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

A promising route to novel quantum technologies are hybrid quantum systems, which combine the advantages of several individual quantum systems. We have realized a hybrid atomic-mechanical experiment consisting of a SiN membrane oscillator cryogenically precooled to 500 mK and optically coupled to a cloud of laser cooled Rb atoms. Here, we demonstrate active feedback cooling of the oscillator to a minimum mode occupation of n = 16 corresponding to a mode temperature of T = 200 {\mu}K. Furthermore, we characterize in detail the coupling of the membrane to the atoms by means of sympathetic cooling. By simultaneously applying both cooling methods we demonstrate the possibility of preparing the oscillator near the motional ground state while it is coupled to the atoms. Realistic modifications of our setup will enable the creation of a ground state hybrid quantum system, which opens the door for coherent quantum state transfer, teleportation and entanglement as well as quantum enhanced sensing applications.


Introduction
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][2][3][4][5][6], semiconductor quantum dots [7][8][9], superconducting circuits [10][11][12][13][14] as well as ensembles of cold atoms and single ions [15][16][17][18][19][20][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][23][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 Si 3 N 4 membrane oscillator coupled to laser cooled 87 Rb 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 87 Rb 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 membranein-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 occupationn m = 16 ± 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 quantum hybrid system 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.

Active feedback cooling
Feedback cooling of mechanical motion [30,31] is capable of reaching the motional ground state even in the unresolved sideband regime [32][33][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] Figure 1. Feedback cooling of the membrane oscillator. a) Scheme of the experimental setup for feedback cooling of a membrane oscillator in a hybrid atomicmechanical system, as described in the text. b) Measured zero-span traces of the membrane temperature T mode (t) during feedback cooling (blue) for different feedback gains g v and fits to the data (red) using equation (1). Every data set is the average of ten individual measurements. c) In-loop PSD S y (f ) measured via Zoom-FFT (blue) for different feedback gains g v . The fits to the data (red) based on the model in [29] have g v as the only free fitting parameter. Inset: zoom into the peak area for g v = 0. d) Extracted mode temperatures T final as a function of the fitted g v from the spectra in c) and a fit to these mode temperatures according to equation (2). We obtain a minimum temperature T min = (203 ± 15) µK and a minimum mode occupationn m = 16 ± 1.
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 .
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] T mode (t) = T bath 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 cool = (Γ m g 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 detection noise floor S xn = 7.4 · 10 −33 m 2 /Hz [29]: 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 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 ofn m = 16 ± 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 xn < 4x 2 zp /(n th Γ m ) [31], where x zp = [h/(2mω m )] 1/2 denotes the zero-point motion of the oscillator and n th ≈ k B T bath /(hω 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 xn will even relax this condition.

Sympathetic cooling
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 HamiltonianĤ eff ∼ g N â † atâm +â † mâ 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 atoms N , the coupling lattice parameters and the laser cooling rate Γ a [27] (r m denotes the membrane's reflectivity and F the cavity finesse): 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 hybrid = 4g 2 N /(Γ a Γ m ) = Γ sym /Γ 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 < ∆ 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: 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).
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 ∆ MOT , intensity I MOT and magnetic field gradient B from ∆ MOT = 2.9 Γ D 2 to 6.2 Γ D 2 , I MOT = 50 mW/cm 2 to 4 mW/cm 2 and B = 5 G/cm to 45 G/cm, 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 timingrelated 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.
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 atoms in the lattice volume 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. We obtain N res from the fitted Γ sym and plot it 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 (∆ lat < 0) and for a MOT cooling with a blue detuned lattice (∆ lat > 0). As these two measurements were performed with exactly the same lattice configuration (except for the sign of ∆ 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  figure 3. The x-axis was calibrated by the fit in b). b) Extracted Γ sym and N res for the best cooling curves in a). Solid lines are fitted, ensemble-integrated cooling rates as described in the text the value of ω a , which in this case was also used to calibrate the x-axis of figure 4 a. N res was calculated from Γ sym as described above.
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 · 10 5 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.

Feedback-assisted sympathetic cooling
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 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 sym = −χ −1 sym x ≈ imωΓ sym x [27], while the feedback force also includes the detector noise contribution x n : F fb = −χ −1 fb (x + x n ) [31,37]. In the case of velocityproportional feedback, the feedback transfer function is given by χ −1 fb = −imωΓ m g v . Inserting this into the equation of motion (5) yields: Here, we have introduced the effective susceptibilities for combined cooling χ eff,sf = χ m [Γ m = Γ m (1 + g v + g s )] and for sympathetic cooling with the effective sympathetic cooling gain g s = Γ sym /Γ m . From equations (6) and (7) we deduce the measured in-loop PSD S sf y (ω) and the real out-of-loop PSD S sf x (ω): S sf y (ω) = |y(ω)| 2 = |χ eff,sf | 2 S F th (ω) + |χ eff,s | −2 S xn (ω) .
By integrating equation (8) we obtain the final steady state temperature T sf final of combined feedback and sympathetic cooling: 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 final (g 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 sf final (g v , g s ) given by equation (10). Both steady state temperatures T final and T sf final were obtained by acquiring Zoom-FFT spectra S sf y (ω). 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 sf y (ω) 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 sf final (g v , g 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 a) b) c) Figure 5. Combined sympathetic and feedback cooling. a) Zero-span traces of the membrane temperature T mode (t) during the experimental sequence of combined cooling, as described in the text. b) In-loop PDS S y (f ) (blue) measured via zoom-FFT during the combined cooling slot and spectral fits to the data (red) using equation (9) for different feedback gains g v . Inset: zoom into the peak area for g v = 0. c) Extracted mode temperatures T final as a function of the fitted g v from the spectra in b) and a fit to these temperatures according to equation (10).
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 v > g 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]. 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 ∆ lat .

Summary and experimental prospects
In this paper, we have demonstrated active feedback cooling of the fundamental lowfrequency mode of a Si 3 N 4 membrane oscillator to a final occupation number of n m = 16 ± 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 hybrid = 150 ± 10. By combining both cooling methods, we have shown that we are able to feedback cool the membrane into the quantum regime 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]. Realsitic 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 F ≈ 160 to an optimal value F ≈ 850 [26] in our hybrid system [35] further increases C hybrid by a factor of 30, since C hybrid ∼ 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][48][49].