Theory of an optomechanical quantum heat engine
Coherent interconversion between optical and mechanical excitations in an optomechanical cavity can be used to engineer a quantum heat engine. This heat engine is based on an Otto cycle between a cold photonic reservoir and a hot phononic reservoir [Phys. Rev. Lett. 112, 150602 (2014)]. Building on our previous work, we (i) develop a detailed theoretical analysis of the work and the efficiency of the engine, and (ii) perform an investigation of the quantum thermodynamics underlying this scheme. In particular, we analyze the thermodynamic performance in both the dressed polariton picture and the original bare photon and phonon picture. Finally, (iii) a numerical simulation is performed to derive the full evolution of the quantum optomechanical system during the Otto cycle, by taking into account all relevant sources of noise.
pacs:05.70.-a, 42.50.Wk, 07.10.Cm, 42.50.Lc
Optomechanical systems have witnessed spectacular developments in the last decade and can now operate deep in the quantum regime (see e.g. Refs RMP (); Meystre (); DSK () for recent reviews). Conventional cryogenic cooling for mechanical oscillators of relatively high frequencies in the gigahertz range or higher Oconnell (), and alternatively sideband cooling at lower mechanical frequencies Teufel (); Painter () have succeeded in bringing mechanical oscillators close to their quantum mechanical ground state. Also, quantum entanglement and squeezed states of photons and phonons have been demonstrated in these systems Palomaki13 (); Naeini13 ().
These developments pave the way to the creation of a generation of quantum interfaces between light and mechanical systems with broad potential for applications in quantum technology. One example is the coherent interconversion between optical and mechanical excitations, which was proposed and analyzed in a number of earlier theoretical studies interface (), and it has been used for the experimental realization of optomechanical light storage and readout interfaceEX (). These effects are based on the mixing of photons and phonons into polariton normal modes. Importantly in the context of the present paper, these excitations are in contact with the thermal reservoirs of the cavity mode and the mechanics which may have a large temperature difference. It is then possible to envision a quantum heat engine whose “working fluid” is a polariton mode of the optomechanical system ourPRL (). In this heat engine the properties of the polariton are controlled by the cavity-pump detuning: by adiabatically switching between the phonon side and the photon side, and enabling thermalization with the corresponding reservoirs, we may realize a quantum Otto cycle. Such a heat engine working deep in the quantum regime may have the potential for challenging the classic law of thermodynamics Scullyengine (); ionengine (). Furthermore, consideration of the fast development in nano- and microelectromechanical systems (NEMS and MEMS) suggests that a quantum engine based on an optomechanical system may prove attractive in terms of manipulation, integration, and application.
This paper presents a detailed theoretical analysis of the work and the efficiency of the optomechanical heat engine and investigates the quantum thermodynamics involved. It compares the thermodynamics in the normal mode and in the bare mode pictures, the associated interpretations of the physics bringing to the fore subtle aspects of the role of quantum correlations. It concludes by presenting results of numerical simulations of the full evolution of the Otto cycle, including dissipation and noise, with parameters within reach of existing technology.
The paper is organized as follows. Section II outlines the quantum model of the optomechanical system and analyzes key features of the polariton modes for various values of the cavity-pump detuning. Section III describes the four stages of the Otto cycle and derives expressions for heat exchanged with reservoirs and work delivered. Section IV discusses the limit of the thermal efficiency of the engine at maximum work. Section V analyzes the effective master equation for the polaritons and discusses implications of the fact that they are coupled to squeezed baths when the bare modes of the system are coupled to thermal reservoirs. Section VI turns the thermodynamics of the whole system and of its subsystems. An intuitive physical picture of the optomechanical engine is also suggested. Finally, Sec. VII presents selected results of full numerical simulations of the engine, and Sec. VIII is a summary and outlook.
Ii The optomechanical system
We consider a generic optomechanical system consisting of a Fabry-Pérot resonator with a compliant end mirror of effective mass and frequency driven by the radiation pressure from a single-mode intracavity field. We assume the system has reached a mean-field steady state characterized by a classical intracavity field and corresponding normalized mirror displacement , where is the zero-point mirror displacement. For small optical damping rates we have , where is the amplitude of the pump and .
This system is described by the linearized optomechanical Hamiltonian steady ()
Here is the photon annihilation operator for the quantum fluctuations of the optical mode of frequency driven by a classical pump of frequency , is the operator describing the quantum fluctuations of the mechanics,
is the effective detuning between the optical pump and the cavity mode, and is the single-photon optomechanical coupling. Finally the linearized effective optomechanical coupling is
We take it to be real and positive in this work without loss of generality.
The Hamiltonian (1) can be diagonalized in terms of two uncoupled bosonic normal modes, or polaritons, with annihilation operators and as
with corresponding eigenfrequencies
(see Fig. 1).
We consider the red-detuned regime () where the beam-splitter interaction term plays a dominant role and the stability condition of the linearized optomechanical system () gives
To second order in and for , the normal mode frequencies reduce to
For we have and , so that describes a photonlike excitation and a phononlike excitation. In contrast, for we have
The polariton is then phononlike while is photonlike as . At the avoided crossing , we have
which shows that the minimum frequency difference between branches , is proportional to .
Iii Otto Cycles
So far, we have only discussed the coherent contribution for the dynamics of the optomechanical system. Accounting in addition for optical and mechanical dissipation allows one to exploit the two thermal reservoirs to engineer a heat engine working between the “hot” thermal bath responsible for the relaxation of the phonon mode and the “cold” thermal bath due to the damping of the optical mode. As discussed in Ref. ourPRL (), it is then possible to operate the optomechanical system as a quantum Otto cycle QuOtto () by varying the detuning while keeping the intracavity optical field constant. Provided that nonadiabatic transitions between the two polariton branches can be avoided, each band can be associated with a different Otto cycle. We now turn to a detailed discussion of these cycles.
We consider a situation where the optomechanical system is initially in thermal equilibrium at large red detuning, , so that the phononlike lower polariton branch is in thermal equilibrium with a reservoir at effective temperature — for all practical purposes the temperature of the phonon heat reservoir. Similarly, the photonlike upper polariton branch is in thermal equilibrium with a reservoir at temperature , an excellent approximation at optical frequencies. Since we have the initial polariton population
The first stroke of the cycle is an adiabatic change of from its initial value to a final value ; this step has to be fast enough that the interaction of the system with the thermal reservoirs can be largely neglected, yet slow enough that nonadiabatic transitions between the two polariton branches are negligible. Ideally, at the end of the stroke the lower-branch polariton becomes photonlike. It is then allowed to reach thermal equilibrium with a reservoir at temperature , the temperature of the photon reservoir, while the upper polariton branch relaxes to the temperature of the phonon reservoir. This is the second stroke. The third stroke of the cycle involves sweeping the detuning back to its initial large negative value. Again, this step has to be fast enough to avoid thermalization, but slow enough to avoid nonadiabatic transitions. The final stroke is the rethermalization at fixed detuning : this leads to essentially the temperature of the phonon reservoir, , for the lower polariton branch and to for the upper branch. We stress that the amplitude of the driving classical field needs to be adjusted during the detuning changes so that the intracavity amplitude is kept constant. The Otto cycles associated with the two polariton branches are sketched in Fig. 2.
Denoting by , and the energies of the system at the four nodes of these cycles, we have that the heat exchanged and the work performed during each stroke are given by
According to the Hamiltonian (4), is dependent on the detuning and the expectation value of the polariton number, so the total work on the two polariton cycles is
Here and , with , are the frequencies and thermal mean populations of the two polariton modes at the initial and the final detunings, respectively. Since and we have that . Similarly, and so that . That is, in the thermal cycle operated along the polariton branch work is performed on the system, associated with the release of heat. In contrast, in the cycle along branch heat is absorbed by the system with corresponding work performed by the system. Cycle thus represents an Otto heat engine: it receives heat from the high temperature reservoir and partially converts it to work, while releasing the remaining heat to the low temperature reservoir. This is the process we are interested in.
The efficiency of cycle is defined by the ratio between the total work and the input heat ionengine ()
Figure 3 shows and as a function of and the dimensionless interaction strength . (We do not show their dependence on because the energy spectrum of mode is weakly dependent on it for large negative values.) The efficiency is independent of the thermal mean polariton number and is maximized for , which is precisely the stability condition of the system, see Eqs. (6) and (7). This means that the thermal efficiency could be large even for large optomechanical interaction strengths and final detuning far from zero, provided that they are close to the instability threshold (7), . However, this is not the case for the total work that reaches its maximum value at small and with a near-resonant .
We finally note that the cycle is a reversed engine whose efficiency is defined by the ratio between the total work and the output heat
In the limiting case , , and , we find . In order to maximize the work extracted from the system we should therefore avoid the occurrence of nonadiabatic transitions between the two cycles.
Iv Thermal efficiency
We now proceed with a more quantitative description of the Otto cycle along branch . For the situations considered here, where the polariton system is adiabatically switched from the phononlike side to the photonlike side and back, it is convenient to work in the bare mode picture, rather than with the dressed polariton modes. There are however potential issues with this approach. These are discussed in the following section.
iv.1 Normal modes and bare modes
We proceed by expressing the annihilation and creation operators of the polaritons in terms of the bare modes via the Bogoliubov transformation
where and are submatrices that satisfy the relationships
with the inverse transformation
In the limit of small dimensionless optomechanical couplings and for detunings we find, to second order in ,
from which the steady-state mean population of the polariton modes can be expressed in terms of mean photon and phonon occupations, second-order photon-phonon correlations, and a term associated with squeezing. For small optomechanical coupling strengths and far from the sideband resonance at , we can neglect these correlations and squeezing, and furthermore approximate the mean photon and phonon numbers as the mean thermal occupations of the optical reservoir ( for optical frequencies) and of the mechanical reservoir , respectively. More precisely, the steady populations of the polariton modes are approximated by the first lines of Eqs. (23) and (24). In the limiting case the populations of the polaritons and approach the thermal photon number and the thermal phonon number, respectively. For detunings , the expressions for the operators and are simply interchanged.
Clearly these simplifications cease to hold for larger and near the sideband resonance, in which case optomechanical entanglement and optomechanical cooling effects can play a significant role. In particular, quantum correlations between photons and phonons can significantly reduce the phonon number from , leaving the bare photon and phonon modes, as well as the polariton modes, out of their thermal equilibrium Genes2008 (); Genes2008b (). We investigate these features in some detail in Sec. V.
Using the approximate expressions of and it is straightforward to evaluate the efficiency of the heat engine based on the polariton mode . We assume that and keep the constant term to second order in the dimensionless optomechanical coupling : this term affects the energy value at each node but it has no influence on the total work and the efficiency of the Otto cycle. Assuming that adiabatic transitions between the two polariton branches can be ignored, the Hamiltonian evolution is governed solely by
As already discussed , so the lower polariton branch is initially essentially phononlike and in thermal equilibrium with the phonon-dominated reservoir at temperature , with mean thermal excitation
Adiabatically changing the detuning to the new value with the energy of the polariton mode then becomes
where the population remains unchanged. After the system reaches its new thermal equilibrium with the photon-dominated reservoir at temperature and mean thermal excitation
its energy becomes
At this point, the detuning is changed back to and the system adiabatically returns to its phononlike nature, but still keeping the population (29), so that
Finally, after thermalization with the phonon-dominated bath, the energy returns to its initial value . Combined with Eqs. (12), (14), and (15) this allows to determine the efficiency and total work of the cycle.
We first consider the limiting case . In this case the adiabaticity condition requires an infinite amount of time for the change in detuning to avoid the coupling of the two polariton branches, a condition in conflict with the requirement that thermalization remains insignificant during that stroke. Nonetheless this limit provides useful insights into the physics of the system. We now have , , and . Taking then the effective temperature of the photon reservoir to be K yields for the total work and efficiency (remember, )
If we further assume , also an unrealistic situation, we then find that the thermal energy of the phonon can be fully converted into work.
A more realistic estimate, consistent with the requirement to change the detuning adiabatically, can be obtained by evaluating these quantities to second order in . Again, we take to be large and negative, and to be a small negative detuning close to zero, so that and ; the thermal efficiency of the Otto cycle is then
which is a maximum for . However, the total work
reaches its minimum (remember, ) for
where we have assumed a phonon temperature high enough that
This yields the efficiency at maximum power
which, with the help of a simple inequality, gives
With a quantum-classical energy correspondence for the zero point energy of the cavity mode in the frame rotating at the cavity pump frequency, , we obtain the quantum version of the classical Curzon-Ahlborn efficiency limit, CAlimit (). Its upper limit is reached for which, according to Eq. (36), corresponds to the ideal situation .
V Master equation for the polariton
When the optomechanical coupling is small but finite, all terms in Eqs. (23) and (24) contribute, and the steady-state polariton occupation will deviate from thermal equilibrium. To investigate this effect we derive the effective master equation for the normal mode below.
For the high- mechanical oscillator that we consider it is safe to use the familiar Lindblad superoperator to describe the effect of Brownian thermal motion on the mechanics Mari2012 (). In the bare mode picture, the master equation of the system is then
where and are the mean photon and phonon numbers in their respective thermal reservoirs, and are their decay rates,
and is the linearized optomechanical Hamiltonian, Eq. (1).
can be diagonalized via the Bogoliubov transformation (20); however, the two polariton modes remain coupled via the Lindblad superoperators and it is not possible to define two uncoupled master equations for the normal modes and . In the following we assume for simplicity that the population of the normal mode vanishes throughout the Otto cycle and we approximate the density matrix of the full system as
In this case it is possible to obtain an effective master equation for the normal mode only,
where we have introduced the new superoperator
and the effective decay rate of the normal mode
where and are the elements of the submatrices and of the Bogoliubov transformation.
whose steady-state expectation values are easily found from the master equation (43),
familiar from squeezed reservoirs. From the uncertainty relation
we also find ( is taken to be real for simplicity in the following )
with maximum squeezing reached for the equal sign.
The presence of a squeezed reservoir implies that the steady state of mode is not a thermal state. Rather, it is a state that is in some sense “hotter” than the corresponding thermal reservoir. Its steady state population is
This property was recently exploited in the ion heat engine scheme of Ref. ionengine2 (), where the use of a squeezed reservoir was proposed to reach an efficiency that violates the familiar Carnot limit. In our case both the cold reservoir (photonlike side) and the hot reservoir (phononlik side) are squeezed and due to the small coupling strength the squeezing effect is also very weak, making it a challenge to break the Carnot limit.
The exact expressions for the steady population and effective decay rate are too cumbersome to be reproduced here. We give instead their approximate forms to second order in . For the case and we find
which show that not only the steady population but also the effective decay rate of the polariton are close to the phonon case for . On the other side where , the expressions become
which tend to the photon case as .
As a final note, a comparison of Eqs. (58) and (26) shows the important role of the ratio between the decay rates of the photons and the mechanics: In general, the terms in the second line of Eq. (24) result in steady-state polariton populations that deviate from the thermal equilibrium result (26). Neglecting the build up of correlations between the optical and phonon modes and of squeezing effects is strictly valid only for values of close to unity, corresponding to equal decay rates of the photon and the phonon. In practice, though, we found that, even for , these effects are weak due to the assumption of small optomechanical coupling strength . A similar situation occurs on the photonlike side, but with a reversed factor ; compare Eqs. (60) and (29).
Vi Quantum thermodynamics analysis
So far, our discussion of the Otto cycle has been based on the polariton modes. This representation or, more precisely, the energy representation of the whole system is naturally required for the study of its thermodynamical properties. However, the thermodynamics of the subsystems, in this case, the photon and the phonon modes, is also of interest as it provides a more direct intuitive understanding of the underlying physics at play. With this in mind this section compares and contrasts the thermodynamics of the heat engine in the polariton and the bare mode pictures.
vi.1 Work and heat exchange
In classical thermodynamics, the expression of the first law is
where , , and , are energy, heat, and work, respectively. This law states that the energy exchanged by a system in a transformation is divided between work and heat . To obtain a quantum version of this expression, we express the average energy in terms of the eigenstates of the Hamiltonian as
where is the energy of the eigenstate with corresponding occupation probability . An infinitesimal change in energy is then given by
and one can identify the first term on the right-hand side as the infinitesimal heat transferred, and the second as the infinitesimal work performed firstlaw (),
The heat transferred to or from a quantum system corresponds to a change in the populations without change of the energy eigenvalues, while the work done on or by a quantum system corresponds to a redistribution of the energy eigenvalues. These quantum expressions of the infinitesimal heat and work are consistent with their definitions in classical thermodynamics and statistical physics. That is, the heat exchange results in a change in the statistical distribution of the microstates of different energies while the work is a change in the energy structure of the system.
One can also obtain more general expressions for and in terms of the density operator and the time-dependent Hamiltonian :
If we take the temporal derivative
then the general quantum definitions of and are
Let us then consider the first stroke of the optomechanical Otto cycle, with the detuning adiabatically changed from a large negative value to a value close to zero so that the nature of the polariton changes from phononlike to photonlike and the system outputs work. When considered in the polariton picture the adiabatic evolution ensures that the stroke is an isentropic process. But that interpretation only holds in the normal mode picture: While thermodynamical adiabaticity does mean that the system as a whole has no heat exchange with the environment, heat can of course be exchanged between its subsystems, resulting in a change in the populations of their energy levels.
To show how this works, we analyze the hierarchical structure of our system. As sketched in Fig. 4 the first level is a bare mode picture, described by the Hamiltonian of Eq. (1) with the photon and phonon modes coupled by the linearized optomechanical interaction
The second level is the dressed picture, where the system is described in terms of the noninteracting normal modes (polaritons) and . Here we ignore the polariton , whose population remains negligible throughout the cycle, so that for all practical purposes the system is then described by the Hamiltonian (25).
The third level, finally, includes the external controls. In our case they are the driving optical field, the steady cavity field , and the normalized displacement . The temperatures of the photon and the phonon reservoirs should also be present at this level, but we ignore them during the isentropic stroke.
In both the bare modes and polariton pictures the change in average energy of the system is of course the same,
but the interpretation of the thermodynamics is different. Specifically, in the polariton picture we have
where is the density matrix of the normal mode . Since the transformation is adiabatic, we have so that
Moreover, as the detuning is changed from a large negative value to zero, decreases, so that , indicative of the fact that work is produced by the heat engine.
In contrast, in the bare picture we have
where is the density matrix of the two-mode system and and are the reduced density matrices of the photon and phonon mode, respectively. Since and are constant, according to the quantum definitions of work and heat, we find
By considering the change of the populations of the photon and phonon modes in the first stroke, we have
indicating that, in the bare picture, the evolutions of the photonic and phononic subsystems are neither adiabatic nor isentropic. Furthermore, since the initial population of the photon mode is zero, we have
where the last term is the change of the quantum correlations between the photon and phonon fields: This is initially zero for a product of thermal states, but becomes finite as a result of the optomechanical coupling. This term is much smaller than and for the weak optomechanical couplings considered here, and interestingly, does not have a corresponding classical thermodynamical quantity.
Summarizing, in the dressed picture the first stroke of the heat engine adiabatically switches the polariton from a phononlike to a photonlike excitation and it performs work on the external control field. In the bare mode picture, the phonon mode releases heat, part of which is then absorbed by the photon field, a small amount contributing to quantum correlation, and the rest being absorbed by the external control field. Similar results can also be obtained for the second stroke of the Otto cycle, with in Eq. (83) replaced by , and , , corresponding to a process dominated by photon exothermic reaction (dissipation). We stress that from Eqs. (71) and (72), thermodynamics may be defined for the eigenstates of the system, namely the polaritons, while the application to the bare modes is only qualitative.
vi.2 Physical picture
Following these considerations, one can gain a simple physical understanding of the engine cycle. As shown in Fig. 5, the change in cavity length (corresponding to the value of steady amplitude of the phonon field, ) represents the effect of the work performed by the engine. The vibration amplitudes of the cavity mirror and the cavity field represent the population of the phonon and photon mode, respectively.
During stroke 1 the detuning is varied so that is brought closer to resonance with the cavity mode frequency , while simultaneously changing the pumping rate so as to keep mean intracavity amplitudes and , and hence the coupling frequency , constant. As this happens the phononlike thermal excitations, which are initially large due to the contact with a thermal reservoir that is essentially at the temperature of the mechanics, are transformed into photonlike excitations. This occurs at a rate characterized by the coupling frequency . During this step the amplitude of vibrations of the mechanics decreases and the excess energy is transferred to the intracavity field. As a result the resonator length increases slightly due to the increased radiation pressure. It is at this point that the mechanical work on the oscillator is produced by the optomechanical heat engine. However, this work is very small due to the disproportion between the steady amplitudes and the quantum fluctuations of the photon and phonon fields. During the thermalization step of stroke 2 the population of the photonlike excitations decays at rate (for a photon reservoir at zero temperature) with the cavity length unchanged. In stroke 3 the remaining photonlike polariton excitations are turned back into phonon-like quanta by adjusting . This costs a small amount of work, resulting in a small contraction of the cavity length. The population of the phonon-like polariton excitation finally grows up to its initial value via thermal contact with the hot mechanical reservoir during stroke 4. The small polariton number in stroke (3) ensures that the total output work of the Otto cycle is positive.
Vii Numerical simulations
vii.1 Time scales
The adiabatic strokes 1 and 3 of the Otto cycle involve changes of the cavity detuning from to . To ensure the adiabaticity of the transformation, their times and must be much longer than the characteristic time of the transition between the two polariton branches . From Eq. (10) we have
where we used the weak coupling condition Additionally, in order to avoid heat exchange between the normal modes and the reservoirs during the adiabatic strokes we also need much shorter than the characteristic interaction time between the system and the reservoirs,