## Abstract

Time-resolved Raman spectroscopy techniques offer various ways to study the dynamics of molecular vibrations in liquids or gases and optical phonons in crystals. While these techniques give access to the coherence time of the vibrational modes, they are not able to reveal the fragile quantum correlations that are spontaneously created between light and vibration during the Raman interaction. Here, we present a scheme leveraging universal properties of spontaneous Raman scattering to demonstrate Bell correlations between light and a collective molecular vibration. We measure the decay of these hybrid photon-phonon Bell correlations with sub-picosecond time resolution and find that they survive over several hundred oscillations at ambient conditions. Our method offers a universal approach to generate entanglement between light and molecular vibrations. Moreover, our results pave the way for the study of quantum correlations in more complex solid-state and molecular systems in their natural state.

## INTRODUCTION

In the hierarchy of nonclassical states, the Bell correlated states represent an extreme. When two parties share such a state, information can be encoded exclusively in the quantum correlations of the random outcomes of measurements between them (*1*, *2*). The strength of such correlations is quantified by Bell inequalities, whose violation demarcates Bell correlated states from less entangled ones (*3*).

Experimental realizations of Bell correlated states—whether between polarization states of light (*4*, *5*), individual atomic systems (*6*–*8*), in atomic ensembles (*9*–*11*), superconducting ciruits (*12*, *13*), or solid-state spins (*14*, *15*)—call for isolated systems that strongly interact with a well-characterized probe. Even mesoscopic acoustic resonators have been engineered to exhibit Bell correlations (*16*) thanks to long coherence times (achieved at milli-kelvin temperatures) and strong interaction with light (by integration with an optical microcavity).

Recent experiments have shown that high-frequency vibrations of bulk crystals (*17*–*22*) or molecular ensembles (*23*–*25*) can mediate nonclassical intensity correlations between inelastically scattered photons under ambient conditions (i.e., at room temperature and atmospheric pressure). In the pioneering work of Lee *et al*. (*17*), two phonon modes in spatially separated bulk diamonds had been entangled with each other by performing coincidence measurements and postselection on the Raman-scattered photons. Recently, leveraging a new two-tone pump-probe method (*22*), it became possible to follow the birth and death of an individual quantum of vibrational energy (i.e., Fock state) excited in a single spatiotemporal mode of vibration in a bulk crystal (*26*).

These experiments did not necessitate specially engineered subjects; they reveal fundamental quantum properties of naturally occurring materials. Together, these developments raise new questions: Are the correlations spontaneously created between light and vibration during Raman scattering strong enough to violate Bell inequalities? How is the vibrational coherence time reflected in the dynamics of the hybrid light-vibration quantum correlations?

In this study, we demonstrate Bell correlations arising from the Raman interaction between light and mechanical vibration at ambient conditions and use them to resolve the decoherence of the vibrational mode mediating these correlations. While this proof-of-principle experiment is realized on a vibrational mode in a bulk diamond crystal, the effect that is revealed should be universally observable in Raman-active molecules and solids. Our scheme for producing hybrid photon-phonon entanglement is agnostic to sample details and is passively phase-stabilized, while our two-color pump-probe technique can address Raman-active vibrations irrespective of any polarization selection rules—all of which differ from earlier work (*17*). Our results demonstrate the strongest form of quantum correlations and is thus a powerful generalization of techniques deployed in atomic physics to study the decoherence of entanglement (*27*).

## MATERIALS AND METHODS

The inelastic scattering of light off an internal vibrational mode—vibrational Raman scattering—is analogous to the radiation-pressure interaction between light and a mechanically compliant mirror (*28*). Specifically, the Raman interaction consists of two processes. In the Stokes process, a quantum of vibrational energy ℏΩ* _{v}* (a phonon) is created together with a quantum of electromagnetic energy ℏω

*(a Stokes photon); in the anti-Stokes process, a phonon is annihilated while an anti-Stokes photon is created at angular frequency ω*

_{s}*. Energy conservation demands that ω*

_{a}_{s, a}± Ω

*= ω*

_{v}_{in}, respectively, where ω

_{in}is the frequency of the incoming photon.

In our experiment, a diamond sample—grown along the [100] direction by a high-pressure, high-temperature method, about 300 μm thick and polished on both faces along the (100) crystallographic plane—is excited with femtosecond pulses from a mode-locked laser through a pair of high numerical aperture objectives (NA = 0.8) (the effective length over which the Raman interaction takes place is of the order of 2 μm). Since the pulses are shorter than the coherence time of the Raman-active vibration, but longer than its oscillation period, there exists perfect time correlation between the generation (annihilation) of a vibrational excitation and the production of a Stokes (anti-Stokes) photon. In the following, we show how to leverage this time correlation to generate time-bin entanglement (*29*) between two effective photonic qubits that reveal properties of the mediating phonon mode and quantify the strength of the quantum correlations using the Clauser-Horne-Shimony-Holt (CHSH) form of the Bell inequality (*4*).

The scheme (Fig. 1) starts when a pair of laser pulses, labeled “write” and “read,” impinge on the sample. Each is a classical wave packet with ∼10^{8} photons per pulse. Their central frequencies are independently tunable, which allows spectral filtering of the Stokes field generated by the write pulse and the anti-Stokes field generated by the read pulse. The delay between them, Δ*t*, is adjustable to probe the decoherence of the vibrational mode. Each pulse passes through an unbalanced Mach-Zehnder interferometer and is split in two temporal modes separated by Δ*T*_{bin} ≫ Δ*t*, which we label the “early” and “late” time bins. Δ*T*_{bin} ≃ 3 ns is chosen to be much longer than the expected vibrational coherence time, which ensures that there can be no quantum-coherent interaction between the two time bins mediated by the vibrational mode.

At room temperature, the thermal state of the vibrational mode (*26*) (at 39.9 THz) has as a mean occupancy of 1.5 × 10^{−3}. The initial state of the vibration in the two time bins is therefore very well approximated by the ground state ∣0* _{v}*⟩ ≡ ∣0

_{v, E}⟩ ⊗ ∣0

_{v, L}⟩, where the subscripts

*E*and

*L*stand for the early and late time bins, respectively. The Stokes (

*s*) and anti-Stokes (

*a*) fields are also in the vacuum state at the start of the experiment, denoted by ∣0

*⟩ ≡ ∣0*

_{s}_{s, E}⟩ ⊗ ∣0

_{s, L}⟩ and ∣0

*⟩ ≡ ∣0*

_{a}_{a, E}⟩ ⊗ ∣0

_{a, L}⟩.

The interaction of the write pulse (split into the two time bins) with the vibrational mode generates a two-mode squeezed state of the Stokes and vibrational fields (*26*) in each time bin. A read pulse delayed by Δ*t* (also split into the two time bins) maps the vibrational state in the respective time bins onto its anti-Stokes sideband.

Since we perform the experiment in the regime of very low Stokes scattering probability and postselect the outcomes where exactly one Stokes photon and one anti-Stokes photon were detected (see the Supplementary Materials for the treatment of triple coincidence), our scheme can be described in a subspace of the full Hilbert space that contains one vibrational excitation only, shared by the early and late time bin. We therefore introduce the shortened notation *E _{s}*⟩, ∣

*L*⟩} ⊗ {∣

_{s}*E*⟩, ∣

_{v}*L*⟩} = {∣

_{v}*E*,

_{s}*E*⟩, ∣

_{v}*E*,

_{s}*L*⟩, ∣

_{v}*L*,

_{s}*E*⟩, ∣

_{v}*L*,

_{s}*L*⟩}. In this sense, we can speak of vibrational and photonic qubits encoded in the time bin basis.

_{v}Within each time bin, the read pulse implements (with a small probability ∼0.1%) the map ∣*E _{s}*,

*E*⟩ → ∣

_{v}*E*,

_{s}*E*⟩ and ∣

_{a}*L*,

_{s}*L*⟩ → ∣

_{v}*L*,

_{s}*L*⟩, where we have defined

_{a}By passing the Stokes and anti-Stokes photons through an unbalanced interferometer identical to the one used on the excitation path (Fig. 1B and the Supplementary Materials), “which-time” information is erased. Moreover, the use of polarizing beam splitters in the interferometer maps the time bin–encoded Stokes and anti-Stokes photonic qubits onto polarization-encoded qubits after they are temporally overlapped, ∣*E _{s}*,

*E*⟩ → ∣

_{a}*V*,

_{s}*V*⟩ and ∣

_{a}*L*,

_{s}*L*⟩ → ∣

_{a}*H*,

_{s}*H*⟩, where

_{a}*H*and

*V*refer to two orthogonal polarizations of the same temporal mode. We thus prepare the heralded Bell correlated state

To prove Bell correlations mediated by the room temperature macroscopic vibration, we send the Stokes and anti-Stokes signals to two independent measurement apparatus labeled Alice and Bob, respectively, who perform local rotations of the Stokes and anti-Stokes states before making a projective measurement in the two-dimensional basis {∣*V _{s}*⟩, ∣

*H*⟩} and {∣

_{s}*V*⟩, ∣

_{a}*H*⟩}, respectively. Each party will obtain one of two outcomes, which we label “+” or “−”. The number of coincident events where Alice obtains the outcome

_{a}*x*∈ { + , − } and Bob obtains the outcome

*y*∈ { + , − } is denoted

*n*. We then define the normalized correlation parameter

_{xy}*E*

_{θ, φ}= 1 while perfectly anticorrelated events yield

*E*

_{θ, φ}= − 1. The CHSH parameter (

*2*)

*S*∣ > 2. In particular, for our scenario, where we target the Bell correlated state (Eq. 1), a maximal violation is expected for

## RESULTS

### Observation of Bell correlations

Figure 2 shows the CHSH parameter (Eq. 3) measured for a varying write-read delay. Our data demonstrate a clear violation of the Bell inequality (whose classical bound is marked as the white region) that persists for more than 5 ps, about 50 times longer that the write and read pulse duration. While this time scale is consistent with the phonon lifetime in diamond, the dynamics of Bell correlations in fact strongly depends on experimental noise and nonidealities, as explained in sections S4 to S6. At a time delay of 0.66 ps, for which there is vanishing temporal overlap of the write and read pulses within the sample and correlations are only mediated by the vibration, we measure *S* = 2.360 ± 0.025. This confirms Bell correlations mediated by the vibration that acts as a room temperature quantum memory (*30*–*34*).

A detailed analysis of the event statistics (see section S6) enables us to make a more precise claim concerning the violation of the Bell inequality (*35*), without assuming that our data are independent and identically distributed. From this analysis, we can claim with a confidence level of 1 × 10^{−7} to 6 × 10^{−7} that the postselected Stokes–anti-Stokes state features Bell correlations with a minimum value of the CHSH parameter *S*_{min} = 2.23.

Note that we rely on the fair sampling assumption (*36*) since the overall detection efficiency in our experiment is not high enough to test a Bell inequality without postselection of events where at least one detector clicks on each side (Alice and Bob). However, it can be shown (*37*) that when all detectors are equally efficient—a condition well approximated in our experiment—the postselected data are faithful to that from an ideal experiment where lossless devices measure a state obtained by quantum filtering the actual Stokes–anti-Stokes state. By reporting a CHSH value higher than 2, we show that this filtered state is Bell correlated.

To gain further insight into the nature of the Bell correlated state prepared in the experiment, and the reasons why the quantum bound (

Figure 3B shows two-photon interference for various settings of Bob’s measurement angle for two fixed values of Alice’s measurement angle, θ = 0, π/2, and a fixed write-read delay of 0.66 ps. The interference is consistent with a model (see section S3) where the Stokes interaction creates a two-mode light-vibration squeezed state and that anti-Stokes scattering implements a beam splitter interaction (*26*).

The curve for the setting θ = 0 (Fig. 3B, blue trace) reveals how accurately we can prepare and distinguish the two states ∣*E _{s}*,

*E*⟩ and ∣

_{a}*L*,

_{s}*L*⟩. At a given delay, the visibility has an upper limit related to the strength of Stokes–anti-Stokes photon number correlations,

_{a}*27*), where

*22*) (see the Supplementary Materials). The value extracted from the fit is

*V*

_{θ = 0}= 93 ± 1%, in agreement with the independently measured value of

The coincidence curve for *V*_{θ = π/2} = 76% from the fit to the experimental data, which is reproduced by the model for a standard deviation σ = 0.31 rad (equivalent to a ±0.18 fs timing uncertainty maintained over ∼4 min). Ultimately, we are able to predict all measured quantities from independently characterized parameters, namely, the Raman scattering probability, the overall Raman signal detection efficiency, and the dark count rate of the detectors (see section S4).

### Decoherence dynamics of the phonon mode

From the temporal behavior of the CHSH parameter, we can extract the rate of pure dephasing of the vibrational mode mediating the Bell correlations. In the absence of pure dephasing, the CHSH parameter decays with the collective vibrational mode. Pure dephasing, in contrast, scrambles the phase ϕ of the superposition in state (Eq. 1). We model it as a random walk of the phase at the characteristic time scale γ^{−1}, so that the standard deviation of the phase ϕ increases with the write-read delay (in addition to technical fluctuations) as *38*).

## DISCUSSION

We have produced Bell correlations between two photons through their interaction with a common Raman-active phonon at room temperature and probed their decay with sub-picosecond resolution. Our data show that Bell correlations are preserved for more than 200 oscillation periods at room temperature, evidencing a mechanical coherence time at par with the state of the art for microfabricated resonators under high vaccum (*39*). Optical phonons in diamond indeed exhibit a room temperature “Q-frequency product” of ∼4 × 10^{16} Hz, making them attractive resonators for ultrafast quantum technologies.

Such highly coherent vibrational modes, together with the toolset of time-resolved single photon Raman spectroscopy that we have demonstrated here, should allow one to entangle two vibrational qubits via entanglement swapping (*40*) or to perform optomechanical conversion between photonic qubits at different frequencies (*41*), among other possible applications. Much longer vibrational coherence times could be achieved with ensembles of molecules that are decoupled from the phonon bath by surface engineering (*42*) or optical trapping and cooling (*43*). Besides, molecules in the gas phase exhibit more complex mechanical degrees of freedom, including rotational and rovibrational modes (*44*), with increased coherence time and rich opportunities for quantum information processing (*45*). In the future, our scheme could be applied to individual molecules free of heterogeneous broadening using the enhancement of light-vibration coupling offered by electronic resonances (*46*), plasmonic nanocavities (*47*), or optical microcavities (*48*).

In addition to being a benchmark for the robust generation of optomechanical Bell correlations at room temperature, our work suggests a new class of techniques able to probe the role of phonon-mediated entanglement in quantum technologies (*49*), chemistry (*50*), or even biology (*51*).

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/51/eabb0260/DC1

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**We thank T. J. Kippenberg and J.-P. Brantut for valuable discussion and A. Geraci, N. Lusardi, and F. Garzetti for providing the custom FPGA-based correlation electronics.

**Funding:**This work was funded by the Swiss National Science Foundation (SNSF) (project no. PP00P2-170684) and the European Research Council’s (ERC) Horizon 2020 research and innovation programme (grant agreement no. 820196). N.S. acknowledges funding by the Swiss National Science Foundation (SNSF), through grant PP00P2-179109, by the Army Research Laboratory Center for Distributed Quantum Information via the project SciNet and from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 820445 and project name Quantum Internet Alliance.

**Author contributions:**S.T.V., N.S., and C.G. designed the experiment; S.T.V. performed the measurements and analyzed the data; S.T.V., V.S., and N.S. developed theoretical models; S.T.V., V.S., and C.G. wrote the manuscript, and all authors discussed the results and the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. The data related to this paper are available via a Zenodo repository at https://zenodo.org/record/4084706#.X8lJahNKhQI. Additional data related to this paper may be requested from the authors.

- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).