Pulsed Quantum Optomechanics

M. R. Vannera,1, I. Pikovskia, G. D. Colea, M. S. Kimb, Č. Bruknera,c, K. Hammererc,d, G. J. Milburne, and M. Aspelmeyera

aVienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria; bQOLS (Quantum Optics and Laser Science Group), Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom; cInstitute for Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy of Sciences, A-1090 Vienna and A-6020 Innsbruck, Austria; dInstitute for Theoretical Physics and Albert Einstein Institute, University of Hannover, Callinstrasse 38, D-30167 Hannover, Germany; and eSchool of Mathematics and Physics, University of Queensland, Saint Lucia 4072, Australia

Edited by Mikhail Lukin, Harvard University, Cambridge, MA, and accepted by the Editorial Board July 21, 2011 (received for review April 1, 2011)

Studying mechanical resonators via radiation pressure offers a distribution of phase noise of strong pulses of light at various rich avenue for the exploration of quantum mechanical behavior times throughout a mechanical period. We show that the same in a macroscopic regime. However, quantum state preparation experimental tools used for quantum state tomography can also and especially quantum state reconstruction of mechanical oscilla- be used for squeezed state preparation and state purification, tors remains a significant challenge. Here we propose a scheme which thus provides a complete experimental framework. Our to realize quantum state tomography, squeezing, and state purifi- scheme does not require “cooling via damping” (11–13) and cation of a mechanical resonator using short optical pulses. The can be performed within a single mechanical cycle thus signifi- scheme presented allows observation of mechanical quantum fea- cantly relaxing the technical requirements to minimize thermal tures despite preparation from a thermal state and is shown to be contributions from the environment. experimentally feasible using optical microcavities. Our framework Using a pulsed interaction that is very short compared to the thus provides a promising means to explore the quantum nature of period of an oscillator to provide a back-action-evading measure- massive mechanical oscillators and can be applied to other systems ment of position was introduced in the seminal contributions of such as trapped ions. Braginsky and coworkers (36, 37), where schemes for sensitive force detection were developed. In our work, the quantum nature optomechanics ∣ quantum measurement ∣ squeezed states of a mechanical resonator is itself the central object of inves- tigation. Here, the pulsed interaction is used to provide an ex- oherent quantum mechanical phenomena, such as entangle- perimentally feasible means to generate and fully reconstruct Cment and superposition, are not apparent in the macroscopic quantum states of mechanical motion. The proposed experimen- realm. It is currently held that on large scales quantum mechan- tal setup is shown in Fig. 1. A pulse of duration much less than the ical behavior is masked by decoherence (1) or that quantum mechanical period is incident upon an optomechanical Fabry- mechanical laws may even require modification (2–5). Despite Pérot cavity which we model as being single sided. Due to the substantial experimental advances, see for example ref. 6, probing entanglement generated during the radiation-pressure interac- this regime remains extremely challenging. Recently however, it tion, the optical phase becomes correlated with the mechanical has been proposed to utilize the precision and control of quantum position while the optical intensity imparts momentum to the optical fields in order to investigate the quantum nature of mechanical oscillator. Time-domain homodyne detection (15) is massive mechanical resonators by means of the radiation-pres- then used to determine the phase of the field emerging from the sure interaction (7–13). Quantum state preparation and the cavity, and thus to obtain a measurement of the mechanical posi- ability to probe the dynamics of mechanical oscillators, the most tion. For each pulse, the measurement outcome PL is recorded, rigorous method being quantum state tomography, are essential which for Gaussian optical states has mean and variance for such investigations. These important elements have been h i¼χh in i σ2 ¼ σ2 þ χ2σ2 [1] P X ; in in ; L M PL P X experimentally realized for various quantum systems, e.g., light L M (14, 15), trapped ions (16, 17), atomic ensemble spin (18, 19), in respectively. X M is the mechanical position quadrature immedi- and intracavity microwave fields (20). By contrast, an experiment in ately prior to the interaction and PL describes the input phase of realizing both quantum state preparation and tomography of light. The position measurement strength χ is an important para- a mechanical resonator is yet to be achieved. Also, schemes meter in this work as it quantifies the scaling of the mechanical that can probe quantum features without full tomography [e.g., position information onto the light field. A derivation of Eq. 1 (9, 10, 21)] are similarly challenging. In nanoelectromechanics, including an optimization of χ by determining the input pulse cooling of resonator motion and preparation of the ground state envelope to gain the largest cavity enhancement is provided in have been observed (22, 23) but quantum state reconstruction the Appendix. (24) remains outstanding. In cavity optomechanics significant In order to describe and quantify the pulse interaction and experimental progress has been made towards quantum state measurement we use the nonunitary operator Υ that determines control over mechanical resonators (11–13), which includes clas- ρout ∝ Υρin Υ† the new mechanical state via M M . This operator is sical phase-space monitoring (25, 26), laser cooling close to the mechanical state independent and can be determined from the ground state (27, 28), and low noise continuous measurement of probability density of measurement outcomes mechanically induced phase fluctuations (29–31). Still, quantum ð Þ¼ ðΥ†Υρin Þ [2] state preparation is technically difficult primarily due to thermal Pr PL TrM M : bath coupling and weak radiation-pressure interaction strength, For pure optical input, it takes the form and quantum state reconstruction remains little explored. Thus far, a common theme in proposals for mechanical state reconstruction is state transfer to and then read-out of an auxillary Author contributions: M.R.V., I.P., G.D.C., M.S.K., Č.B., K.H., G.J.M., and M.A. designed quantum system (32–35). This technique is a technically demand- research; M.R.V., I.P., G.D.C., M.S.K., Č.B., K.H., G.J.M., and M.A. performed research; ing approach and remains a challenge. and M.R.V., I.P., G.D.C., M.S.K., Č.B., K.H., G.J.M., and M.A. wrote the paper. In this paper we introduce an optomechanical scheme that The authors declare no conflict of interest. provides direct access to all the mechanical quadratures in order This article is a PNAS Direct Submission. M.L. is a guest editor invited by the Editorial Board. to obtain full knowledge about the quantum state of mechanical Freely available online through the PNAS open access option. motion. This quadrature access is achieved by observing the 1To whom correspondence should be addressed. E-mail: [email protected].

16182–16187 ∣ PNAS ∣ September 27, 2011 ∣ vol. 108 ∣ no. 39 www.pnas.org/cgi/doi/10.1073/pnas.1105098108 Downloaded by guest on September 29, 2021 Our scheme provides a means for precision measurement A B of the mechanical quadrature marginals, thus allowing the mechanical quantum state to be determined. Specifically, given a ρin θ ¼ ω mechanical state M , harmonic evolution of angle M t provides access to all the quadratures of this mechanical quantum state which can then be measured by a subsequent pulse. Thus, reconstruction of any mechanical quantum state can be performed. The optical phase distribution Eq. 2, including this harmonic evolution, becomes Z dX M −ð −χ Þ2 − θ θ ð Þ¼ pffiffiffi PL XM h j i nρin i nj i [4] Pr PL e X M e e X M ; Fig. 1. (A) Schematic of the optical setup to achieve measurement based π M quantum state engineering and quantum state tomography of a mechanical which is a convolution between the mechanical marginal of inter- resonator. An incident pulse (in) resonantly drives an optomechanical cavity, χ where the intracavity field a accumulates phase with the position quadrature est and a kernel that is dependent upon and the quantum phase XM of a mechanical oscillator. The field emerges from the cavity (out) and noise of light. The effect of the convolution is to broaden the balanced homodyne detection is used to measure the optical phase with a marginals and to smooth any features present. local oscillator pulse (LO) shaped to maximize the measurement of the me- Let us consider the specific example of a mechanical resonator B chanical position. ( ) Scaled envelopes of the optimal input pulse, its corre- in a superposition of two coherent states, i.e., jψ δi ∝ jiδiþj− iδi. sponding intracavity field and the optimal local oscillator as computed in the Appendix The X M marginal of this mechanical Schrödinger-cat state . contains oscillations on a scale smaller than the ground state. ð − χ Þ2 2 −1 PL X M Υ ¼ðπ2σ Þ 4 Ω − [3] The convolution scales the amplitude of these oscillations by in exp i X M 2 ; 2δ2 PL 4σ ð− Þ χ in exp χ2þ1 and thus for small they become difficult to resolve PL in the optical phase noise distribution. Shown in Fig. 2 are where Ω quantifies the momentum transfer to the mechanics due marginals of the mechanical state jψ δi and the optical phase to the pulse mean photon number. Υ can be readily understood distributions that would be observed according to Eq. 4. Scaling by considering its action on a mechanical position wavefunction. ∕χ This operator selectively narrows the wavefunction to a width the phase distribution by using the variable PL provides an −2 approximation to the mechanical marginals, which becomes more scaling with χ about a position which depends upon the mea- accurate with increasing χ and may even show the interference surement outcome. Moreover, the quantum non-demolition-like Υ features in a superposition state. Indeed, the limiting case of in- nature of allows for back-action-evading measurements of X M , χ i.e., the back-action noise imparted by the quantum measurement finite corresponds to a von-Neumann projective measurement of the mechanical position, such that the distribution obtained process occurs in the momentum quadrature only*. Other meth- ∕χ ods, such as the continuous variational measurement scheme for PL becomes identical to the mechanical marginals. How- ever, the mechanical marginals can be recovered even for small (38), which has recently been considered for gravitational-wave χ detectors (39, 40), also allow for back-action-evading measure- measurement strength . This recovery is achieved as follows: ments. However, using short pulses offers a technically simpler First, by fixing the length of the cavity the optical phase distribu- route for quantum state tomography and is readily implementation can be observed without contributions from mechanical ble, as will be discussed below. position fluctuations. This rigidity allows measurement of the σ2 ¼ 1∕2 convolution kernel for a particular χ (determined by the proper- In the following, we consider coherent drive i.e., in . PL We first address the important challenge of how to experimentally determine the motional quantum state of a mechanical resonator. We then discuss how such a measurement can be used for quantum state preparation and finally we provide details for a physical implementation and analyze a thorough list of potential experimental limitations. Mechanical Quantum State Tomography Of vital importance to any experiment aiming to explore quantum mechanical phenomena is a means to measure coherences and complementary properties of the quantum system. Such measurement is best achieved by complete quantum state tomography, which despite being an important quantum optical tool has received very little attention for mechanical resonators†. Any measurement made on a single realization of a quantum state cannot yield sufficient information to characterize that quantum Fig. 2. The scheme presented here provides an experimentally feasible state. The essence of quantum state tomography is to make mea- means to obtain direct access to the marginals of a quantum state of a me- surements of a specific set of observables over an ensemble of chanical resonator. Shown are complementary quadrature marginals of the identically prepared realizations. The set is such that the mea- mechanical coherent state superposition jψ δi ∝ jiδiþj− iδi,forδ ¼ 1.5 (blue surement results provide sufficient information for the quantum dashed lines with fill, plotted with XM). The mechanical ground state is state to be uniquely determined. One such method is to measure shown for comparison in gray dashed lines. The two population components the marginals hXje−iθnρeiθnjXi, where n is the number operator, are seen for the quadrature angle θ ¼ π∕2 and the quantum interference for all phase-space angles θ, see refs. 14, 15, 42 and e.g., ref. 43. fringes for θ ¼ 0. A coherent optical pulse is used to probe the mechanical state where its phase quadrature becomes the convolution between the intrinsic phase noise, with variance scaling with χ−2, and the mechanical *No mechanical position noise is added as our measurement operator commutes with the marginal (red solid lines, plotted with PL∕χ where χ ¼ 2), see Eq. 4. The con- mechanical position. This is because the mechanical evolution can be neglected during volution kernel can be observed by using a fixed length cavity, shown in the PHYSICS the short optomechanical interaction. θ ¼ 0 plot (red dashed line with fill, fixed length with XM ¼ −4), which allows †During the submission process of this manuscript a scheme to perform tomography of the for accurate recovery of the mechanical marginals even for a weak measure- motional state of a trapped particle using a time-of-flight expansion was proposed (41). ment strength χ.

Vanner et al. PNAS ∣ September 27, 2011 ∣ vol. 108 ∣ no. 39 ∣ 16183 Downloaded by guest on September 29, 2021 ties of the mechanical resonator of interest, cavity geometry, and There is currently significant interest in the preparation of pulse strength, see Eq. 14). With χ and the kernel known one low entropy states of mechanical resonators as a starting point can then perform deconvolution to determine the mechanical for quantum experiments, e.g., refs. 22, 23, 27, 28. The two main marginals. The performance of such a deconvolution is limited methods being pursued in optomechanics (11–13) are “passive by experimental noise in the calibration of χ and the measure- cooling” which requires the stable operation of a (usually cryo- ð Þ “ ” ment of Pr PL . However, it is expected that these quantities genically compatible) high-finesse cavity, and active cooling can be accurately measured as quantum noise limited detection which requires precision measurement and feedback. Closer in is readily achieved. spirit to the latter, our pulsed measurement scheme provides a third method to realize high-purity states of the mechanical re- Mechanical Quantum State Engineering and sonator. We quantify the state purity after measurement via an ¯ Characterization effective mechanicalq thermalffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi occupation neff , which we define We now discuss how the measurement affects the mechanical 1 þ 2¯ ¼ 4σ2 σ2 through neff θ¼0 θ¼π∕2. When acting on an initial state. First, we consider Υ acting on a mechanical coherent state jβi. By casting the exponent of Υ in a normal ordered form, one thermal state, the measurement dramatically reduces uncertainty in the X M quadrature, but leaves the thermal noise in the PM can show that the resulting mechanical state, which is conditioned ð1Þ N Υjβi¼ ð Þ ðμ Þj0i 6 ¯ ≫ 1 ¯ ≃ on measurement outcome PL,is β S r D β . Here, pquadratureffiffiffiffiffiffiffiffiffiffiffiffi unchanged: use of Eq. for n yields neff N β ¯∕2χ2 β is a -dependent normalization,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD is the displacement opera- n . The purity can be further improved by a second pulse, pffiffiffi θ ¼ ω ¼ π∕2 μ ¼ð 2β þ Ω þ χ Þ∕ 2ðχ2 þ 1Þ which is maximized for pulse separation M t , where tor for β i PL , and S is the squeez- 2 −2r the initial uncertainty in the momentum becomes the uncertainty ing operator, which yields the position width 2σ ¼ e ¼ ‡ XM 2 −1 in position. Such a sequence of pulses is represented in Fig. 3, ðχ þ 1Þ . where the resulting state was obtained akin to Eq. 5. The effective In most experimental situations, the initial mechanical state occupation of the final state after two pulses is ρ ¼ 1 ∫ 2β −jβj2∕n¯jβihβj sffiffiffiffiffiffiffiffiffiffiffiffiffi is in a thermal state n¯ πn¯ d e , quantified by its average phonon occupation number n¯. The marginals of the ð2Þ 1 1 n¯ ≃ 1 þ − 1 ; [7] resulting state after the action of Υ are eff 2 χ4 ðX − hX θ iÞ2 ð2Þ h j −iθnΥρ Υ† iθnj i ∝ − M M [5] which is also independent of initial occupation. For χ > 1, n¯ X M e n¯ e XM exp 2 ; eff 2σθ is well below unity and therefore this scheme can be used as an alternative to “cooling via damping” for mechanical state pur- where χP ification. hX θ i¼ L cosðθÞ − Ω sinðθÞ; “ M χ2 þ 1 Following state preparation, one can use a subsequent read- 1þ2n¯ out” pulse after an angle of mechanical free evolution θ to per- 1 2ðθÞ 1 form tomography. During state preparation however, the random σ2 ¼ cos þ ðχ2 þ 1 þ 2¯Þ 2ðθÞ [6] θ 2 χ2 þ 1 2 n sin measurement outcomes will result in random mechanical means 1þ2n¯ Eq. 6. This randomness can be overcome by recording and utiliz- are the mean and variance of the resulting conditional state, ing the measurement outcomes. One can achieve unconditional respectively. For large initial occupation (provided thermal fluc- state preparation with use of appropriate displacement prior to tuations are negligible during the short interaction), the resultant the read-out pulse. Or, use postselection to analyze states pre- h θ¼0i ≃ ∕χ position quadrature of the mechanics has mean X M PL pared within a certain window. Alternatively, one may com- 2σ2 ≃ χ−2 and width θ¼0 . Thus, squeezing in the X M quadrature pensate during data analysis by appropriately adjusting each below the ground state is obtained when χ > 1 and is independent measurement outcome obtained during read-out. We now look of the initial thermal occupation of the mechanics. We have thus at the latter option and consider a Gaussian mechanical state pre- shown how the remarkable behavior of quantum measurement pared by a prior pulsed measurement. The position distribution – – σ2 h ðpÞi (also used in refs. 18 20, 44 47) can be experimentally applied has variance to be characterized and has a known mean X M , to a mechanical resonator for quantum state preparation. which is dependent upon the random measurement outcome. ð Þ ∝ The read-out pulse will then have the distribution Pr PL ½ð−ð − χh ðpÞiÞ2Þ∕ð1 þ χ22σ2Þ exp PL X M . For each read-out pulse, j ¼ − χh ðpÞi by taking PL p PL X M one can obtain the conditional 2 variance σ j for all θ to characterize the noise of the prepared PL p Gaussian state. We note that this concept of compensating for a random but known mean can also be used to characterize non-Gaussian states. Experimental Feasibility We now provide a route for experimental implementation, dis- Fig. 3. Wigner functions of the mechanical state (above) at different times cussing potential limitations and an experimentally feasible para- (indicated by arrows) during the experimental protocol (below). From left: meter regime. To ensure that the interaction time be much less Starting with an initial thermal state n¯ ¼ 10, (this is chosen to ensure the fig- than mechanical time scales the cavity decay rate κ must be much ure dimensions are reasonable,) a pulsed measurement is made with χ ¼ 1.5 larger than the mechanical frequency. To this end, we consider Pð1Þ ¼ 4χ X and outcome L obtained, which yields an M quadrature squeezed the use of optical microcavities operating at λ ¼ 1;064 nm, length state. The mechanical state evolves into a PM quadrature squeezed state fol- 4λ and finesse of 7,000, which have an amplitude decay rate lowing free harmonic evolution of 1∕4 of a mechanical period prior to a sec- ð2Þ κ∕2π ≃ 2.5 GHz. Such short cavity devices incorporating a micro- ond pulse with outcome PL ¼ −3χ yielding the high-purity mechanical squeezed state. The effective thermal occupation of the mechanical states ’ during the protocol is indicated. The final state s occupation can be reduced ‡We note that strong squeezing of an oscillator can also be achieved by using rapid below unity even for large initial occupation, see Eq. 7 of the main text. modifications to the potential at quarter period intervals (48). However, we would like Dashed lines indicate the 2σ-widths and the dotted lines show the ground to emphasize that the squeezing we are discussing here does not arise from a parametric state (n¯ ¼ 0) for comparative purposes. The displacement Ω is not shown. process, see e.g., ref. 49, rather it is due to the nonunitary action of measurement.

16184 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1105098108 Vanner et al. Downloaded by guest on September 29, 2021 mechanical element as one of the cavity mirrors have previously A been fabricated for tunable optical filters, vertical-cavity surface- emitting lasers and amplifiers (see for example ref. 50), but are yet to be considered for quantum optomechanical applications. Typically, these devices employ plane-parallel geometries, which places a severe constraint on the minimum lateral dimensions of the suspended mirror structure in order to minimize diffraction losses (51). Geometries using curved mirrors are required to re- duce diffraction losses for the practical realization of high-finesse cavities. Presently, all realizations use a curved suspended mirror, B F see e.g., refs. 52, 53. However, in order to allow for enhanced freedom in the construction of the mechanical resonator, parti- C cularly with respect to the development of ultra-low loss mechanical devices (54), a flat suspended mirror is desired. In Fig. 4 our proposed fabrication procedure for such a device is shown. The small-mode-volume cavity considered here provides the band- D G width necessary to accommodate the short optical pulses and additionally offers a large optomechanical coupling rate. One technical challenge associated with these microcavities is fabrication with sufficient tolerance to achieve the desired optical re- E H sonance (under the assumption of a limited range of working wavelength), however this can be overcome by incorporating electrically controlled tunability of the cavity length (50, 52, 53). ω ∕2π ¼ For a mechanical resonator with eigenfrequency M Fig. 4. Our proposed design and fabrication procedure for high-finesse op- 500 kHz and effectivepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mass m ¼ 10 ng, the mechanical ground- tomechanical microcavities: Using microcavities provides optomechanical ¼ ℏ∕ ω ≃ 1 8 coupling rates many orders of magnitude larger than current millimeter state size is x0 m M . pfmffiffiffi and optomechanical cou- ∕2π ¼ ω ð ∕ 2 Þ∕2π ≃ 86 ω or centimeter length scale implementations of optomechanical Fabry-Pérot pling proceeds at g0 c x0 L kHz, where c cavities and can provide sufficient radiation-pressure interaction to resolve is the mean cavity frequency and L is the mean cavity length. The the small scale quantum properties of the mechanical resonator. (A) Cross- primary limitation in measurement strength is the optical inten- sectional view with a quarter of the device removed. Uppermost (colored sity that can be homodyned before photodetection begins to sa- green) is the mechanical resonator supported by auxiliary beams as was con- ¼ 108 sidered in ref. 54. The optical field is injected into the device from below turate. Using pulses of mean photon number Np , which can through a transparent handle (colored blue) and the curved rigid input mir- be homodyned, yields Ω ≃ 104 for the mean momentum transfer§ χ ≃ 1 5 χ ror (colored pink) and then resonates in the vacuum-gap between this and and . . For this , the action of a single pulse on a large ther- the mechanical device before being retroreflected. The design is a layered 2 mal state reduces the mechanical variance to σ ≃ 0.2, i.e., less structure, fabricated in the following steps: (B) The base consists of a XM than half the width of the ground state. With a second pulse after high-reflectivity distributed Bragg reflector (DBR) and an etch stop layer de- C 7 ¯ð2Þ ≃ 0 05 posited on a suitable handle substrate. ( ) First, a sacrificial film is deposited mechanical evolution the effective occupation [ ]isneff . . atop the DBR. (D) Next, a microlens pattern is transferred into the sacrificial In order to observe mechanical squeezing, i.e., σ2 < 1∕2, the XM layer through a reflow and reactive ion etching process. The radius of cur- σ2 σ2 þ χ2∕2 vature of this structure is designed to match the phase front of the optical conditional variance must satisfy P j < Pin , where addi- L p L mode to minimize diffraction loss. (E) Following the microlens fabrication tional noise sources that do not affect the mechanical state, e.g., process a high reflectivity dielectric DBR is deposited over the sample surface. 2 σ (F) The structure is then flipped and bonded to a transparent handle using a detector noise, can be subsumed into Pin . It is therefore neces- L G sary to have an accurate experimental calibration of χ to quantify suitable low-absorption adhesive (e.g., spin on glass or UV-curable epoxy). ( ) Ω After mounting, the original growth substrate and etch stop are removed via the mechanical width. (Similarly, must also be accurately chemo-mechanical etching. (H) Finally, the mechanical resonator is patterned known to determine the conditional mean, see Eq. 6). This cali- and subsequently released via selective removal of the underlying sacrificial bration can be performed in the laboratory as follows: For a fixed film. We remark that these integrated structures provide a platform for length cavity and a given pulse intensity, the length of the cavity is “on-chip” hybridization with other quantum systems. adjusted by a known amount (by a calibrated piezo for example) and the proportionality between the homodyne measurement comes and yields a spurious broadening of the tomographic outcomes and the cavity length is determined. The pulses are results for the mode of interest. In practice however, one can then applied to a mechanical resonator and χ is determined with minimize these contributions by engineering mechanical devices knowledge of x0 of the resonator. With χ known Ω can then also with high effective masses for the undesired modes and tailoring be measured by observing the displacement of the mechanical the intensity profile of the optical spot to have good overlap with state after one-quarter of a period. a particular vibrational profile (55). Finally we discuss practical limitations. Firstly, finite mechan- Coupling to a Thermal Bath ical evolution during the interaction decreases the back-action- For our tomography scheme the mechanical quantum state must evading nature of the measurement, which is described in the −1 not be significantly perturbed during the time scale ω .To Appendix. Such evolution is not expected to be a severe limitation M ω ∕κ ≃ estimate the effect of the thermal bath following state prepara- in the proposed implementation considered here as M 10−4 η tion we consider weak and linear coupling to a Markovian bath of . Secondly, the optical measurement efficiency , affected harmonic oscillators. For this model, assuming no initial correla- by optical loss, inefficient detection, andpffiffiffi mode mismatch, yields tions between the mechanics and the bath, the rethermalization a reduced measurement strength χ → ηχ. And thirdly, in many ¯γ γ scales with n M , where M is the mechanical damping rate. It situations coupling to other mechanical vibrational modes is follows that an initially squeezed variance ðχ > 1Þ will increase expected. This coupling contributes to the measurement out- to 1∕2 on a time scale

PHYSICS pffiffiffi Q 1 1 § ¯ τ ¼ 1 − [8] This momentum is comparable to the width of a thermal state, i.e., Ω∕ n < 10 for room ¯ω 2 χ2 : temperature. Thus the mechanical motion remains harmonic. n M

Vanner et al. PNAS ∣ September 27, 2011 ∣ vol. 108 ∣ no. 39 ∣ 16185 Downloaded by guest on September 29, 2021 Thus, for the parameters above and mechanical quality The mechanicalpffiffiffi position and momentumpffiffiffi quadratures are ¼ ω ∕γ ≃ 105 ≲ 1 ¼ð þ †Þ∕ 2 ¼ ð † − Þ∕ 2 Q M M a temperature T K is required for the X M b b and PM i b b , respectively, the observation of squeezing during one mechanical period. cavity (and its input/output) quadratures are similarly defined ~ ~ ∕~ The state purification protocol, as shown in Fig. 3, is affected via a (ain aout). The statistics of the optical amplitude quadrature by rethermalization between the two pulsed measurements. This are unaffected by the interaction, however, the phase quadra- thermal process increases the effective thermal occupation and ture contains the phase dependent upon the mechanical posi- 7 [ ] is modified to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tion. The output phase quadrature emerging from the cavity is pffiffiffiffiffiffi 0 outð Þ¼g0 φð Þ in þ 2κ −κt∫ t 0 κt inð 0Þ − inð Þ ð2Þ 1 1 πn¯ PL t κ Np t X M e −∞dt e PL t PL t , where ¯ ð Þ ≃ 1 þ þ − 1 [9] 3 −κ 0 κ 0 0 neff T 4 2 : φð Þ¼ð2κÞ2 t∫ t t αð Þ 2 χ Qχ t e −∞dt e t describes the accumulation of ð2Þ phase, Xin is the mechanical position prior to the interaction, and For the above system parameters n¯ ðT ¼ 1 KÞ ≃ 0.15. Thus, M eff the last two terms are the input phase noise contributions. Pout is mechanical state purification by measurement is readily attain- pffiffiffi L ¼ 2∫ α ð Þ outð Þ able even at a modest bath temperature. measured via homodyne detection, i.e., PL dt LO t PL t . Moreover, we note that the position measurements of this To maximize the measurement of the mechanical position the lo- α ð Þ¼N φð Þ N scheme can be used to probe open system dynamics and thus cal oscillator envelope is chosen as LO t φ t , where φ in provide an empirical means to explore decoherence and bath ensurespffiffiffi normalization.pffiffiffiffiffiffi The contribution of X M in PL scales with coupling models (56). χ ¼ 2 1 g0 N , which quantifies the mechanical position Nφ κ p Conclusions measurement strength. The mean and variance of PL are given in Eq. 1 for pure Gaussian optical input and together with Ω We have described a scheme to overcome the current challenge 2 Υ 3 of quantum state reconstruction of a mechanical resonator, which and Eq. are used to determine , as given in Eq. . We have provides a means to explore quantum mechanical phenomena on thus arrived, for our physical setting, at an operator which is a macroscopic scale. Our experimental protocol allows for state known from generalized linear measurement theory (see for example ref. 57). Also, we note that Eq. 3 is equivalent to purification, remote preparation of a mechanical squeezed state, Ω χ Υ ¼ i XM h j i XLXM j0i and direct measurements of the mechanical marginals for quan- e PL e , though the nonunitary process of tum state reconstruction, thus providing a complete experimental cavity filling and decay is not explicit. We also remark that the Υ framework. The experimental feasibility has been analyzed and construction of can be readily generalized to include non- we have shown that with the use of optomechanical microcavities Gaussian operations. pffiffiffi χ α ð Þ¼ κ −κjtj this scheme can be readily implemented. The optomechanical The maximum is obtained for the input drive in t e . −2 2 entanglement generated by the pulsed interaction may also be This maximization can be seen by noting that Nφ ¼ ∫ dtφ ðtÞ, N−2 ∝∫ ωðω2 þ κ2Þ−2jα ðωÞj2 a useful resource for quantum information processing. Moreover, which in Fourier space is φ d in . Hence, the framework we have introduced can be built upon for further for such cavity-based measurement schemes, the optimal drive applications in quantum optomechanics and can be generalized has Lorentzian spectrum. This drive, αðtÞ obtained from Eq. 12 to other systems, such as nanoelectromechanics and supercon- and the local oscillator are shown in Fig. 1B. The resulting ducting resonators, or used with dispersive interaction to study optimal measurement strength is given by the motional state of mechanical membranes, trapped ions, or pffiffiffi qffiffiffiffiffiffi g0 particles in a cavity. χ ¼ 2 5 [14] κ Np; Appendix α2 Ω ¼ p3ffiffi g0 and the mean momentum transfer due to is 2 κ Np. Model. The intracavity optomechanical Hamiltonian in the We note that this optimization of the driving field may also rotating frame at the cavity frequency is be applied to cavity-enhanced pulsed measurement of the spin ¼ ℏω † − ℏ † ð þ †Þ [10] H M b b g0a a b b ; of an atomic ensemble (18, 19, 58) or the coordinate of a trapped ion/particle (59–61). Particularly in the latter case, this approach where a (b) is the optical (mechanical) field operator. The cavity will broaden the repertoire of measurement techniques available field accumulates phase in proportion to the mechanical position and may lead to some interesting applications. and is driven by resonant radiation via the equation of motion da pffiffiffiffiffi ¼ ð þ †Þ − κ þ 2κ [11] Finite Mechanical Evolution During Interaction. ig0 b b a a ain; In the model used dt above we have assumed that the mechanical position remains κ where is the cavity decay rate and ain describes the optical constant during the pulsed optomechanical interaction. Including input including drive and vacuum. During a pulsed interaction finite mechanical evolution, the intracavity field dynamics Eq. 13 κ−1 ≪ ω−1 of time scale M the mechanical position is approximately must be determined simultaneously with the mechanical dy- constant. This constancy allows decoupling of Eq. 11 from the namics. In the mechanical rotating frame with the conjugate corresponding mechanical equation of motion and during the quadratures X M ;PM these dynamics are solved to first order in ∕ ≃ † ω ∕κ short interaction we have db dt ig0a a, where we neglect M resulting in the input-output relations: the mechanical harmonic motion,pffiffiffiffiffiffi mechanical damping, and noise out ¼ in þ Ω þ N χ ð Þ¼ α ð Þþ~ ð Þ α ð Þ PM PM 1 X C1; processes. We write ain t Np in t ain t , where in t is ω ω out ¼ in − M ξ Ω − M χN the slowly varying envelope of the drive amplitude with X M X M 1 2X C2; ∫ α2 ¼ 1 κ κ dt in andpffiffiffiffiffiffiNp is the mean photon number per pulse and si- ω ω ω † in in M in M 2 M ¼ αð Þþ~ð Þ ð þ Þ~ P ¼ P þ χðX þ ξ2P Þþχ ξ3Ω þ χ N3X 3;[15] milarly a Np t a t . Neglecting ig0 b b a and approx- L L M κ M κ κ C imating α as real, Eq. 11 becomes the pair of linear equations: N where PL still represents the measurement outcome, 1;2;3 and dα pffiffiffiffiffi ξ ¼ 2κα − κα; [12] 1;2;3 are input drive-dependent dimensionless parameters of or- dt in der unity, the former normalizing the nonorthogonal amplitude ~ qffiffiffiffiffiffi pffiffiffiffiffi da ¼ ð þ †Þα þ 2κ~ − κ~ [13] quadrature temporal modes X C1;2;3. The main effects of the finite ig0 Np b b ain a: dt mechanical evolution can be seen in PL.(i) The mechanical quadrature measured has been rotated, which in terms of the nonro- ω ~ð Þ e ≃ þ M ξ After solving for a t , the outputpffiffiffiffiffi field is then found by using the tating quadratures is X M X M κ 2PM . Such a rotation poses ~ ¼ 2κ~ − ~ input-output relation aout a ain. no principle limitation to our scheme however this must be taken

16186 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1105098108 Vanner et al. Downloaded by guest on September 29, 2021 χ κ ≫ ω into account for the measurement of a particular mechanical of will lie much below this optimum point, however, as M quadrature. (ii) Each pulsed measurement now has a nonzero for the parameters we consider, the broadening due to finite mean proportional to Ω. This mean can be experimentally char- evolution is small and strong squeezing can be achieved. acterized and appropriately subtracted from the outcomes. (iii) PL now includes a term proportional to the optical amplitude ACKNOWLEDGMENTS. The kind hospitality provided by the University of noise. This term decreases the back-action evading quality of Gdańsk (I.P.), the University of Queensland (M.R.V.) and the University of the measurement and has arisen due to mechanical momentum Vienna (G.J.M.) is acknowledged. We thank S. Hofer, N. Kiesel and noise gained from the optical amplitude quadrature evolving into M. Żukowski for useful discussion. We thank the Australian Research Council position noise. The conditional variance of the rotated mechan- (ARC) (Grant FF0776191), United Kingdom Engineering and Physical Sciences ical quadrature including these effects, for large initial occupa- Research Council (EPSRC) and the Royal Society, European Research Council tion, is (ERC) (StG QOM), European Commission (MINOS, Q-ESSENCE), Foundational Questions Institute (FQXi), Fonds zur Förderung der wissenschaftlichen 1 1 ω 2 σ2 ≃ þ ζ2χ2 M [16] Forschung (FWF) (L426, P19570, SFB FoQuS, START), Centre for Quantum ~ 2 ; XM 2 χ κ Engineering and Space-Time Research (QUEST), and an ÖAD/MNiSW ζ program for support. M.R.V. and I.P. are members of the FWF Doctoral where is another drive-dependent parameter of order unity. Programme CoQuS (W 1210). M.R.V. is a recipient of a DOC fellowship of The two competing terms here give rise to a minimum variance the Austrian Academy of Sciences. G.D.C. is a recipient of a Marie Curie ζω ∕κ χ2 ¼ κ∕ðζω Þ of M when M . Experimentally reasonable values Fellowship of the European Commission.

1. Zurek WH (2003) Decoherence, einselection, and the quantum origins of the classical. 31. Clerk AA, Girvin SM, Marquardt F, Schoelkopf RJ (2010) Introduction to quantum Rev Mod Phys 75:715–775. noise, measurement, and amplification. Rev Mod Phys 82:1155–1208. 2. Ghirardi GC, Rimini A, Weber T (1986) Unified dynamics for microscopic and macro- 32. Zhang J, Peng K, Braunstein S (2003) Quantum-state transfer from light to macroscopic scopic systems. Phys Rev D 34:470–491. oscillators. Phys Rev A 68:013808. 3. Diosi L (1989) Models for universal reduction of macroscopic quantum fluctuations. 33. Woolley M, Doherty A, Milburn G, Schwab K (2008) Nanomechanical squeezing with Phys Rev A 40:1165–1174. detection via a microwave cavity. Phys Rev A 78:062303. 4. Pearle P (1989) Combining stochastic dynamical state-vector reduction with sponta- 34. Singh S, Meystre P (2010) Atomic probe Wigner tomography of a nanomechanical neous localization. Phys Rev A 39:2277–2289. system. Phys Rev A 81:041804(R). 5. Penrose R (1996) On gravity’s role in quantum state reduction. Gen Relat Gravit 35. Marek P, Filip R (2010) Noise-resilient quantum interface based on quantum nonde- 28:581–600. molition interactions. Phys Rev A 81:042325. 6. Hackermüller L, Hornberger K, Brezger B, Zeilinger A, Arndt M (2004) Decoherence of 36. Braginsky VB, Vorontsov YI, Khalili FY (1978) Optimal quantum measurements in matter waves by thermal emission of radiation. Nature (London) 427:711–714. detectors of gravitation radiation. JETP Lett+ 27:276–280. 7. Bose S, Jacobs K, Knight PL (1997) Preparation of nonclassical states in cavities with a 37. Braginsky VB, Khalili FY (1992) Quantum Measurement (Cambridge University Press, moving mirror. Phys Rev A 56:4175–4186. United Kingdom). JETP Lett+ 8. Bose S, Jacobs K, Knight PL (1999) Scheme to probe the decoherence of a macroscopic 38. Vyatchanin SP, Matsko AB (1993) Quantum limit on force measurements. – object. Phys Rev A 59:3204–3210. 77:218 221. 9. Marshall W, Simon C, Penrose R, Bouwmeester D (2003) Towards quantum superposi- 39. Mueller-Ebhardt H, et al. (2009) Quantum-state preparation and macroscopic entan- Phys Rev A tions of a mirror. Phys Rev Lett 91:130401. glement in gravitational-wave detectors. 80:043802. 10. Kleckner D, et al. (2008) Creating and verifying a quantum superposition in a micro- 40. Miao H, et al. (2010) Probing macroscopic quantum states with a sub-Heisenberg Phys Rev A optomechanical system. New J Phys 10:095020. accuracy. 81:012114. 11. Kippenberg TJ, Vahala KJ (2008) Cavity optomechanics: back-action at the mesoscale. 41. Romero-Isart O, et al. (2011) Optically levitating dielectrics in the quantum regime: Phys Rev A Science 321:1172–1176. theory and protocols. 83:013803. 12. Marquardt F, Girvin S (2009) Optomechanics. Physics 2:40. 42. Vogel K, Risken H (1989) Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys Rev A 40:2847–2849. 13. Aspelmeyer M, Groeblacher S, Hammerer K, Kiesel N (2010) Quantum optomechanics 43. Dunn TJ, Walmsley IA, Mukamel S (1995) Experimental determination of the quantum- throwing a glance. J Opt Soc Am B 27:A189. mechanical state of a molecular vibrational mode using fluorescence tomography. 14. Smithey DT, Beck M, Raymer MG, Faridani A (1993) Measurement of the Wigner dis- Phys Rev Lett 74:884–887. tribution and the density matrix of a light mode using optical homodyne tomography: 44. Levenson MD, Shelby RM, Reid M, Walls DF (1986) Quantum nondemolition detection application to squeezed states and the vacuum. Phys Rev Lett 70:1244–1247. of optical quadrature amplitudes. Phys Rev Lett 57:24732476. 15. Lvovsky AI, Raymer MG (2009) Continuous-variable optical quantum-state tomogra- 45. La Porta A, Slusher RE, Yurke B (1989) Back-action evading measurements of an optical phy. Rev Mod Phys 81:299–332. field using parametric down conversion. Phys Rev Lett 62:28–31. 16. Leibfried D, et al. (1996) Experimental determination of the motional quantum state 46. Roch J-F, et al. (1997) Quantum nondemolition measurements using cold trapped of a trapped atom. Phys Rev Lett 77:4281–4285. atoms. Phys Rev Lett 78:634–637. 17. Leibfried D, Blatt R, Monroe C, Wineland D (2003) Quantum dynamics of single 47. Hammerer K, Aspelmeyer M, Polzik ES, Zoller P (2009) Establishing Einstein-Poldosky- trapped ions. Rev Mod Phys 75:281–324. Rosen channels between nanomechanics and atomic ensembles. Phys Rev Lett 18. Kuzmich A, Mandel L, Bigelow NP (2000) Generation of spin squeezing via continuous 102:020501. quantum nondemolition measurement. Phys Rev Lett 85:1594–1597. 48. Janszky J, Adam P (1992) Strong squeezing by repeated frequency jumps. Phys Rev A 19. Appel J, et al. (2009) Mesoscopic atomic entanglement for precision measurements 46:6091–6092. beyond the standard quantum limit. Proc Natl Acad Sci USA 106:10960–10965. 49. Walls DF (1983) Squeezed states of light. Nature (London) 306:141–146. 20. Deléglise S, et al. (2008) Reconstruction of non-classical cavity field states with 50. Cole GD, Behymer E, Bond TC, Goddard LL (2008) Short-wavelength MEMS-tunable Nature (London) – snapshots of their decoherence. 455:510 514. VCSELs. Opt Express 16:16093–16103. 21. Armour A, Blencowe M, Schwab K (2002) Entanglement and decoherence of a micro- 51. Riazi A (1981) Resonator equation relating Fresnel number, finesse, and number of Phys Rev Lett mechanical resonator via coupling to a cooper-pair box. 88:148301. beam transits. Appl Optics 20:3469–3470. 22. Rocheleau T, et al. (2010) Preparation and detection of a mechanical resonator near 52. Tayebati P, et al. (1998) Half-symmetric cavity tunable microelectromechanical VCSEL Nature (London) – the ground state of motion. 463:72 75. with single spatial mode. IEEE Photonic Tech L 10:1679–1681. ’ 23. O Connell AD, et al. (2010) Quantum ground state and single-phonon control of a 53. Riemenschneider F, Aziz M, Halbritter H, Sagnes I, Meissner P (2002) Low-cost electro- Nature (London) – mechanical resonator. 464:697 703. thermally tunable optical microcavities based on GaAs. IEEE Photonic Tech L 24. Rabl P, Shnirman A, Zoller P (2004) Generation of squeezed states of nanomechanical 14:1566–1568. Phys Rev B resonators by reservoir engineering. 70:205304. 54. Cole GD, Wilson-Rae I, Werbach K, Vanner MR, Aspelmeyer M (2010) Phonon-tunnel- 25. Tittonen I, et al. (1999) Interferometric measurements of the position of a macroscopic ling dissipation in mechanical resonators. Nature Communications 2:231. body: towards observation of quantum limits. Phys Rev A 59:1038–1044. 55. Pinard M, Hadjar Y, Heidmann A (1999) Effective mass in quantum effects of radiation 26. Briant T, Cohadon PF, Pinard M, Heidmann A (2003) Optical phase-space reconstruction pressure. Eur Phys J D 7:107–116. of mirror motion at the attometer level. Eur Phys J D 22:131–140. 56. Breuer H-P, Petruccione F (2002) The Theory of Open Quantum Systems (Oxford 27. Groeblacher S, et al. (2009) Demonstration of an ultracold micro-optomechanical University Press, United Kingdom). oscillator in a cryogenic cavity. Nature Physics 5:485–488. 57. Caves CM, Milburn GJ (1987) Quantum-mechanical model for continuous position 28. Schliesser A, Arcizet O, Riviere R, Anetsberger G, Kippenberg TJ (2009) Resolved- measurements. Phys Rev A 36:5543–5555. sideband cooling and position measurement of a micromechanical oscillator close 58. Teper I, Vrijsen G, Lee J, Kasevich MA (2008) Backaction noise produced via cavity- to the Heisenberg uncertainty limit. Nature Physics 5:509–514. aided nondemolition measurement of an atomic clock state. Phys Rev A 78:051803(R). 29. Teufel JD, Donner T, Castellanos-Beltran MA, Harlow JW, Lehnert KW (2009) Nanome- 59. Jost JD, et al. (2009) Entangled mechanical oscillators. Nature (London) 459:683–685. chanical motion measured with an imprecision below that at the standard quantum 60. Chang DE, et al. (2010) Cavity opto-mechanics using an optically levitated nanosphere. PHYSICS limit. Nat Nanotechnol 4:820–823. Proc Natl Acad Sci USA 107:1005–1010. 30. Anetsberger G, et al. (2010) Measuring nanomechanical motion with an imprecision 61. Romero-Isart O, Juan ML, Quidant R, Cirac JI (2010) Toward quantum superposition of below the standard quantum limit. Phys Rev A 82:061804(R). living organisms. New J Phys 12:033015.

Vanner et al. PNAS ∣ September 27, 2011 ∣ vol. 108 ∣ no. 39 ∣ 16187 Downloaded by guest on September 29, 2021