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 state. The essence of quantum state tomography is to make measurements of a specific set of observables over an ensemble of identically prepared realizations. The set is such that the measurement results provide sufficient information for the quantum state to be uniquely determined. One such method is to measure the marginals , where n is the number operator, for all phase-space angles θ, see refs. 14, 15, 42 and e.g., ref. 43.

Our scheme provides a means for precision measurement of the mechanical quadrature marginals, thus allowing the mechanical quantum state to be determined. Specifically, given a mechanical state , harmonic evolution of angle θ = ω_Mt 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 [4]which is a convolution between the mechanical marginal of interest and a kernel that is dependent upon χ and the quantum phase noise of light. The effect of the convolution is to broaden the marginals and to smooth any features present.

Let us consider the specific example of a mechanical resonator in a superposition of two coherent states, i.e., . The X_M marginal of this mechanical Schrödinger-cat state contains oscillations on a scale smaller than the ground state. The convolution scales the amplitude of these oscillations by and thus for small χ they become difficult to resolve in the optical phase noise distribution. Shown in Fig. 2 are marginals of the mechanical state and the optical phase distributions that would be observed according to Eq. 4. Scaling the phase distribution by using the variable P_L/χ provides an approximation to the mechanical marginals, which becomes more accurate with increasing χ and may even show the interference features in a superposition state. Indeed, the limiting case of infinite χ corresponds to a von-Neumann projective measurement of the mechanical position, such that the distribution obtained for P_L/χ becomes identical to the mechanical marginals. However, the mechanical marginals can be recovered even for small measurement strength χ. This recovery is achieved as follows: First, by fixing the length of the cavity the optical phase distribution can be observed without contributions from mechanical position fluctuations. This rigidity allows measurement of the convolution kernel for a particular χ (determined by the properties of the mechanical resonator of interest, cavity geometry, and pulse strength, see Eq. 14). With χ and the kernel known one can then perform deconvolution to determine the mechanical marginals. The performance of such a deconvolution is limited by experimental noise in the calibration of χ and the measurement of Pr(P_L). However, it is expected that these quantities can be accurately measured as quantum noise limited detection is readily achieved.

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Fig. 2.

The scheme presented here provides an experimentally feasible means to obtain direct access to the marginals of a quantum state of a mechanical resonator. Shown are complementary quadrature marginals of the mechanical coherent state superposition , for δ = 1.5 (blue dashed lines with fill, plotted with X_M). The mechanical ground state is shown for comparison in gray dashed lines. The two population components are seen for the quadrature angle θ = π/2 and the quantum interference fringes for θ = 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 marginal (red solid lines, plotted with P_L/χ where χ = 2), see Eq. 4. The convolution kernel can be observed by using a fixed length cavity, shown in the θ = 0 plot (red dashed line with fill, fixed length with X_M = -4), which allows for accurate recovery of the mechanical marginals even for a weak measurement strength χ.

Mechanical Quantum State Engineering and Characterization

We now discuss how the measurement affects the mechanical state. First, we consider Υ acting on a mechanical coherent state . By casting the exponent of Υ in a normal ordered form, one can show that the resulting mechanical state, which is conditioned on measurement outcome P_L, is . Here, is a β-dependent normalization, D is the displacement operator for , and S is the squeezing operator, which yields the position width .

In most experimental situations, the initial mechanical state is in a thermal state , quantified by its average phonon occupation number . The marginals of the resulting state after the action of Υ are [5]where [6]are the mean and variance of the resulting conditional state, respectively. For large initial occupation (provided thermal fluctuations are negligible during the short interaction), the resultant position quadrature of the mechanics has mean and width . Thus, squeezing in the X_M quadrature below the ground state is obtained when χ > 1 and is independent of the initial thermal occupation of the mechanics. We have thus shown how the remarkable behavior of quantum measurement (also used in refs. 18–20, 44–47) can be experimentally applied to a mechanical resonator for quantum state preparation.

There is currently significant interest in the preparation of low entropy states of mechanical resonators as a starting point for quantum experiments, e.g., refs. 22, 23, 27, 28. The two main methods being pursued in optomechanics (11–13) are “passive cooling” which requires the stable operation of a (usually cryogenically compatible) high-finesse cavity, and “active cooling” which requires precision measurement and feedback. Closer in spirit to the latter, our pulsed measurement scheme provides a third method to realize high-purity states of the mechanical resonator. We quantify the state purity after measurement via an effective mechanical thermal occupation , which we define through . When acting on an initial thermal state, the measurement dramatically reduces uncertainty in the X_M quadrature, but leaves the thermal noise in the P_M quadrature unchanged: use of Eq. 6 for yields . The purity can be further improved by a second pulse, which is maximized for pulse separation θ = ω_Mt = π/2, where the initial uncertainty in the momentum becomes the uncertainty in position. Such a sequence of pulses^‡ is represented in Fig. 3, where the resulting state was obtained akin to Eq. 5. The effective occupation of the final state after two pulses is [7]which is also independent of initial occupation. For χ > 1, is well below unity and therefore this scheme can be used as an alternative to “cooling via damping” for mechanical state purification.

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Fig. 3.

Wigner functions of the mechanical state (above) at different times (indicated by arrows) during the experimental protocol (below). From left: Starting with an initial thermal state , (this is chosen to ensure the figure dimensions are reasonable,) a pulsed measurement is made with χ = 1.5 and outcome obtained, which yields an X_M quadrature squeezed state. The mechanical state evolves into a P_M quadrature squeezed state following free harmonic evolution of 1/4 of a mechanical period prior to a second pulse with outcome 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 below unity even for large initial occupation, see Eq. 7 of the main text. Dashed lines indicate the 2σ-widths and the dotted lines show the ground state () for comparative purposes. The displacement Ω is not shown.

Following state preparation, one can use a subsequent “read-out” pulse after an angle of mechanical free evolution θ to perform tomography. During state preparation however, the random measurement outcomes will result in random mechanical means Eq. 6. This randomness can be overcome by recording and utilizing the measurement outcomes. One can achieve unconditional state preparation with use of appropriate displacement prior to the read-out pulse. Or, use postselection to analyze states prepared within a certain window. Alternatively, one may compensate during data analysis by appropriately adjusting each measurement outcome obtained during read-out. We now look at the latter option and consider a Gaussian mechanical state prepared by a prior pulsed measurement. The position distribution has variance σ² to be characterized and has a known mean , which is dependent upon the random measurement outcome. The read-out pulse will then have the distribution . For each read-out pulse, by taking one can obtain the conditional variance for all θ to characterize the noise of the prepared 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, discussing potential limitations and an experimentally feasible parameter regime. To ensure that the interaction time be much less than mechanical time scales the cavity decay rate κ must be much larger than the mechanical frequency. To this end, we consider the use of optical microcavities operating at λ = 1,064 nm, length 4λ and finesse of 7,000, which have an amplitude decay rate κ/2π ≃ 2.5 GHz. Such short cavity devices incorporating a micromechanical element as one of the cavity mirrors have previously 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 reduce diffraction losses for the practical realization of high-finesse cavities. Presently, all realizations use a curved suspended mirror, see e.g., refs. 52, 53. However, in order to allow for enhanced freedom in the construction of the mechanical resonator, particularly 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 bandwidth 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 resonance (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).

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Fig. 4.

Our proposed design and fabrication procedure for high-finesse optomechanical microcavities: Using microcavities provides optomechanical coupling rates many orders of magnitude larger than current millimeter or centimeter length scale implementations of optomechanical Fabry-Pérot cavities and can provide sufficient radiation-pressure interaction to resolve the small scale quantum properties of the mechanical resonator. (A) Cross-sectional view with a quarter of the device removed. Uppermost (colored green) is the mechanical resonator supported by auxiliary beams as was considered in ref. 54. The optical field is injected into the device from below through a transparent handle (colored blue) and the curved rigid input mirror (colored pink) and then resonates in the vacuum-gap between this and the mechanical device before being retroreflected. The design is a layered structure, fabricated in the following steps: (B) The base consists of a high-reflectivity distributed Bragg reflector (DBR) and an etch stop layer deposited on a suitable handle substrate. (C) First, a sacrificial film is deposited atop the DBR. (D) Next, a microlens pattern is transferred into the sacrificial layer through a reflow and reactive ion etching process. The radius of curvature of this structure is designed to match the phase front of the optical mode to minimize diffraction loss. (E) Following the microlens fabrication process a high reflectivity dielectric DBR is deposited over the sample surface. (F) The structure is then flipped and bonded to a transparent handle using a suitable low-absorption adhesive (e.g., spin on glass or UV-curable epoxy). (G) After mounting, the original growth substrate and etch stop are removed via chemo-mechanical etching. (H) Finally, the mechanical resonator is patterned and subsequently released via selective removal of the underlying sacrificial film. We remark that these integrated structures provide a platform for “on-chip” hybridization with other quantum systems.

For a mechanical resonator with eigenfrequency ω_M/2π = 500 kHz and effective mass m = 10 ng, the mechanical ground-state size is and optomechanical coupling proceeds at , where ω_c is the mean cavity frequency and L is the mean cavity length. The primary limitation in measurement strength is the optical intensity that can be homodyned before photodetection begins to saturate. Using pulses of mean photon number N_p = 10⁸, which can be homodyned, yields Ω ≃ 10⁴ for the mean momentum transfer^§ and χ ≃ 1.5. For this χ, the action of a single pulse on a large thermal state reduces the mechanical variance to , i.e., less than half the width of the ground state. With a second pulse after mechanical evolution the effective occupation [7] is .

In order to observe mechanical squeezing, i.e., , the conditional variance must satisfy , where additional noise sources that do not affect the mechanical state, e.g., detector noise, can be subsumed into . It is therefore necessary to have an accurate experimental calibration of χ to quantify the mechanical width. (Similarly, Ω must also be accurately known to determine the conditional mean, see Eq. 6). This calibration can be performed in the laboratory as follows: For a fixed length cavity and a given pulse intensity, the length of the cavity is adjusted by a known amount (by a calibrated piezo for example) and the proportionality between the homodyne measurement outcomes and the cavity length is determined. The pulses are then applied to a mechanical resonator and χ is determined with knowledge of x₀ of the resonator. With χ known Ω can then also be measured by observing the displacement of the mechanical state after one-quarter of a period.

Finally we discuss practical limitations. Firstly, finite mechanical evolution during the interaction decreases the back-action-evading nature of the measurement, which is described in the Appendix. Such evolution is not expected to be a severe limitation in the proposed implementation considered here as . Secondly, the optical measurement efficiency η, affected by optical loss, inefficient detection, and mode mismatch, yields a reduced measurement strength . And thirdly, in many situations coupling to other mechanical vibrational modes is expected. This coupling contributes to the measurement outcomes and yields a spurious broadening of the tomographic results for the mode of interest. In practice however, one can minimize these contributions by engineering mechanical devices with high effective masses for the undesired modes and tailoring the intensity profile of the optical spot to have good overlap with a particular vibrational profile (55).

Coupling to a Thermal Bath

For our tomography scheme the mechanical quantum state must not be significantly perturbed during the time scale . To estimate the effect of the thermal bath following state preparation we consider weak and linear coupling to a Markovian bath of harmonic oscillators. For this model, assuming no initial correlations between the mechanics and the bath, the rethermalization scales with , where γ_M is the mechanical damping rate. It follows that an initially squeezed variance (χ > 1) will increase to 1/2 on a time scale [8]Thus, for the parameters above and mechanical quality Q = ω_M/γ_M ≃ 10⁵ a temperature T ≲ 1 K is required for the observation of squeezing during one mechanical period.

The state purification protocol, as shown in Fig. 3, is affected by rethermalization between the two pulsed measurements. This thermal process increases the effective thermal occupation and [7] is modified to [9]For the above system parameters . Thus, mechanical state purification by measurement is readily attainable even at a modest bath temperature.

Moreover, we note that the position measurements of this scheme can be used to probe open system dynamics and thus provide an empirical means to explore decoherence and bath coupling models (56).

Appendix