Abstract
Quantum control of a system requires the manipulation of quantum states faster than any decoherence rate. For mesoscopic systems, this has so far only been reached by few cryogenic systems. An important milestone towards quantum control is the socalled strong coupling regime, which in cavity optomechanics corresponds to an optomechanical coupling strength larger than cavity decay rate and mechanical damping. Here, we demonstrate the strong coupling regime at room temperature between a levitated silica particle and a high finesse optical cavity. Normal mode splitting is achieved by employing coherent scattering, instead of directly driving the cavity. The coupling strength achieved here approaches three times the cavity linewidth, crossing deep into the strong coupling regime. Entering the strong coupling regime is an essential step towards quantum control with mesoscopic objects at room temperature.
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Introduction
Laser cooling has revolutionised our understanding of atoms, ions and molecules. Lately, after a decade of experimental and theoretical efforts employing the same techniques^{1,2,3,4,5,6,7,8}, the motional ground state of levitated silica nanoparticles at room temperature has been reported^{9}. While this represents an important milestone towards the creation of mesoscopic quantum objects, coherent quantum control of levitated nanoparticles^{10,11} still remains elusive.
Levitated particles stand out among the plethora of optomechanical systems^{12} due to their detachment, and therefore high degree of isolation from the environment. Their centre of mass, rotational and vibrational degrees of freedom^{13} make them attractive tools for inertial sensing^{14}, rotational dynamics^{15,16,17,18}, free fall experiments^{19}, exploration of dynamic potentials^{20}, and are envisioned for testing macroscopic quantum phenomena at room temperature^{2,10,21,22}.
Recently, the centreofmass motion of a levitated particle has successfully been 3D cooled employing coherent scattering (CS)^{8,23}. Cooling with CS is less sensitive to phase noise heating than actively driving the cavity^{7,24}, because optimal coupling takes place at the intensity node. Lately, this has enabled phonon occupation numbers of less than one^{9}.
For controlled quantum experiments, such as the preparation of nonclassical, squeezed^{25,26} or entangled states^{27,28}, the particle’s motional state needs to be manipulated faster than the absorption of a single phonon from the environment. A valuable but less stringent condition is the socalled strong coupling regime (SCR), where the optomechanical coupling strength g between the mechanical motion of a particle and an external optical cavity exceeds the particle’s mechanical damping Γ_{m} and the cavity linewidth κ (g ≫ Γ_{m}, κ). The SCR presents one of the first stepping stones towards full quantum control and has been demonstrated in opto and electromechanical systems^{29,30,31}, followed by quantumcoherent control^{32}.
Here, we observe normal mode splitting (NMS) in SCR with levitated nanoparticles^{33}, as originally reported in atoms^{34}. In contrast to previous experiments, we employ CS^{8,23,35,36}. Our tabletop experiment offers numerous ways to tune the optomechanical coupling strength at room temperature, a working regime that is otherwise nearly exclusive to plasmonic nanocavities^{37,38}.
Results
Experimental setup for levitation
Our experimental setup is displayed in Fig. 1. A silica nanoparticle (green) of radius R ≈ 90 nm, mass m = 6.4 × 10^{−18} kg and refractive index n_{r} = 1.45 is placed in a cavity (purple) by an optical tweezers trap (yellow) with wavelength \(\lambda\)_{t} = 2π/k_{t} = 1064 nm, power P_{t} ≃ 150 mW, numerical aperture NA = 0.8, and optical axis (z) perpendicular to the cavity axis (y). The trap is linearly polarised along the axis defined as \({\epsilon }_{\theta }={\epsilon }_{{{x}}}\cos \theta\) (see inset in Fig. 1).
The nanoparticle’s eigenfrequencies Ω_{x,y,z} = 2π × (172 kHz, 197 kHz, 56 kHz) are nondegenerate due to tight focusing. The trap is mounted on a nanopositioning stage allowing for precise 3D placement of the particle inside the low loss, high finesse FabryPérot cavity with a cavity linewidth κ ≈ 2π × 10 kHz, cavity finesse F = 5.4 × 10^{5} and free spectral range Δω_{FSR} = πc/L_{c} = 2π × 5.4 GHz. The relative detuning Δ = ω_{t} − ω_{c} between the trap and the cavity resonance is tunable. The intracavity photon number n_{cav} is estimated from the transmitted cavity power P_{out} (CO in Fig. 1), and the particle position displacement is measured by interfering the scattered light with a copropagating reference beam^{39}. In CS, scattering events from the detuned trapping field, locked at Δ, populate the cavity. This contrasts the approach of actively driving the cavity^{3,7,24}. A particle in free space, solely interacting with the trapping light, Raman scatters photons into free space and the energy difference between incident and emitted light equals ±ℏΩ_{m} with m = x, y, z. In this case, photon up and down conversion are equally probable^{40}. The presence of an optical cavity alters the density of states of electromagnetic modes and enhances the CS into the cavity modes through the Purcell effect. If trap photons are red (blue) detuned with respect to the cavity resonance, the cavity enhances photon up (down) conversion and net cooling (heating) takes place.
Coherent scattering theory
In order to estimate the corresponding optomechanical coupling strength in CS, we follow ref. ^{36}. The interaction Hamiltonian for a polarisable particle interacting with an electric field E(R) is given by \({\hat{{\rm{H}}}}_{{\rm{int}}}=\frac{1}{2}\alpha {{\bf{E}}}^{2}({\bf{R}})\) with the particle polarizability \(\alpha =4\pi {\epsilon }_{0}{R}^{3}\frac{{n}_{{\rm{r}}}^{2}1}{{n}_{{\rm{r}}}^{2}+2}\) and vacuum permittivity ϵ_{0}. The total electric field consists of the trap (\({{\bf{E}}}_{{\rm{tr}}}({\bf{R}})\)), cavity (E_{cav}(R)) and free space electromagnetic modes (E_{free}(R)) yielding the interaction Hamiltonian
where E_{cav}(R) and E_{free}(R) are only populated by scattering events from the particle (n_{trap} ≫ n_{cav} with n_{trap} (n_{cav}) being the number of trap (cavity) photons). As can be seen from Eq. (1), the interaction Hamiltonian consists of six terms of which only the two terms proportional to \({{\bf{E}}}_{{\rm{tr}}}({\bf{R}}){{\bf{E}}}_{{\rm{cav}}}({\bf{R}})\) and E_{cav}(R)^{2} are relevant for the following discussion^{36}. The former one gives rise to the optomechanical coupling by CS, and the latter to the coupling achieved by actively driving the cavity. The term \(\propto {{\bf{E}}}_{{\rm{tr}}}^{2}({\bf{R}})\) gives rise to the trapping potential, while the term \(\propto {{\bf{E}}}_{{\rm{tr}}}({\bf{R}}){{\bf{E}}}_{{\rm{free}}}({\bf{R}})\) causes recoil heating^{36,41}, which can be neglected for the moderate vacuum conditions presented here^{7,41}. The remaining two terms can be safely neglected according to ref. ^{36}.
In the following, we use the simplified interaction Hamiltonian given by Eq. (2) where we separate the parts contributing to the optomechanical coupling due to CS \({\hat{{\rm{H}}}}_{{\rm{CS}}}\), active driving \({\hat{{\rm{H}}}}_{{\rm{DR}}}\), and population of the intracavity field \({\hat{{\rm{H}}}}_{{\rm{CAV}}}\) (see “Interaction Hamiltonian and power spectral densities” section).
For the measurements presented here, the trap is xpolarised with θ = 0 (see inset Fig. 1). This simplifies \({\hat{{\rm{H}}}}_{{\rm{CS}}}\) to \({\hat{{\rm{H}}}}_{{\rm{CS}}}=\hslash [{g}_{{{y}}}({\hat{a}}^{\dagger }+\hat{a})({\hat{{b}_{y}}}^{\dagger }+\hat{{b}_{y}})+{g}_{{{z}}}({\hat{a}}^{\dagger }\hat{a})({\hat{{b}_{z}}}^{\dagger }+\hat{{b}_{z}})]\), where \(\hat{a}\) (\({\hat{a}}^{\dagger }\)) is the photon annihilation (creation) operator and \(\hat{b}\) (\({\hat{b}}^{\dagger }\)) is the phonon annihilation (creation) operator. The CS optomechanical coupling strengths g_{y,z} are
with cavity wavevector k_{c} = 2π/\(\lambda\)_{c}, zeropoint fluctuations \({y}_{{\rm{zpf}}},{z}_{{\rm{zpf}}}=\sqrt{\frac{\hslash }{2m{{{\Omega }}}_{{{y}},{{z}}}}}\) and ϕ = 2πy_{0}/\(\lambda\)_{c}, with y_{0} being the particle position along the cavity axis and y_{0} = \(\lambda\)_{c}/4 corresponding to the intensity minimum.
The optical cavity resonance frequency shift caused by a particle located at maximum intensity of the intracavity standing wave is \({G}_{\perp }=\alpha {E}_{0}\sqrt{\frac{{\omega }_{{\rm{c}}}}{2\hslash {\epsilon }_{0}{V}_{{\rm{c}}}}}\) with cavity mode volume \({V}_{{\rm{c}}}=\pi {{\rm{w}}}_{{\rm{c}}}^{2}{L}_{{\rm{c}}}/4\), cavity waist w_{c}, cavity length L_{c}, and ω_{c} = 2πc/\(\lambda\)_{c} The trap electric field is \({E}_{0}=\sqrt{\frac{4{P}_{{\rm{t}}}}{\pi {\epsilon }_{0}c{{{\mathrm{w}}}}_{{{x}}}{{{\mathrm{w}}}}_{{{y}}}}}\) with trap waists w_{x} and w_{y}.
Due to the intracavity standing wave, the optomechanical coupling strength has a sinusoidal dependence on y_{0} with opposite phase for g_{y} and g_{z}. In contrast, g_{x} = 0 if θ = 0.
For clarity, we limit the discussion to coupling along the cavity axis (y), such that Ω_{m} = Ω_{y} and g = g_{y}. Similar results can be obtained for the other directions x, z with the same level of control.
The maximum expected coupling strength from CS is \({g}_{{{y}}}^{\max }={G}_{\perp }{k}_{{\rm{c}}}\,{y}_{{\rm{zpf}}}=2\pi \times 31.7\ {\rm{kHz}}\) for our parameters. However, we displace the particle by δz ≈ 40 μm from the cavity centre for better experimental stability. Hence, our expected optomechanical coupling strength is reduced by ≈30% down to \({g}_{{{y}}}^{{\rm{th}}}=2\pi \times 22.6\ {\rm{kHz}}\), enabling the SCR with g_{y} > κ. Despite the fact that this value is a factor of ≈3 lower than previously reported^{9}, the deep SRC with g > κ remains unaccomplished.
Transition to the SCR
In the weak coupling regime g < κ, the Lorentzianshaped spectra of our mechanical oscillator displays a single peak at its resonance frequency Ω_{m}. When g increases, the energy exchange rate between optical and mechanical mode grows until the SCR is reached at g > κ/4 (ref. ^{33}). In the SCR, the optical and mechanical mode hybridise, which gives rise to two new eigenmodes at shifted eigenfrequencies Ω_{±} (see Eq. (9)). At this point, the energy exchange in between the optical and mechanical mode is faster than the decoherence rate of each individual mode. The hybridised eigenmode frequencies
experience an avoided crossing, the socalled NMS, which reaches a maximum of Ω_{+} − Ω_{−} = 2g_{y} at the optimal detuning Δ = − Ω_{m}. The linewidth of the hybrid modes at this detuning is (κ + Γ_{m})/2. Therefore, Γ_{m} needs to be smaller or comparable to κ to resolve the NMS of 2g_{y}.
As can be seen from Eq. (3), we control g_{y} through various parameters like the trap power P_{t}, the particle position y_{0} and the polarisation angle θ. The optical coupling rate Γ_{opt} depends additionally on the trap detuning Δ and is maximised at Δ = − Ω_{m} to \({{{\Gamma }}}_{{\rm{opt}}}=4{g}_{{{y}}}^{2}/\kappa\) (refs. ^{7}^{[,12}). While P_{t} and Δ only influence the magnitude of the coupling strength, y_{0} and θ change also the nature of the coupling from 1D to potentially 3D^{36}. For simplicity, we focus on varying Δ and y_{0} in the following measurements and keep P_{t}, θ and Γ_{m} = 2π × 0.8 kHz, corresponding to p = 1.4 mBar, fixed (see “Experimental setup” section). The range of Δ is limited due to instabilities in the experiment.
Observation of strong coupling
Figure 2 left panel displays the experimental position power spectral density (PSD) versus Δ for different y_{0}. Throughout the remaining part of the manuscript, we fit our PSD to Eq. (9), if not stated differently. From this fit, we can extract the hybridised modes Ω_{±} that are separated by 2g_{y}. We cover a total distance of δy_{0} ≈ \(\lambda\)_{c}/4 and change the optomechanical coupling strength, and therefore also the NMS, from (a) g_{y}/2π = 22.8 kHz, (b) 15.4 kHz, (c) 4.6 kHz and (d) 0 kHz, exploring the entire range from strong coupling to zero coupling. The right panel shows the fit, which is in good agreement with the data. We observe two eigenmodes Ω_{±} with an exceptional NMS of 2g_{y} ≈ 4.6κ at Δ = − Ω_{m}, corresponding to 20% of the bare mechanical eigenfrequency, once the system enters the SCR at g > κ/4 (ref. ^{33}). For g_{y} = 0, we observe only the mechanical mode with slightly increased frequency Ω_{m} = 2π × 200 kHz due to the additional trapping potential supplied by the cavity field (see Fig. 2d). In Fig. 2a and b, we observe an additional NMS in the ymode, which stems from a second cross polarised optical mode. Note that, throughout all our measurements (see Figs. 2–4), the second NMS is the largest source for discrepancies between experiment and theory (for more details, see Supplementary Information). We also attribute the NMS of the xmode at Ω/Ω_{m} = 0.89 to the second optical mode as observed in Fig. 2d, since the xmode should be decoupled from the first mode (g_{x} = 0 if θ = 0).
Figure 3a–c displays the particle’s position PSD at different Δ while it is located at the intensity minimum, corresponding to the position of maximum coupling g_{y} = 2.3κ, displayed in Fig. 2a. Our theory (yellow) captures the data (purple) well. In Fig. 3a, the optical mode and mechancial mode begin to hybridise into new eigenmodes at Δ = −1.5Ω_{m} which is confirmed by a second peak appearing at Ω ≈ 2π × 300 kHz. The hybridisation becomes stronger as Δ approaches the cavity resonance and the NMS is maximised at Δ ≈ −Ω_{m} as shown in Fig. 3c. The dependence of the new eigenmodes Ω_{±} on Δ is shown in Fig. 3d, displaying clearly the expected avoided crossing of 2g_{y}. The solid line is a fit to Eq. (4). The edges of the shaded area represent the upper and lower limit of the fit, which we obtain by fitting only the upper branch (yellow) or the lower branch (purple), respectively.
As already discussed previously, our experiment allows to change the optomechanical coupling by changing various experimental parameters, which stands in contrast to many other experimental platforms. Figure 4 displays this flexibility to reach the SCR by demonstrating the position dependence of g_{y} at optimal detuning Δ ≈ −Ω_{m} extracted from the data Fig. 2a–d. The experimental and theoretical position PSDs versus y_{0} are depicted in Fig. 4a and b. The mode at Ω/Ω_{m} ≈ 0.89 corresponds to the decoupled xmode. The dashed line highlights the theoretical frequency of the eigenmodes Ω_{±}/Ω_{m} following Eq. (4). In both experiment and theory, we observe the expected sinusoidal behaviour predicted by Eq. (3). Figure 4c depicts ∣g_{y}∣ = (Ω_{+} − Ω_{−})/2 (circles) extracted from Fig. 4a. The dashed line represents the fit to the absolute value of Eq. (3) yielding \({g}_{{{y}}}^{\exp }=2\pi \times (22.8\pm 0.2)\)kHz which coincides well with the theoretical value of \({g}_{{{y}}}^{{\rm{th}}}=2\pi \times 22.6\) kHz. The measured period coincides with the expected period of \(\lambda\)_{c}/4. The shaded area corresponds to 3σ_{std} of the fit.
Discussion
As a figure of merit to assess the potential of our system for quantum applications, we use the quantum cooperativity, which yields here \({C}_{{\rm{CS}}}={(2{g}_{{{y}}}^{\max })}^{2}/(\kappa {{{\Gamma }}}_{{\rm{m}}}({n}_{{\rm{th}}}+1))=8\times 1{0}^{6}\) at a pressure p = 1.4 mbar and promises a value as large as C_{CS} ≈ 36 at p = 3 × 10^{−7} mbar, since Γ_{m} ∝ p. At this low pressure, the photon recoil heating rate Γ_{rec}^{41} equals our mechanical decoherence rate Γ_{m}(n_{th} + 1)), and therefore halves the reachable C_{CS}. The maximum C_{CS} is ultimately limited by Γ_{rec}, regardless if we reduce the pressure even further. Nevertheless, this predicted value of C_{CS} is many orders of magnitude larger than what has been achieved in levitation setups by actively driving the cavity^{7,24} and larger than achieved in ref. ^{9}. More importantly it enables coherent quantum control at g ≫ κ, Γ_{m} ⋅ n_{th} at pressure levels p ≤ 10^{−6} mbar, a pressure regime commonly demonstrated in numerous levitation experiments^{7,9,41}.
Furthermore, our experimental parameters promise the possibility of motional ground state cooling in our system^{9}, which in combination with coherent quantum control enables us to fully enter the quantum regime with levitated systems and to create nonclassical states of motion and superposition states of macroscopic objects in free fall experiments^{10,11} in the future.
Methods
Experimental setup
The experimental setup is displayed in Fig. 5. A silica nanoparticle is loaded at ambient pressure into a long range single beam trap and transferred to a more stable, short range optical tweezers trap^{42} (with wavelength \(\lambda\)_{t} = 1064 nm, power P ≃ 150 mW, focusing lens NA = 0.8) inside a vacuum chamber. Due to the tight focusing, the nanoparticle nondegenerate eigenfrequencies are Ω_{x,y,z} = 2π × (172 kHz, 197 kHz, 56 kHz), respectively. The optical tweezers are mounted on a 3D nanometre resolution piezo system allowing for precise 3D positioning inside a high finesse FabryPerot cavity (with cavity finesse F = 540,000, free spectral range FSR = 2π × 5.4 GHz).
In order to control the detuning Δ = ω_{t} − ω_{c} between the cavity resonance ω_{c} and the trap field ω_{t}, we use a weak cavity field for locking the cavity via the PoundDreverHall technique (PDH) on the TEM01 mode minimising additional heating effects through the photon recoil heating of the cavity lock field. The PDH errorsignal acts on the internal laser piezo and an external AOM (not shown). We separate lock and trap light in frequency space by one free spectral range (FSR) such that the total detuning between lock and trap yields ω_{t} = ω − FSR − Δ. The variable EOM modulation FSR + Δ is provided by a signal generator. The intracavity power can be deduced from the transmitted cooling light observed on a photodiode behind the cavity (CO).
All particle information shown is gained in forward balanced detection interfering the scattered light field and the noninteracting part of the trap beam as shown in Fig. 5. The highly divergent trap light is collected using a lens (NA = 0.8). We use three balanced detectors (FS) to monitor the oscillation of the particle in all three degrees of freedom.
The data time traces are acquired at 1 MHz acquisition rate. Each particle position PSD is obtained by averaging over N = 25 samples of which each one is calculated from individual 40 ms time traces, corresponding to a total measurement time of t = 1 s.
We keep the pressure stable at p = 1.4 mbar. The thermal bath couples as
where Q_{m} = Ω_{m}/Γ_{m} is the mechanical quality factor, \({n}_{{\rm{th}}}=\frac{{k}_{{\mathrm{B}}}T}{\hslash {{{\Omega }}}_{{{m}}}}\) the thermal occupation number, R the particle radius, p the surrounding gas pressure and \({v}_{{\rm{gas}}}=\sqrt{3{k}_{{\mathrm{B}}}T/{m}_{{\rm{gas}}}}\).
In the measurements presented, we cool our particle’s centre of mass motion to T = 235, corresponding to a reduction of the phonon occupation by roughly 20%. The theoretically expected heating rate due to the residual gas accounts fully for the experimentally observed heating rate.
Interaction Hamiltonian and power spectral densities
Following ref. ^{36}, the relevant contributions to the CS interaction Hamiltonian for θ = 0 are given by
The single photon optomechanical coupling strength achieved by actively driving the cavity is \({g}_{{{y}}}^{{\rm{dr}}}=\frac{\alpha {\omega }_{{\rm{c}}}}{2{\epsilon }_{0}{V}_{{\rm{c}}}}{k}_{{\rm{c}}}{y}_{{\rm{zpf}}}\sin (2\phi )=2\pi \times 0.05\;{\rm{Hz}}\sin (2\phi )\). This value is enhanced by the intracavity photon number n_{cav} = 1.6 × 10^{8}, inferred from the transmitted cavity power P_{out}. At optimal conditions, we achieve \({g}_{{{y}}}^{{\rm{rp}}}\sqrt{{n}_{{\rm{c}}}}=2\pi \times 0.6\) kHz. Thus, the optomechanical coupling strength is about 40 times larger for CS, since the photons contributing to the CS interaction are confined in a much smaller volume due to the much smaller trap waist \({{{w}}}_{{\mathrm{t}}}\times {{{w}}}_{{\mathrm{c}}}\ll {{{w}}}_{{\mathrm{c}}}^{2}\).
The mechanical susceptibility χ is given as^{33}
with the effective (optical) damping Γ_{eff} (Γ_{opt}) and the optomechanical spring effect δΩ_{m}. We fit the three mechanical modes Ω_{x,y,z} to Eq. (9) where g_{y}, κ, Γ_{m} and the relative mode amplitudes are chosen as free fit parameters.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
The authors thank J. Gieseler and C. GonzálezBallestero for stimulating discussions. The project acknowledges financial support from the European Research Council through grant QnanoMECA (CoG64790), Fundació Privada Cellex, CERCA Programme/Generalitat de Catalunya, and the Spanish Ministry of Economy and Competitiveness through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV20150522). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement No. 713729.
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A.d.l.R.S. performed the measurements, N.M. evaluated the data and wrote the manuscript, R.Q. supervised the study.
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de los Ríos Sommer, A., Meyer, N. & Quidant, R. Strong optomechanical coupling at room temperature by coherent scattering. Nat Commun 12, 276 (2021). https://doi.org/10.1038/s41467020204192
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DOI: https://doi.org/10.1038/s41467020204192
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