Heterodyne photodetection measurements on cavity optomechanical systems: Interpretation of sideband asymmetry and limits to a classical explanation

Kjetil Børkje

Phys. Rev. A 94, 043816 – Published 12 October 2016

Abstract

We consider a system where an optical cavity mode is parametrically coupled to a mechanical oscillator. A laser beam driving the cavity at its resonance frequency will acquire red- and blue-shifted sidebands due to noise in the position of the mechanical oscillator. In a classical theory without noise in the electromagnetic field, the powers of these sidebands are of equal magnitude. In a quantum theory, however, an asymmetry between the sidebands can be resolved when the oscillator's average number of vibrational excitations (phonons) becomes small, i.e., comparable to 1. We discuss the interpretation of this sideband asymmetry in a heterodyne photodetection measurement scheme and show that it depends on the choice of detector model. In the optical regime, standard photodetection theory leads to a photocurrent noise spectrum given by normal- and time-ordered expectation values. The sideband asymmetry is in that case a direct reflection of the quantum asymmetry of the position noise spectrum of the mechanical oscillator. Conversely, for a detector that measures symmetric, nonordered expectation values, we show that the sideband asymmetry can be traced back to quantum optomechanical interference terms. This ambiguity in interpretation applies not only to mechanical oscillators, but to any degree of freedom that couples linearly to noise in the electromagnetic field. Finally, we also compare the quantum theory to a fully classical model, where sideband asymmetry can arise from classical optomechanical interference terms. We show that, due to the oscillator's lack of zero-point motion in a classical theory, the sidebands in the photocurrent spectrum differ qualitatively from those of a quantum theory at sufficiently low temperatures. We discuss the observable consequences of this deviation between classical and quantum theories.

Received 5 August 2016
Revised 14 September 2016

DOI:https://doi.org/10.1103/PhysRevA.94.043816

Physics Subject Headings (PhySH)

Optomechanics Quantum measurements Quantum optics

Homodyne & heterodyne detection

Atomic, Molecular & Optical

Authors & Affiliations

Kjetil Børkje

Royal Norwegian Naval Academy, Postboks 83, Haakonsvern NO-5886 Bergen, Norway

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Issue

Vol. 94, Iss. 4 — October 2016

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Images

Figure 1
The experimental setup that we will have in mind. An optical cavity mode's resonance frequency depends on the motion of a mechanical oscillator, which is here depicted as a thin dielectric membrane. A probe laser beam at frequency $ω_{d}$ is sent into the cavity. The light emanating the cavity is combined with a local oscillator beam at frequency $ω_{lo}$ . The two beams are detected by a photomultiplier. (a) Measurement in transmission where the cavity output and the local oscillator are combined on a beam splitter before arriving at the photomultiplier. (b) Measurement in reflection where both beams are sent towards the cavity. The probe beam enters the cavity, whereas the local oscillator is promptly reflected. The beams are then sent to the photomultiplier by means of a circulator.
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Figure 2
An overview of the angular frequencies involved. The probe laser beam at frequency $ω_{d}$ will acquire a red-shifted (blue-shifted) sideband at $ω_{d} - ω_{m} (ω_{d} + ω_{m})$ as a result of modulation by the mechanical oscillator. The noise in the probe laser is converted to noise in the photocurrent around the intermediate frequency $ω_{if}$ due to beating between the probe beam and the local oscillator. Note that the mixing down makes the sidebands switch place when $ω_{lo} > ω_{d}$ , i.e., the red (blue) sideband is found at $ω_{if} + ω_{m} (ω_{if} - ω_{m})$ .
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Figure 3
Origin of sideband asymmetry with the two different detector models in absence of classical noise in the electromagnetic field. Upper panels: the SCL detector model. Lower panels: the QUA detector model. Left: the optomechanical contribution $S^{(o m)} [ω_{if} + \tilde{ω}]$ (dashed line) and the mechanical contribution $S^{(m)} [ω_{if} + \tilde{ω}]$ (dotted line) at the red sideband. Right: the optomechanical contribution $S^{(o m)} [ω_{if} - \tilde{ω}]$ (dashed line) and the mechanical contribution $S^{(m)} [ω_{if} - \tilde{ω}]$ (dotted line) at the blue sideband. Center: the sidebands with the noise floor subtracted. The red sideband $S_{r r} [\tilde{ω}] - {| Z |}^{2}$ is taller than the blue $S_{b b} [\tilde{ω}] - {| Z |}^{2}$ . All functions are presented in units of $4 p {\bar{κ}}_{ext} {| Z |}^{2}$ .
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Figure 4
Classical optomechanical correlations can lead to a negative blue sideband in the low-temperature regime, i.e., noise squashing. This does not occur in the quantum theory, where the blue sideband height will asymptotically approach $p$ from above as the temperatur decreases. Left: the optomechanical contribution $S^{(o m)} [ω_{if} + \tilde{ω}]$ (dashed line) and the mechanical contribution $S^{(m)} [ω_{if} + \tilde{ω}]$ (dotted line) at the red sideband. Right: the optomechanical contribution $S^{(o m)} [ω_{if} - \tilde{ω}]$ (dashed line) and the mechanical contribution $S^{(m)} [ω_{if} - \tilde{ω}]$ (dotted line) at the blue sideband. Center: the sidebands with the noise floor subtracted. Here, the red sideband $S_{r r} [\tilde{ω}] - {| \tilde{Z} |}^{2}$ is positive, whereas the blue $S_{b b} [\tilde{ω}] - {| \tilde{Z} |}^{2}$ is negative. All functions are presented in units of $4 p {\bar{κ}}_{ext} {| \tilde{Z} |}^{2}$ .
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Figure 5
The blue sideband height above the noise floor in units of $4 p {\bar{κ}}_{ext} {| Z |}^{2}$ as a function of inverse temperature for the quantum model (solid curve) and the classical model (dashed curve). We have chosen $α = 1$ and $p = 0.1$ here. We see that for all $T$ where ${\tilde{n}}_{\inf} > p$ , the magnitude of the derivative of ${\tilde{n}}_{\inf}$ is above a finite value.
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