Quantum Synchronisation in Nanoscale Heat Engines
Noufal Jaseem Michal Hajdušek Vlatko Vedral Rosario Fazio Leong-Chuan Kwek Sai Vinjanampathy






noufal@iitb.ac.in
Department of Physics, Indian Institute of Technology-Bombay, Powai, Mumbai 400076, India


cqtmich@nus.edu.sg
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore


Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore
Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, UK


Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore
ICTP, Strada Costiera 11, 34151 Trieste, Italy
NEST, Scuola Normale Superiore & Instituto Nanoscienze-CNR, I-56126 Pisa, Italy


Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore
Institute of Advanced Studies, Nanyang Technological University, Singapore 639673
National Institute of Education, Nanyang Technological University, Singapore 637616


sai@phy.iitb.ac.in
Department of Physics, Indian Institute of Technology-Bombay, Powai, Mumbai 400076, India
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore





	Owing to the ubiquity of synchronization in the classical world, it is interesting to study its behavior in quantum regime systems. Though quantum synchronisation has been investigated in many systems, a clear connection to quantum technology applications is lacking.
	Hence a clear experimental demonstration of quantum synchronization is lacking. We  bridge this gap and show how that nanoscale heat engines are a natural platform to study quantum synchronization and always possess a stable limit cycle.
	We demonstrate how the power output of the quantum engine shows signatures of synchronization which can be readily observed with current technologies.
	Furthermore, we demonstrate an intimate relationship between the power of a heat engine and its phase-locking properties by proving that synchronization places an upper bound on the achievable steady-state power of the engine.
	Finally, we show that the efficiency of the engine sets a point in terms of the bath temperatures where synchronization vanishes.
	We link the physical phenomenon of synchronization with the emerging field of quantum thermodynamics by establishing quantum synchronization as a mechanism of stable phase coherence. 
 
Quantum Synchronization in Nanoscale Heat Engines
    Sai Vinjanampathy
    December 30, 2023
=================================================






Introduction.—
Synchronization has been observed and studied in a multitude of naturally occurring as well as man-made systems, finding different applications in fields ranging from engineering to medicine <cit.>. Recently the universality of this phenomenon has also been embraced in the quantum realm. Thus far all the efforts have been aimed at characterising synchronization in quantum systems<cit.>.
It is compelling to ask if/where this phenomenon may play a role in the functionality itself of quantum devices.
What is the impact of synchronization on quantum technology platforms?
A natural playground to  explore this question is quantum thermodynamics <cit.>.
Nanoscale heat engines can be modelled as  multi-level atoms that  produce work when cyclically coupled to two or more heat baths <cit.>.
In this manuscript we highlight a deep connection between quantum synchronization and the performance in of nanoscale heat engines.
This relation is particularly evident in the emitted power of the engine displaying an Arnold tongue, a distinct signature of synchronization.
We show that synchronization provides an upper bound on the achievable magnitude of the steady-state power of the heat engine.
Finally, we determine that the efficiency of the heat engine η sets the ratio of bath temperatures where synchronization vanishes.
This points at a fundamental relation between the operational regime of a thermal machine and its synchronization properties.
 An important outcome of this relationship is to bring the study of quantum synchronisation closer to quantum technology applications.



A basic prerequisite for synchronization is the existence of a stable limit cycle <cit.>.
Once a limit cycle has been established in a non-linear dynamical system, we can synchronize the given system to either an external frequency standard [synchronisation is usually reserved for the adjustment of rhythms between two or more physical systems, whereas the adjustment of a systems rhythm to an external frequency is known as entrainment. Following existing literature in quantum synchronisation, we will use the same word for both phenomena.].
We show that thermal atoms possess a limit cycle by constructing the quasi-probability distribution for arbitrary three-level atoms.
This allows us to perform a synchronization analysis of a thermal three-level system and investigate its phase-locking properties in the context of the engine's performance. 
We then investigate phase locking of thermal atoms to an external field and show how the power spectral density of the emitted radiation exhibits signatures of synchronization.
Finally, we show an intimate relationship between synchronization and the performance of thermal three-level masers as a heat engine.


 
Thermal three-level atoms.—
We consider the  three-level maser model introduced by Scovil and Schulz-Dubois as an example of quantum synchronization <cit.>.
In view of the connection to  heat engines, we will examine its properties when coupled to two baths kept at different temperatures.
Though we consider a specific model to develop the results presented here, the techniques are transferrable to generic coherent quantum technologies.

We consider thermal three-level atoms to study synchronization.
The evolution for the system depicted in Fig. (<ref>) can be written (ħ=1) as

    ρ̇=-i[H,ρ]+ℒ_h[ρ]+ℒ_c[ρ],

where H=H_0+V represents the sum of the bare Hamiltonian H_0 and the drive V=ε(e^iω_d tσ_23+e^-iω_d tσ_32), with σ_ij≡|i⟩⟨j|. We model the baths as single-mode thermal fields resonant with the respective atomic transitions that they couple.
The bath frequency ω, its temperature T and its average photon number n̅ are related by e^-ħω/k_B T = n̅/(n̅ + 1) <cit.>.
The Lindbladian dissipators are given by ℒ_h[ρ]≡γ_h n̅_h 𝒟[σ_31]ρ + γ_h (1+n̅_h) 𝒟[σ_13] ρ, incoherently coupling the |1⟩↔|3⟩ transition at temperature T_h, and ℒ_c[ρ]≡γ_c n̅_c 𝒟[σ_21] ρ+ γ_c  (1+n̅_c) 𝒟[σ_12]ρ representing a cold bath at temperature T_c and incoherently coupling the |1⟩↔|2⟩ transition.
The Markovian master equation is written in the standard Lindblad form <cit.> as 𝒟[O]ρ=Oρ O^†-1/2{O^†O,ρ}.

The system is weakly driven with small detunings, which ensures that (a) the limit cycles presented below are not deformed by the drive, (b) the populations in the steady states are dominated by the dissipation rates and (c) the adiabatic definitions of heat and work are valid in this regime.

Points (a) and (b) are related and ensure that we are in a regime where it is sensible to study synchronization <cit.>.
Large driving strength also produces phase locking.
However, due to deformation of the limit cycle this regime is known as suppression of natural dynamics <cit.> and is beyond the scope of this manuscript.
In general, the power of an engine can be attributed to two parts, one that accounts for the change in the instantaneous eigenvalues and another which accounts for the change in eigenvectors.
For perturbative driving the first term does not contribute (indeed our eigenvalues are constant in time).

The definition of heat (δ Q={ Hδρ}) and work (δ W={δ Hρ}) due to Alicki <cit.> are known to (a) not saturate in steady state and (b) be frame dependent quantities.
This motivated Boukobza and Tannor to define heat current as Q̇ =tr{ℒ[ρ] H_0} and power as P = -i  tr{ [H, ρ] H_0 } <cit.>, which are known to behave appropriately.
The analytical steady state of this system, ρ^ss can be calculated and is discussed in the supplemental material alongside formulae for heat currents and power for completeness.
We note that since our driving is perturbative, the two definitions only differ by a small number of the order of ε.



Synchronization in heat engines.— A stable limit cycle is defined as a phase trajectory that attracts nearby orbits.
Such a limit cycle arises as a compromise between dissipation and gain in the system.
Another defining property of the limit cycle is that an observable phase is free.
It is this freedom of the phase combined with the stability of the limit cycle that allows a weak external signal to influence the phase distribution of the oscillator.
To study limit cycles in thermal three-level systems, we define the Husimi-Kano Q-representation function <cit.>, given for a generic three-level atom as Q(θ,ξ,φ_1,φ_2)≡(6/π^2)⟨n_3|ρ|n_3⟩ where |n_3⟩≡(cosθ,  e^i φ_1cosξsinθ,  e^i φ_2sinξsinθ)^T is the SU(3) coherent state <cit.>.

From the definition of |n_3⟩ we can see that (φ_1,φ_2) and (θ,ξ) carry information about the coherences and the populations of the system, respectively.
Under the influence of a bare Hamiltonian written in the energy eigenbasis, the SU(3) coherent state evolves as |n_3⟩=|θ,ξ,φ_1,φ_2⟩→|θ,ξ,φ_1-ω_21t,φ_2-ω_31t⟩, where ω_ij≡ω_i-ω_j.
This implies that the angles (φ_1,φ_2) are the relevant dynamical phases to the study of synchronization.

Similarly to a spin-1 atom considered in <cit.> the system becomes synchronized when the distribution of φ_1 and φ_2 becomes localized.
The difference between the localized and uniform distributions serves as a measure of synchronization, 

    𝒮(φ_1,φ_2) ≡∫ dΩ Q(θ,ξ,φ_1,φ_2) - 1/4π^2,

where ∫ dΩ≡∫_0^π/2 dθcosθsin^3 θ∫_0^π/2 dξcosξsinξ.
The measure 𝒮(φ_1,φ_2) quantifies the deviation of the phase distribution from the uniform distribution and hence measures the phase locking of the angles (φ_1,φ_2).
This is a generalisation to the full SU(3) group of a previous definition presented for SO(3) subgroup of SU(3) <cit.>.

Under the influence of the bare Hamiltonian and the dissipators, the system settles into a stable limit cycle given by no phase preference in the angles (φ_1,φ_2).
When the system is driven, it develops a phase preference characterised by localization of Q(θ,ξ,φ_1,φ_2) in the relevant angles (φ_1,φ_2).
The synchronization measure 𝒮(φ_1,φ_2) in the steady state is evaluated to be

    𝒮(φ_1,φ_2)    =   1/8π{Re[ e^iφ_1ρ^ss_12] + Re[ e^iφ_2ρ^ss_13] 
       +   Re[ e^i(φ_2-φ_1)ρ^ss_23] }.

Here ρ^ss is the steady-state density matrix.
For the evolution given in Eq. (<ref>), this synchronization measure simplifies to 𝒮(φ_1,φ_2)=Re[ e^i(φ_2-φ_1)ρ_23]/8π.
We are primarily concerned with how much the phases are localised in the steady state, given by the maximum of 𝒮(φ_1,φ_2) which can be readily computed, 

    𝒮^max≡max_φ_1,φ_2𝒮(φ_1, φ_2) = 1/16πC_l_1(ρ^ss),

where C_l_1(ρ)≡∑_i≠ j|ρ_ij| is the l_1-norm of coherence <cit.>.
We note that Eq. (<ref>) holds for the evolution given by Eq. (<ref>) and for more general evolution the right-hand-side of Eq. (<ref>) becomes an upper bound on 𝒮^max.
The synchronization measure 𝒮^max is shown in Fig. <ref>(a) as a function of the drive's detuning Δ=ω_32-ω_d and its strength ε.
In analogy to classical synchronization, note the Arnold tongue-like behaviour of the figure indicating the phase locking of the atom to the external drive.
The range of detuning for which the atom displays phase locking to the external drive increases with the strength of the drive.

Engine Performance & Power Arnold tongue.— Efficiency and power are the characteristics that most often define the operation of an engine <cit.>.
Scovil and Schulz-Dubois <cit.> showed that a maser can be operated as a heat engine whose efficiency is bounded by Carnot efficiency, namely η=ω_32/ω_31≤η_c≡1-T_c/T_h.
This bound justifies our choice of driving a generic three-level atom, whose Husimi-Kano Q function is defined in terms of SU(3) coherent states as opposed to considering an equally spaced three-level atom of <cit.>.
Our generalisation reveals the complex dynamics inherent to the full SU(3) group while simultaneously allowing the efficiency to be varied by the choice of ω_ij.

A natural question arises, namely if the aforementioned thermodynamical characteristics depend on the underlying dynamical properties of the system, undergoing a transition from synchronized to unsynchronized regime.
The efficiency only depends on the transition frequencies of the system and is independent of any dynamical properties.
This is because the efficiency tracks the energy transactions from the bath to the piston via the working medium, and is unconcerned about the dynamical processes involved in the perturbative driving regime. 

On the other hand, the power of an engine is intimately related with dynamics <cit.>.
To operate an engine at the Carnot efficiency, one must perform adiabatic strokes which will not generate any heat.
Since adiabatic strokes take infinite time, such an engine does not generate power.
Likewise in the quantum regime, adiabatic strokes are transitionless and do not generate any excitations, allowing the engine to follow the instantaneous eigenstate of the total Hamiltonian.
Adiabatic driving has to thermalize with the bath at each instant in time, also implying the slowing down of dynamics <cit.>. Furthermore, perturbative driving does not change eigenvalues, and hence the contribution to power cannot arise from such terms.
Such a scenario hence precludes generating power from diagonal density matrices. To generate power, the density matrix hence has to be off-diagonal in the Hamiltonian eigenbasis <cit.>. Any definition of power has to satisfy all these constraints and has to be frame independent, saturate in the steady state and satisfy the second law of thermodynamics for all values of detuning Δ unlike the definition presented in <cit.>.
This intuition is captured by the steady-state power P^ss=-i  tr{[H, ρ^ss]H_0} due to Tannor and Boukobza <cit.> and satisfies all the conditions stated above.
A detailed analysis of the definitions of heat and work, alongside the engine performance is presented in the supplemental material for completeness.
Our drive is perturbative, since entrainment (synchronization) is best studied as an adjustment of rhythms that happens in the presence of perturbative driving (coupling).
Likewise the thermodynamics of driven systems is uncontroversial in the perturbative regime.
In general, the power of an engine can be attributed to two parts, one that accounts for the change in the instantaneous eigenvalues and another which accounts for the change in eigenvectors.
Since we are driving our quantum system perturbatively, the first term does not contribute (indeed our eigenvalues are constant in time).

The magnitude of steady-state power |P^ss| is presented in Fig. <ref>(b) as a function of Δ and ε, and displays an Arnold tongue-like behaviour.
To understand this, we note that the power is non-zero only when the steady state density matrix is off-diagonal with respect to the bare Hamiltonian.
This means that to observe a powerful quantum engine, the system has to be driven externally by a Hamiltonian that does not commute with the bare Hamiltonian.
Power is only produced in the region where we observe phase locking to the external drive.
If the system is sufficiently detuned at a given value of the driving strength ε, negligible power is emitted by the maser.
The underlying mechanism for this Arnold tongue-like behaviour is synchronization which produces coherence in the Hamiltonian eigenbasis.

To make this connection between synchronization and power quantitative we begin by calculating the steady-state power explicitly,

    P^ss = 2 εω_32Im[ ρ^ss_23].

Using | Im[ ρ^ss_23] | ≤| ρ^ss_23| along with Eq. (<ref>) leads to a synchronization bound on the magnitude of the steady-state power of the engine,

    | P^ss| ≤ 16 π| ε| ω_32𝒮^max.

The bound becomes an equality when the engine is driven resonantly, Δ=0, because in this case ρ^ss_23 becomes pure imaginary.
This can be seen by computing ρ_23^ss,

    ρ_23^ss = i εΓ_23^* γ_c γ_h ( n̅_c - n̅_h ) / β,

where Γ_23 = [γ_h (n̅_h + 1) + γ_c (n̅_c + 1)]/2 - iΔ and β is a real number whose exact form is given in the supplemental material.
Inequality (<ref>) is our main result.
It directly links the power of a heat engine with the amount of synchronization in the system.
Synchronization sets an upper bound on the achievable power of an engine for a fixed driving strength ε.

We explore this connection further and study how the synchronization bound in (<ref>) varies with bath temperatures. Fig. <ref>(c) shows the magnitude of the steady-state power |P^ss| (solid lines) as a function of the ratio of bath temperatures T_c/T_h, controlled by varying n̅_c and keeping n̅_h fixed.
Strong power is achieved for vanishing T_c/T_h which is expected and then decreases as T_c/T_h increases.
We see that |P^ss| saturates the synchronization bound (dashed lines) when the engine is driven resonantly while in the case of finite detuning |P^ss| is lower than the synchronization bound.
As T_c/T_h increases both |P^ss| and the synchronization bound decrease until we reach an interesting point where n̅_c=n̅_h and the system stops behaving like an engine.
It can be shown that this point corresponds to T_c/T_h=ω_21/ω_31=1-η.
This shows that the engine efficiency η determines the point where the synchronization measure 𝒮^max and therefore |P^ss| vanish as can be seen from Eq. (<ref>) and is also shown in the inset of Fig. <ref>(c).
As the temperature ratio increases past T_c/T_h=1-η the synchronization bound becomes finite again and so does |P^ss|, and the system enters a new regime where it behaves like a fridge, where heat currents and power reverse sign and the energy starts flowing from the cold bath to the hot one as detailed in the supplemental material.
Note that the synchronization bound in (<ref>) holds true regardless of whether the system is operated as an engine or a fridge.












Discussion.— 
We have demonstrated an intimate connection between synchronization and nanoscale heat engines and showed how the properties of the output power can be understood as a consequence of synchronization developing in the working fluid of the engine. 
An explicit example of this connection is the case of three-level atoms connected to two thermal reservoirs.
We have derived an upper bound on the engine power in terms of the measure of synchronization, 𝒮^max.
The maximum amount of synchronization as measured by 𝒮^max can be operationally understood as the maximum magnitude of steady-state power of a three-level engine when it is driven resonantly.
Finally, we have showed that the engine's efficiency η determines the ratio of the bath temperatures where synchronization vanishes. 

This connection offers, in addition, a very important route towards the experimental observation of quantum synchronization since masers are mature quantum platforms.
Three-level atoms have been experimentally coupled to thermal baths using magneto-optical traps <cit.> pumped by incoherent laser beams.
We envisage however that the connection is far more general and it applies to any generic multi-level quantum engine.
We hence anticipate that signatures of quantum synchronization can be observed experimentally.
We note that the underlying principle of synchronization of multi-level systems with linear baths is universal and can be applied wherever a stable phase relationship between atomic levels is needed.
This will pave the way for future quantum technologies based on quantum synchronization.



§ ACKNOWLEDGEMENTS

Centre for Quantum Technologies is a Research Centre of Excellence funded by the Ministry of Education and the National Research Foundation of Singapore. This research is supported by the National Research Foundation, Prime Minister's Office, Singapore under its Competitive Research Programme (CRP Award No. NRF-CRP14-2014-02). SV also acknowledges support from an IITB-IRCC grant number 16IRCCSG019 and a DST-SERB Early Career Research Award (ECR/2018/000957).













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  author author K. Nemoto, @noop  journal journal
  Journal of Physics A: Mathematical and General volume
  33, pages 3493 (year 2000)NoStop
[Alicki(1979)]Alicki_1979
  author author R. Alicki, @noop  journal journal
  Journal of Physics A: Mathematical and General volume
  12, pages L103 (year 1979)NoStop











§ SUPPLEMENTARY MATERIAL



Measure of synchronization.—	










In this section we extend the synchronization measure to a general three-level system using SU(3) coherent states.
We follow the SU(3) coherent state analysis due to Nemoto <cit.> where the SU(3) coherent state is 

    n_3 ⟩ = (cosθ,  e^i φ_1cosξsinθ,  e^i φ_2sinξsinθ)^T,

where 0≤θ, ξ≤π/2, and 0≤φ_1, φ_2 ≤ 2 π.








The SU(3) group measure is given by <cit.>

    dv = dθ dξ dφ_1 dφ_2cosθsin^3 θcosξsinξ,
	
which leads to the completeness relation for the coherent state |n_3⟩,

    ∫_0^2π dφ_1  ∫_0^2π dφ_2∫_0^π/2 dθcosθsin^3 θ∫_0^π/2 dξcosξsinξ|n_3⟩⟨n_3|  = π^2/6𝕀.

The steady state Husimi-Kano Q function can be written as

    Q^ss   =   6/π^2⟨ n_3 |ρ^ss| n_3 ⟩
       =   6/π^2( ρ^ss_33cos^2 θ + ρ^ss_22sin^2 θcos^2 ξ + ρ^ss_11sin^2 θsin^2 ξ) 
       +   6/π^2( sin 2θcosξ Re[e^iφ_1ρ^ss_12]  + sin 2θsinξ Re[e^iφ_2ρ^ss_13] + sin^2 θsin 2 ξ Re[e^i(φ_2-φ_1)ρ^ss_23] )

 One can see that in the absence of off-diagonal elements, Q^ss is independent of the phases φ_1 and φ_2, and hence the Eq. (<ref>) establishes the limit cycle for a system evolving under thermal Lindbladians.
 
Under the action of the bare Hamiltonian,

    H_0 =  ([ ω_1   0   0;   0 ω_2   0;   0   0 ω_3 ]),

 the coherent state evolves as









e^-i H_0 t n_3(θ,ξ, φ_1,φ_2)⟩ =  n_3(θ,ξ, φ_1 - ω_21 t,φ_2 - ω_31t)⟩, where ω_ij≡ω_i-ω_j.
There are two parameters, phases φ_1 and φ_2, that are evolving with time,  thus one has to look at the system's preference for both of these phases.
We can define the synchronization measure for a general three-level system as

    𝒮(φ_1,φ_2)    =   ∫_0^π/2 dθcosθsin^3 θ∫_0^π/2 dξcosξsinξ  Q^ss(θ,ξ,φ_1,φ_2) - 1/4π^2
       =   1/8 π{Re[e^iφ_1ρ^ss_12 ]+ Re[e^iφ_2ρ^ss_13 ] +  Re[ e^i(φ_2-φ_1)ρ^ss_23 ]}.

We can now maximize 𝒮(φ_1,φ_2) over the two phases,

    𝒮^max   =   max_φ_1,φ_21/8π{Re[ e^iφ_1ρ^ss_12] + Re[ e^iφ_2ρ^ss_13] + Re[ e^i(φ_2 - φ_1)ρ^ss_23] }
       =   max_φ_1, φ_21/8π[ cos(φ_1+ϑ_12) |ρ^ss_12| + cos(φ_2 + ϑ_13) |ρ^ss_13| + cos(φ_2 - φ_1 + ϑ_23) |ρ^ss_23| ] 
       ≤   1/8π( |ρ^ss_12| + |ρ^ss_13| + |ρ^ss_23| ) 
       =   1/16π C_l_1(ρ^ss),

where we expressed the coherences in polar coordinates ρ^ss_ij=e^iϑ_ij|ρ^ss_ij|.
𝒮^max saturates this bound only when ϑ_23=ϑ_13-ϑ_12 or when two of the three coherences vanish.
In the main text, 𝒮(φ_1,φ_2)=Re[e^i(φ_2 - φ_1)ρ^ss_23] / 8π and 𝒮^max=C_l_1(ρ^ss)/16π because two of the coherences vanish in the steady state, namely ρ^ss_12=ρ^ss_13=0.






Heat and Work for Quantum Engines.—	
Let H̃ be the Hamiltonian in an arbitrary frame obtained by the unitary U= e^-i x t, ie. H̃ = U^† H U, where H is the Hamiltonian in the Schrödinger picture.
As the expectation values of an operator is independent of the frame, we have ⟨H̃⟩=⟨ H⟩.
By taking the derivative of ⟨H̃⟩ with respect to time, we get 

    d/dt⟨H̃⟩ = tr{d H̃/dtρ̃} +tr{H̃dρ̃/dt}.

In the Schrödinger picture this becomes  d/dt⟨ H ⟩ = P + Q̇, where Q̇ is the heat current and P is the power which are respectively given by <cit.>

    Q̇ = tr{ H ∂ρ/∂ t}  and   P = tr{∂ H/∂ tρ}.

Following the above definitions, if we define heat currents and power in an arbitrary frame as 

    Q̇̃̇ = tr{H̃dρ̃/dt}  and  P̃ = tr{d H̃/dtρ̃},

then we have

	
	
    Q̇̃̇   =   tr{H̃dρ̃/dt} = i tr{ [H, x ] ρ} + tr{∂ρ/∂ t H}, 
    P̃   =   tr{d H̃/dtρ̃} = -i tr{ [H, x ] ρ} + tr{∂ H/∂ tρ}.


From Eq. (<ref>) and Eq. (<ref>), we can see that the heat current and power are frame dependent.
Authors of reference <cit.> pointed out a further complication and showed that at certain values of detuning Eq.(<ref>) violates the second law of thermodynamics.
To circumvent these problems, Boukobza and Tannor considered the system energy alone and redefined the heat currents and power in terms of the energy currents of the bare Hamiltonian <cit.>.
The energy flux for the system alone is given by

    d/dttr{ρ H_0 }    =   tr{dρ/dtH_0 }
       =    -i  tr{ [H, ρ] H_0 }+ tr{ℒ[ρ] H_0},

where the first term is due to the Hamiltonian part of the evolution which is defined as the power and the second term is due to the dissipation part of the evolution which is defined as the heat current. 



Steady States, Heat and Work for a 3-level Quantum Engines.—	
The Hamiltonian describing our system in the Schrödinger picture can be written as a sum of the bare Hamiltonian and a time-dependent drive,

    H = H_0 + V(t).

The drive V(t) has the following form, 

    V(t) = ε(σ_23e^iω_d t + σ_32 e^-iω_d t),

where σ_ij≡ |i⟩⟨ j|.
The Hamiltonian may be written in time-independent form by transforming into a rotating frame via unitary,

    U(t) = e^-iω_1tσ_11 + e^-iω_2tσ_22+ e^-i(ω_2+ω_d)tσ_33.

In the rotating frame the Lindblad master equation becomes

    ρ̇̃̇ = -i [H̃, ρ̃] + ℒ_h[ρ̃] + ℒ_c[ρ̃],

with

	
    H̃   =   Δσ_33 + ε( σ_23 + σ_32), 
    ℒ_h[ρ̃]    =   γ_h(n̅_h + 1) 𝒟 [σ_13] ρ̃ + γ_h n̅_h 𝒟 [σ_31] ρ̃, 
    ℒ_c[ρ̃]    =   γ_c(n̅_c + 1) 𝒟 [σ_12] ρ̃ + γ_c n̅_c 𝒟 [σ_21] ρ̃,


where Δ≡ω_32 - ω_d.
The equations of motion for the density matrix elements are

	
    ρ̇̃̇_11   =   γ_h ( n̅_h + 1) ρ̃_33 + γ_c ( n̅_c + 1 ) ρ̃_22 - (γ_h n̅_h + γ_c n̅_c) ρ̃_11, 
    ρ̇̃̇_22   =    iε ( ρ̃_23 - ρ̃_32 ) - γ_c ( n̅_c + 1 ) ρ̃_22 + γ_cn̅_c ρ̃_11, 
    ρ̇̃̇_33   =    -iε ( ρ̃_23 - ρ̃_32 ) - γ_h ( n̅_h + 1 ) ρ̃_33 + γ_hn̅_h ρ̃_11, 
    ρ̇̃̇_12   =    -Γ_12ρ̃_12 + iερ̃_13, 
    ρ̇̃̇_13   =    -Γ_13ρ̃_13 + iερ̃_12, 
    ρ̇̃̇_23   =    -Γ_23ρ̃_23 - iε( ρ̃_33 - ρ̃_22),


where the dissipation rates are defined by

	
    Γ_12   ≡   1/2[ γ_hn̅_h + γ_c (2n̅_c + 1) ], 
    Γ_13   ≡   1/2[ γ_h (2n̅_h  + 1) + γ_cn̅_c ] - iΔ, 
    Γ_23   ≡   1/2[ γ_h(n̅_h + 1) + γ_c(n̅_c + 1) ] - iΔ.


Solving for the steady state by setting ρ̇̃̇_ij =0, we get

    ρ̃^ss_11 = α_1/β,     ρ̃^ss_22 = α_2/β,     ρ̃^ss_33 = α_3/β,     ρ̃^ss_23 = α_4/β,

with the remaining coherences vanishing, ρ̃^ss_12=ρ̃^ss_13=0, and where

	
	
    α_1    =    2ϵ ^2 Re [Γ_23] [ γ_c (n̅_c + 1) + γ_h (n̅_h + 1) ] + |Γ_23|^2 γ_c γ_h (n̅_c+1) (n̅_h+1), 
    α_2    =    2ϵ ^2 Re [Γ_23] (γ_c n̅_c+γ_h n̅_h)+ |Γ_23|^2 γ_c γ_h n̅_c (n̅_h+1), 
    α_3    =    2ϵ ^2 Re [Γ_23] (γ_c n̅_c+γ_h n̅_h)+ |Γ_23|^2 γ_c γ_h (n̅_c+1) n̅_h, 
    α_4    =    i  Γ_ 23^* γ_c γ_h ϵ  (n̅_c-n̅_h), 
    β   =    2ϵ ^2 Re [Γ_23] [ γ_c (3n̅_c + 1) + γ_h (3n̅_h + 1) ]+ |Γ_23|^2 γ_c γ_h [ n̅_c (3 n̅_h+2)+2 n̅_h+1 ].


The power and heat currents in the steady state are

	
    P^ss   ≡    -i  tr{ [H̃, ρ̃^ss] H_0 } = 2 εω_32Im[ ρ̃^ss_23], 
    Q̇^ss_h    ≡   tr{ℒ_h[ρ̃^ss]H_0 } = γ_hn̅_hω_31 (ρ̃^ss_11 - ρ̃^ss_33) - γ_hω_31ρ̃^ss_33, 
    Q̇^ss_c    ≡   tr{ℒ_c[ρ̃^ss]H_0 } = γ_cn̅_cω_21 (ρ̃^ss_11 - ρ̃^ss_22) - γ_cω_21ρ̃^ss_22.


Using Eq. (<ref>) the heat currents can be rewritten,

	
    Q̇^ss_h    =   2γ_hγ_cε^2ω_31Re[Γ_23] (n̅_h - n̅_c)/β, 
    Q̇^ss_c    =    -2γ_hγ_cε^2ω_21Re[Γ_23] (n̅_h - n̅_c)/β.


We see that for n̅_h>n̅_c we have Q̇^ss_h>0 and Q̇^ss_c<0 meaning the energy flows from the hot bath to the cold bath and the system behaves like an engine.
On the other hand when n̅_h<n̅_c the signs of the heat currents and power reverse and the system behaves like a fridge.





