Optomechanical Bell Test

Delft University of Technology

Marinković, Igor; Wallucks, Andreas; Riedinger, Ralf; Hong, Sungkun; Aspelmeyer, Markus; Gröblacher, Simon DOI 10.1103/PhysRevLett.121.220404 Publication date 2018 Document Version Final published version Published in Physical Review Letters

Citation (APA) Marinković, I., Wallucks, A., Riedinger, R., Hong, S., Aspelmeyer, M., & Gröblacher, S. (2018). Optomechanical Bell Test. Physical Review Letters, 121(22), [220404]. https://doi.org/10.1103/PhysRevLett.121.220404 Important note To cite this publication, please use the final published version (if applicable). Please check the document version above.

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Optomechanical Bell Test

† Igor Marinković,1,* Andreas Wallucks,1,* Ralf Riedinger,2 Sungkun Hong,2 Markus Aspelmeyer,2 and Simon Gröblacher1, 1Department of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft University of Technology, 2628CJ Delft, Netherlands 2Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, A-1090 Vienna, Austria (Received 18 June 2018; published 29 November 2018)

Over the past few decades, experimental tests of Bell-type inequalities have been at the forefront of understanding quantum mechanics and its implications. These strong bounds on specific measurements on a physical system originate from some of the most fundamental concepts of classical physics—in particular that properties of an object are well-defined independent of measurements (realism) and only affected by local interactions (locality). The violation of these bounds unambiguously shows that the measured system does not behave classically, void of any assumption on the validity of quantum theory. It has also found applications in quantum technologies for certifying the suitability of devices for generating quantum randomness, distributing secret keys and for quantum computing. Here we report on the violation of a Bell inequality involving a massive, macroscopic mechanical system. We create light-matter entanglement between the vibrational motion of two silicon optomechanical oscillators, each comprising approx. 1010 atoms, and two optical modes. This state allows us to violate a Bell inequality by more than 4 standard deviations, directly confirming the nonclassical behavior of our optomechanical system under the fair sampling assumption.

DOI: 10.1103/PhysRevLett.121.220404

Bell’s theorem [1] predicts that any local realistic theory scale. Recent experiments have demonstrated quantum is at variance with quantum mechanics. It was originally control of such mechanical systems, including mechanical conceived to settle an argument between Einstein [2] and squeezing [24], single-phonon manipulation [25–28],as Bohr [3] on locality in physics, and to investigate the well as entanglement between light and mechanics [29] and axioms of quantum physics. First tests of the Clauser- entanglement between two mechanical modes [30–32]. Horne-Shimony-Holt (CHSH) inequality [4], an experi- However, explaining the observed results in these experimentally testable version of Bell’s original inequality, were ments required assuming the validity of quantum theory at performed with photons from cascaded decays of atoms some level. A Bell test, in contrast, is a genuine test of [5,6] and parametric down-conversion [7–9]. Subsequent nonclassicality without quantum assumptions. experiments reduced the set of assumptions required for the Here we report on the first Bell test using correlations falsification of classical theories, closing, e.g., the locality between light and microfabricated mechanical resonators, [10] and detection loopholes [11], first individually and which constitute massive macroscopic objects, hence recently simultaneously [12–15]. In addition to the funda- verifying nonclassical behavior of our system without mental importance of these experiments, the violation of a relying on the quantum formalism. Bell tests do not require Bell-type inequality has very practical implications—in assumptions about the physical implementation of a quan- particular, it has become the most important benchmark for tum system such as the dimension of the underlying Hilbert thrust-worthily verifying entanglement in various systems space or the fundamental interactions involved in state [16,17], including mesoscopic superconducting circuits preparation and measurement [33]. The violation of a Bell [18], for certifying randomness [19,20], secret keys [21], inequality is hence the most unambiguous demonstration and quantum computing [22]. of entanglement with numerous important implications. While the standard form of quantum theory does not From a fundamental perspective, the robust entanglement impose any limits on the mass or size of a quantum system between flying optical photons and a stored mechanical [23], the potential persistence of quantum effects on a state rules out local hidden variables, which can be used for macroscopic scale seems to contradict the human experi- further tests of quantum mechanics at even larger mass ence of classical physics. Over the past years, quantum scales [34,35]. From an application perspective, the pre- optomechanics has emerged as a new research field, sented measurements also imply that optomechanics is a coupling mechanical oscillators to optical fields. While promising technique to be used for quantum information these systems are very promising for quantum information processing tasks including teleportation, quantum memo- applications due to their complete engineerability, they also ries, and the possibility of quantum communication with hold great potential to test quantum physics on a new mass device-independent security [21].

(a) (b) A blue detuned, ∼40 ns long laser pulse with frequency νb ¼ νo þ νm (νo optical resonance, νm mechanical resonance) generates photon-phonon pairs. The interaction in ˆ pffiffiffiffiffi † ˆ † this case is described by Hb ¼ −ℏg0 nbaˆ b þ H:c:, with the intracavity photon number nb, the optomechanical (c) single photon coupling g0 and the optical (mechanical) creation operators aˆ † (bˆ †). This correlates the number of mechanical and optical excitations in each of the arms of the interferometer as jψi ∝ ½j00i þ ϵj11i þ Oðϵ2Þ ð Þ om om ; 1 where o denotes the optical and m the mechanical mode, while p ¼ ϵ2 is the excitation probability. For small p ≪ 1, FIG. 1. (a) Schematic of the setup: blue detuned drive pulses states with multiple excitations are unlikely to occur, and A B interact with the mechanical resonators (devices and ) can therefore be neglected in the statistical analysis. producing entangled photon-phonon pairs. The light-matter entanglement is in the path basis (A or B), corresponding to Driving the devices simultaneously and postselecting on the device in which the Stokes scattering event took place. The trials with a successful detection of both the Stokes photon generated photons are detected in single-photon detectors giving and the phonon, we approximate the combined state as the measurement results ab1 and ab2. The detection of the 1 ϕ phonons is done by transferring their states to another optical jΨi¼pffiffiffi ðj11i j00i þ i b j00i j11i Þ A B e A B mode by using a red drive after some time Δτ and, subsequently, 2 obtaining the results a 1 and a 2. Note that for technical reasons r r 1 ϕ ¼ pffiffiffi ðjAAi þ ei b jBBi Þ; ð2Þ the photons created by the blue and red drives are detected on the 2 om om same pair of detectors, but with a time delay Δτ ¼ 200 ns. Therefore we have time separation of the two parties of the Bell again neglecting higher order excitations. Here ϕ is the test instead of space separation (as commonly done). BS1=2 b phase difference that the blue drives acquire in the two represent beam splitter 1=2. (b) Scanning electron microscope image of one of the optomechanical devices, represented with a interferometer paths A and B, including the phase shift of the star symbol in (a) and (c), next to the coupling waveguide (top). first beam splitter. Expressing the state in a path basis j i ¼j10i (c) Illustration of our experimental sequence: one party of the Bell A x AB, where x is o for the photonic and m for test measures in which detector path the Stokes photon is found at the phononic subsystem in arms A and B,allowsusto time t ¼ 0, while the other performs the same measurement for identify the Bell state, similarly to polarization entanglement the anti-Stokes photon after a time t ¼ Δτ. We probe their in optical down-conversion experiments. Unlike the two correlations in order to violate the CHSH inequality. Since the mode entangled mechanical state in Ref. [31],thisfour-mode two photons never interacted directly (only through the mechan- entangled optomechanical state allows us to realize a Bell ics), the observed correlations are a direct consequence of the measurement of the type suggested by Horne, Shimony, correlations between the Stokes photons and phonons. and Zeilinger [39] and first demonstrated by Rarity and Tapster [8] involving two-particle interference between four The optomechanical structures used in this work are two different modes. In order to access interferences between the photonic crystal nanobeams on two separate chips. They mechanical modes, we convert the phonons into photons are designed to have an optical resonance in the telecom using a red-detuned laser pulse (duration ∼40 ns, drive ν ¼ ν − ν band that is coupled to a colocalized, high-frequency frequency r o m). This realizes an optomechanical mechanical mode [36]. Each device is placed in one of beam splitterp interactionffiffiffiffi which allows for a state transfer ˆ ¼ −ℏ ˆ † ˆ þ the arms of an actively stabilized fiber interferometer (see (Hr g0 nra b H:c:, with the intracavity photon Ref. [31] and the Supplemental Material [37] for additional number nr). Note that this can also be described as a classical details). The resonators are cryogenically cooled close to mapping process. The optical readout fields in the interfer- their motional ground state inside a dilution refrigerator. ometer arms are again recombined on a beam splitter, after Our entanglement creation and verification protocol con- which the state of Stokes or anti-Stokes field is sists of two optical control pulses that give rise to linearized 1 optomechanical interactions, addressing the Stokes and ðϕ þϕ Þ † † † † jΦi¼ pffiffiffi ½ð1 − ei b r Þðaˆ aˆ − aˆ aˆ Þ anti-Stokes transitions of the system (see Fig. 1). Both 2 2 r1 b1 r2 b2 types of interactions result in scattered photons that are ðϕ þϕ Þ † † † † þ ið1 þ ei b r Þðaˆ aˆ þ aˆ aˆ Þj0000i: ð3Þ resonant with the cavity and can be efficiently filtered from r1 b2 r2 b1 the drive beams before being detected by superconducting Here we express the detected fields in terms of their nanowire single photon detectors (SNSPDs). creation operators with labels b (r) for photons scattered

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At the base temperature of the dilution refrigerator of around 12 mK we obtain the phonon temperature of the mechanical modes by performing sideband asymmetry measurements [40]. The measured thermal occupations ≤ 0 09 for both devices is ninit . . We determine the lifetimes of the phonons in our structures to be τA ¼ 3.3 0.5 μs and τB ¼ 3.6 0.7 μs using a pump-probe type experiment in which we excite the devices and vary the delay to the readout pulse. To reinitialize the devices in their ground states prior to each measurement trial, we repeat the drive sequence every 50 μs, leaving more than 10 times their lifetime for thermalization with the environment. Furthermore, we set the delay between the blue and red detuned pulses to Δτ ¼ 200 ns. The pulse energies for the Bell inequality experiment are chosen such that the FIG. 2. Correlation coefficients for various phase settings. We excitation probability is 0.8% (1%), while the readout efficiency is 3% (4.1%) for device A (device B). These set the blue phase parameter ϕb to 0 (orange) and 0.5π (green), probabilities match the number of optomechanically gen- while we scan the red pulse’s phase setting ϕr over more than 2π. The optimal angles to test the CHSH inequality are shown with erated photons for each device at the beam splitter. different symbols. The associated measured values can be found To characterize the performance of the devices, we first in Table I. perform cross-correlation measurements of the photons scattered from blue and red drives on each individual optomechanical system. With the above-mentioned set- from the blue (red) drive and 1 (2) for the two detectors tings, we obtain normalized cross-correlation values of (cf. Fig. 1). Furthermore ϕ is the phase difference that r ð2Þ ¼ 9 3 0 5 ð2Þ ¼ 11 2 0 6 the red detuned pulse photons acquire in the two arms of gbr;A . . and gbr;B . . [40]. We can use the interferometer. Since experimentally the mechanical this to estimate the expected interferometric visibility for Δν ¼½ ð2Þ − 1 ½ ð2Þ þ 1 frequencies of the devices differ by a small offset m the experiments below as Vxpcd g br = g br (see below), the state acquires an additional phase [41]. As there is a small mismatch in the observed Ω ¼ ΔνmΔτ, where Δτ is the delay between the blue cross-correlations of the two devices, we use the smaller and red pulses. In all data below, however, we keep Δτ value of device A, which results in an expected visibility of Ω ¼ 0 ¼ 81% fixed such that we can treat it as constant and set . around Vxpcd . Typically, Bell experiments are done by rotating the In order to experimentally test a Bell inequality, we then measurement basis in which each particle is detected. drive the two devices simultaneously in a Mach-Zehnder Equivalently, the state itself can be rotated, while keeping interferometer (see Fig. 1 and the Supplemental Material the measurement basis fixed. In our experiment we [37]). We define the correlation coefficients choose the latter option, as this is simpler to implement in our setup. We achieve this by applying a phase shift n11 þ n22 − n12 − n21 Eðϕb; ϕrÞ¼ : ð4Þ with an electro-optical modulator (EOM) in arm A of the n11 þ n22 þ n12 þ n21 interferometer, with which we can vary ϕb and ϕr independently (see Supplemental Material [37]). This Here nij represents the number of detected coincidences allows us to select the relative angles between the scattered from blue (i) and red (j) pulses on the two photonic and phononic states. detectors (i, j ¼ 1, 2), such that, e.g., n21 is the number In our experiment, the optical resonances are at a of trials where the blue drive resulted in an event on wavelength of λ ¼ 1550.4 nm with a relative mismatch detector 2, whereas the consecutive red drive on detector 1. of Δνo ≈ 150 MHz. The mechanical modes have frequen- The visibility V is given as the maximum correlation ν ¼ 5 101 ¼j ðϕ ϕ Þj cies of m . and 5.099 GHz for device A and B, coefficient V E b; r max. We measure the correlation respectively. The bare optomechanical coupling rate g0=2π coefficients for various phase settings for the blue (ϕb) and is 910 kHz for device A and 950 kHz for device B. While red (ϕr) pulses, as shown in Fig. 2. Strong correlations the optical mismatch is much smaller than the linewidth in the detection events by photons scattered from blue and Δνo ≪ κ ∼ 1 GHz such that the devices are sufficiently red pump pulses can be seen, of which the latter are a identical, the mechanical mismatch requires optical com- coherent mapping of the mechanical state of the resonator. pensation. This is realized using the EOM in arm A of the This sweep demonstrates that we are able to independently interferometer to ensure that the scattered photons from shift the phases for the Stokes and anti-Stokes states. each arm interfere with a well-defined phase on the second The visibility V ¼ 80.0 2.5% we obtain from fitting beam splitter (see also Supplemental Material [37]). the data matches the prediction from the individual

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(a) (b) TABLE I. Correlation coefficients for the optimal CHSH angles. The violation of the inequality can be computed according ¼ 2 174þ0.041 to Eq. (5) and results in a S value of S . −0.042 , corresponding to a violation of the classical bound by more than 4 standard deviations.

ϕi ½π ϕj½π ðϕ ϕ Þ Settings i, j b r E b; r 0 561þ0.019 (1,1) 0.0 0.087 . −0.020 0 550þ0.020 (1,2) 0.0 0.587 . −0.022 0 542þ0.018 (2,1) 0.5 0.087 . −0.021 −0 523þ0.021 (2,2) 0.5 0.587 . −0.021 cross-correlation measurements very well. The interference furthermore shows the expected periodicity of 2π. To test possible local hidden-variable descriptions of our correlation measurements we use the CHSH inequality [4], FIG. 3. Visibility as a function of generation rate and state a Bell-type inequality. Using the correlation coefficients transfer probability. We sweep the power of the blue pulse while Eðϕ ; ϕ Þ, it is defined as keeping the red state transfer probability fixed, inducing absorp- b r tion of the optical field in the silicon structure, and see that for ¼j ðϕ1 ϕ1Þþ ðϕ1 ϕ2Þþ ðϕ2 ϕ1Þ − ðϕ2 ϕ2Þj ≤ 2 excitation probabilities up to around 3% the measured visibility S E b; r E b; r E b; r E b; r : exceeds the threshold to violate the CHSH inequality (left). When ð5Þ increasing only the red pump power (right) a similar behavior can be observed, allowing us to increase the state transfer probability A violation of this bound allows us to exclude a potential beyond 14%, while still being able to overcome the classical bound (orange shaded region). The visibilities V ¼ local realistic theory from describing the optomechanical jEðϕ ; ϕ Þj are measured in a single phase setting at the state that we generate in our setup. The maximal violation b r max pffiffiffi optimal angles ϕb ¼ 0 and ϕr ¼ 0.337π. ¼ 2 2 ϕi ¼½0 π 2 SQM V is expected for settings b ; = and j ϕr ¼½−π=4 þ ϕc; π=4 þ ϕc, with i, j ¼ 1,2[42]. Here ϕ ¼ 0 337π thermal excitations. As observed in previous experiments c . is an arbitrary, fixed phase offset that is [31,40], optical pumping of the devices creates a thermal inherent to the setup. Our experimentally achieved visibil- population of the mechanical modes with timescales on the ity exceeds the minimal requirementpffiffiffi for a violation of the order of several hundreds of nanoseconds (see also the ≥ 1 2 ≈ 70 7% classical bound V = . . We proceed to Supplemental Material [37]). While we keep the delay to directly measure the correlation coefficients in the four the readout pulse short (Δτ ¼ 200 ns), we cannot fully ¼ 2 174þ0.041 settings, as indicated in Fig. 2, and obtain S . −0.042 avoid these spurious heating effects. Hence the decrease in (cf. Table I). This corresponds to a violation of the CHSH visibility with increased pulse energy, as seen in Fig. 3(a), inequality by more than 4 standard deviations, clearly can be attributed mostly to this direct absorption heating. confirming the nonclassical character of our state. From the To further test the heating dynamics of our state, we also observed visibility of V ¼ 80.0%, we would expect a sweep the red pulse energies while keeping the excitation slightly stronger violation with S ≈ 2.26. The reduction energy fixed at the value used in the main experiment in our experimentally obtained value for S can be attributed (pb ¼ 0.8% and 1%). As expected, the increased readout to imperfect filtering of drive photons in front of one of the pulse energies lead to substantial heating of the devices SNPSDs, which gives rise to varying amounts of leak [27]. However, even for relatively large optical powers photons at different phase settings (see discussion in the corresponding to ∼14% readout efficiency, the correla- Supplemental Material [37]). tion coefficient is above the threshold for violating a For quantum network applications it is also important to Bell inequality under the fair sampling assumption, see analyze the quality of the detected optomechanical entan- Fig. 3(b). glement with regard to the detection rate. In our measure- Our system is fully engineered and hence we have ments we can achieve this by changing the energies of the complete control over the resonance frequencies and drive beams to alter the optomechanical interaction possibilities to integrate with other systems. While in strengths. An increase in the blue pulse energy is accom- our current structures we intentionally cap the mechanical panied by two mechanisms that decrease the state fidelity. quality factors to keep the measurement time short [43], First, the probability for higher order scattering events recent experiments with very similar devices have observed Oð 2 Þ pA;B is increased. Second, higher pulse energies also lifetimes larger than 1 s [44]. Long lived nonclassical states result in more absorption, degrading the state through of large masses are interesting for fundamental studies of

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