Work measurement in an optomechanical quantum heat engine

Ying Dong, Keye Zhang, Francesco Bariani, and Pierre Meystre

Phys. Rev. A 92, 033854 – Published 28 September 2015

Abstract

We analyze theoretically measurement schemes of the mean output work and its fluctuations in a recently proposed optomechanical quantum heat engine [Zhang et al., Phys. Rev. Lett. 112, 150602 (2014)]. After showing that this work can be operationally determined by continuous measurements of the intracavity photon number, we discuss both dispersive and absorptive measurement schemes and analyze their back-action effects on the efficiency of the engine. Both measurements are found to reduce the efficiency of the engine, but their back-action is both qualitatively and quantitatively different. For dispersive measurements the efficiency decreases as a result of the mixing of photonic and phononic excitations, while for absorptive measurements, its reduction arises from photon losses due to the interaction with the quantum probe.

Received 11 April 2015
Revised 4 August 2015

DOI:https://doi.org/10.1103/PhysRevA.92.033854

Authors & Affiliations

Ying Dong^1,2, Keye Zhang^3,1, Francesco Bariani¹, and Pierre Meystre¹

¹B2 Institute, Department of Physics and College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA
²Department of Physics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China
³Quantum Institute for Light and Atoms, State Key Laboratory of Precision Spectroscopy, Department of Physics, East China Normal University, Shanghai 200241, China

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Vol. 92, Iss. 3 — September 2015

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Images

Figure 1
(a) Frequencies of the two normal modes (polaritons) of the optomechanical system for $G / ω_{m} = 0.1$ in the red-detuned case $Δ < 0$ . The dashed lines correspond to the frequencies of the bare photon and phonon modes. The plot also illustrates that for large negative detunings the B polariton is phononlike and it is photonlike for small negative detunings. (b) Intuitive physical picture of the Otto cycle. Initially (top left corner), the mechanics undergoes relatively large thermal fluctuations due to its coupling to a hot thermal reservoir. At the end of the first adiabatic step (top right corner), the polariton has, however, become photonlike. Its unchanged mean occupation, converted into photons, results in added radiation pressure force on the mechanics. Thermalization at rate $κ$ to the temperature of the radiation reservoir $T \approx 0$ then significantly reduces the polariton occupation number (bottom right corner). At the end of the second adiabatic step, the polariton has regained its phononic nature, but now with small occupation number and hence a small adjustment of the mirror position (bottom left corner). Finally, the phononlike polariton thermalizes to the temperature of the hot phonon thermal bath in the final step, regaining the initial thermal occupation number at rate $γ$ (back at top left corner).
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Figure 2
Schematic setup for a classical measurement of the output work, whereby the work performed by radiation pressure acting on the mirror of large mass $M$ can be stored in the control system. See text for details.
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Figure 3
Schematic setup for a continuous quantum measurement of the output work with a beam of two-level atoms. (a) Absorptive measurement: the cavity field is resonant with the atomic transition, and the coupling induces real oscillations in the atomic population, which result in the loss of intracavity photons. (b) Dispersive measurement: the cavity-mode frequency is far off resonant from the two-level atom transition frequency, resulting in a dispersive interaction that only modifies the phase of the atomic ground-state wave function. See text for details.
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Figure 4
Time evolution of the mean excitations ${\bar{N}}_{B}$ and ${\bar{N}}_{A}$ of the polariton modes B (solid lines) and A (dashed lines) during the first stroke of the heat engine, averaged over 20 000 trajectories of the stochastic Schrödinger equation. Black lines marked by squares: no measurement. Red lines marked by triangles: dispersive measurement with $λ_{d} = 0.04 ω_{m}$ . Blue lines with circles: absorptive measurements with $λ_{a} = 0.04 ω_{m}$ . Other parameters: $G = 0.2 ω_{m}, Δ_{i} = - 3 ω_{m}, Δ_{f} = - 0.4 ω_{m}, κ = 5 \times 10^{- 3} ω_{m}$ , and $γ = 10^{- 4} ω_{m}$ . The inset shows the time dependence of the pump-cavity detuning $Δ (t)$ in units of $ω_{m}$ . Time is in units of $1 / ω_{m}$ .
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Figure 5
(a) Expectation value and (b) variance of the output work, in units of $ℏ ω_{m}$ , for the full cycle as a function of the measurement strength $λ_{a, d}$ (in units of $ω_{m})$ . The points are results of numerical simulations, and the lines serve to guide the eye. In both figures, the squares (red solid line) and circles (blue dashed line) stand for the dispersive and absorptive measurement schemes, respectively. The statistic is performed upon 20 000 trajectories. The stroke times are $τ_{1} = 40 ω_{m}^{- 1}, τ_{2} = 400 ω_{m}^{- 1}, τ_{3} = 40 ω_{m}^{- 1}, τ_{4} = 4 \times 10^{4} ω_{m}^{- 1}$ . All other parameters are as in Fig. 4.
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Figure 6
(top) Linear and (bottom) logarithmic plots of the probability distribution of the output work $W_{out}$ (in units of $ω_{m})$ for dispersive (left column) and absorptive (right column) measurement in the absence of measurement and for two measurement strength values, which are labeled on the axes. All parameters are the same as in Fig. 4.
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