Generation of mechanical interference fringes by multi-photon counting

For simplicity we first describe our scheme for two-port photon counting before generalizing to the multi-port case. The two-port case describes our experimental results and can be implemented with Mach–Zehnder-type interferometers. Figure 1(a) shows a schematic of our experiment. A weak coherent state is injected into the interferometer and interacts with a mechanical resonator in one of the interferometer arms. The two optical fields inside the interferometer then interfere on a beamsplitter and photon counting is performed on the two output ports. A single-photon click on one of the detectors gives an event-ready signal that the mechanics had interacted with the optical path-entangled number state $(| 10\rangle +| 01\rangle )/\sqrt{2}$. If, instead, both detectors register a single photon then, due to second-order quantum interference, the optical state that interacted with the mechanical resonator within the interferometer must have been the 2-photon NOON-state $(| 20\rangle -| 02\rangle )/\sqrt{2}$. In this case, the mechanical resonator was subject to a superposition of the identity operation and a two-photon radiation–pressure displacement, thus enhancing the size of the superposition by a factor of two compared with the single-photon detection case.

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This operation can be described using the measurement- (or Kraus-) operator approach, which allows us to compute the mechanical state after the interaction and photon-counting measurement as well as the measurement outcome probabilities. Consider a mechanical resonator in the pure initial state $| {\psi }_{\mathrm{in}}{\rangle }_{{\rm{M}}}$ and coherent states $| \alpha {\rangle }_{1}| \alpha {\rangle }_{2}$ in the interferometer arms 1 and 2, respectively (i.e. the state after the initial 50/50 beamsplitter in figure 1(a)). The state immediately after the interaction with mode 1 of the interferometer is then given by ${{\rm{e}}}^{{\rm{i}}\mu {a}_{1}^{\dagger }{a}_{1}{X}_{{\rm{M}}}}| \alpha {\rangle }_{1}| \alpha {\rangle }_{2}| {\psi }_{\mathrm{in}}{\rangle }_{{\rm{M}}}$, where μ is the single-photon optomechanical coupling strength, which quantifies the mechanical momentum transfer per photon in units of the mechanical quantum noise. For a cavity optomechanical system $\mu \propto {g}_{0}/\kappa =4{ \mathcal F }{x}_{0}/\lambda$, and is thus enhanced by the cavity finesse ${ \mathcal F }$. For a single prompt reflection, μ = 4πx₀/λ. Here, ${x}_{0}=\sqrt{{\rm{\hslash }}/m\omega }$ is the mechanical ground state size (m; effective mass, ω; mechanical angular frequency), ${X}_{{\rm{M}}}=x/{x}_{0}$ is the mechanical position operator in units of x₀, and a_1,2 are the annihilation operators for the interferometer arms 1 and 2. The two optical fields then interfere on a beamsplitter and photon counting is performed on the two output ports where m and n photons are detected in modes 1 and 2, respectively. The mechanical state after this interaction and measurement is $| {\psi }_{\mathrm{out}}{\rangle }_{{\rm{M}}}\propto {}_{2}\langle n| {}_{1}\langle m| {{B}{\rm{e}}}^{{\rm{i}}\mu {a}_{1}^{\dagger }{a}_{1}{X}_{{\rm{M}}}}| \alpha {\rangle }_{1}| \alpha {\rangle }_{2}| {\psi }_{\mathrm{in}}{\rangle }_{{\rm{M}}}$, where B is the beamsplitter operator. Assuming a 50:50 beamsplitter, and including a static phase-shift ϕ in mode 2 of the interferometer, we obtain the measurement operator,

$$\begin{eqnarray}&&{\rm{\Upsilon }}=\displaystyle \frac{{{\rm{e}}}^{-| \alpha {| }^{2}}}{\sqrt{m!n!}}\displaystyle \frac{{\alpha }^{m+n}}{{(\sqrt{2})}^{m+n}}{({{\rm{e}}}^{{\rm{i}}\mu {X}_{{\rm{M}}}}+{{\rm{e}}}^{{\rm{i}}\phi })}^{m}{({{\rm{e}}}^{{\rm{i}}\mu {X}_{{\rm{M}}}}-{{\rm{e}}}^{{\rm{i}}\phi })}^{n}.\end{eqnarray} \tag{ 1 }$$

For the click event {m, n} = {1, 0}, this operation corresponds to a superposition of a mechanical displacement ${{\rm{e}}}^{{\rm{i}}\mu {X}_{{\rm{M}}}}$ and the identity operation with a controllable phase e^iϕ. Using this measurement operator, the mechanical state after the operation is ${\rho }_{{\rm{M}}}^{{\rm{out}}}={\rm{\Upsilon }}{\rho }_{{\rm{M}}}^{{\rm{in}}}{{\rm{\Upsilon }}}^{\dagger }/{ \mathcal P }$, where ${\rho }_{{\rm{M}}}^{{\rm{in,out}}}$ are the input and output mechanical density matrices, respectively, and $P={\int }_{-{\rm{\infty }}}^{{\rm{\infty }}}{\rm{d}}{X}_{{\rm{M}}}\,{{\rm{{\Upsilon }}}}^{\dagger }{\rm{{\Upsilon }}} \langle {X}_{{\rm{M}}}| {\rho }_{{\rm{M}}}^{{\rm{in}}}| {X}_{{\rm{M}}} \rangle$ is the probability for obtaining the photon counting outcomes m and n. See the supplementary material for further details of our theoretical model, which is available online at stacks.iop.org/NJP/20/053042/mmedia.

The mechanical position probability distribution of the state after the interaction and measurement is $\langle {X}_{{\rm{M}}}| {\rho }_{{\rm{M}}}^{{\rm{out}}}| {X}_{{\rm{M}}}\rangle \propto {{\rm{\Upsilon }}}^{\dagger }{\rm{\Upsilon }}\langle {X}_{{\rm{M}}}| {\rho }_{{\rm{M}}}^{{\rm{in}}}| {X}_{{\rm{M}}}\rangle$. The function ϒ^†ϒ is oscillatory in ${X}_{{\rm{M}}}$, as obtaining a click in our interferometer gives information periodic in ${X}_{{\rm{M}}}$, and can be interpreted as a filter acting on the initial mechanical position probability distribution. This oscillatory behaviour is intrinsically linked with the cubic nature of the full optomechanical radiation–pressure interaction ${a}_{1}^{\dagger }{a}_{1}{X}_{{\rm{M}}}$ and the nonlinearity of photon counting. This allows our scheme to generate non-Gaussian states of motion, which is not possible in the more commonly considered linearized regime with quadratic interactions and linear measurements. It is also useful to note here that for pure states the position and momentum probability distributions are Fourier transforms of one another. Thus, the bimodal momentum probability distribution generated by the superposition of identity and displacement operations gives rise to the interference fringes in the position quadrature.

In our experiment, figure 1(b), we use a high-stress 1.7 × 1.7 mm Si₃N₄ membrane [37] embedded in a 10 × 10 mm Si-frame. The membrane has a thickness of 50 ± 2.5 nm and, at our state-preparation-field wavelength of 795 nm, has a measured reflectivity of $23.0\pm 0.5 \%$, while the frame has a reflectivity of $20.5\pm 0.2 \%$. The noise power spectrum of the fundamental drum mode at ${\omega }_{{\rm{M}}}/2\pi =105.64\pm 0.02\,\mathrm{kHz}$ is shown in figure 1(c). At room temperature and at atmospheric pressure the mechanical line-width (FWHM) was measured to be $\delta {\omega }_{{\rm{M}}}/2\pi =3.10\pm 0.05\,\mathrm{kHz}$ and the effective mass is on the order of 100 ng, which comprises approximately 10¹⁶ atoms. In order to probe the regime where the optomechanical phase shifts are large, i.e. ${\mu }^{2}\langle {X}_{{\rm{M}}}^{2}\rangle \gtrsim 1$, we use a ring-piezo to drive the membrane motion with random (thermal) fluctuations up to an RMS motion of 200 nm, see supplementary material for details.

The membrane is mounted in a way that allows optical access from both sides, and forms the central part of two folded Mach–Zehnder interferometers (MZI), see figure 1(b). One interferometer is used for mechanical state preparation with photon counting, as illustrated in figure 1(a), while the other is used for mechanical position readout using a balanced detector and a ∼100 μW laser at a wavelength of 632.8 nm. For simplicity, both the state-preparation and the readout laser are operated continuous-wave for this proof-of-concept experiment. In the ideal case, they would both be pulsed, one after the other, however, as the state-preparation field is very weak, i.e. the probability of having a single photon present per mechanical period is low, an effective pulsed operation is implemented a posteriori by the photon counters that have a timing resolution much smaller than a mechanical period. Our setup employs a compact polarization interferometer design that does not require active phase stabilization [38]. The two arms of the MZI in figure 1(a) are represented by orthogonal polarizations and the role of the beamsplitter is achieved by a half-wave plate, which allows for precise control of the splitting ratio. For the interaction with the membrane the two polarizations are separated using a calcite beam displacer and recombined after reflection from the mechanical device with one polarization gaining a mechanical position dependent phase shift.

An APD click—either one of the detectors or a coincidence event within a 7.8 ns window—triggers the balanced detector and recording of a 50 μs long trace at a sampling rate of 100 MS s⁻¹, see figure 1(d). The mechanical quadratures X and P, defined in units of the interferometer readout range (${\lambda }_{{\rm{r}}}/4$), are then extracted from a fit to this time trace. For small mechanical displacements the time trace is almost sinusoidal, but becomes overmodulated as the resonator displacement surpasses ∼100 nm. In addition, for larger mechanical displacements we observe a mechanical-position-dependent amplitude modulation of the interferometer signal. We have taken the first order corrections due to this amplitude modulation into account when fitting the time traces, see supplementary material. For each type of click event we record ∼3000 such time traces to create a phase-space histogram of the mechanical motional state (100 bins of width 0.034 (0.08) in the weak (strong) drive regime, in units of the interferometer readout range). By using a combination of spectral and polarization filters the read-out beam transmitted through the membrane is suppressed below the dark-count level of our single-photon detectors of ∼150 Hz. The effect of these dark counts is a negligible decrease in fringe visibility, as discussed in the supplementary material.

Figure 2 shows the measured mechanical phase-space distributions prepared via one- and two-photon measurements on a piezo-driven initial Gaussian thermal-state (figure 2(a)) with RMS position fluctuations of 198 ± 2 nm. This corresponds to the regime of large optical phase-shifts, i.e. ${\mu }^{2}\langle {X}_{{\rm{M}}}^{2}\rangle \gtrsim 1$. For single-photon detection, low frequency fringes are observed with a π phase-shift between the detection events {m, n} = {0, 1} (figure 2(b)) and {1, 0} (figure 2(c)). Moreover, we observe the start of the second fringe peak in the tails of the Gaussian envelope, which is on the right in figure 2(b) and the left in figure 2(c). In the case of a two-photon detection event {m, n} = {1, 1} (figure 2(d)), the mechanical resonator interacted with an effective two-photon NOON state. Consequently, we observe phase super-resolution in the mechanical interference pattern at twice the fringe-frequency of the single photon cases. We would like to highlight that our scheme maps the fringe pattern onto the state of another bosonic mode and thus uses a time-reversed programme of quantum metrology for quantum-state-engineering applications. Thus, the super-resolution achieved here is a resource for state preparation. In a quantum regime, the fringe pattern observed can then be interpreted as either the quantum interference in the superposition state or as a consequence of the filter of the quantum measurement. This measurement-based technique provides considerable advantages to generate quantum superposition states of mechanical motion and can be employed in other quantum optical systems. Note that all the states prepared (figures 2(b–d)) feature interference fringes in the position distribution, while the momentum quadrature remains close to the initial Gaussian distribution. The conditional mechanical states shown here were prepared with the phase set to ϕ = π/2, which gives a fringe maximum in the centre of the distribution for the {1, 1} event.

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We would like to additionally note here that our method can be utilized to determine the dimensionless optomechanical coupling strength μ by fitting to the fringe pattern observed. This technique requires a well calibrated position axis in units of the mechanical ground state size and can be performed for any mechanical thermal occupation.

Our scheme can also generate non-Gaussian states of motion in the regime of small optomechanical phase shifts, i.e. ${\mu }^{2}\langle {X}_{{\rm{M}}}^{2}\rangle \ll 1$. Indeed, mechanical non-classicality, in the form of Wigner negativity, can be generated independent of the coupling strength, providing a promising route to generate and explore macroscopic mechanical quantum states even for weak single-photon optomechanical coupling. For small μ, applying our scheme to the mechanical ground state for {m, n} = {0, 1} with ϕ = 0 yields $({{\rm{e}}}^{{\rm{i}}\mu {X}_{M}}-1)| 0\rangle \simeq ({\rm{i}}\mu {X}_{M})| 0\rangle ={\rm{i}}\mu {2}^{-1/2}| 1\rangle$. Here, $| 1\rangle$ denotes a single phonon Fock state and the mechanical position operator in terms of the phonon annihilation (b) and creation (b^†) operators is ${X}_{{\rm{M}}}={2}^{-1/2}(b+{b}^{\dagger })$. This state is a result of quantum interference and cannot be described classically [39–41]. Note that in this regime the detection event {m, n} = {1, 1} generates a similar state. The Wigner function of this mechanical single-phonon Fock state and its two conjugate quadrature distributions are shown in figure 3(b). A mechanical Fock state has a rotationally invariant distribution in phase-space and has the quadrature distribution ${\rm{\Pr }}({X}_{{\rm{M}}})=2{\pi }^{-1/2}{X}_{{\rm{M}}}^{2}{{\rm{e}}}^{-{X}_{{\rm{M}}}^{2}}$ for all quadrature angles. For ${\mu }^{2}\langle {X}_{{\rm{M}}}^{2}\rangle \ll 1$ the filter for these detection events is ${{\rm{\Upsilon }}}^{\dagger }{\rm{\Upsilon }}\propto {\mu }^{2}{X}_{{\rm{M}}}^{2}$. Thus, we note that the position probability distribution after this operation has the same functional form as a single-phonon Fock state for any thermal occupation, not just the ground state. More specifically, the position probability distribution is a product of a Gaussian with ${X}_{{\rm{M}}}^{2}$ in both cases, where the initial thermal occupation sets the overall width of the Gaussian envelope of the distribution. This comparison identifies that there is no qualitative transition in the position probability distribution between application of our scheme in the large thermal noise regime and application on the mechanical ground state. Thus, careful calibration and full tomography becomes more important in order to certify mechanical non-classical behaviour. These points will be discussed further in the final section of this manuscript.

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To observe this type of fringe pattern we use the detection event {m, n} = {1, 1} on a mechanical thermal state with RMS position fluctuations of 91 ± 1 nm. The observed mechanical phase-space distribution and quadrature distributions are shown in figure 3(a). The measured position probability distribution is in good agreement with the theoretical prediction. Note that the momentum distribution remains Gaussian due to the thermal nature of the initial state.

The approach presented here has the additional versatility that multiple measurements in time can be used to generate more complex states of mechanical motion. As an example, figure 3(c) shows the state resulting from an experimental sequence of two {m, n} = {1, 0} events, separated by a quarter period of mechanical harmonic evolution. Notably, this state reproduces the qualitative form of the marginal probability distribution of the Fock state not only in position, but also in momentum, thus highlighting the importance of full state reconstruction. When applied to a low temperature state this procedure would generate quantum compass states [42], which are superpositions of four coherent states at cardinal positions in phase space, see figure 3(d). These highly non-classical states have very sharp phase-space features and are of considerable interest for quantum information and quantum sensing applications.

Our scheme has the advantage that it can be easily extended to generate mechanical superposition states with increasing separation size by changing only the optical heralding measurement. This extension requires two coherent states, one of which interacts with the mechanical system, while the other acquires a static phase shift, together with N − 2 vacuum ancilla modes injected into an optical N-port interferometer, see figure 4(a). N-fold single-photon coincidence detection at the output of the N-port then provides an event-ready signal that the mechanical resonator interacted with an optical NOON-state. The mechanical resonator is thus subject to a superposition of a radiation–pressure interaction with 0 or N photons which enhances the separation in the superposition state, see figure 4(b). Here, the high-frequency phase-superresolving fringes of the optical NOON-state are mapped onto the mechanical position probability distribution. In a cavity optomechanics experiment, one would use a number of photon counters N that provide a practical experimental duration given the heralding probability and repetition rate of the experiment. The heralding probability for this operation including projection onto NOON states is ${P}_{N}{(\bar{n})={\textstyle \tfrac{2}{{N}^{N}}}{{\rm{e}}}^{-2| \alpha {| }^{2}}| \alpha {| }^{2N}(1-(-1}^{N})\exp [-{\textstyle \tfrac{1}{4}}{\mu }^{2}(1+2\bar{n}){N}^{2}]\cos (N\phi ))$. See the supplementary material for further details and a mathematical description for this NOON state projection technique.

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Importantly, the states generated by our scheme can exhibit strong negativity of the Wigner quasi-probability distribution independent of the optomechanical coupling strength μ. As shown in figure 4(c), this negativity approaches zero from below asymptotically with increasing $\bar{n}$, and thus the scheme is resilient against finite initial thermal occupation. Furthermore, we would like to highlight that for large μN this negativity scales with ${(1+2\bar{n})}^{-1}$. To the best of our knowledge, this is the most resilient scaling of Wigner negativity with $\bar{n}$ reported thus far. Operations generating Wigner negativity when applied to thermal states have been reported previously, and the scaling found here should be compared to the previous best result of single-quanta addition [39], which was applied in the quantum optics community for single photon addition to an optical thermal state. In that work the Wigner negativity generated goes with ${(1+2\bar{n})}^{-2}$. Though this latter scheme has the advantage of operating in the weak coupling regime, our work demonstrates that more resilient scaling is achievable for large μN. Please refer to the supplementary material for more quantitative details. Moreover, since low amplitude optical coherent states are used, our scheme is robust against optical loss and inefficiency as the single-photon event-ready signals selects the cases where no light was lost from the system.