# Quantum optomechanical straight-twin engine

###### Abstract

We propose a realization of a quantum heat engine in a hybrid microwave-optomechanical system that is the analog of a classical straight-twin engine. It exploits a pair of polariton modes that operate as out-of-phase quantum Otto cycles. A third polariton mode that is essential in the coupling of the optical and microwave fields is maintained in a quasi-dark mode to suppress disturbances from the mechanical noise. We also find that the fluctuations in the contributions to the total work from the two polariton modes are characterized by quantum correlations that generally lead to a reduction in the extractable work compared to its classical version.

## I Introduction

Research in quantum heat engines has progressed rapidly in recent years, due in part to its potential to investigate aspects of stochastic quantum thermodynamics at the microscopic and mesoscopic scales. This development has benefited in particular from emerging experimental technologies in AMO and NEMS systems, as evidenced in particular by the single ion heat engine theoretically proposed Abah2012 () and experimentally demonstrated Rossnagel2016 (). Additional approaches being considered involve spin systems Thomas2011 (); Altintas2015 (), qubit systems Brunner2012 (), ultracold atoms Fialko2012 (), cavity-assisted atom-light systems Scully2003 (), nanomechanical resonators Tercas2016 (), and optomechanical systems Zhang2014a (); Zhang2014b (); Mari2012 (); Mari2015 (); Elouard2015 (); Kurizki2015 ().

Quantum heat engines are typically based on implementations of quantum Carnot, Otto Quan2007 (), Brayton Quan2009 (); Huang2013 (), Diesel Quan2009 (); Dong2013 () or Stirling cycles Wu1998 (); Huang2014 (), with a succession of strokes that alternatively extract work and exchange heat with the hot and cold reservoirs. As a result the generation of work is an intermittent process. In classical thermodynamics, a transition to continuous work production can be achieved with multi-cylinder structures. In this paper we propose an implementation of a twin-cylinder optomechanical quantum heat engine. It is a hybrid microwave-opto-mechanical system with two working fluids – or two “cylinders”. The first one is a normal mode excitation (polariton) whose nature changes from microwave-like to optical-like, depending on the externally controlled position of the optomechanical oscillator (the analog of a piston) the engine being driven by the effective temperature difference between the microwave and the optical cavity modes. At the same time a second polariton mode that switches its nature in reversed sequence serves as a second working fluid that provides an additional work output channel operating out of phase with the first one. This is similar to the situation in classical straight-twin engines, but with the important difference that the work produced by the two polariton fluids can develop quantum correlations.

## Ii Model

As shown in Fig. 1(a) the engine consists of a pair of electromagnetic resonators that share a common optomechanical end mirror of oscillation frequency and damping rate . The first one is an optical cavity of frequency and damping rate and the second one a microwave resonator of frequency and damping rate Andrews2014 (). They are driven by coherent fields at frequencies and , respectively. We assume that the intracavity fields are strong enough that they can be described as the sum of large mean classical amplitudes and small quantum fluctuations. In the frame rotating at the pump frequencies the system Hamiltonian can then be linearized as

(1) |

where the bosonic annihilation operators , , and account for the quantum fluctuations of the optical cavity field, microwave resonator field, and mechanical mode around their mean amplitudes , , and , taken to be real without loss of generality. The effective optomechanical couplings, , , and and are the single-photon optical and microwave couplings noteG (). In the following we assume that and are opposite in sign. Finally

(2) |

where we included the additional shift due to the classically controlled mean dimensionless displacement of the compliant mirror, the “” accounting for the implied opposite change in length of the two cavities.

The Hamiltonian can be diagonalized in terms of three uncoupled bosonic normal modes, or polaritons, with annihilation operators , and as

(3) |

The polariton modes are superpositions of the optical, microwave, and mechanical modes with relative amplitudes that depend on external parameters such as the detunings . The fact that these constituents are coupled to thermal reservoirs at different temperatures allows for the realization of polaritonic heat engines as previously discussed in Refs. Zhang2014a (); Zhang2014b (). The present scheme is the simultaneously uses two polariton modes to realize a quantum analog of a classical two-cylinder straight-twin engine.

The two “cylinders” of the engine are the polariton modes and , whose nature and frequencies are controlled via by changing , the power source being the temperature difference between the microwave reservoir and the optical reservoir, which is effectively at zero temperature. Ideally the compliant mirror should merely allow for the conversion between microwave and optical photons so as to avoid unwanted energy losses through mechanical excitations. This suggests exploiting the “polaritonic dark mode” and externally varying to achieve the adiabatic conversion between the microwave and optical photons free of the mechanical noise Wang2012a (); Wang2012b (); Tian2012 (); Dong2012 (). However the realization of a perfect dark mode requires , preventing the extraction of work in the adiabatic conversion. But we find that if the detunings remain symmetric about the mechanical frequency throughout the thermodynamical cycle the polariton remains a quasi-dark mode with nearly constant frequency . This choice of detunings is optimal in keeping the mechanical excitations out of the heat-work conversion note1 ().

As an example we consider the case and , so that and . Taking further and dropping the anti-rotating terms in gives a three-mode Bogoliubov transformation

(4) | |||||

where and the upper and lower signs in the third equation correspond to the positive and negative value of , respectively. The polariton frequencies are , , and . In the limit , Eqs. (4) give and , while is approximately phonon-like. For , and exchange their properties.

Fig. 1(b) shows the polariton and bare modes frequencies as functions of the mean oscillator displacement for a symmetric arrangement of the optomechanical parameters of the optical and microwave fields. As is changed from large negative to large positive values the polaritons and switch their natures from microwave-like to optical-like and from optical-like to microwave-like, respectively, with an avoided crossing of frequency at . As already mentioned remains phonon-like.

## Iii Twin Otto cycle

The heat engine is driven by the temperature difference between the microwave and optical reservoirs, with the polariton modes and undergoing asynchronous quantum Otto cycles. This is illustrated in Figs. 1(c), which shows that when is in its isentropic compression stroke then is in the isentropic expansion stroke, and vice versa. This is analogous to the classical straight-twin-cylinder heat engine with a crankshaft angle of Fig. 1(d), with one cylinder expanding with the other compressing.

We assume that initially and the system is in thermal equilibrium with the optical, microwave and phonon modes at the temperature of their respective reservoirs, with mean occupation numbers , and . For polariton the first stroke, an adiabatic change of from to , corresponds to a change from microwave-like with and to optical-like. 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 remain negligible. We then have , so that if and are such that that stroke is an isentropic expansion of polariton , resulting in an energy loss and a work output . In contrast for polariton that stroke is an isentropic compression since , resulting in an energy gain, hence a work input . However its value is negligible since the temperature of the optical reservoir is effectively zero.

Following the second stroke, where the optical, microwave and phonon have reached thermal equilibrium again, the third stroke consists in adiabatically changing back to . Here the polariton suffers an isentropic compression, requiring a negligible work input, , while undergoes an isentropic expansion with work output, . In the fourth and final stroke the system is left to reach thermal equilibrium again. Importantly during the full Otto cycle the polariton contributes negligible work since its frequency remains constant. Ideally the total work output of the engine is therefore

This simplified expression would indicate that maximizing the work requires an asymmetry and right-skewed working range with is large and positive and close to zero. However this choice results in an imperfect microwave-optical conversion of the polariton modes, the appearance of non-negligible contributions and to the total work, and the onset of quantum correlations between and . We return to this point later on.

## Iv Physical picture

The operation of the engine can be understood in the bare mode picture by considering the combined effects of the radiation pressure forces from the quantum fluctuations of the microwave and optical fields and the classical control of the position of the compliant mirror. Its dynamics can be described by a differential equation for the covariance matrix with Markovian-correlated quantum noise sources Rogers2012 ().

Fig. 2 shows the evolution of the population of each mode for a full cycle of the engine obtained in this way. In this example the compliant mirror frequency is GHz and its temperature mK, resulting in Bochmann2013 (). For the optical cavity and for the microwave resonator . The linear optomechanical coupling is and the normalized optical, microwave and mechanical decay rates are , , and , respectively. In the first and third isentropic-like strokes the compliant mirror is moved linearly between and and back. The duration of these strokes is to satisfy the adiabatic requirement . The duration of the thermalizing second and fourth strokes is .

Initially the microwave and optical modes are thermally populated, but with the latter one is essentially empty due to the near-zero effective temperature of the optical reservoir. During the first stroke the compliant mirror is classically displaced from to and the radiation pressure force from the microwave photons produces work. (We choose the convention where work produced by the engine is negative.) For the resonant condition is gradually approached and microwave photons convert into optical photons through the phonon-mediated interaction Andrews2014 (); Bochmann2013 () with a radiation pressure force acting against to the mirror motion, see Fig. 2(a). The asymmetry of and about results in that stroke ending before the positive work done by the microwave photons is offset by the negative work done by the optical photons. In the second stroke the optical field and microwave field rethermalize, again with . In the third stroke the microwaves convert back to optical photons, see Fig. 2(c), and the optical radiation pressure force, which is along the displacement of the mirror in this stroke, becomes dominant and produces work. Finally the system is thermalized once more in stroke 4.

Adiabatic conversion from the microwave to the optical mode occurs both in the first and the third stroke, as evidenced by the nearly constant population of the polariton modes and . Importantly, as illustrated in Figs. 2(a) and 2(c) the unequal distance of and from results in a larger initial detuning and a harder conversion and thus leaves a relatively long time for the microwave field to produce work. In contrast, during the third stroke the conversion occurs earlier so the population of the optical mode and the associated force are dominant most of the time. Note however that since is not quite large enough for the far-off-resonance condition to be fully satisfied there is a non-negligible coupling between the three bare modes as they are thermalized in the second stroke. This results in a finite thermal equilibrium population of the optical mode and a small deviation between the evolution of the polaritons and the bare modes. This does not occur in the last stroke due to the far-off-resonant . Except near resonance, where the phonons are directly involved in the optical-microwave conversion, their evolution coincides with the polariton mode whose population remains almost constant during the full engine cycle.

## V Output work and correlation

In the quantum adiabatic limit, the average work produced by polariton during stroke 3 is

(6) |

where are the steady-state density matrix of the system and the population of polariton at , and similarly for with . The work produced during that stroke is , and its variance . The non-classicality of the quantum twin engine is charactered by a nonzero correlation coefficient Gerry2005 ()

(7) |

with . The expressions for stroke 1 are identical with and .

To evaluate these quantities we introduce the quantum work operator , where and are the Hamiltonians of the heat engine at the beginning and end of the isentropic strokes footnoteW (), and is the time evolution operator

(8) |

where and are the initial and final times of the stroke and the time-orderd product.

For we have that , , and . With Eqs. (4) and the associated polariton frequencies this gives

(9) | |||||

The largest population difference, , occurs for . However in that limit, so that there is no net work. Extracting work from the heat engine requires therefore that neither nor be perfectly optical-like or microwave-like. Hence the influence of the phonon mode and the resulting correlations between the and polaritons can not be ignored.

The total work and quantum correlations can be estimated more accurately from the solution of the master equation in the polariton basis. In that basis the transformed Lindblad super-operators reveal the coupling of the polariton modes, implying effectively multi-mode correlated reservoirs (See the detailed derivation in the appendix). Fig. 3(a) shows the dependence of the total work on on the mean thermal phonon number . The dependence illustrates the detrimental effect of phonon reservoir temperature on . This is because at higher temperature the optomechanical coupling increases the steady-state photon population, thereby decreasing the effective difference in temperatures of the polariton during the Otto cycle. For large the total work can even change sign when is near , as the polariton is then phonon-like rather than optical-like, and is warmer than the microwave-like side. Conversely, when is chosen large and negative the polariton becomes more and more optical-like, weakening the dependence of on .

Figure 3(b) shows for comparison a quasi-classical result obtained by neglecting the correlations. As in the quantum case the maximum work is reached near . The difference between the quantum and quasi-classical cases is most significant in the region of large and small , where the correlations between and become larger, resulting in decreased work. As shown in Fig. 3(c), when the populations of and are both dominated by the thermalization of the phonon mode, resulting in negative work correlation. Note however that becomes positive for due to resonantly-enhanced conversion (the work correlation is opposite to polariton correlation).

A similar situation also occurs in the dependence of the work on the optomechanical coupling , as shown in Figs. 3(d)-(f), which illustrate the increase in work and decrease in quantum correlations as is decreased. Note finally that the validity of the linearized model imposes that . The positive and negative work correlation near this boundary are caused by the anti-rotating terms in the optomechanical coupling noteRWA ().

## Vi Conclusion

To summarize, we have proposed and analyzed a quantum optomechanical heat engine driven by the effective temperature difference between microwave and the optical fields, and with a classical-like twin-cylinder four-stroke structure that dramatically decreases the intervals between work extractions in the quantum Otto cycle and brings quantum correlation between work from each cylinder. Future work will extend this idea to structures with more cylinders, include an autonomous engine design Garcia2016 (); Roulet2016 (), and explore the applications of work correlation.

###### Acknowledgements.

We acknowledge enlightening discussions with P. Meystre, S. Singh, H. Pu and L. Zhou. KZ is supported by NSFC Grants No. 11574086 and No. 11654005, and the Shanghai Rising-Star Program 16QA1401600. WZ is supported by the National Key Research Program of China under Grant No. 2016YFA0302001 and NSFC Grants No. 11234003 and No. 11129402. *## Appendix A The master equation in the polariton basis

The master equation in the bare mode basis writes

(10) |

where and are the decay rate and the mean thermal occupation number of bare mode , is the linearized optomechanical Hamiltonian

(11) |

and the Lindblad superoperators are

(12) |

When , the analytical form of the inverse Bogoliubov transformation can be derived under the rotating wave approximation (RWA), which is

(13) |

Substituting it into Eq. (A.1) we can rewrite the master equation in polariton basis,

where and . In addition to the ordinary dissipation terms with effective polariton decay rates and mean thermal occupation numbers

(15) | |||||

the transformed Lindblad superoperators of the bare modes bring out several coupling terms of the polariton modes in the form of the superoperators

(16) |

with the coupling coefficients

(17) | |||||

The expressions for are similar with the replacement . These coupling terms cause polariton correlation in the final steady state of the thermalization strokes and then result in work correlation in the isentropic strokes.

In the case corresponding to the limit , the correlations disappear due to the negligible values of correlation coefficients and . Then the steady state of the three polariton modes are uncorrelated thermal states with the mean occupation numbers , respectively, and and . For the opposite limit , the correlation is also near zero, and exchange their values. However, to other cases, the non-vanishing correlation will affect the steady population of the polariton modes. For example, from the steady solution of the master equation (LABEL:masterABC) we have

(18) | |||||

The exact dependence of the populations and correlations on and are obtained by the numerical simulation; specifically, for a large value of the RWA is invalid and the analytical form of the polariton transform is unavailable. The results are displayed in the Fig. 3 of the main text where we compared the exact work with that obtained by neglecting all coupling terms in the master equation (LABEL:masterABC) to show the influence of the correlation.

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