Optomechanical generation of a mechanical catlike state by phonon subtraction

We propose a scheme to prepare a macroscopic mechanical oscillator in a catlike state, close to a coherent state superposition. The mechanical oscillator, coupled by radiation-pressure interaction to a field in an optical cavity, is first prepared close to a squeezed vacuum state using a reservoir engineering technique. The system is then probed using a short optical pulse tuned to the lower motional sideband of the cavity resonance, realizing a photon-phonon swap interaction. A photon number measurement of the photons emerging from the cavity then conditions a phonon-subtracted catlike state with a negative Wigner distribution exhibiting separated peaks and multiple interference fringes. We show that this scheme is feasible using state-of-the-art photonic crystal optomechanical system.

Ref. 43 studied nonclassical state generation through various combinations of pulsed position measurements or measurement-induced mechanical squeezing, with singlephonon subtraction or addition, using two optical modes.
Our approach combines methods adopted from continuous quantum control with number-resolved photon counting, both available in a single sideband-resolved optical mode, to generate a phonon-subtracted squeezed arXiv:1909.10624v3[quant-ph] 12 Mar 2020 mechanical state (Fig. 1).In optomechanics, it is possible to realize a beamsplitter-type interaction between a cavity photon (frequency ω c ) and a mechanical excitation (a phonon of frequency Ω m ) within the resolved-sideband regime Ω m κ, where κ is the cavity linewidth [9].This interaction occurs when the cavity is driven with frequency ω l on the lower motional sideband, ω c − ω l = −Ω m , through cavity-enhanced anti-Stokes scattering of drive photons by the oscillator (Fig. 1d), and is the basis of sideband cooling of mechanical motion [9] and coherent photon-phonon swap [60,61].In our scheme, shown in Fig. 1, the mechanical oscillator is first prepared close to a squeezed vacuum state [19][20][21][22][23], and then one or several phonons are swapped with photons which proceed to emerge from the cavity.Light from the cavity is optically filtered on the resonance frequency in order to detect only anti-Stokes scattered photons.Conditioned on subsequent number-resolved photon detection [62], a mechanical phonon-subtracted squeezed state is generated.The state can be subsequently analyzed by mechanical tomography, for example using single-quadrature QND measurements, demonstrated in the optical domain [18], or by state swap followed by homodyne detection [61].
Optomechanical crystals [63] are an especially promising platform for our scheme.They operate in the resolved sideband regime, and cooling to the ground state with strong driving has been demonstrated [11] (note that cooling and squeezing are here combined in a single step [19]).Additionally, they can show extremely long coherence times of more than a second [64], making them attractive for studying nonclassical states of motion.We note that the individual components of our scheme have both been separately implemented.Squeezing was successfully demonstrated in several optomechanical systems [20][21][22][23], and photon counting was applied to optomechanical crystals prepared in the ground state to generate single-phonon and entangled [25-27, 65, 66] mechanical states.
Squeezing of the mechanical state.-Thefirst stage of our protocol is squeezing of the mechanical oscillator by reservoir engineering [19].The optomechanical system in the resolved-sideband regime is driven with two tones tuned to the upper (+) and lower (−) mechanical sidebands, with coupling rates g ± = g 0 √ n± (Fig. 1c), where g 0 is the single-photon optomechanical coupling rate and n± is the mean intracavity photon number due to each drive.When g + = g − , a QND measurement of a single mechanical quadrature X1 = ( b † + b)/ √ 2 is performed [67,68], with b being the phonon annihilation operator.When g − > g + , however, both quadratures X1 and X2 = i( b † − b)/ √ 2 are equally damped by the cavity field while the fluctuations associated with the damping are distributed unequally.This results in a squeezed thermal state characterized by a squeezing parameter r and purity n eff , where tanh r = g + /g − and . The advantage of this scheme is that it allows arbitrarily strong squeezing (limited by drive power), in particular exceeding the 3 dB limit of 0.0 0.5 1.0 1.5 2.0 Squeezing parameter r The cooperativity required to achieve a given purity vs. r.The steady-state in optomechanical dissipative squeezing, in particular the variance of the squeezed quadrature but also the thermal component n eff , results from a trade-off between optical damping and ratio of the drives.The two working points with n eff = 0.02 used in this work are indicated in both panels.parametric driving.While Ref. 19 focused on maximum squeezing (minimum variance in one quadrature) for a given drive power characterized by the cooperativity C = 4g 2 − /κγ, this comes at the expense of increased n eff (although for optimal squeezing, n eff → 0.2 in the limit of high cooperativity [19]).State purity, however, is important for engineering quantum states, and in this work we relax the demand for optimal squeezing in favor of purity.For a given cooling strength C, there is a trade-off between the state purity n eff and the amount of squeezing r related to the imbalance of the drives [19].Figure 2a shows the state purity n eff vs. the squeezing parameter r for different cooperativities, and Fig. 2b shows the required cooperativity vs. the squeezing parameter for different purities.In Fig. 2, we assume that the mechanical oscillator is coupled to a bath with mean thermal occupancy nth = 2.For conciseness, in this work we consider two working points, both with n eff = 0.02: (1) r = 0.5 (4.3 dB squeezing), C 200, compatible with recent high fidelity ground state cooling experiments in optomechanical crystals [11] and (2) r = 1 (8.7 dB), C 1000.Note that mechanical squeezing of 4.7 dB has been reported [23].Accordingly, we will assume in the following that the oscillator is prepared in the desired squeezed state.
Conditional phonon subtraction.-Followingthe squeezing stage, we apply a weak pulse tuned to the lower motional sideband (Fig. 1d), realizing a beamsplitter interaction, Ĥint = g(â †b + b † â), where â is the photon annihilation operator in a frame displaced by the mean cavity field ā, and g = g 0 ā is the coupling rate enhanced by ā [69].The relation between the mechanical mode b(t) and the cavity output field Âout (t) assuming weak coupling g κ is (see the appendix) where cos θ ≡ e −gt is the beamsplitter amplitude "transmission" with g ≡ 2g 2 /κ being the interaction strength and Âin (t) being the optical input in the second "port" of the beamsplitter, which is vacuum noise in the displaced frame.If the initial mechanical state is squeezed vacuum (n eff = 0) Ŝ(r)ρ 0 Ŝ † (r) where ρ0 = |0 0| and Ŝ(r) = e r( b2 − b †2 )/2 is the squeezing operator, the final mechanical state ρ(m) out conditioned on the detection of m photons in the output field can be calculated analytically [58].It is parametrized by m and by cos 2 θ tanh r, with the initial squeezing degraded by the transmission cos 2 θ due to mixing with the optical vacuum noise Âin .Increasing the transmission however also reduces the probability to herald m subtracted phonons.Importantly, as we discuss below, non-negligible values of θ can be easily obtained with pulse durations satisfying κ −1 t pulse (n th Γ m ) −1 , with nth Γ m being the thermal decoherence rate; thus, we can neglect decoherence of the mechanical state during the pulse.The Wigner distribution of the conditioned mechanical state appears as two displaced peaks with an intermediate oscillating region, similar to a cat state.The peak separation increases with m and initial squeezing r [58].
Squeezed Fock and and thermal states have also been treated in the literature [70][71][72][73][74] but yield complicated expressions.Instead we solve numerically for the mechanical output state when the input is a squeezed thermal state, with parameters m, r, n eff , and θ [69].As in Refs.44 and 47, we characterize the quantum nature of the output state using two measures based on the Wigner distribution W (x, p).The macroscopicity [75] assesses W (x, p) through the amplitude and frequency of its interference fringes, with higher values indicating higher nonclassicality.For any state with a given mean excitation number n , the maximum possible value of I is n .In particular, this maximum is attained both for cat states and for phonon-subtracted squeezed vacuum states, but also for squeezed vacuum [76].We also consider the Wigner negativity [77] which is simply the phase-space volume of the negative part of W (x, p). Figure 3a,b shows these measures vs. the squeezing parameter r for different initial mechanical state purities n eff and for detection of m = 1, 2 or 3 photons.As expected, the nonclassicality of the final mechanical state is degraded by initial state impurity, but can be increased by stronger squeezing.Figure 3a,b also shows that for highly impure initial states or very weak squeezing, more subtractions actually decrease nonclassicality.
Figure 3c-e shows the Wigner distributions for r = 1, n eff = 0.02, and m = 1, 2, 3, indicating the achieved macroscopicity I. Figure 3 assumes no additional optical losses and thus gives maximum nonclassicality for the given parameters.Even with optical losses, this maximum can be maintained by reducing the interaction strength at the expense of heralding probability, such that any photon lost will prevent heralding.In an actual experiment, a balance must be struck between constraints on experiment duration and decoherence of the mechanical state due to optical losses.We extend the previous analysis by including a beam splitter in the optical path to account for a finite optical detection efficiency η. Figure 4a,b shows the effect on the heralding probability and macroscopicity, respectively, in the case n eff = 0.02 for r = 0.5, 1 and m = 1, 2, 3.
Experimental realization.-Toestimate the experimental feasibility of our scheme we consider an optomechanical crystal, similar to that used in our recent QND [18] and ground-state cooling [11] experiments, operating in the resolved sideband regime, with mechanical fre- quency Ω m /2π = 5.2 GHz, intrinsic mechanical linewidth Γ m /2π = 100 kHz, and optical cavity at telecommunication wavelengths with linewidth κ/2π = 1 GHz of which κ ex /2π = 800 MHz output coupling and the rest intrinsic dissipation [78].We assume cryogenic operation at temperature T bath = 0.5 K, yielding bath occupation nth = 2 and thermal decoherence time of (n th Γ m ) −1 ≈ 1 µs, much longer than the cavity lifetime and mechanical period, and thus we can safely neglect thermal decoherence in the analysis.Note that optomechanical crystals with decoherence times above 1 s have been demonstrated, albeit at mK temperatures [64].With a typical single-photon optomechanical coupling rate g 0 /2π ∼ 1 MHz, beamsplitter reflections of the order of a few percent as used here can be realized using, e.g., 10 ns pulses of low input power ∼ 10 µW, much weaker than in typical cooling experiments.
We assume total detection efficiency η = 20%, which is feasible assuming almost 100% outcoupling of light from the cavity into an optical fiber, as was demonstrated in Refs.79-81; 80% cavity efficiency κ e /κ; and additional 25% efficiency due to the transmission of the optical filter and other components, and photon-counter detection efficiency.This gives heralding probabilities in the range 10 −4 -10 −7 from Fig. 4a.Squeezing of the initial thermal state occurs at a rate CΓ m ≈ 2π × 20 MHz for C = 200, and thus reducing the initial occupancy nth = 2 to n eff = 0.02 can be done in a timescale of ∼ 100 ns.As noted above, the subtraction pulse duration can be ∼ 10 ns.We assume next a tomography of the final mechanical state to take ∼ 100 ns, given the mechanical period of ∼ 30 ps.Overall we conservatively assume a repetition rate ∼ 10 µs.Thus even with a heralding probability of 10 −7 , we expect 1 event every 100 s, resulting in a feasible experiment duration of several hours.Note that similar photon-counting experiments were done on a time scale of 100 hrs [25].
Conclusion.-We presented a scheme to prepare a macroscopic mechanical oscillator in a cat-like state by combining reservoir-engineering techniques, phononphoton swap operations, and photon counting.A key feature of our scheme is its simplicity.It does not require preparation of nonclassical states of light, and is similar to methods used to generate macroscopic Fock states [24,25], differing essentially in the squeezing step.We have used experimental parameters that are currently available in optomechanical crystals.While in this work we considered phonon subtraction from a squeezed state, phonon addition may equally well be performed, by applying a pulse tuned to the upper motional sideband, providing additional avenues for generating nonclassical mechanical states [43,48].Generation of such states will enable the study of quantum theory in macroscopic objects, and is a first step in using highly coherent and scalable mechanical platforms for continuous variable quantum information applications [54].

FIG. 1 .
FIG.1.Optomechanical scheme for generation of a mechanical catlike state.(a) Illustration of a cavity-optomechanical system.A mechanical oscillator (frequency Ωm, energy dissipation rate Γm, displacement x) forms part of an optical cavity (frequency ωc, energy dissipation rate κ).Cavity photons couple to the oscillator through radiation-pressure interaction, and the output light from the cavity is analyzed.(b) Time-domain picture of the scheme.The oscillator is first prepared in a squeezed state by driving the cavity on the upper and lower motional sidebands.Then, a short pulse on the lower motional sideband drives an anti-Stokes photon-phonon scattering (beamsplitter interaction), subtracting phonons from the mechanical state, which can be analyzed after a variable wait time.(c,d) Frequency-domain pictures of the squeezing (c) and subtraction (d) stages.The Wigner distribution of the mechanical state is also shown.

2 FIG. 2 .
FIG. 2. State purity in optomechanical dissipative squeezing.(a) State purity n eff vs. squeezing parameter r for different cooperativities C. (b)The cooperativity required to achieve a given purity vs. r.The steady-state in optomechanical dissipative squeezing, in particular the variance of the squeezed quadrature but also the thermal component n eff , results from a trade-off between optical damping and ratio of the drives.The two working points with n eff = 0.02 used in this work are indicated in both panels.

1 FIG. 4 .
FIG. 4. Effect of optical losses on generated catlike mechanical state.(a) Heralding probability (successful detection of m photons) vs. total optical detection efficiency η, shown for initial squeezing r = 0.5 (blue) and r = 1 (orange), and for m = 1 (solid), 2 (dashed), and 3 (dash-dotted) subtracted phonons.This probability incorporates the weak optomechanical beam splitter interaction θ, chosen as θ = 0.05 for m = 1 cases and θ = 0.1 for m = 2 cases.(b) The macroscopicity I corresponding to the same curves in panel (a).(c-f) Wigner distributions and macroscopicities of the heralded mechanical state, assuming η = 0.2, for the points indicated in panels (a) and (b).