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Quantum correlations from a room-temperature optomechanical cavity

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Science  23 Jun 2017:
Vol. 356, Issue 6344, pp. 1265-1268
DOI: 10.1126/science.aag1407

Quantum effects in ambient conditions

Quantum back action—the “reaction” of a quantum mechanical object to being measured—is normally observed at cryogenic temperatures, where it is easier to distinguish from thermal motion. Purdy et al. managed to tease out the effects of quantum back action at room temperature by using a mechanical oscillator and probing it with light (see the Perspective by Harris). The fluctuations of the force produced by the light probe caused correlated changes to the motion of the oscillator and the properties of the transmitted light. These correlations revealed the effects of the back action, which allows the system to be used as a quantum thermometer.

Science, this issue p. 1265; see also p. 1232

Abstract

The act of position measurement alters the motion of an object being measured. This quantum measurement backaction is typically much smaller than the thermal motion of a room-temperature object and thus difficult to observe. By shining laser light through a nanomechanical beam, we measure the beam’s thermally driven vibrations and perturb its motion with optical force fluctuations at a level dictated by the Heisenberg measurement-disturbance uncertainty relation. We demonstrate a cross-correlation technique to distinguish optically driven motion from thermally driven motion, observing this quantum backaction signature up to room temperature. We use the scale of the quantum correlations, which is determined by fundamental constants, to gauge the size of thermal motion, demonstrating a path toward absolute thermometry with quantum mechanically calibrated ticks.

Quantum effects in macroscopic mechanical systems are typically difficult to observe under ambient conditions. At room temperature and atmospheric pressure, such effects are dominated by thermal and other noise sources. However, measuring the system must still imprint at least a small, telltale quantum signature, exemplified in its simplest form by Heisenberg’s microscope thought experiment, where optical probing necessarily perturbs the motion of the object being probed (1, 2). Focusing on harmonic oscillators, a variety of quantum backaction effects have been observed for nanomechanical systems in cryogenic environments (314) and cold atom optomechanical systems (15). However, measurements under ambient conditions have remained an experimental challenge. Here, we describe a method to tease out the small quantum correlations that are produced when optical-measurement–induced motion is in turn written back onto the light probing a nanomechanical resonator. Despite a scale several orders of magnitude smaller for these correlations compared with the thermal and measurement noise, we clearly observe this distinctly quantum signature all the way up to room temperature. By comparing the magnitude of thermally induced mechanical vibrations, which scales in proportion to the temperature, to the size of the quantum correlation, we demonstrate a potentially wide-range, chip-integrated Brownian motion thermometer the scale of which is referenced to fundamental constants, providing a path toward quantum primary thermometry (16).

Primary thermometers are the rare type whose temperature can be directly calculated from a readout variable using a known equation of state or physical law, without reference to another thermometer. Although such systems are often slow and technically sophisticated, they ultimately provide absolute accuracy. Primary thermometers find application in defining temperature scales over wide ranges; in measuring the Boltzmann constant, Embedded Image [of much interest in light of the proposed redefinition of the kelvin (17)]; and in obtaining temperature measurements where other thermometers cannot be deployed or accurately maintained, such as turning long spans of fiber-optic cable into distributed temperature sensors (18) or in deep cryogenic environments (19, 20).

Our system consists of a nanophotonic cavity coupled to a nanomechanical resonator, where vibrations of the mechanical mode modulate the optical resonance frequency and radiation pressure from an optical probe perturbs the mechanics (Fig. 1, A and B). Considering the dynamics and noise in this optomechanical system reveals a simple picture of the origin of the measurement backaction and quantum correlations. The nanomechanical resonator is driven to vibrate randomly in a band of frequencies around its resonance in response to thermal forces from its environment. This Brownian motion modulates the optical resonance frequency, imprinting phase modulation on a resonant optical probe. Along with this displacement measurement, the random quantum intensity fluctuations of the probe drive the mechanics with so-called radiation pressure shot noise (RPSN), constituting the quantum measurement backaction. The motion from backaction is also imprinted as phase fluctuations on the output light. If Embedded Image and Embedded Image are the quantum fluctuations of the optical amplitude and phase quadratures of the input probe, respectively, then, for a probe resonant with the optical cavity and assuming a small overall amplitude of motion, the optomechanical interaction leaves Embedded Image unchanged, whereas Embedded Image, where the first term represents the input optical vacuum fluctuations; the second represents the transduction of thermal force fluctuations, Embedded Image; and the third represents the effect of RPSN. Thus, a quantum correlation is established between amplitude and phase fluctuations when Embedded Image acquires a term proportional to Embedded Image. Such correlations are the basis of phenomena including the generation of optomechanically squeezed light (5, 21, 22), optomechanical Raman sideband asymmetry (3, 6, 7, 9, 11, 14, 15), and the application of correlated squeezed light to enhance displacement sensing below the standard quantum limit (23, 24). Several methods have been explored to directly measure quantum-backaction–induced correlations (4, 2528) [our method is similar to the proposal in (27)]), but the measurement setup and data processing techniques described below allow us to resolve quantum correlations between optical signals that, for a room-temperature device, are dominated by classical fluctuations.

Fig. 1 Measurement backaction in an optomechanical system.

(A) Effective optomechanical system for device pictured in (B), a harmonically bound mirror forming one end of an optical cavity. (B) Simulated electric field distribution (top) and simulated displacement field distribution (bottom) of a suspended Si3N4 nanobeam optomechanical crystal (33, 34). (C) Quantum limited displacement sensing involves the phase-quadrature shot-noise measurement floor (dashed purple) and the quantum measurement backaction caused by radiation pressure shot noise (dashed orange). The product of these two fundamental noise sources (green) is constant as dictated by a Heisenberg measurement–disturbance uncertainty relation. The sum of the two fundamental noise sources (black) is minimized at the standard quantum limit (SQL). Brownian motion of the mechanical resonator is shown in red. Our experiment takes place in the weak-probe power regime (dotted blue vertical), where measurement backaction is small. (Inset) Simulated optomechanical cross-correlations. The thermal correlation, Embedded Image, is shown in red. The quantum correlation, Embedded Image, is shown in blue (real part) and green (imaginary part). The ratio of the thermal motion peak, A, to the quantum correlation size, B, allows us to extract the temperature of the device. (D) Experimental setup. Optomechanical signal is detected via balanced heterodyne with local oscillator-probe detuning of Embedded Image. (Inset) Coordinate system for optical quadratures.

After passing through the optomechanical system, the probe laser acquires sidebands (Fig. 2A) from the optomechanical interaction. These sidebands can be thought of as the Stokes and anti-Stokes Raman scattering generated when a probe photon is down- or upshifted in energy while a vibrational quantum from the mechanical resonator is created or annihilated. In this picture, quantum correlations manifest as an asymmetry between the sidebands (6, 29). However, it is most natural to consider our heterodyne detected optical signal in terms of fluctuations of optical quadratures, Embedded Image, for quadrature angle Embedded Image. Although quantum signals are not immediately evident in the spectrum of any individual quadrature, for pairs of optical quadratures, we can compute the Fourier transformation of the two-time cross-correlation of the form Embedded Image, normalized to shot noise [see (30) for derivation of Eqs. 1 to 3]. We designate the amplitude-phase cross-correlation, Embedded Image, as the quantum correlation (Fig. 2G).

Fig. 2 Optical correlation spectra.

(A) Power spectral density (PSD) of heterodyne photodetector signal, normalized to shot noise (SN) (yellow). Stokes and anti-Stokes mechanical sidebands are shown. (B) PSD of the amplitude quadrature [Embedded Image, brown] displaying shot noise and the phase quadrature [Embedded Image, purple] showing the mechanical signal and shot noise. (C and D) Cross-correlation spectra, Embedded Image, for Embedded Image MHz (C) and Embedded Image MHz (D). From top to bottom curve of each panel, Embedded Image varies from –0.008 to 0.008 in 0.004 steps (orange); fits are shown in black. (E) Embedded Image for two Embedded Image’s differing by about 0.0005 Embedded Image (cyan and magenta); fits are shown in black. (F) Spectrum of thermal cross-correlation, Embedded Image (red) and Lorentzian fit (black). (G) Spectrum of quantum cross-correlation, real part [Embedded Image, blue] and imaginary part [Embedded Image, green]; fits are shown in black. All traces are generated from the same 300 s of time-domain data. Embedded Image K, Embedded Image for these data. See (30) for extracted fit parameters.

Embedded Image(1)

Embedded Image represents the strength of the optomechanical transduction, with Embedded Image representing the effects of optical loss and finite detection efficiency, Embedded Image the average intracavity photon occupation, Embedded Image the optomechanical coupling constant, and Embedded Image and Embedded Image the optical decay rate and resonance frequency, respectively, and Embedded Image is the Planck constant. We define the optomechanical cooperativity as Embedded Image, with Embedded Image. The quantum correlation represents a measurement of the linear response of the mechanics to the applied force of RPSN, characterized by a complex mechanical susceptibility, Embedded Image, where Embedded Image, Embedded Image, and Embedded Image are the mechanical resonance frequency, effective mass, and damping rate, respectively. Hence, the quantum correlation is complex, with real and imaginary parts following the dispersive and dissipative parts of a damped harmonic oscillator response, respectively. Thermal signals and measurement noise backgrounds, all of which are uncorrelated between amplitude and phase, average to zero. The characteristic scale of the quantum correlation is set by the Heisenberg measurement-disturbance uncertainty relation and is equivalent to the scale of mechanical vacuum fluctuations, when expressed as an effective displacement noise (as opposed to the shot-noise-normalized form of Eq. 1). In these units the quantum correlation is independent of probe power (see Fig. 3D), evident by noting that Embedded Image is fundamentally a product of backaction-driven motion (Embedded Image) and shot noise (Embedded Image) similar to the Heisenberg limited measurement backaction–shot noise imprecision product shown in Fig. 1C.

Fig. 3 Quantum correlation thermometry.

(A and B) Quadrature cross-correlations are shown for operation at room temperature, Embedded Image K, and atmospheric pressure, Embedded Image [(A), red], and Embedded Image [(B), blue]. Solid black curves are fits to data. The sharp feature at 3.611 GHz is an artifact of the heterodyne local oscillator, which is excluded from the fits and removed from (B). See (30) for fit residuals and extracted fit parameters. (C) Temperature measured via quantum correlation thermometry is plotted (red circles) as the system temperature is varied. Dashed gray line represents agreement between quantum correlation thermometry and a resistive thermometer measuring the sample temperature. The statistical error bars represent the estimated standard deviation. (Inset) Allan deviation of the temperature measurement for 294 K (orange squares) and 40 K (green diamonds). Fit curves (black) correspond to 640 K/Embedded Image and 30 K/Embedded Image for the 294 K and 40 K data, respectively. (D) Estimated spectral area of Lorentzian imaginary quantum correlation. Units are such that 1 corresponds to the motion of an excitation of 0.25 vibrational quanta. (E) The results of quantum correlation thermometry as the probe power is varied. Dashed gray line represents the expected result, including the estimated self-heating. In (D) and (E), data are taken with cryostat held at 40 K (red circles) or 80 K (green squares), and statistical error bars represent the estimated standard deviation.

To measure thermal signals, we could simply compute the power spectrum of the phase quadrature and subtract any unwanted noise backgrounds. However, a more elegant approach to isolate the thermal signal is to consider the cross-correlationEmbedded Image(2)where Embedded Image is the average mechanical occupation of the mechanical bath at temperature Embedded Image, coupled to the resonator. We designate Embedded Image as the thermal correlation. It retains the thermal motion contribution of the phase quadrature power spectrum but has zero background (Fig. 2, B and F). The form of the thermal correlation is governed by a fluctuation-dissipation theorem (31), evident from Eq. 2, where the imaginary part of the mechanical susceptibility links the mechanical dissipation to thermal fluctuations. Embedded Image is independent of Embedded Image and is purely quantum when the probe-cavity detuning, Embedded Image, is zero. From Eq. 2 we have

Embedded Image(3)

This relationship allows us to measure temperature from the ratio of components of cross-correlations without detailed knowledge of the optomechanical device parameters, detection efficiency, and mechanical and optical susceptibilities. We note that the above analysis only holds for a coherent state probe laser with no excess classical noise and with Embedded Image. Our Brownian motion thermometer is similar in concept to other types of noise measurement and thermometry—such as Johnson noise in electrical systems, blackbody radiation, or optical fiber Raman scattering—that can be calibrated with fundamental quantum noise (18, 19, 20, 32).

Our measurements employ a Si3N4 optomechanical crystal (33, 34) (Fig. 1B), a suspended Si3N4 nanobeam patterned with a series of holes that produces a mechanical resonance, frequency Embedded Image GHz, which varies over 10 MHz from cryogenic to room temperature, and an optical resonance near 991 nm, coupled at a rate Embedded Image kHz. The mechanical decay rate varies from Embedded Image MHz at room temperature to an order of magnitude lower at cryogenic temperatures. We chose the optomechanical crystal geometry (35) for its large optomechanical coupling at high mechanical frequency, facilitating high signal-to-noise readout of mechanical motion at relatively low Embedded Image (tens to thousands from cryogenic to room temperature), as well as for its potential for device integration and atmospheric pressure operation. Probe light is evanescently coupled to the optical resonance through a tapered optical fiber and detected with Embedded Image. Although the intrinsic optical decay rate is Embedded Image GHz, we intentionally overcouple the device to the tapered fiber to achieve an overall optical decay rate Embedded Image GHz and output coupling to the transmitted fiber port Embedded Image. The probe wavelength is actively stabilized near optical resonance, with a residual detuning Embedded Image at the percent level. All of our measurements are in the weak-probe regime, with C typically in the range of Embedded Image to Embedded Image (typical probe power ~10 μW), which offers many technical advantages and simplifications over experiments that seek the strong quantum backaction regime. The effects of dynamical backaction, photothermal or optomechanical instabilities, and RPSN-induced motion are minimized. Importantly, we avoid the possibility of classical noise on the probe laser interacting with the mechanics, which induces additional optomechanical cross-correlations. The excess laser noise on our weak probe is measured to be far below the shot noise level. [See (30) for more experimental details.]

To obtain a purely quantum correlation signal at large Embedded Image, we must carefully understand the effects of finite probe detuning and imperfect quadrature determination. Residual probe detuning mixes the amplitude and phase quadratures at the output of the cavity (Fig. 2E where Embedded Image), adding to Embedded Image, a thermal contribution that becomes dominant when Embedded Image [see (30) for details]. Several methods have been explored that ease this stringent stability requirement by reducing the amplitude of thermal motion, including active feedback damping (14), optical damping (3, 6, 7, 9, 11), or only analyzing signal frequencies far away from mechanical resonance (28). However these techniques generally require strong lasers (C >> 1) and introduce extra systematic uncertainties for thermometry. We counter the effects of finite detuning by rotating the analyzed pair of orthogonal quadratures by Embedded Image. This postprocessing undoes the cavity-induced optical quadrature rotation for the real component of the cross-correlation (Fig. 2C). By toggling the sign of the heterodyne local oscillator (LO)–probe detuning, Embedded Image, so that Embedded Image, we perform a type of lock-in detection, where the quantum correlation inverts (Fig. 2, C and D), whereas thermal correlations resulting from imperfections and dispersion in the optical quadrature determination remain unchanged. With these two tools, we are able to reject thermal signals from the real component of the quantum correlation at all temperatures and from the imaginary component of the cross-correlation only at low temperatures, as the demands on the stability of Embedded Image grow with increasing Embedded Image (30).

Figure 3B shows the quantum correlation obtained for our device at room temperature. Here the RPSN-driven motion is estimated to be about Embedded Image smaller than the thermal motion, whereas the quantum correlation in Fig. 3B is about Embedded Image smaller than the thermal correlation in Fig. 3A. Given our ability to resolve Embedded Image over a wide temperature range, we look to apply it to Brownian motion thermometry. We could formally apply a Kramers-Kronig transformation to express Embedded Image in terms of Embedded Image in Eq. 3, reconstructing the imaginary part of the mechanical susceptibility from the real part, assuming Embedded Image is analytic. We use a simple approximation of this idea, valid for a weakly damped simple harmonic oscillator. We fit curves such as those shown in Fig. 3, A and B, to obtain the height of the Lorentzian thermal correlation, A, and the peak-to-peak value of the real part of the quantum correlation, B, which is also equal to the peak height of the imaginary quantum correlation (Fig. 1C). The ratio Embedded Image is our approximate measure of the temperature of the mechanical resonator.

In absolute terms, we expect our quantum correlation signal to be independent of probe power and temperature, when expressed as an effective, mean-square-average displacement noise. In these units, the area of the Lorentzian imaginary quantum correlation, when Embedded Image, corresponds to the motion produced by Embedded Image worth of energy. In Fig. 3D, we estimate this area from measurements of the real part of the quantum correlation and see agreement.

Figure 3C shows agreement at the few percent level between our quantum correlation thermometry and the independently measured cryostat temperature over the range of 10 to 294 K. Allan deviations for the measurement are shown in the inset of Fig. 3C. At higher temperatures, the statistical noise in determining the quantum correlation is dominated by thermal fluctuations, whereas at lower temperatures, the shot-noise measurement floor is dominant. Each data point in Fig. 3C includes between several hundred and one thousand seconds of integration time. The dominant systematic effect at low temperatures is self-heating owing to optical absorption of the probe light, spurred by low thermal conductivity of silicon nitride thin films, which drops dramatically below a few tens of kelvin (36). To account for this effect, we probe our device with a range of optical power and extrapolate the temperature at zero probe power (30). We have applied this correction to the data points at 40 K and below in Fig. 3C. At higher temperatures, the self-heating is negligible compared to the statistical uncertainty. We also perform quantum correlation thermometry at multiple probe powers and see consistent results (Fig. 3E), a valuable check against many potential systematic effects (30).

We have demonstrated a proof-of-principle Brownian motion thermometer calibrated with optomechanical quantum correlations. We use optical quantum force fluctuations as a standard reference scale for measuring thermal force fluctuations. With improved devices—for example, using alternate geometries or materials to increase the optomechanical coupling (35), thermal conductivity, and mechanical resonance frequency (37)—we believe that this technique can attain metrologically relevant levels of accuracy over a wide temperature range. Such a device could ultimately improve temperature determination in practical applications where in situ or on-demand calibration is required. The sensor’s small size and fiber-optic readout make it attractive for long-distance remote-monitoring applications, operation in harsh electromagnetic environments, and permanently embedded applications. More generally, this glimpse of quantum effects in a mechanical system strongly coupled to a room-temperature environment represents a step toward room-temperature quantum sensing and information applications.

Supplementary Materials

www.sciencemag.org/content/356/6344/1265/suppl/DC1

Materials and Methods

Figs. S1 to S9

Tables S1 and S2

References (3843)

References and Notes

  1. Materials and methods are available online as supplementary materials.
  2. Acknowledgments: We thank K. C. Balram and M. Davanço for technical assistance and acknowledge useful discussions with Z. Ahmed, N. Klimov, and G. Strouse. Requests for data should be addressed to the corresponding author.
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