. However, in a real-world implementation, baths will undergo external disturbances, such as continuous measurement of thermal baths. These perturbations can significantly impact the system’s behavior and need to be accounted for in any comprehensive analysis. Noise models and stochastic approaches where randomness is introduced to the system equations78,88 is one such alternative description that can be used to study the effects of perturbations accurately.

In electronics, if we directly connect multiple diodes in series, it will still perform as a diode. However, we cannot expect similar behavior from our model of thermal diode with thermal reservoirs at either end, as we analyzed here. We cannot directly connect two thermal diode models, as the energy levels that mediate the energy transfer from one diode to the other will be off-resonant to each other. Previously, we have tackled this scenario for interconnected quantum thermal transistors by introducing another quantum system between the two diodes, which will bridge the gap in energy levels.42,43 We can follow a similar strategy to obtain diode-like thermal energy transfer from multiple series connected thermal diode models. This is left as future work.

Physical realization of the proposed model is possible using superconducting circuits. Very much like the quantum thermal transistor design proposed by Majland et al.,59 we can use two single transmons coupled via a resonator to design the proposed quantum thermal diode. Figure 8 shows its lumped element representation, where transmons are modeled as an-harmonic circuits with Josephson junctions.

FIG. 8.
Lumped element representation of the quantum thermal diode implementation using superconducting circuits. The circuit encapsulated in red is the transmon qutrit coupled to the thermal bath BL, and the blue circuit is the transmon qubit coupled to the thermal bath BR. The symbol ⊠ represents the Josephson junction in transmons.
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Lumped element representation of the quantum thermal diode implementation using superconducting circuits. The circuit encapsulated in red is the transmon qutrit coupled to the thermal bath BL, and the blue circuit is the transmon qubit coupled to the thermal bath BR. The symbol ⊠ represents the Josephson junction in transmons.

FIG. 8.
Lumped element representation of the quantum thermal diode implementation using superconducting circuits. The circuit encapsulated in red is the transmon qutrit coupled to the thermal bath BL, and the blue circuit is the transmon qubit coupled to the thermal bath BR. The symbol ⊠ represents the Josephson junction in transmons.
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Lumped element representation of the quantum thermal diode implementation using superconducting circuits. The circuit encapsulated in red is the transmon qutrit coupled to the thermal bath BL, and the blue circuit is the transmon qubit coupled to the thermal bath BR. The symbol ⊠ represents the Josephson junction in transmons.

Close modal

To obtain the required behavior, the upper transmon circuit should truncate to have a Hilbert space of three dimensions (qutrit), and the lower transmon circuit should truncate to have a Hilbert space of two dimensions (qubit).

In this paper, we propose a quantum thermal diode model developed using a cascade type qutrit and a qubit. We derived the Hamiltonian of the combined system and used a transition between degenerate states to derive the interaction Hamiltonian between the qutrit and qubit. This interaction Hamiltonian eliminated the requirement of additional sources to create interaction between the qutrit and qubit. We found the states and their respective energies of the complete system using eigenvalue decomposition. Then, allowed state transitions were derived by looking at the complete system states and interaction with thermal baths.

Thereafter, we derived the reduced system dynamics using the quantum master equation with Markov approximation, Born approximation, and rotating-wave approximation. It resulted in six non-zero differential equations, which correspond to the pure states in the density matrix. We can find the time evolution of the pure states of the reduced system by solving this set of differential equations with given initial conditions. However, we focused more on the steady state behavior of the system and derived an equation for the thermal energy inflows to the system. We found that our model is compliant with the 1st law of classical thermodynamics as expected.

Afterward, we used numerical simulations to study the characteristics of our model with given system parameters. We found that the proposed model has a diode like behavior with the selected parameters and has good rectification within a broad temperature range. We used state transition diagrams to understand the behavior of the diode.

Later, we discussed the limitations and conditions the model should satisfy to act as a thermal diode and possible improvements. Finally, we remarked on how the proposed model can be implemented using superconducting circuits and presented a basic configuration that will help to develop this kind of device.

See the supplementary material for the WOLFRAM MATHEMATICA V14 code for simulating the quantum thermal diode model at https://github.com/AnuradhiRajapaksha/Coupled-Qutrit-Qubit-Quantum-Thermal-Diode.git

A.R. would like to thank all members of AχL at Monash University for their encouragement and insightful discussions. The work of A.R. is supported by the Monash University Graduate Research Scholarship.

The authors have no conflicts to disclose.

Anuradhi Rajapaksha: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Software (lead); Validation (equal); Writing – original draft (lead); Writing – review & editing (lead). Sarath D. Gunapala: Conceptualization (equal); Supervision (equal). Malin Premaratne: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – original draft (supporting); Writing – review & editing (lead).

The data that support the findings of this study are available within the article and the supplementary material.

We follow the procedure in the literature61,66,89 to derive the quantum master equation for our model. Figure 9 shows a block diagram of the complete system of interest.

FIG. 9.
Block diagram of the complete system. S is our quantum diode model, and BL and BR thermal baths create the environment for S. Ĥ(t) is the complete system Hamiltonian.
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Block diagram of the complete system. S is our quantum diode model, and BL and BR thermal baths create the environment for S. Ĥ(t) is the complete system Hamiltonian.

FIG. 9.
Block diagram of the complete system. S is our quantum diode model, and BL and BR thermal baths create the environment for S. Ĥ(t) is the complete system Hamiltonian.
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Block diagram of the complete system. S is our quantum diode model, and BL and BR thermal baths create the environment for S. Ĥ(t) is the complete system Hamiltonian.

Close modal
Block S is our quantum thermal diode model, and blocks BL and BR create the environment for our system S. Ĥ(t) represents the closed system Hamiltonian, and it follows Hamiltonian dynamics. If Ĥsys is the system Hamiltonian, ĤBath is the bath Hamiltonian, and ĤInt is the interaction Hamiltonian between the system and baths, we can write the total system Hamiltonian as
(A1)
where α is the intensity of interaction between system and baths (we let α = 1 i.e., full interaction in our derivation for simplicity) and ÎB and ÎS are the identities in Hilbert space of baths and systems, respectively. Here,
(A2)
and
(A3)
Then, we use the interaction picture Liouville–von Neumann equation given in Eq. (12) to derive the quantum Markovian master equation. Integrating Eq. (12) from 0 to t will result in
(A4)
and we can substitute Eq. (A4) back in Eq. (12) as
(A5)
Taking the partial trace of Eq. (A5) over both baths degrees of freedom, i.e., TrL,R{·}, with assumption TrL,R[HInt(t),ρ̂T(0)]=0 results in the following integro-differential equation for the reduced density matrix of the system:
(A6)
Here, ρ̂(t)=TrL,R{ρ̂T(t)} is the reduced density matrix of the system.
To remove ρ̂T(s) from Eq. (A6), we use the Born approximation, assuming a weak coupling between the system and bath. This approximation implies that the influence of the system on baths is small and, hence, baths can be treated as approximately constant. This results,
(A7)
Next, we perform Markov approximation, assuming the time development of the states of the system at time t only depends on the present state ρ̂(t)⁠. This modifies Eq. (A7) to the following:
(A8)
Equation (A8) is known as the “Redfield Equation,” but it is still not a Markovian master equation as the time evolution of the reduced density matrix depends on the initial preparation at time t = 0. We replace s by t − s in Eq. (A8), invoke Markov approximation, and let the upper limit of the integral approach ∞,
(A9)
We can write the Schrödinger picture interaction Hamiltonian as
(A10)
where ÂP and B̂P are the Hermitian operators acting on systems and baths, respectively. The definition in Eq. (17) implies that the operators ÂP(ω) and ÂP†(ω) are the eigenoperators of Ĥsys belonging to the frequencies ±ω, respectively. Therefore, summing Eq. (17) overall energy differences results in
(A11)
and, therefore, the interaction Hamiltonian becomes
(A12)
Transforming Eq. (A12) into interaction picture gives
(A13)
where interaction picture operators of the baths are given by
(A14)
Expanding the commutators in Eq. (A9) and then substituting eigenoperator decomposition with
(A15)
for P ∈ {L, R} allow us to apply secular approximation, and then we follow the procedure extensively discussed in Ref. 61 to obtain the Scrödinger picture master equation,
(A16)
To derive the equation for thermal energy flow, we follow the procedure in Refs. 29 and 74. According to the first law of thermodynamics, we can write the law of conservation of energy to our system as
(B1)
where E is the energy of the system and Q is the heat supplied to the system by its surroundings. However,
(B2)
and, therefore,
(B3)
Since there are two heat sources that interact with system S, we can write the thermal current into the system as
(B4)
By equating Eqs. (B3) and (B4) according to Eq. (B1) results,
(B5)
which can be further simplified using Eq. (13). Then, we can obtain two equations for the thermal current flow into system S as
(B6)

To compare the differences in diode characteristics and performance between our qutrit–qubit based diode model and the commonly used qubit–qubit based diode models, we select the quantum thermal diode developed by Ordonez-Miranda et al.30 This diode model consists of two qubits coupled individually to two thermal baths, and the interaction between the qubits is of Ising-type.29 

To enable a fair comparison, we use the system parameters shown in Fig. 10, which feature identical interaction energies and similar energy gaps between the ground state and the highest energy state. These parameters allow for an equitable assessment of the two models.

FIG. 10.
Selection of system parameters for comparison. (a) Qutrit–Qubit based model. (b) Qubit–Qubit based model.
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Selection of system parameters for comparison. (a) Qutrit–Qubit based model. (b) Qubit–Qubit based model.

FIG. 10.
Selection of system parameters for comparison. (a) Qutrit–Qubit based model. (b) Qubit–Qubit based model.
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Selection of system parameters for comparison. (a) Qutrit–Qubit based model. (b) Qubit–Qubit based model.

Close modal

The variation of the steady state thermal energy flows with the above system parameters for the two models is shown in Fig. 11. It is obvious that our qutrit–qubit based model results in a higher thermal flow than the qubit–qubit based model by Ordonez-Miranda et al.30 Consequently, the proposed qutrit–qubit configuration significantly outperforms the established qubit–qubit based diode model, highlighting its advantage in performance.

FIG. 11.
Variation of steady state thermal energy flows in the (solid) qutrit–qubit model and the (dashed) qubit–qubit model. Blue curves show the variation in JL when TR is fixed at 150 mK and TL is varying along the T axis. Orange curves show the variation of JL when TL is fixed at 150 mK and TR is varying along the T axis.
View largeDownload slide

Variation of steady state thermal energy flows in the (solid) qutrit–qubit model and the (dashed) qubit–qubit model. Blue curves show the variation in JL when TR is fixed at 150 mK and TL is varying along the T axis. Orange curves show the variation of JL when TL is fixed at 150 mK and TR is varying along the T axis.

FIG. 11.
Variation of steady state thermal energy flows in the (solid) qutrit–qubit model and the (dashed) qubit–qubit model. Blue curves show the variation in JL when TR is fixed at 150 mK and TL is varying along the T axis. Orange curves show the variation of JL when TL is fixed at 150 mK and TR is varying along the T axis.
View largeDownload slide

Variation of steady state thermal energy flows in the (solid) qutrit–qubit model and the (dashed) qubit–qubit model. Blue curves show the variation in JL when TR is fixed at 150 mK and TL is varying along the T axis. Orange curves show the variation of JL when TL is fixed at 150 mK and TR is varying along the T axis.

Close modal
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