ARTICLES Resolved-sideband cooling of a micromechanical oscillator
A. SCHLIESSER, R. RIVIERE,` G. ANETSBERGER, O. ARCIZET AND T. J. KIPPENBERG* Max-Planck-Institut f ¨urQuantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany *e-mail: [email protected]
Published online: 13 April 2008; doi:10.1038/nphys939
In atomic laser cooling, preparation of the motional quantum ground state has been achieved using resolved-sideband cooling of trapped ions. Here, we report the first demonstration of resolved-sideband cooling of a mesoscopic mechanical oscillator, a key step towards ground-state cooling as quantum back-action is sufficiently suppressed in this scheme. A laser drives the first lower sideband of an optical microcavity resonance, the decay rate of which is twenty times smaller than the eigenfrequency of the associated mechanical oscillator. Cooling rates above 1.5 MHz are attained, three orders√ of magnitude higher than the intrinsic dissipation rate of the mechanical device that is independently monitored at the 10−18 m/ Hz level. Direct spectroscopy of the motional sidebands of the cooling laser confirms the expected suppression of motional increasing processes during cooling. Moreover, using two-mode pumping, this regime could enable motion measurement beyond the standard quantum limit and the concomitant generation of non- classical states of motion.
In atomic laser cooling, the lowest temperature that can be attained a reduction of its translational energy by one quantum, leading to for a trapped ion (or atom) with an optical decay rate κ is cooling. The lowest occupancy that can be attained is given by6 ∼ 1 h i ≈ 2 2 given by TD = h¯ κ/4kB, the Doppler temperature . If the harmonic n κ /16Ωm 1, implying that the particle can be found in the trapping frequency Ωm is smaller than κ, the minimum average ground state most of the time. Hence, by making the energy scale occupation number hni in the harmonic trapping potential is h¯ κ set by the spontaneous emission small in comparison with the hni ≈ κ/4Ωm > 1, that is, the ion’s harmonic motion cannot be level spacing of the trapped ion or atom, h¯Ωm, ground-state cooling cooled to the quantum ground state1. This fundamental limit can can be achieved. This powerful cooling technique, first proposed5 be viewed as a direct consequence of the Heisenberg uncertainty in 1975 has been called ‘cooling by motional sideband excitation’ or relation2: a spontaneous emission, which occurs over a timescale ‘sideband cooling’ and, fifteen years later, has led to the remarkable 1/κ, implies an energy uncertainty of 1E ∼ h¯ κ. As the average demonstration of ground-state cooling of trapped ions7,8. energy of the oscillator hEi = h¯Ωm (hni+1/2) cannot be lower Importantly, many of these considerations in atomic laser than its uncertainty, this implies that the ground state is not cooling also apply to electromechanics and the emerging field of 9 reached when κ Ωm. In a different perspective, this fundamental cavity optomechanics , which study the dynamics of a mechanical temperature limit can also be interpreted as the quantum back- oscillator parametrically coupled to an electrically or optically action3 that light exerts on the ion. Owing to the discrete and resonant device. Owing to the dynamical back-action3,10 of the stochastic nature of photons, scattering events occur randomly in electrical or optical field on the mechanical oscillator in the form time and lead to a fluctuation of the radiation-pressure force that of a Coulomb or radiation-pressure force, it is possible to cool the entails heating of the ion and prevents ground-state cooling. mechanical mode when the resonant system is excited in a detuned Ground-state cooling is however possible in the resolved- manner. As in atomic laser cooling, this effect is classical in nature, sideband regime. Resolved-sideband cooling4,5 requires a as opposed to the quantum back-action previously discussed. harmonically trapped dipole such as an atom or ion exhibiting Elemental back-action cooling was first observed experimentally a trapping frequency Ωm much larger that the optical resonance in 2006 for various micromechanical oscillators constituting the linewidth κ, thereby satisfying the so-called ‘strong binding boundary of very high-finesse optical cavities11–13, thus enabling condition’1. The physics of resolved-sideband cooling can be radiation-pressure cooling. A similar phenomenon based on understood in a simple manner. Owing to its harmonic motion, thermal effects has been reported earlier14. Back-action cooling a spatially oscillating excited ion will emit phase-modulated was also reported for a nanomechanical oscillator acting as the radiation. Consequently, the emission spectrum consists of a series gate electrode of a superconducting single-electron transistor15. of sidebands at frequencies ω0 −jΩm, where j = ±1,±2,... and ω0 Subsequently, back-action cooling in optical experiments at a larger is the unperturbed transition frequency. Inversely, the absorption scale16, involving very-high-Q mechanical nanomembranes17, and spectrum as probed by an observer in the laboratory frame will electromechanical cooling of a cantilever forming one plate of consist of a series of absorption lines, broadened owing to the a capacitor of a resonant electrical circuit18 were demonstrated. upper state’s decay rate (Fig. 1a). Cooling can be achieved by Other anticipated embodiments such as mechanical oscillators tuning the incident laser radiation to one of the energetically coupled to an optical quantum dot19, trapped ions20,21, a microwave 22,23 24 lower-lying sidebands, for example, ωL = ω0 − Ωm. As the ion resonator , a superconducting quantum interference device or 25 absorbs a photon of energy h¯ ωL = h¯ (ω0 −Ωm ) whereas it emits a resonant electronic circuit with an embedded Cooper pair box 1 on average a photon of energy h¯ ω0 (neglecting recoil ), this entails are predicted to exhibit similar phenomena. nature physics VOL 4 MAY 2008 www.nature.com/naturephysics 415
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ω ω a L L a κ Ω κ Ω >> m < m
ω ωωω 1 μm 0 0 ≈ κ Ω ≈κ 2 Ω 2 nmin /4 m >> 1 nmin /16 m < 1
| 〉 10 μm 10 μm b 1,n+1 1 μm |1,n〉 h– Ω |1,n–1〉 m κ c –ω b h 0 Optical WGM Mechanical RBM ω ω – Ω ω κ ( ΩΓ , ) L L m ( 0, ) m m ω Ω L+ m |0,n+1〉 β = 0.94 |0,n〉 |0,n–1〉
β = 1.47
Figure 1 Cooling a mechanical oscillator. a, Cooling proceeds by exciting the (a.u.) Transmission resonant (red curve) system coupled to the oscillator with a detuned source, for example, a laser (blue line). When the resonant system’s decay rate κ exceeds the β = 1.75 mechanical oscillator’s resonance frequency Ωm, ground-state cooling is impossible –200 –100 0 +100 +200 (left panel) owing to the quantum back-action. If however Ωm κ, the motional Detuning (MHz) sidebands are resolved and cooling can proceed to occupation far below unity (right panel). b, The underlying transitions, showing the ladder of the motional Fock states of the mechanical oscillator coupled to the resonant system’s transitions |0i → |1i Figure 2 Resolved-sideband regime of a mesoscopic optomechanical (ref. 19). Cooling proceeds by predominant absorption on the red sideband (blue oscillator. a, Scanning electron microscope image of the system used, consisting of arrows), giving rise to a blue-shifted photon carrying away the energy from an a silica microtoroidal optical cavity supporting both ultrahigh-finesse optical annihilated phonon. The opposite process (red arrows) leads to heating, and is resonances and high-Q radial breathing modes (RBMs) (Q = 30,000) held by a sufficiently well suppressed only in the resolved-sideband regime. ‘needle’ pillar. An image of an intentionally broken cavity structure is also shown, revealing the ultrathin silicon support pillar with a diameter of 500 nm, which reduces the coupling to the pillar and thus enables high mechanical Q factors. Interestingly however, for virtually all studied systems, b, Schematic diagram of the radiation-pressure coupling between the optical and theoretical work15,18,19,25–28 has identified a cooling limit equivalent mechanical modes in the toroidal microcavity. c, Cavity transmission spectrum of a to the Doppler temperature in atomic physics, caused by the microtoroid when its mechanical degree of freedom is excited with a coherent drive random momentum kicks of the discrete particles (photons, (using an auxiliary laser beam, see the Supplementary Information) of different electrons or charged quasiparticles) involved in the cooling amplitudes. The optical cavity decay rate corresponds to κ/2π = 3.2 MHz, whereas = mechanism. In analogy to atomic physics, dynamic back-action the mechanical breathing mode exhibits a frequency of Ωm /2π 73.5 MHz, cooling of the oscillator to its motional ground state is impossible thereby placing the system deeply into the resolved-sideband regime. The weights of the sidebands follow the Bessel function expansion (solid line is a fit). if the mechanical frequency Ωm does not exceed the decay rate κ of the electromagnetic resonance. Remarkably, in spite of the prevalent awareness of the Doppler limit, no electro- or optomechanical experiment so far has been able to enter the resolved-sideband regime. the mechanical mode, leading to its cooling. This process competes Here, we present resolved-sideband cooling of a mechanical with the reverse process where the phonon occupation is increased device, the frequency Ωm of which exceeds the decay rate κ of n → n + 1 by off-resonant absorption on the fist upper sideband. the optical resonator it is coupled to by more than twenty times. Rigorous quantum mechanical calculations27,28 show that motional The system therefore operates in the resolved-sideband regime, increasing and decreasing processes occur with rates proportional also termed the ‘good-cavity limit’. Similar to the ion’s case, the to (n + 1) · A+ and n · A−, respectively, where n is the phonon ± 2 2 −1 parametric modulation of the resonance condition of the optical occupation number and A ∝ ((κ/2) + (∆∓Ωm ) ) , with the cavity through the harmonic motion of its boundary gives rise to laser detuning ∆ = ωL − ω0 from the cavity resonance. Evidently, optical sidebands in the cavity absorption spectrum at frequencies if the laser is tuned to the lower sideband ∆ = −Ωm, cooling ω0 − jΩm where j = ±1,±2,... and ω0 is the unperturbed cavity is resonantly enhanced. In the case of well-resolved sidebands resonance frequency. The relative weights of the sidebands are κ Ωm, the motional decreasing transitions remain dominant 2 determined by Bessel functions |Jj (β)| , with β = ω0 x/Ωm R in the also for very small n, as required for ground-state cooling, and optomechanical case, R being the radius of the device. We note the minimum achievable phonon number reproduces the result h i ≈ 2 2 that in the optomechanical case usually β 1 making it sufficient n κ /16Ωm 1 known from atomic physics. to consider only first-order sidebands. In an energy picture, the Silica whispering-gallery mode (WGM) resonators, such sidebands correspond to transitions in which the state of the cavity as toroidal microcavities29 have previously been shown is changed (a photon is added) simultaneously with a change in to exhibit strong optomechanical coupling and dynamical the occupation of the mechanical oscillator (Fig. 1b). Motional back-action effects30–33. By optimizing microfabrication, we obtain decreasing transitions imply a reduction of phonon occupation, toroidal structures operating in the ‘strong binding’ condition, n → n − 1, when the photon is scattered into the cavity. As, on accommodating high-quality-factor optical and mechanical average, the escape of the photon from the cavity does not change modes in the same device (Fig. 2b). The measured parameters the motional state, the photon carries away the gained energy from Ωm /2π = 73.5 MHz and κ/2π = 3.2 MHz, corresponding to
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a Shot-noise-limited Hansch–Couillaud¨ polarization spectroscopy technique34 to the displacement sensor present experiment (see the Supplementary Information) enables Nd:YAG Frequency locking the monitoring laser to the centre of the resonance which, monitoring lock Spectrum PBS in conjunction with low power levels, ensures that the readout laser analyser laser’s effect on the mechanical oscillator motion is negligible. λ/4 λ/2 IF Simultaneously, it provides a shot-noise-limited signal, which FPC Vacuum enables monitoring the displacement noise of the cavity caused by 1,064 nm EOM chamber the thermal excitation of the toroid’s different mechanical modes FPC Displacement (Fig. 4a,b). The√ displacement sensitivity of this measurement WDM WDM − calibration 980 nm reaches 10 18 m/ Hz and is among the best values achieved so μ-toroid far35. It would in principle enable observing the radiation-pressure b quantum back-action once the sample is cooled to sufficiently low Far-detuned 1,064 nm λ/2 Cooling temperature (see the Supplementary Information), and eventually ELO diode cooling laser lock approaching the standard quantum limit. laser Turning on the cooling laser (while keeping the readout laser Ecav Fibre Toroid ϕ Detuning unchanged), a clear reduction of the displacement fluctuations is control observed, characteristic of cooling. Note that our previous work has already demonstrated unambiguously that cooling in toroidal c Monitoring Cooling microcavities is solely due to radiation pressure13 and thermal laser – laser Monitoring A+ A Cooling contributions14 play negligible role. We thus observe, for the first cavity cavity time, resolved-sideband cooling of a micromechanical oscillator. mode mode κ Simultaneous measurements on other radially symmetric modes Optical Ω Ω at lower frequencies reveal that these remain unaffected (Fig. 4c). m m frequency This selectivity to a single mechanical mode is specific to the Δ = 0 ΔΩ = – m regime of resolved-sideband cooling. In contrast to the ‘weak binding’ case, where the κ-wide absorption sidebands of different mechanical modes overlap, resolved-sideband cooling can provide Figure 3 Schematic diagram of the experiment. a, The optomechanical system highly targeted cooling of only one mechanical mode. As shown (toroidal microcavity) is held in an evacuated chamber, and is simultaneously excited in Fig. 4d, the highest attained cooling rate Γc /2π = 1.56 MHz was by a 980 nm diode laser that serves for cooling and a 1,064 nm Nd:YAG laser for achieved in the first sample (Ωm /2π=73.5 MHz, κ/2π=3.2 MHz) shot-noise-limited monitoring of the displacement of the cavity’s mechanical modes. for less than 3 mW of input power. Note that only a fraction 2 −1 −4 b, The monitoring laser is polarized such that only a small fraction of its power ∼(4(Ωm /κ) +1) ≈ 5 × 10 of the launched power in the couples into the (polarization non-degenerate) WGM. The stronger orthogonal field fibre is coupled into the cavity (1.5 µW) owing to the highly component serves as a local oscillator in a polarization-sensitive detection scheme, detuned working point. Thus, efficient cooling is achieved with enabling shot-noise-limited displacement monitoring. The absolute calibration of the extremely low powers coupled to the cavity. Combining such measured displacements is derived from a phase modulation of the monitoring laser high cooling rates with the lowest achieved reservoir heating with a known modulation depth (see the Supplementary Information). c, By means rates of Γm /2π = 1.3 kHz, it seems feasible to attain a ratio of ∼ 3 of a frequency modulation technique, the cooling laser is locked far outside the initial to final occupancy nR /nf = (Γc +Γm )/Γm exceeding 10 . 2 2 cavity’s WGM resonance to a detuning ∆ = −Ωm, such that anti-Stokes scattering = With the demonstrated 16Ωm /κ 7,700, this value would be of photons into the cavity mode is resonantly enhanced. The independent monitoring sufficient to reach nf < 0.5 when starting at a cryogenic lHe ¨ 27,28 laser is locked to the centre of a different resonance using the Hansch–Couillaud temperature of 1.8 K, while still satisfying nR /nf < Q ≡ Ωm /Γm signal (see the Supplementary Information). FPC: fibre polarization controller, WDM: and Γc < κ. Starting from room temperature as reported wavelength division multiplexer, IF: interference filter, PBS: polarizing beam splitter, here, this would lead to nf < 100. However, analysis of the l/2, l/4: optical retarder plates. integrated calibrated displacement noise spectra through the R 2 relation nf h¯Ωm = meffΩ Sx (Ω ) dΩ indicates significantly higher nf. This discrepancy is attributed to heating by excess phase noise on the cooling laser beam. Using an√ independent measurement, 5 −1/2 a finesse of F = 4.4 × 10 , place the system deeply into the it was measured to be as high as Sϕ ≈ 4 µrad Hz even at resolved-sideband regime. The repercussions of this regime on the radiofrequencies close to Ωm (see the Supplementary Information). cavity transmission are most strikingly observed when an auxiliary The resulting classical radiation-pressure√ back-action limits the ∼ laser is used to drive the mechanical motion (using blue-detuned achievable occupancy to nmin = 2kB TmeffΓm Sϕ RΩm /h¯ ω0 ≈ 5,200 light31, see the Supplementary Information). Indeed, when tuning for the parameters of the second sample, with which the lowest over the driven cavity, a series of optical resonances spaced by occupancies were achieved. Note that in this case the (classical) the mechanical frequency can be observed, satisfying Ωm /κ ≈ 22 correlations between the laser noise and the induced displacement (Fig. 2c), which convincingly proves that the device satisfies the fluctuations can cause ‘squashing’36 artefacts if the diode laser ‘strong binding’ condition. were also used for mechanical readout. In contrast, the use of To implement resolved-sideband cooling, a grating-stabilized the independent Nd:YAG laser provides a faithful displacement laser diode is coupled to a high-finesse WGM near 970 nm monitoring with which these induced fluctuations can be revealed. of a second sample (Ωm /2π = 40.6 MHz, κ/2π = 5.8 MHz, Such an analysis yields a final occupancy of nf = 5,900, in good Γm /2π = 1.3 kHz, meff = 10 ng), and locked to the first lower agreement with the above estimate. In future experiments, this motional sideband (∆ = −Ωm) using a frequency modulation technical limitation can be overcome by using quantum-noise- technique and fast feedback (see Supplementary Information limited, tunable solid-state lasers such as a Ti:sapphire laser. and Fig. 3a). The cooling caused by this laser is independently A direct consequence of the resolved-sideband regime is the and continuously monitored by a Nd:YAG laser (l = 1,064 nm) strong suppression of motional increasing processes, which should coupled to a different cavity mode. An adaptation of the lead to a suppression of the red sideband in the spectrum of the light nature physics VOL 4 MAY 2008 www.nature.com/naturephysics 417
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ω ω Ω a +50 3 a L L– AOM AOM (fm) 0 x –50 200 MHz Heterodyne 0 1 BS μ-toroid detection Time (μs) ω BS L Ω Ω b m m 3 1,000 ω L laser Cooling Ω ± Ω
) 1 2 AOM m –1/2 100 b m Hz –18
(10 –10 x 10 δ
PSD (dB) –20 1
010203040 Frequency (MHz) 159.35 159.40 159.45 240.55 240.60 240.65 Frequency (MHz)
c ) 300 1 2 3 × –1/2 0.25 200 Figure 5 Motional sideband spectroscopy. a, Experimental set-up used to resolve m Hz upper and lower motional sidebands generated during interaction of the cooling –18 100 laser with the cavity, similar to the spectroscopy of the resonance fluorescence of a (10 x 37 δ 0 laser-cooled ion . The cooling laser interacts with the optical microcavity, the 14.95 15.00 28.55 28.60 40.55 40.60 transmission of which is subsequently superimposed with a second laser beam Frequency (MHz) retrieved from the same cooling laser but down-shifted by 200 MHz using an acousto-optic modulator. The beating of the two signals is recorded using a balanced
d ) P = 0 mW ) cool Pcool = 2.7 mW heterodyne detector, yielding spectral components of the lower and upper sidebands
–1/2 100 –1/2 100 at 200 MHz−Ωm /2π and 200 MHz+Ωm /2π, respectively. BS: beam splitter.
m Hz m Hz b, Beat signals of the upper (anti-Stokes) and lower (Stokes) motional sidebands, for
–18 10 –18 10 ∼ ∆ = 0 (red) and a detuning close to ∆ = −Ωm (blue). The plotted electrical noise (10 (10 x x Γ π eff/2 = 1.6 MHz power spectral density (PSD) is proportional to the optical power spectral density in δ δ 73 74 73 74 the sidebands. For zero detuning of the pump with respect to the optical cavity, the Frequency (MHz) Frequency (MHz) motional sidebands are equal in power. By tuning the laser to the lower sideband, which induced cooling, strong reduction of the Stokes sideband is observed.
Figure 4 Resolved-sideband cooling of the radial breathing mode. a, Time-domain trace of the brownian motion of the radial breathing mode as experiment, by beating the cooling laser with a local oscillator observed by the monitoring laser in a 2 MHz spectral bandwidth around (Fig. 5a), derived by down-shifting part of the cooling laser light Ωm /2π = 40.6 MHz. For observation times that are short compared with the using an acousto-optic modulator at ΩAOM /2π = 200 MHz. The coherence time of the mechanical oscillator, a sinusoidal oscillation is observed. beat of the local oscillator and the cooling laser produces a b, Full spectrum of the displacement fluctuation spectral density δx at room modulation at Ω , whereas the motional sidebands’ signals now temperature, recorded with the Nd:YAG laser (red). The different peaks appearing in AOM appear at ΩAOM ±Ωm, thereby enabling measuring their individual the spectrum represent the mechanical eigenmodes that can be identified using weights. Figure 5b shows the result of this measurement for two three-dimensional finite-element analysis. The modes denoted by (1,2,3) are different laser detunings. Whereas for excitation on the cavity line rotationally symmetric mechanical modes, the strain (colour code) and deformed centre (∆ = 0) the sideband intensities are equal (A−n ≈ A+(n+1) shapes of which are shown in the insets. The background of the measurement (grey) because n 1 and A− = A+), detuning the laser to the lower is due to shot noise; its frequency dependence results from the reduced sideband ∆ = −Ωm should lead to a strong suppression of the red- displacement sensitivity (for the same measured noise level) at frequencies − + ∼ 2 2 sideband beat by a factor of A /A = 16Ωm /κ . In the experiment exceeding the cavity’s bandwidth. A signal-to-background ratio close to 60 dB in the with the 40.6 MHz sample, the detuning is chosen such that noise power spectral density is achieved at room temperature. c, Resolved-sideband the red sideband is still discernible above the laser phase noise, cooling with the cooling laser tuned to the lower sideband of the radially symmetric corresponding to a suppression of more than 15 dB. Optimizing radial breathing mode (3). As evident, only mode (3) is cooled, whereas all other the laser detuning, the red emission sideband could be reduced modes (of which 1 and 2 are shown) remain unaffected. Circles represent noise even further. It is important to note that the ability to measure µ spectra with the cooling laser off (red) and running at 300 W (blue). Lines are the individual sidebands separately—as demonstrated here—is lorentzian fits. d, Cooling rates exceeding 1.5 MHz obtained with the 73.5 MHz radial important for future experiments that venture into the quantum breathing mode of a different sample. regime. As theoretically predicted27—and in analogy to trapped ions7,38—the weights of the sidebands enable inferring the average motional occupation number27 for low occupancies by measuring emerging from the cavity (as analysed theoretically in ref. 27). To the ratio between the red and blue sidebands. confirm this aspect, the motional sidebands generated during the The regime of resolved sidebands has another important— cooling cycles were probed, similar to spectroscopy of the resonance and counterintuitive—benefit, because the cooling rate is indeed fluorescence of a cooled ion37. This is achieved with a heterodyne higher in comparison with the unresolved case. Keeping the
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launched power P as well as Ωm and R fixed, an increase in 17. Thomson, J. D. et al. Strong dispersive coupling of a high finesse cavity to a micromechanical the cavity finesse increases the cooling rate, until it saturates in membrane. Nature 452, 72–75 (2008). 18. Brown, K. R. et al. Passive cooling of a micromechanical oscillator with a resonant electric circuit. the highly resolved-sideband case and approaches an asymptotic Phys. Rev. Lett. 99, 137205 (2007). value (see the Supplementary Information). The circulating power 19. Wilson-Rae, I., Zoller, P. & Imamoglu, A. Laser cooling of a nanomechanical resonator mode to its quantum ground state. Phys. Rev. Lett. 92, 075507 (2004). however continues to decrease, mitigating undesired effects such 20. Wineland, D. et al. Experimental issues in coherent quantum-state manipulation of trapped atomic as photothermal14- or radiation-pressure-induced bistability39 or ions. J. Res. Natl Inst. Standards Technol. 103, 259–328 (1998). 21. Tian, L. & Zoller, P. Coupled ion-nanomechanical systems. Phys. Rev. Lett. 93, 266403 (2004). absorption-induced heating. It is important to note however, that 22. Braginsky, V. B. Measurement of Weak Forces in Physics Experiments (Univ. Chicago Press, the final occupancies for very high optical finesse are bound by the Chicago, 1977). 28 23. Regal, C. A., Teufel, J. D. & Lehnert, K. W. Measuring nanomechanical motion with a microwave cavity decay rate . cavity interferometer. Preprint at
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