Combined feedback and sympathetic cooling of a mechanical oscillator coupled to ultracold atoms

Within the growing field of quantum technologies, quantum hybrid systems composed of a mechanical oscillator coupled to an atom-like microscopic quantum object such as spins [1–6], semiconductor quantum dots [7–9], superconducting circuits [10–14] as well as ensembles of cold atoms and single ions [15–21] are of special interest. The general idea behind these hybrid quantum systems is to pave the way for technologically valuable multi-tasking capabilities by combining the individual advantages of the different constituents. These include high scalability and a plethora of different interaction mechanisms on the mechanical side and the realization of qubits with long coherence times and fast high-fidelity quantum gates on the atom side. Technological prospects range from quantum computation and quantum communication to quantum enhanced sensing.

During the last decade, ground state cooling of a variety of mechanical oscillators via cryogenic or so-called resolved sideband cooling methods has become possible [11, 22–24] offering promising prospects for a whole wealth of novel quantum hybrid systems.

We have recently realized a specific hybrid experiment consisting of a cryogenically cooled Si₃N₄ membrane oscillator coupled to laser cooled ⁸⁷Rb atoms via long-range light interactions [25]. Beyond the possibility to work with laser cooled atoms we routinely prepare Bose–Einstein condensates in a magnetic trap, dipole trap or 3D optical lattice. For the purposes presented here we cool the ⁸⁷Rb atoms in a magneto-optical trap (MOT) or optical molasses. The atom–membrane interaction is generated by an optical lattice, which resonantly couples the atomic motion in the lattice wells to the fundamental mode of the membrane [26]. We generate the lattice by retro-reflecting a laser beam off the optomechanical system based on a fiber Fabry–Pérot membrane-in-the-middle (MiM) cavity. This cavity enhances the hybrid coupling as well as the sensitivity of balanced homodyne detection which we use to measure the motion of the membrane. The homodyne signal is also used to prepare the oscillator close to the quantum ground state by active feedback cooling with a dedicated feedback laser beam.

In this paper we report on the first successful combination of active feedback cooling and laser mediated atom–membrane hybrid coupling, which marks an important step towards the realization of a ground state atomic-mechanical hybrid quantum system. Specifically, we demonstrate feedback cooling of the membrane oscillator to a minimum mode occupation ${\bar{n}}_{{\rm{m}}}=16\pm 1$. Furthermore, we demonstrate the robustness of the hybrid coupling mechanism in our experiment with respect to all crucial experimentally controllable parameters. We determine a hybrid cooperativity C_hybrid = 150 ± 10 through sympathetic cooling of the membrane oscillator by laser cooling the atoms [27]. Finally, we show that the realization of a true quantum hybrid system with both constituents in its respective ground state is within reach by coupling a feedback cooled membrane oscillator near the quantum ground state to the atomic cloud. Our measurements represent the first successful combination of sympathetic cooling and feedback cooling [28].

The paper is organized as follows. At first, the feedback cooling setup is presented and the experimental results are discussed. Subsequently, the hybrid coupling scheme and the sympathetic cooling measurements are presented. Finally, the principles and results of combined feedback and sympathetic cooling are discussed, followed by concluding remarks on our results and future prospects of our experiment.

Feedback cooling of mechanical motion [30, 31] is capable of reaching the motional ground state even in the unresolved sideband regime [32–34], where optomechanical self-cooling [22, 23] is extremely inefficient. Strong coupling demands within our hybrid coupling scheme require to operate our fiber cavity [25, 35] far in the unresolved sideband regime, which makes feedback cooling the ideal technique to reach the quantum ground state for low-frequency oscillators. We generate feedback by digitally phase-shifting the homodyne signal [36] and feeding it to a fast fiber EOM, which modulates the intensity of a dedicated laser beam. This beam is then coupled into the MiM system and exerts a feedback force on the membrane via radiation pressure, as shown in figure 1(a). We generate velocity-proportional feedback at gain g_v by digitally adjusting the effective phase delay to Φ_eff = π/2, which provides optimal cooling [31, 37]. By changing the output gain of the digital feedback control we adjust g_v.

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When feedback is applied, the membrane quickly reaches a steady state temperature far below its bath temperature T_bath. As shown in figure 1(b), the temporal behavior of the cooldown can be described well by the model [38]

$$\begin{eqnarray}&&{T}_{\mathrm{mode}}(t)=\displaystyle \frac{{T}_{\mathrm{bath}}}{1+{g}_{{\rm{v}}}}(1+{g}_{{\rm{v}}}\,{{\rm{e}}}^{-{{\rm{\Gamma }}}_{{\rm{m}}}(1+{g}_{{\rm{v}}})t}),\end{eqnarray} \tag{ 1 }$$

where Γ_m = 2π × 24.5 mHz is the natural linewidth of the ground mode ω_m = 2π × 264 kHz of our membrane oscillator [25]. After a few cooldown times ${t}_{\mathrm{cool}}={({{\rm{\Gamma }}}_{{\rm{m}}}{g}_{{\rm{v}}})}^{-1}$ the final temperature T_final is reached. T_final just depends on the feedback gain g_v, as long as the measured signal is well above our shot noise limited detection noise floor ${S}_{{{\rm{x}}}_{{\rm{n}}}}=7.4\times {10}^{-33}\,{{\rm{m}}}^{2}\text{}{\mathrm{Hz}}^{-1}$ [29]:

$$\begin{eqnarray}&&{T}_{\mathrm{final}}=\displaystyle \frac{{T}_{\mathrm{bath}}}{1+{g}_{{\rm{v}}}}\,+\left[\displaystyle \frac{m{\omega }_{{\rm{m}}}^{3}}{4{k}_{{\rm{B}}}{Q}_{{\rm{m}}}}\,\displaystyle \frac{{g}_{{\rm{v}}}^{2}}{1+{g}_{{\rm{v}}}}\right]{S}_{{{\rm{x}}}_{{\rm{n}}}}.\end{eqnarray} \tag{ 2 }$$

Here, m = 76 ng denotes the effective mass of the membrane and Q_m = ω_m/Γ_m its quality factor. At very large feedback gains, we observe strong noise-squashing [39] of our in-loop measured power spectral density (PSD) ${S}_{{\rm{y}}}(f)$, as shown in figure 1(c). We obtain the mode temperature by fitting S_y(f) and subsequently calculating the real out-of-loop PSD S_x(f) and finally integrating these spectra [37]. Analytically, this is equivalent to equation (2) using g_v from the spectral fits [29]. Figure 1(d) shows a fit to these final temperatures yielding a minimum temperature of T_min = (203 ± 15) μK which corresponds to a minimum mode occupation of ${\bar{n}}_{{\rm{m}}}=16\pm 1$. The error is mainly given by the systematic calibration error of our homodyne detection, which we obtained by sweeps of the cryostat temperature [31] and which agrees well with our optomechanical calibration method presented in [25].

Ground state feedback cooling is possible if ${S}_{{{\rm{x}}}_{{\rm{n}}}}\lt 4{x}_{\mathrm{zp}}^{2}/({n}_{\mathrm{th}}{{\rm{\Gamma }}}_{{\rm{m}}})$ [31], where ${x}_{\mathrm{zp}}={[{\hslash }/(2m{\omega }_{{\rm{m}}})]}^{1/2}$ denotes the zero-point motion of the oscillator and ${n}_{\mathrm{th}}\approx {k}_{{\rm{B}}}{T}_{\mathrm{bath}}/({\hslash }{\omega }_{{\rm{m}}})$ its thermal phonon occupation. We expect ground state cooling in our setup for a ten times larger quality factor Q_m and a ten times lower mass of the oscillator, which can easily be met by using new types of mechanical oscillators [40, 41]. Reducing T_bath or improving the detection noise floor ${S}_{{{\rm{x}}}_{{\rm{n}}}}$ will even relax this condition.

The coupling mechanism for sympathetic cooling can be easily understood on the basis of two coupled harmonic oscillators, one being represented by the membrane oscillator ω_m, the other by atoms oscillating in the individual potential wells of a deep optical lattice with frequency ω_a [26]. Atoms are loaded in a 1D optical lattice whose retroreflection mirror is replaced by a membrane oscillator inside an optical cavity. The vibrating motion of the membrane slightly changes the cavity's resonance condition leading to a phase modulation of the back-reflected light. As a consequence, the lattice wells in which the atoms are trapped are periodically displaced, which can excite the atoms to higher levels in the corresponding harmonic oscillator potential of each individual lattice well. In turn, a displacement of the atoms in the optical lattice potential wells leads to a redistribution of photons between the two counterpropagating lattice beams related to the optical dipole force [42] and modulates the light intensity and consequently the corresponding radiation pressure acting on the membrane. In this way, a bidirectional resonant coupling is realized if ω_a = ω_m as described in detail in [26].

The effective coupling Hamiltonian ${\hat{H}}_{\mathrm{eff}}\sim {g}_{{\rm{N}}}({\hat{a}}_{\mathrm{at}}^{\dagger }{\hat{a}}_{{\rm{m}}}+{\hat{a}}_{{\rm{m}}}^{\dagger }{\hat{a}}_{\mathrm{at}})$ is of beam-splitter type and allows to exchange energy between the two systems on the single phonon level at rate g_N. If the atoms are laser cooled, sympathetic cooling of the oscillator can be realized with the potential for ground state cooling even for low-frequency oscillators.

The sympathetic cooling rate Γ_sym depends on many different parameters. Besides the MiM properties, these are mainly the number of coupled atoms N, the coupling lattice parameters and the laser cooling rate Γ_a [27] (r_m denotes the membrane's reflectivity and ${ \mathcal F }$ the cavity finesse):

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{sym}}[N,{\omega }_{{\rm{a}}}]=\displaystyle \frac{{g}_{{\rm{N}}}^{2}{{\rm{\Gamma }}}_{{\rm{a}}}}{{({\omega }_{{\rm{a}}}-{\omega }_{{\rm{m}}})}^{2}+{({{\rm{\Gamma }}}_{{\rm{a}}}/2)}^{2}},{g}_{{\rm{N}}}=| {r}_{{\rm{m}}}{| }^{2}{\omega }_{{\rm{a}}}\sqrt{\displaystyle \frac{{{Nm}}_{{\rm{a}}}{\omega }_{{\rm{a}}}}{m{\omega }_{{\rm{m}}}}}\displaystyle \frac{2{ \mathcal F }}{\pi }.\end{eqnarray} \tag{ 3 }$$

We performed extensive characterization measurements of Γ_sym which will be presented in the following. Through these measurements we also determine the hybrid cooperativity ${C}_{\mathrm{hybrid}}=4{g}_{{\rm{N}}}^{2}/({{\rm{\Gamma }}}_{{\rm{a}}}{{\rm{\Gamma }}}_{{\rm{m}}})={{\rm{\Gamma }}}_{\mathrm{sym}}/{{\rm{\Gamma }}}_{{\rm{m}}}$, which summarizes the system's capabilities to operate in the strong coupling regime.

We start by preparing samples of laser cooled atoms. Subsequently, we quickly ramp up the coupling lattice within 1 ms to a variable final value of ω_a that we calibrate separately using Kapitza–Dirac diffraction of a Bose–Einstein condensate [43]. The optical lattice is near-detuned with −2 GHz < Δ_lat/ 2π < 2 GHz for the measurements presented here. During this sequence we continuously monitor the temperature of the membrane T_mode as a function of time. Exemplary time traces of T_mode are shown in figure 2(b) for MOT cooling and in 2(c) for optical molasses cooling. The sympathetic cooling leads to minimum mode temperatures T_min ≈ 20 mK within approximately 100 ms. We obtain the corresponding Γ_sym from T_min using:

$$\begin{eqnarray}&&{T}_{\min }={T}_{\mathrm{bath}}\,{(1+{{\rm{\Gamma }}}_{\mathrm{sym}}/{{\rm{\Gamma }}}_{{\rm{m}}})}^{-1}.\end{eqnarray} \tag{ 4 }$$

As the time dependence of sympathetic cooling can be described in the same formalism as feedback cooling [28], the cooldown data can be fitted well with equation (1).

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For MOT cooling as shown in figure 2(b), we observe that T_mode reaches a quasi steady state at T_min before the decay of the MOT leads to an increase of T_min on the time scale of several seconds (barely visible in this figure). We have optimized our MOT for maximum cooling performance, which is achieved when quickly ramping the laser cooling parameters to new values before switching on the coupling lattice in the following way. We simultaneously ramp the MOT detuning Δ_MOT, intensity I_MOT and magnetic field gradient $B^{\prime}$ from ${{\rm{\Delta }}}_{\mathrm{MOT}}=2.9\,{{\rm{\Gamma }}}_{{{\rm{D}}}_{2}}$ to $6.2\,{{\rm{\Gamma }}}_{{{\rm{D}}}_{2}},{I}_{\mathrm{MOT}}=50\,\mathrm{mW}\,{\mathrm{cm}}^{-2}$ to $4\,\mathrm{mW}\,{\mathrm{cm}}^{-2}$ and $B^{\prime} =5\,{\rm{G}}\,{\mathrm{cm}}^{-1}$ to 45 G cm⁻¹, correspondingly. We have checked with independent OD measurements, that this produces particularly dense atomic samples.

For molasses cooling we observe that T_min increases exponentially with time as seen in figure 2(c), This is due to atomic diffusion out of the lattice volume, which can be fitted well with an exponentially decreasing atom number N in Γ_sym (see equation (3)) using equation (4). We iteratively optimize intensity and detuning of the optical molasses to find the maximal Γ_sym and observe the expected behavior [27].

All sympathetic cooling measurements presented in the following were performed with these optimized MOT and molasses parameters. It is important to note that once the high density MOT or the optical molasses is generated, the requirements on timing-related lattice parameters like ramping speed or wait time before the ramp up are very relaxed, which makes sympathetic cooling in our system extremely robust.

In order to characterize the coupling mechanism further, we varied ω_a around ω_m by changing the lattice detuning at a given lattice power P_lat. The resulting minimum temperatures T_min were measured as described in figure 2 and the result is shown in figure 3(a). As expected the lowest temperature T_min is achieved for ω_a > ω_m [27]. Moreover the measurements show that the resonance condition of the sympathetic cooling mechanism around ω_m is not very critical and allows for robust operation.

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Unlike the predicted monotonous increase of Γ_sym for larger ω_a, we observe that the sympathetic cooling rate decreases if ω_a is increased beyond a certain point. This decrease in cooling rate is more pronounced for small lattice detunings as seen in figure 3(a). We explain this by increased light scattering for near-detuned optical lattice light.

Another important parameter in the hybrid coupling scheme is the number of coupled atoms N, which was altered systematically by changing the MOT loading time and calculating Γ_sym from the measured T_min according to equation (4). According to equation (3) the sympathetic cooling rate depends linearly on N, which we confirm in figure 3(b). We observe this behavior for molasses cooling and for MOT cooling with a blue detuned lattice. The qualitative value of N was obtained by measuring the optical density along the lattice beam using a mode matched, weak probe beam. The theoretically expected number of coupled atoms N_theo was extracted from the measured Γ_sym according to equation (3) and is plotted as the second y-axis of figure 3(b).

All tested laser cooling methods with experimental parameters individually optimized are summarized in figure 4. We observe the best sympathetic cooling down to T_min ≈ 20 mK with optical molasses cooling using a red detuned lattice (Δ_lat < 0) and for a MOT cooling with a blue detuned lattice (Δ_lat > 0). As these two measurements were performed with exactly the same lattice configuration (except for the sign of Δ_lat), the faster increase of T_min(ω_a) for molasses cooling can not be explained only by the light scattering mentioned above. Probably, the additional parasitic effect is caused by the known instability of the hybrid coupling mechanism [44], which is predicted to be larger for red lattice detunings [45, 46]. This assumption is in agreement with our observation that sympathetic cooling is drastically reduced and turned into heating for MOT cooling with a red detuned lattice. Sympathetic molasses cooling is less efficient for a blue detuned lattice as the repulsive potential of the lattice beam leads to a reduction of the number of atoms N in the coupling volume. For the two best cooling curves in figure 4(a) we extracted the corresponding Γ_sym using equation (4), as shown in figure 4(b). We find that our measured Γ_sym agrees very well with the theoretically expected cooling rate based on an ensemble-integrated model [27]. Furthermore, the fit allows to extract the value of the laser cooling rate Γ_a = (0. 11 ± 0.03)ω_m for molasses cooling and Γ_a = (0. 24 ± 0.07)ω_m for MOT cooling. The fit can also be used to precisely calibrate the value of ω_a, which in this case was also used to calibrate the x-axis of figure 4(a). N_theo was calculated from Γ_sym as described above.

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The maximum sympathetic cooling rate we could achieve was measured independently with molasses cooling using the same parameters as quoted in figure 4(b) at the optimal point ω_a = 1. 25 ω_m. For this, we further increased the atom number in the optical molasses leading to a maximum cooling rate of Γ_sym = 23.3 ± 1.4 Hz. This corresponds to a hybrid cooperativity C_hybrid = 150 ± 10. As C_hybrid ≪ n_th ≈ 2 × 10⁵ in our current setup, we operate far outside the strong coupling regime [26]. However, as discussed in more detail in the conclusion realistic improvements on our optomechanical MiM system will enable us to enter the strong coupling regime.

The realization of a strongly coupled hybrid quantum system in our setup demands for cooling and coupling at the same time. In the following we study the prospects of reaching the strongly coupled regime by experimentally characterizing simultaneous sympathetic and feedback cooling and comparing the results to a classical model for the measured in-loop PSD S^sf_y(ω) of combined sympathetic and feedback cooling, which we use to fit the measured spectra.

Consider the equation of motion for the membrane in frequency-domain

$$\begin{eqnarray}&&x={\chi }_{{\rm{m}}}[{F}_{\mathrm{th}}+{F}_{\mathrm{fb}}+{F}_{\mathrm{sym}}].\end{eqnarray} \tag{ 5 }$$

Here, χ_m is the mechanical susceptibility of the membrane, F_th denotes the random thermal force acting on the membrane due to its coupling to the environment, F_fb is the feedback force and F_sym the sympathetic cooling force given by the hybrid coupling mechanism. Note that all frequency dependencies were omitted for clarity and that all quantum noise contributions are neglected. The sympathetic cooling force can be approximated by ${F}_{\mathrm{sym}}=-{\chi }_{\mathrm{sym}}^{-1}x\approx {\rm{im}}\omega {{\rm{\Gamma }}}_{\mathrm{sym}}x$ [27], while the feedback force also includes the detector noise contribution x_n: ${F}_{\mathrm{fb}}=-{\chi }_{\mathrm{fb}}^{-1}(x+{x}_{{\rm{n}}})$ [31, 37]. In the case of velocity-proportional feedback, the feedback transfer function is given by ${\chi }_{\mathrm{fb}}^{-1}=-{\rm{im}}\omega {{\rm{\Gamma }}}_{{\rm{m}}}{g}_{{\rm{v}}}$. Inserting this into the equation of motion (5) yields:

$$\begin{eqnarray}&&({\chi }_{{\rm{m}}}^{-1}+{\chi }_{\mathrm{fb}}^{-1}+{\chi }_{\mathrm{sym}}^{-1})\,x\ \equiv \ {\chi }_{\mathrm{eff},\mathrm{sf}}^{-1}\,x={F}_{\mathrm{th}}-{\chi }_{\mathrm{fb}}^{-1}\,{x}_{{\rm{n}}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\iff \,({\chi }_{{\rm{m}}}^{-1}+{\chi }_{\mathrm{fb}}^{-1}+{\chi }_{\mathrm{sym}}^{-1})(x+{x}_{{\rm{n}}})\equiv {\chi }_{\mathrm{eff},\mathrm{sf}}^{-1}\,y={F}_{\mathrm{th}}+\mathop{\underbrace{({\chi }_{{\rm{m}}}^{-1}+{\chi }_{\mathrm{sym}}^{-1})}}\limits_{{\chi }_{\mathrm{eff},{\rm{s}}}^{-1}}\,{x}_{{\rm{n}}}.\end{eqnarray} \tag{ 7 }$$

Here, we have introduced the effective susceptibilities for combined cooling ${\chi }_{\mathrm{eff},\mathrm{sf}}={\chi }_{{\rm{m}}}[{{\rm{\Gamma }}}_{{\rm{m}}}^{{\prime} }={{\rm{\Gamma }}}_{{\rm{m}}}(1+{g}_{{\rm{v}}}+{g}_{{\rm{s}}})]$ and for sympathetic cooling ${\chi }_{\mathrm{eff},{\rm{s}}}={\chi }_{{\rm{m}}}[{{\rm{\Gamma }}}_{{\rm{m}}}^{{\prime} }={{\rm{\Gamma }}}_{{\rm{m}}}(1+{g}_{{\rm{s}}})]$ with the effective sympathetic cooling gain ${g}_{{\rm{s}}}={{\rm{\Gamma }}}_{\mathrm{sym}}/{{\rm{\Gamma }}}_{{\rm{m}}}$. From equations (6) and (7) we deduce the measured in-loop PSD ${S}_{{\rm{y}}}^{\mathrm{sf}}(\omega )$ and the real out-of-loop PSD ${S}_{{\rm{x}}}^{\mathrm{sf}}(\omega )$:

$$\begin{eqnarray}&&{S}_{{\rm{x}}}^{\mathrm{sf}}(\omega )=\langle \,| x(\omega ){| }^{2}\rangle =| {\chi }_{\mathrm{eff},\mathrm{sf}}{| }^{2}[{S}_{{{\rm{F}}}_{\mathrm{th}}}(\omega )+| {\chi }_{\mathrm{fb}}{| }^{-2}{S}_{{{\rm{x}}}_{{\rm{n}}}}(\omega )],\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{S}_{{\rm{y}}}^{\mathrm{sf}}(\omega )=\langle \,| y(\omega ){| }^{2}\rangle =| {\chi }_{\mathrm{eff},\mathrm{sf}}{| }^{2}[{S}_{{{\rm{F}}}_{\mathrm{th}}}(\omega )+| {\chi }_{\mathrm{eff},{\rm{s}}}{| }^{-2}{S}_{{{\rm{x}}}_{{\rm{n}}}}(\omega )].\end{eqnarray} \tag{ 9 }$$

By integrating equation (8) we obtain the final steady state temperature ${T}_{\mathrm{final}}^{\mathrm{sf}}$ of combined feedback and sympathetic cooling:

$$\begin{eqnarray}&&{T}_{\mathrm{final}}^{\mathrm{sf}}=\displaystyle \frac{{T}_{\mathrm{bath}}}{(1+{g}_{{\rm{v}}}+{g}_{{\rm{s}}})}\,+\displaystyle \frac{{k}_{{\rm{m}}}{\omega }_{{\rm{m}}}}{4{k}_{{\rm{B}}}Q}\,\displaystyle \frac{{g}_{{\rm{v}}}^{2}}{(1+{g}_{{\rm{v}}}+{g}_{{\rm{s}}})}\,{S}_{{{\rm{x}}}_{{\rm{n}}}}.\end{eqnarray} \tag{ 10 }$$

In order to validate this model we adjusted an experimental sequence of sympathetic MOT cooling and systematically altered the feedback gain g_v, as shown in figure 5(a). At t = 0 we apply feedback and the membrane temperature T_mode reaches a new steady state ${T}_{\mathrm{final}}({g}_{{\rm{v}}})$ given by equation (2). At t = 3 s sympathetic MOT cooling is applied in addition, as described in the previous section. The combined cooling leads to a new steady state temperature T_final^sf(g_v, g_s) given by equation (10). Both steady state temperatures T_final and ${T}_{\mathrm{final}}^{\mathrm{sf}}$ were obtained by acquiring Zoom-FFT spectra ${S}_{{\rm{y}}}^{\mathrm{sf}}(\omega )$. The spectral fits for combined cooling are shown in figure 5(b) using expression (9) with g_v as the only free fitting parameter and g_s = 170, which was extracted from the trace with g_v = 0 in figure 5(a). Similar to the pure feedback cooling spectra in figure 1(c), the combined cooling PSD ${S}_{{\rm{y}}}^{\mathrm{sf}}(\omega )$ can be fitted very well for all feedback gains g_v. The additional noise peak left from the mechanical resonance is related to parasitic light from the coupling lattice beam entering the homodyne detection and it is also visible if no atoms are loaded into the MOT. We obtain ${T}_{\mathrm{final}}^{\mathrm{sf}}({g}_{{\rm{v}}},{g}_{{\rm{s}}})$ using equation (10) and g_v from the spectral fits in figure 5(b), as described above. The result is shown in figure 5(c). In good agreement with the behavior expected from equation (10), we observe that for low feedback gains g_v < g_s the feedback leads to significantly lower temperatures compared to pure sympathetic cooling. For large feedback gains ${g}_{{\rm{v}}}\gt {g}_{{\rm{s}}}$ the temperature of combined cooling is mostly determined by g_v and the effect of additional sympathetic cooling effect becomes very small. Hence, the lowest achievable temperature of combined cooling at the optimal feedback gain differs only by a few percent from the temperature of pure feedback cooling. However, the measurement shows that the hybrid coupling mechanism is compatible with feedback cooling of the membrane. Note that for different experimental conditions, where sympathetic and optimal feedback gain are comparable, the combined cooling rate is significantly enhanced compared to each of the individual cooling methods as proposed in [28]. This could be achieved by increasing C_hybrid through realistic changes of our experimental parameters, as discussed further in the following section. It is worth noting that the minimum achievable membrane temperature T_min is 50% larger than the value for pure feedback cooling. By directly comparing the feedback cooling performance with and without coupling lattice (and without atoms) we confirmed that this effect is caused by laser heating of the membrane through the coupling lattice. This can be significantly reduced by using mechanical oscillators with a lower resonance frequency, which allows for generating the resonance condition ω_a = ω_m with much lower lattice powers P_lat at comparable lattice detunings Δ_lat.

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In this paper, we have demonstrated active feedback cooling of the fundamental low-frequency mode of a Si₃N₄ membrane oscillator to a final occupation number of ${\bar{n}}_{{\rm{m}}}=16\pm 1$. Furthermore, we performed sympathetic cooling measurements of the membrane in our hybrid atomic-mechanical system [25], which we used to determine the maximum hybrid cooperativity ${C}_{\mathrm{hybrid}}=150\pm 10$. By combining both cooling methods, we have shown that we are able to feedback cool the membrane to a thermal occupation of only a few quanta while it is coupled to the atoms. Moreover, our measurements represent the first realization of combined sympathetic and feedback cooling, which was proposed as a novel approach of ground state cooling in hybrid atomic-mechanical experiments [28].

Realistic improvements of our setup will allow creating a quantum hybrid system with the mechanical oscillator in the quantum ground state which is strongly coupled to an atomic ensemble in the quantum many-body ground state. This can be achieved by replacing the simple square membrane with more elaborated types of mechanical oscillators [40, 41], which can be easily performed in our cryogenic optomechanical setup [25]. We expect feedback cooling into the quantum ground state for an oscillator with a ten times larger quality factor Q_m and a ten times lower oscillator mass m. This will also enhance the hybrid cooperativity C_hybrid by a factor of 100, as can be seen from equation (3). Increasing the cavity finesse from the current value ${ \mathcal F }\approx 160$ to an optimal value ${ \mathcal F }\approx 850$ [26] in our hybrid system [35] further increases C_hybrid by a factor of 30, since ${C}_{\mathrm{hybrid}}\sim {{ \mathcal F }}^{2}$. With these improvements our hybrid system would satisfy the strong coupling condition C_hybrid > n_th, offering the possibilities of coherent quantum state transfer, teleportation and entanglement [47–49].

We acknowledge experimental support in the early stage of the experiment from Christina Staarmann, Andreas Bick, Gotthold Fläschner and Ortwin Hellmig. Financial support by the DFG (grant No. BE 4793/2-1, SCHW 780/8-1, SE 717/9-1, WI 1277/29-1) is gratefully acknowledged.