Ground-State Cooling of Levitated Magnets in Low-Frequency Traps
GGround-State Cooling of Levitated Magnets in Low-Frequency Traps
Kirill Streltsov, Julen S. Pedernales, and Martin B. Plenio Institut f¨ur Theoretische Physik und IQST, Albert-Einstein-Allee 11, Universit¨at Ulm, D-89081 Ulm, Germany (Dated: February 8, 2021)We present a ground-state cooling scheme for the mechanical degrees of freedom of mesoscopicmagnetic particles levitated in low-frequency traps. Our method makes use of a binary sensor andsuitably shaped pulses to perform weak, adaptive measurements on the position of the magnet. Thisallows us to precisely determine the position and momentum of the particle, transforming the initialhigh-entropy thermal state into a pure coherent state. The energy is then extracted by shifting thetrap center. By delegating the task of energy extraction to a coherent displacement operation weovercome the limitations associated with cooling schemes that rely on the dissipation of a two-levelsystem coupled to the oscillator. We numerically benchmark our protocol in realistic experimentalconditions, including heating rates and imperfect readout fidelities, showing that it is well suitedfor magneto-gravitational traps operating at cryogenic temperatures. Our results pave the way forground-state cooling of micron-scale particles.
Introduction. –
Cooling the center-of-mass (COM) mo-tion of a massive oscillator down to its minimum energyis a convenient way of transferring it from a classical ther-mal state into a pure quantum state for applications inquantum technologies. The ability to operate massiveparticles in the quantum regime is predicted to bringexceptional enhancements in sensitivity for metrologicalapplications [1–5], and to have relevant implications forour understanding of Nature [6–12]. For this enterprise,levitated optomechanics stands out as a promising plat-form, where nano- and micro-scale solids suspended invacuum behave as massive mechanical oscillators. Re-cently, ground-state (GS) cooling of an optically levitatedmesoscopic particle has been demonstrated [13, 14], con-stituting a remarkable milestone for the field. However,optical setups suffer from significant dissipation rates dueto recoil and absorption of photons from the trappingfields, which makes progress beyond GS cooling, such asthe generation of non-classical states, daunting.In the light of these challenges, levitation with pas-sive fields, e. g. magneto-gravitational levitation [15–21], which promises extended coherence times, has at-tracted renewed interest. In these setups the presenceof superconductors and the cryogenic operating temper-atures [19–21] make the use of high-intensity laser fieldschallenging, thereby limiting the applicability of opticalcooling schemes. A number of works propose to engi-neer a low-temperature environment for the oscillator bycoupling it to a two-level system (TLS) that is repeat-edly initialized in its GS [22–25]. However, in each ini-tialization cycle at most one phonon can be extracted,leading to a linear reduction in energy over time anda diminishing cooling rate [26] with increasing initialphonon numbers. Therefore, this approach becomes pro-hibitively slow in passive traps that operate in the low-frequency regime (100 Hz) with initial thermal occupan-cies of ¯ n T =300 K ≈ for the COM mode at room tem-perature. Another class of TLS-based cooling schemesrelies on the projection of the oscillator onto the GS [27– ! M M eq. z R R ! TLS + B M ( r z ) r z r FIG. 1.
Setup sketch.
A levitated magnetic particle ofradius R and magnetization M follows a harmonic motion inthe Z-direction with frequency ω M . A magnetically sensitiveTLS (e.g. an NV center in diamond) of bare energy splitting ω TLS is placed in the axis of oscillation at a distance r fromthe center of the harmonic potential. The energy splittingof the TLS is a function of the position of the magnet r z .The position of the particle is initially described by a ther-mal distribution (indicated as the purple blurred area). Theequilibrium position of the trap can be shifted. a r X i v : . [ qu a n t - ph ] F e b sidering that the entropy of the initial thermal state is S ≈ log (¯ n T =300K ) ≈
33 bits, which indicates that, un-der ideal conditions, a pure state can be reached afterjust ∼
33 measurements with a binary sensor. This isin stark contrast to the 10 measurements required forthe previously mentioned proposals. To achieve this re-duction in entropy within a minimal time and number ofmeasurements we designed an adaptive sensing schemethat ensures that the maximal amount of information isgained in each measurement. We numerically show therobustness of our method to imperfect readout fidelitiesand realistic heating rates. Our adaptive sensing schemeis directly applicable to DC field sensing tasks and differsfrom previous schemes [30–32] in its focus on maximal in-formation gain in each measurement and the possibilityto mitigate back-action effects. The use of a TLS sen-sor also distinguishes our protocol from existing feedbackschemes for optical setups, which operate in the contin-uous measurement regime with infinitesimal informationgain and simultaneous feedback [14, 33–35] as well as inthe pulsed regime [36]. Moreover, the presence of a TLSprovides a non-linearity that can be employed in sub-sequent stages of the experiment for non-classical statepreparation or advanced sensing protocols [37–41]. Setup and protocol. –
We consider a spherical mag-netic particle of radius R , mass m and magnetization M levitated in vacuum, such that the dynamics of its COMis well approximated by three uncoupled harmonic os-cillators. We assume that the magnetic dipole momentis aligned with the Z -direction and that a magneticallysensitive TLS, with bare energy splitting ω TLS , is alsoaligned in the same axis at a distance r from the equi-librium position of the trap, see Fig. (1). The magneticmoment generates a field at the position of the TLS thatcan be expanded to first order in terms of the displace-ment of the magnet from its equilibrium position, r z , as B M ( r z ) ≈ B + Gr z , provided that the position vari-ance of the magnet ∆ r z fulfills the condition ∆ r z (cid:28) r .Here, B = 2 µ M R / (3 r ), with µ the vacuum perme-ability, and G = − µ M R /r . While the B leads toan overall detuning of the TLS, G provides a coupling g = γGa between the TLS and the position of the os-cillator, where γ is the gyromagnetic ratio of the TLS,and a = p ~ / ( mω M ). We require the ultra-strong cou-pling regime g (cid:29) ω M , with ω M the trap frequency, whichallows us to run our protocol in timescales where the har-monic dynamics of the oscillator are negligible and thesystem is well described by the Hamiltonianˆ H = ~ g ˆ z ˆ σ z ~ Ω( t ) ˆ σ x , (1)which is stated in the rotating frame of the TLS. Thelast term represents a driving on the TLS, with time-dependent Rabi frequency Ω( t ) and ˆ z = ˆ r z /a denotesthe dimensionless position operator.Our protocol consists of three main parts: first, (1) we reduce the position uncertainty of the oscillator to thatof the GS, then (2) we allow for a quarter of a rotation inphase space that maps the momentum quadrature ontothe position quadrature and subsequently repeat the firstpart to again reduce the uncertainty in the position (be-fore momentum) quadrature. This leaves the system ina coherent state. Finally, (3) we displace the center ofthe trap to the position of this coherent state, cooling thesystem to the GS. We achieve the reduction of positionuncertainty in part (1) and (2) by performing a sequenceof adaptive measurement steps on the TLS.Each step starts with the TLS in its GS |↓i and theoscillator in the mixed state ρ M , resulting in the totalstate of the system ρ = |↓i h↓| ⊗ ρ M . The operator ˆ z in-troduces a shift of the bare TLS frequency, which allowsus to correlate the TLS state with the particle positionby performing a π -pulse that only inverts the TLS statefor a specific range of detunings. A subsequent measure-ment of the TLS provides information about the particleposition. To compute the spatial probability distributionassociated with the measurement outcome i = {↑ , ↓} , wenote that the Hamiltonian is diagonal in the spatial basis,which allows us to express the total evolution operatoras ˆ U = R ˆ U ( z ) | z ih z | d z . The spatial probability distri-bution then follows by applying the projection operatorˆ P = | z ih z | ⊗ | i ih i | to the evolved state and taking thetrace. The resulting expression is the familiar Bayesianupdate rule P n +1 ( z | i ) = I n ( z | i ) p n ( i ) P n ( z ) . (2)Here, n denotes the measurement step, p n ( i ) is the prob-ability of outcome | i i , and I n ( z | ↑ ) = | h↑| ˆ U n ( z ) |↓i | =1 − I n ( z | ↓ ) is the inversion profile of the pulse. Thelatter is a function of the eigenvalue of the dimensionlessposition operator ˆ z , and its specific shape is determinedby the pulse amplitude modulation Ω( t ). Provided that P n ( z ) is known, Ω( t ) can be designed to guarantee thatafter each measurement of the TLS the widths of thespatial distributions associated with each outcome are re-duced, and with them the entropy of the oscillator. Forour protocol, we use a Gaussian form for Ω( t ) which re-sults in a Gaussian-shaped inversion profile whose widthand mean are adapted in every step to yield the maxi-mal reduction in entropy, see the Supplemental Material(SM) [42]. This choice ensures that the probability dis-tribution P n ( z ) remains close to a Gaussian throughoutthe measurement sequence. After each step, the measure-ment outcome is used to update the probability distribu-tion using Eq. (2), which in turn allows us to computethe parameters for the next pulse. This computation canbe performed on a real-time computer with minimal re-sources, because the analytical form of I n ( z | i ) is knownand no simulation of the quantum dynamics is required.Our protocol also accounts for the measurement backaction on the oscillator. For perfect readout fidelities andGaussian-shaped π -pulses, we find that the back actionamounts to a displacement of the momentum probabilitydistribution for the |↓i outcome while no back action oc-curs for |↑i . This is shown in the lower panel of Fig. (2).The displacement only depends on the pulse duration andis therefore known for every pulse [42]. Although it doesnot affect the spatial probability distribution, it must bekept track of, as it becomes relevant in the part of the pro-tocol where the momentum quadrature is mapped ontothe position quadrature. The fact that we are measur-ing a quantum system that can suffer back action effectsprecludes an application of existing Ramsey based adap-tive sensing schemes [30–32], because their back actioninevitably leads to a broadening of the momentum dis-tribution [27].If the TLS readout is imperfect, with fidelity f < F ( o | ↑ ), with o = { , } denoting the outcome. In consequence, thespatial probability distribution after the measurementcontains contributions from the oscillator state associ-ated with TLS-state up and TLS-state down. This leads,assuming equal readout fidelities for both TLS states F ( o | ↑ ) = F ( o | ↓ ), to a modified update rule [42] P n +1 ( z |
1) = 1 p n (1) [ f I n ( z | ↑ ) + (1 − f ) I n ( z | ↓ )] P n ( z ) , (3)which is stated here for outcome o = 1 and which has ananalogous form for o = 0. Such measurements still leadto a narrowing of the distribution in the position quadra-ture, but they introduce a broadening in the momentumquadrature, because the displacement is different for eachof the two outcomes. To prevent this we introduce a hard π -pulse, which inverts the TLS state independently of itsdetuning. This swaps the TLS state that is associated toeach of the two parts of the oscillator state and a sub-sequent free evolution leads to a displacement of bothparts. The final displacement of both parts of the oscil-lator state is non-zero but equal, preventing the broaden-ing of the momentum distribution and making this caseequivalent to that of f = 1. The corresponding sequenceis shown in Fig. (2). Numerical simulations. –
To demonstrate the perfor-mance of our protocol, we numerically simulate the fullquantum dynamics of a thermal initial state [42]. We usethe GS fidelity as a figure of merit instead of the meanphonon number because, due to the properties of our pro-tocol, any deviations from the GS are of a non-thermalnature. Hence, GS fidelities are significantly higher thanthose of a thermal state with the same mean phononnumber. Figure (3a) shows that high GS fidelities canbe reached even for readout fidelities significantly belowunity and high initial phonon numbers. In fact, as ourprotocol makes no assumption on the initial phonon num-
ZP Z Z Z
Free evolutionfor t = π / ( ω M ) t Gaussian π -pulse Hard π -pulse Readout A B C D ZP A Z B |↓i |↑i Z C |↓i |↑i Z D |↑i FIG. 2.
Cooling scheme.
The upper panels show theWigner function of the particle from its initial state till theend of the second part of the protocol. The panels correspondto the: initial state; low entropy state in the first quadrature;state after a quarter of an oscillation; final coherent state.The middle section shows the amplitude profile of the pulsein each step. The lower section shows the evolution of theWigner function during a single step of the protocol and high-lights the displacement that is induced by the pulse. ber and Fig. (3b) shows no deterioration of the perfor-mance with increasing initial thermal occupancies, we ex-pect a similar performance also in cases starting at roomtemperature, for which numerical simulations become in-feasible. This is a key distinguishing factor to previousTLS-based cooling proposals [22–24, 29, 43].The time needed by our protocol to reach the minimalwidth in each quadrature is dominated by the last fewmeasurements, and is, therefore, almost independent ofthe initial phonon number as shown in Fig. (3c). Thisproperty follows from the fact that the duration of eachstep is inversely proportional to the width of the spatialdistribution in that step. Notice, that we are neglect-ing the readout time, as this is highly dependent on theexperimental implementation. For readout fidelities of0.9 and moderate initial phonon numbers of ¯ n ≈ , thenumber of measurements for each quadrature can be keptbelow 75. Moreover, the total protocol duration is set bythe trap frequency, which in turn determines the coolingrate. Figure (3d) shows that GS fidelities of 0.5 can bereached for heating rates on the order of Γ ≈ ω M /
10. Infuture work, the update rules in Eq. (2) and (3) can bemodified to account for the effect of heating on the prob-ability distributions, which should allow higher heatingrates.
Experimental feasibility. –
Our protocol works underthe assumption that the coupling between TLS and the
Case
R ω M / (2 π ) d d NV g A B Two parameter regimes with NV centers. R is the particle radius, ω M the trap frequency, d the distancebetween particle center and NV, d NV the implantation depthof the NV center, and g the coupling between the magnet andthe NV. Distances are stated in µ m and frequencies in kHz. particle, g, fulfills two conditions: (i) it is larger than theinverse of the TLS coherence time, g > /T , and (ii)it is in the ultra-strong coupling regime g (cid:29) ω M . Ad-ditionally, numerical simulations show that the protocolperforms best when (iii) the readout fidelity of the TLS isabove 0.8 (in order to reach GS fidelities above 0.5) and(iv) the motional heating rate Γ is lower than the trapfrequency, Γ < ω M . We analyze the experimental feasi-bility of these requirements by considering a particle witha density of 7 · kg/m [19], and a TLS implementationbased on a single Nitrogen-Vacancy (NV) center [44] attwo different implantation depths: shallow (case A ) anddeep (case B ). The specific parameters are provided inTab. I. Our protocol requires coherence times of 10 and200 µ s for cases A and B , respectively, which is wellwithin reported values. For shallow NV centers, room-temperature coherence times as long as 250 µ s have beendemonstrated [45], as well as fast single-shot readout atcryogenic temperatures with fidelities of 78 . ± .
5% [46].For a scenario like that of case B , coherence times of upto 2 . − µ m radius, in contrast to reportedGS cooling experiments [13, 14] with particles one orderof magnitude smaller. The upper limit on the particlesize in our scheme is set by the coupling, which decreaseswith particle size for point-like TLS sensors and leads toa violation of the ultra strong coupling requirement or aninsufficient cooling rate.Our protocol is well suited for the experimental setupin [19], which can operate in the ultra-strong couplingregime, and where the authors expect to reach couplingsof 2 . . −
100 times lower thanthe quality factor of the resonator [26], which necessitatesadditional precooling mechanisms. Remarkably, our pro- F (a) ¯ n Initial Phonon Number0.40.60.81.0 F (b) f: Initial Phonon Number10305070 T i m e [ / g ] (c) f: − − − Heating Rate Γ [ ω M ]0.00.20.40.60.8 F (d) f: FIG. 3.
Numerical simulations. (a) GS fidelity F at theend of the protocol versus readout fidelity for different initialphonon numbers. (b) Same data as (a) plotted as a functionof the initial phonon number. These results were obtainedby simulating the von Neumann equation of the closed sys-tem consisting of the TLS and the particle initially in a ther-mal state. (c) Time needed to reduce the uncertainty of onequadrature in units of inverse coupling 1 /g as a function ofinitial phonon number, for different readout fidelities. (d) GSfidelity F as a function of the heating rate for different readoutfidelities and an initial state with ¯ n = 100 that is in thermalequilibrium with the environment. The data was generatedby simulating the Lindblad equation with ω M = 2 πg/ F = Tr hp √ σρ √ σ i , where σ and ρ are the density matrices of the target GS and the finalstate in the simulation of our protocol. tocol does not have such a limitation, which makes it ap-plicable even in room-temperature environments. How-ever, the fact that the heating rate is dependent on thetemperature Γ = k b T / ( ~ Q ) sets a constraint between thequality factor Q and the tolerable initial temperature.Yet, unlike in previous schemes, our protocol reaches afinal phonon number that is below the thermal equilib-rium, even when the heating rate is higher than the cool-ing rate.We note that for particles with radii above 10 µ m incryogenic environments, an implementation based on fluxqubits may be advantageous as their size and couplingcan be scaled accordingly [24]. Conclusion. –
By splitting the process of cooling intotwo steps: entropy reduction and energy extraction, ourprotocol is able to overcome the limitations present indissipative TLS-based cooling schemes when operatingat low-frequencies. Thus, it constitutes, to the best ofour knowledge, the first viable GS-cooling scheme formicron-scale particles in low-frequency traps. In partic-ular, an implementation of our ideas using NV centers incryogenic environments can reach the required parame-ter regime. Moreover, the ideas presented here can beindependently applied as an adaptive sensing scheme,suitable for scenarios where the prior information on theunknown parameter is limited, or where the back actionon the sensed system becomes relevant [32]. Our protocolcan be improved further by a rigorous optimization of theparameters of the adaptive algorithm, and by extendingit to multiple sensors. More involved TLS drivings canbe used to lift the requirement of the ultra-strong cou-pling regime and to allow for even higher heating rates.Finally, our ideas can also be used for the generation ofsqueezing or superposition states.
Acknowledgments.—
We thank J.F. Haase, B. D’Anjouand M. Korzeczek for helpful comments on themanuscript. We acknowledge support by the ERC Syn-ergy grant HyperQ (Grant No. 856432), the EU projectsHYPERDIAMOND (Grant No. 667192) and AsteriQs(Grant No. 820394), the QuantERA project NanoSpin(13N14811), the BMBF project DiaPol (13GW 0281C),the state of Baden-W¨urttemberg through bwHPC, theGerman Research Foundation (DFG) through Grant No.INST 40/467-1 FUGG, and the Alexander von HumboldtFoundation through a postdoctoral fellowship. [1] J. Millen, T. S. Monteiro, R. Pettit, and A. N. Vamivakas,Optomechanics with levitated particles, Rep. Prog. Phys. , 026401 (2020).[2] D. C. Moore and A. A. Geraci, Searching for newphysics using optically levitated sensors, (2020),arXiv:2008.13197.[3] M. Rademacher, J. Millen, and Y. L. Li, Quantum sens-ing with nanoparticles for gravimetry: When bigger isbetter, Advanced Optical Technologies , 227 (2020).[4] T. Weiss, M. Roda-Llordes, E. Torrontegui, M. As-pelmeyer, and O. 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I. SHAPED INVERSION PROFILES
In this section, we present a detailed derivation of the Bayesian update rule in Eq. (2) of the main text, elucidateour choice of the Gaussian inversion profile, describe the modulation of the drive amplitude that is needed to generatethis profile and explain the properties of such a pulse, including the back action that it induces on the particle. TheHamiltonian in Eq. (1) of the main text ˆ H = ~ g ˆ z ˆ σ z ~ Ω( t ) ˆ σ x , (1)is the starting point of our analysis. It is diagonal in the spatial basis, allowing us to express the evolution operatoras ˆ U = R ˆ U ( z ) | z i h z | d z , where U ( z ) acts in the TLS subspace and is the unitary evolution operator associated toHamiltonian (1) after substituting the dimensionless position operator ˆ z by the eigenvalue z of the correspondingeigensate | z i . The goal of our protocol is to reduce the widths of the spatial and momentum probability distributionsto the width of a coherent state. The effect of a single evolution and measurement step on the spatial probabilitydistribution can be easily computed for each measurement outcome i = {↑ , ↓} P n +1 ( z | i ) = 1 p n ( i ) h z | h i | ˆ U n |↓i h↓| ⊗ ρ n ˆ U † n | i i | z i = | h i | ˆ U n ( z ) |↓i | p n ( i ) P n ( z ) = I n ( z | i ) p n ( i ) P n ( z ) . (2)The index n indicates the measurement number, p n ( i ) the probability of outcome i, and P n ( z ) = h↓| h z | ρ n | z i |↓i thespatial probability distribution at the beginning of the step. The initial probability distribution P ( z ) correspondsto that of a thermal state and is known before the start of the experiment. To compute the post-measurementdistribution only the knowledge of the inversion profile I n ( z | ↑ ) = | h↑| ˆ U n ( z ) |↓i | ≤
1, which corresponds to thetransition probability as a function of z , is required. This depends on the specific form of the amplitude modulationΩ( t ) and has no general analytical solution. Nevertheless, to avoid its computation in each step of the protocol, itsfunctional form can be precomputed and fitted to a parametrized curve, such that this analytic approximation canbe used in the update rule (2) during the experiment. Provided that this parametrized curve fits the real inversionprofile well, this will not corrupt the functioning of our protocol. Thus, to obtain the probability distribution aftereach measurement, the approximated inversion profile simply has to be multiplied with the previous probabilitydistribution. Of course, this procedure becomes exact when the analytical solution of the inversion profile is known.For example, consider a pulse with constant Rabi frequency Ω( t ) ≡ Ω and duration τ , such that R τ Ωd t = π , i.e. asquare π -pulse; the analytical form of the inversion profile is I ( z | ↑ ) = Ω Ω + ∆( z ) sin p Ω + ∆( z ) Ω π ! , (3)which is parametrized by the Rabi frequency Ω and where ∆( z ) = gz . Even though square pulses are not used in ourprotocol we note that the width of this inversion profile is dependent on the amplitude Ω and, therefore, the durationof the pulse, which is a common feature of all pulses.One of the most important requirements when choosing the appropriate inversion profile is that the measurementmust yield a significant reduction in entropy. Furthermore, this reduction must be achievable not only for the initialdistribution, but also for all intermediate distributions. This requirement is fulfilled for a rectangular inversion profilethat correlates half of the distribution with the TLS-state up and the other half with TLS-state down as shown inFig. (1a). This means that the probabilities of measuring either TLS state are 1 / f < a r X i v : . [ qu a n t - ph ] F e b − − P ( z ) (a) − − z I ( z | ↑ ) (b) FIG. 1. (a) The Gaussian prior distribution is shown in blue and the shaded area indicates the part covered by the rectangularinversion profile. They are aligned such that the inversion profile covers half of the distribution. (b) Comparison of the inversionprofile generated by a Gaussian modulation (blue) of the driving amplitude and a Gaussian function with the parameters statedin the text (orange). to its rectangular (discontinuous) form. For our protocol, we choose to use a Gaussian inversion profile parameterizedby the mean µ I and the variance σ I as I ( z | ↑ ) = exp (cid:18) − ( z − µ I ) σ (cid:19) . (4)Given that our initial (thermal state) and target (coherent state) distributions are Gaussian and that a product oftwo Gaussians (prior times inversion profile) yields another Gaussian, we expect that upon measuring outcome |↑i the distribution will remain Gaussian. However, we expect slight deviations because the inversion profile for the |↓i outcome is given by I n ( z | ↓ ) = 1 − I n ( z | ↑ ), which is not Gaussian. It is shown in the next section that despite thisfact the deviations are typically small.To the best of our knowledge, the inversion profile associated to a Gaussian modulation of the pulse amplitudeΩ( t ) has no analytic solution. However, it has been shown that it closely resembles a Gaussian profile [1], which issufficient for the proper functioning of our protocol. In particular, for a modulation of the formΩ( t ) = π q πσ exp (cid:26)(cid:18) − ( t − τ / σ (cid:19)(cid:27) , (5)the numerically computed inversion profile can be fitted to a Gaussian of variance σ I = 1 / (2 gασ p ). Here, the factorof 2 is a consequence of the square in the definition of the inversion profile, and the constant α will vary with theduration of the pulse τ , and needs to be numerically determined. For the simulations in this work, we chose a pulseduration of τ = 10 · σ p , for which we find that the value α ≈ .
15 yields an approximation of the inversion profilethat is sufficiently good for the functioning of our protocol. A comparison between the numerically determined profileand the Gaussian approximation with the stated parameters is shown in Fig. (1b). We note that a too-short pulseduration leads to a windowing, effect which results in undesired oscillations on top of the Gaussian form.In addition to correlating the TLS and particle states, the pulse also introduces a displacement of the particle.The numerically determined displacement of the particle is shown in Fig. (2). The displacement of the particle statethat is associated with the TLS-state up outcome is always zero while the state associated with TLS-state down getsdisplaced. Our simulations show that when the relationship between pulse duration τ and pulse width σ p is keptconstant the displacement becomes linearly dependent on the pulse duration. Therefore the imposed displacement isalways known to the experimenter without additional computational effort. II. ADAPTIVE SENSING ALGORITHM FOR GAUSSIAN INVERSION PROFILES
A key ingredient in our protocol is the adaptive sensing scheme that ensures efficient entropy reduction in eachmeasurement and a final distribution that is close to a Gaussian. In this section we describe how the pulse parametersare determined in each step of the algorithm based on the current spatial probability distribution. − − − − D i s p l a c e m e n t [ a ] FIG. 2. Displacement of the particle state during the Gaussian pulse that is associated with TLS-state down (blue), TLS-stateup (orange) and the displacement that would be expected under the action of H = − g ˆ z/ P ( z ) I ( z ) (a) σ I σ µ I | ↓ > | ↑ > (b) FIG. 3. (a) Shows the parameters of the adaptive protocol. (b) Shows the posterior distributions for the two measurementoutcomes. For this plot a Gaussian was used as the prior.
At the beginning of the algorithm we assume that the system is in a thermal state which means that the probabilitydistributions for both quadratures are of the form P ( z ) = 12 π p ¯ n + σ exp (cid:26)(cid:18) − z n + σ ) (cid:19)(cid:27) , (6)with σ = 1 / √ n = e − ~ ωMkBT − e − ~ ωMkBT (7)the thermal occupation number, with k B denoting Boltzmann’s constant and T the temperature. The prior distri-bution is saved as a vector and updated in each step by applying the Bayesian update rule. Our algorithm is aBayesian inference algorithm where P ( z ) corresponds to the prior distribution and the inversion profile I ( z | ↑ ) to thelikelihood function. As in other inference algorithms of this type, the precise form of the prior is not important forthe proper functioning of the algorithm as long as it assigns a non-zero probability to the true outcome. Therefore,precise knowledge of ¯ n is not required. It is safe to assume a too-high initial temperature, because, due to the factthat the number of measurements grows logarithmically with ¯ n , this will not significantly affect the efficiency of theprotocol. However, a too-low ¯ n can lead to systematic errors, because the overlap of the true state with the priordistribution might become too small.Given a Gaussian prior distribution centered at µ and with a variance σ , we need to determine the detuning µ I andthe width σ I of the Gaussian inversion profile. These parameters are visualized in Fig. (3a). We choose the width tobe linearly related to the width of the prior distribution via a multiplicative factor w that is kept fixed throughout ∆ z n θ n P n ( z ) I n ( z | ↑ ) (a) ∆ z n θ n P n ( z ) I n ( z | ↑ ) (b) FIG. 4. (a) The threshold θ n is larger than the maximal point of the side peak. In this case the side peak will grow in the nextmeasurement. (b) The threshold θ n is smaller than the side peak leading to a wide inversion profile and therefore a suppressionof the side peak in the next measurement. the protocol σ I = w · σ. (8)Given σ I , we determine the detuning by fixing the probability of the TLS-state up outcome and inverting the rela-tionship p ↑ = Z ∞−∞ d z √ πσ e − ( z − µ )22 σ e − ( z − µ I)22 σ . (9)This leads to µ I = µ ± vuut − σ + σ ) log p ↑ s σ + σ σ ! . (10)This parametrization was chosen because the outcome probability is a good measure of the equality of the entropyreduction over the two outcomes. For a given σ I , p ↑ = p ↓ = 0 . p ↑ < p ↓ , which would lead to a higher entropy reduction for the spin up outcome. Hence, σ I givesus a handle on how much entropy can be extracted in the measurement and p ↑ determines how this reduction isdistributed over the two possible outcomes. Note that both update rules in Eq. (8) and Eq. (10) assume that theprior distribution has a Gaussian form and require the knowledge of the width σ of that Gaussian.To obtain an approximately deterministic cooling rate, the entropy reduction for both measurement outcomesshould be equal. An alternative approach is to tune the algorithm parameters such that the entropy reductionaveraged over the two outcomes (weighted by their probabilities) is maximal. However, this approach leads to multi-peaked distributions that violate our simplifying assumption that the distribution is close to a Gaussian in eachstep. As shown in Fig. (3b), a small side peak always emerges for the spin down outcome. To ensure that thispeak remains small, we choose the width of the inversion profile to be larger than the width of the prior distribution σ I > σ and the probability of detecting the TLS-state up to be p ↑ < .
5. The prior condition makes the side peakwider and therefore lower, but also reduces the amount of entropy that can be extracted in the measurement. Thelatter condition decreases the absolute size of the side peak and makes the entropy reduction more unequal over themeasurement outcomes. With these two considerations it can be ensured that the distribution remains close to aGaussian. The entropy reduction for the TLS-state down outcome gives a lower bound on the entropy that can beextracted with each measurement. This deterministic decrease in entropy is shown in Fig. (5a).To design the inversion profile, we have assumed that, in every step n , the prior probability distribution is aGaussian, which, as discussed in Sec. I, will not be the case in general. Nevertheless, our algorithm will still work inthis case, albeit with reduced efficiency. To that end, we need to establish a methodology to associate, in each step,a Gaussian width σ n to the actual probability distribution P n ( z ), to be used in the update formulas (8-10). Moreimportantly, we want to do this in a way that avoids the deviations from accumulating as the protocol progresses.With that in mind, we define a threshold value θ n and we look for the width ∆ z n of the region where the probabilitydistribution P n ( z ) exceeds this value, see Fig. (4). We then use the parameters θ n and ∆ z n to assign a Gaussian withwidth σ n to the probability distribution P n ( z ) according to the relation θ n = 1 p πσ n e − (∆ zn/ σ n , (11)obtaining σ n = s − (∆ z n / W − ( − π (∆ z n / θ n ) , (12)where W − is the − θ n , relative to the size of the distribution in each step of the protocol. To do so we rely on the width ofthe previous prior σ n − , which should be close to σ n , and set θ n = 1 q πσ n − e − θ z , (13)with θ z a parameter that is kept fixed over the duration of the protocol, and which determines the relation betweenthe threshold value θ n and the maximum of the Gaussian distribution.Deviations from a Gaussian probability distribution originate in TLS-state down outcomes, whose associated prob-ability distribution displays an additional smaller peak at the tail of the Gaussian, see Fig. (4). The key advantage ofchoosing a threshold θ n to determine the width (instead of for example computing the variance of P n ( z )) is that in thisway side peaks can be automatically detected. In the situation depicted in Fig. (4a), the side peak is initially belowthe threshold value, which leads to its growth independently of the measurement outcome, because both inversionprofiles I n ( z | ↓ ) and I n ( z | ↑ ) have a significant overlap with it. Eventually the situation depicted in Fig. (4b) willarise, where this side peak becomes bigger than θ n which consequentially leads to an increased ∆ z n and σ I . In thiscase, the side peak keeps growing for only one of the two possible outcomes. In fact the outcome, where the side peakgrows is heavily suppressed because the probability mass covered by the associated inversion profile is small, whichis visualized in Fig. (4b). However, even in the case of this unlikely outcome the algorithm does not fail, instead thesuppression of the side peak is simple deferred to a later measurement. This might lead to a reduction in efficiency ofthe algorithm, which is why we introduce another trick to suppress the growth of the side peak even when it is stillsmaller than θ n . To that end, we flip the sign of the detuning after each TLS-state down measurement, i.e. right afterthe emergence of the side peak. Therefore, unlike in Fig. (4a), the I n ( z | ↑ ) profile does not have an overlap with theside peak and in case of a TLS-state up outcome the side peak gets suppressed.Our numerical simulations show that with the algorithm described in this section the distribution P n ( z ) stays closeto a Gaussian whose width is given by σ n and whose mean corresponds to the value of z where P n ( z ) is maximal. Weshow this by computing the Kullback-Leibner divergence between P n ( z ) and a Gaussian with the stated parameters.A typical result is shown in Fig. (5b). As can be seen in this plot, the deviations from the Gaussian form donot accumulate. Furthermore, the Kullback-Leibner divergence is a measure of the distance to a particular targetdistribution (in this case a specific Gaussian). Therefore, non-zero values of the Kullback-Leibner divergence arenot immediately related to deviations from a Gaussian form but can rather mean that the distribution is given by aGaussian with different parameters than the one we are comparing it to. Hence, it sets a more stringent conditionthan would be required to prove our point.The free parameters of our algorithm are the multiplicative factor w in Eq. (8), the target probability p ↑ in Eq. (10)and the threshold θ z in Eq. (13). In the next section, we present the numerical optimization over these parameters. III. OPTIMIZATION OF THE ALGORITHM PARAMETERS
The adaptive algorithm contains three free parameters: the multiplicative factor w in Eq. (8), the target probability p ↑ in Eq. (10) and the threshold θ z in Eq. (13). The algorithm also requires a stop condition, i.e. the desired statewidth. We perform simulations of the Bayesian update rule in Eq. (2) for ranges of these parameters to determine theset that leads to the lowest final entropy in the shortest time. For the stop condition we choose σ = 1 /
2. However,this choice does not affect the optimization of the other parameters. We also subtract the entropy of a Gaussian with σ = 1 /
2, such that we obtain a final entropy of 0 in the optimal case. However, this subtraction can also lead tonegative values of the final entropy. This has no further significance for our analysis as what matters is the deviation B i t Entropy K L Kullback Leibner Divergence
FIG. 5. (left) Entropy reduction in each measurement. The dashed lines show the trajectories of individual simulations, thered line shows the average over these trajectories. The parameters used for this plot are ¯ n = 300, f = 0 .
9. (right)Kullback-Leibner divergence for one of the trajectories in the left plot. from the target value (in this case the entropy of a Gaussian width σ = 1 / w and p ↑ are independent of the readout fidelityand θ z . It is also obvious that there is a trade-off between final entropy and protocol duration. The optimal choice ofthese parameters will depend on the heating rate of the experiment. For the simulations in this work, we chose theparameters w = 1 . p ↑ = 0 .
4. We also see that a higher value for θ z leads to a significantly lower final entropywhile only moderately increasing the protocol duration which is why we chose θ z = 2 .
75 for our simulations.After choosing the above parameters, we performed an optimization of the stop condition by simulating the full-quantum dynamics, see Sec. IV The algorithm requires two stop conditions, one for each quadrature. We observed inour numerical simulations that the GS fidelity increases with the amount of squeezing that is introduced in the firstquadrature. Therefore, a threshold on σ n , that is related to the width of the probability distribution via Eq. (11),should be chosen that achieves the maximal amount of squeezing, while considering the heating rate of the particle andthe dephasing rate of the TLS. These set a bound on the amount of squeezing that can be achieved. We numericallyoptimized the threshold for the second quadrature and the results are shown in Fig. (8). However, the heating rateswere set to zero in these simulations. The optimal threshold depends on the readout fidelity f and comes close to theexpected value of the coherent state width σ = 1 / √ f = 1. w p ↑ f: 0.80 θ z : 2.25 1.25 1.40 1.55 1.70 1.85 2.00 w p ↑ f: 1.00 θ z : 2.251.25 1.40 1.55 1.70 1.85 2.00 w p ↑ f: 0.80 θ z : 2.75 1.25 1.40 1.55 1.70 1.85 2.00 w p ↑ f: 1.00 θ z : 2.750.00.51.01.5 − − − − FIG. 6. The panels show the final entropy as a function of w and the target probability p ↑ for different readout fidelities andthreshold values θ z . While the absolute values of the entropy differ, the scaling with w and p ↑ is equal in each panel. Therelevant quantity for the performance of the algorithm is the deviation of the entropy from the desired target value rather thanthe absolute value of the entropy. In the simulations presented here, the target value is the entropy of a Gaussian distributionwith width σ = 1 /
2. This value is subtracted from the value reached by the algorithm, leading to negative values in certaincases. w p ↑ f: 0.80 θ z : 2.25 1.25 1.40 1.55 1.70 1.85 2.00 w p ↑ f: 1.00 θ z : 2.251.25 1.40 1.55 1.70 1.85 2.00 w p ↑ f: 0.80 θ z : 2.75 1.25 1.40 1.55 1.70 1.85 2.00 w p ↑ f: 1.00 θ z : 2.75100150200250300350 405060708090200250300350 5060708090 FIG. 7. The panels show the total protocol duration, in units of inverse coupling 1 /g , as a function of w and the targetprobability p ↑ for different readout fidelities and threshold values θ z . While the absolute values differ, the scaling with w and p ↑ is equal in each panel. IV. SIMULATIONS
In this section we present the full-quantum simulation that we used to generate the data presented in Fig. (3) ofthe main text. To validate our algorithm we numerically integrated the Lindblad equation˙ˆ ρ = − i h ˆ H ( t ) , ˆ ρ i + L (ˆ ρ ) , ˆ H ( t ) = g z ˆ σ z + Ω( t )2 ˆ σ x , L (ˆ ρ ) = γ (cid:18) ¯ n + 12 (cid:19) (cid:0) z ˆ ρ ˆ z − ˆ z ˆ ρ − ˆ ρ ˆ z + 2ˆ p ˆ ρ ˆ p − ˆ p ˆ ρ − ˆ ρ ˆ p (cid:1) + i γ p ˆ ρ ˆ z − z ˆ ρ ˆ p + ˆ p ˆ z ˆ ρ + ˆ ρ ˆ p ˆ z − ˆ z ˆ p ˆ ρ − ˆ ρ ˆ z ˆ p ) . G S F FIG. 8. Ground-state fidelity (GSF) as a function of the stop threshold for the second quadrature plotted for different readoutfidelities.
Here, we set ~ = p ~ / ( mω M ) = 1. To be able to efficiently simulate high-temperature states, we reformulated theseequations in the phase space of the levitated particle while keeping the Hilbert space formulation for the TLS. Thiswas done by writing the density matrix asˆ ρ = W ↑↑ |↑ih↑| + W ↑↓ |↑ih↑| + W ↓↑ |↓ih↓| + W ↓↓ |↓ih↓| . (14)Here, W ij are Wigner functions multiplied with the probability amplitude of the associated TLS state. We define theWigner functions via the characteristic function [2] χ ( ξ, η ) = Tr (cid:0) ρe iξ ˆ z + iη ˆ p (cid:1) = Z ∞−∞ d µ Z ∞−∞ d νW ( µ, ν ) e iξµ e iην . This leads to the following equations of motion˙ W ↑↑ ( µ, ν ) = − i (cid:18) Ω2 W ↓↑ ( µ, ν ) − Ω2 W ↑↓ ( µ, ν ) (cid:19) + g ∂∂ν W ↑↑ ( µ, ν ) + D [ W ↑↑ ] , (15)˙ W ↑↓ ( µ, ν ) = − i (cid:18) Ω2 W ↓↓ ( µ, ν ) − Ω2 W ↑↑ ( µ, ν ) (cid:19) − igµW ↑↓ ( µ, ν ) + D [ W ↑↓ ] , (16)˙ W ↓↑ ( µ, ν ) = − i (cid:18) Ω2 W ↑↑ ( µ, ν ) − Ω2 W ↓↓ ( µ, ν ) (cid:19) + igµW ↓↑ ( µ, ν ) + D [ W ↓↑ ] , (17)˙ W ↓↓ ( µ, ν ) = − i (cid:18) Ω2 W ↑↓ ( µ, ν ) − Ω2 W ↓↑ ( µ, ν ) (cid:19) − g ∂∂ν W ↓↓ ( µ, ν ) + D [ W ↓↓ ] , (18) D [ W ] = (cid:20) γ (cid:18) ∂∂µ µ + ∂∂ν ν (cid:19) + γ (cid:18) ¯ n + 12 (cid:19) (cid:18) ∂ ∂µ + ∂ ∂ν (cid:19)(cid:21) W. (19)These simulations serve to validate the algorithm, but are not required to determine the measurement parametersin each step.The algorithm requires a prior distribution, which for thermal states and with our conventions corresponds to theGaussian P ( z ) = 1 √ πσ e − z σ , (20)with σ = p ¯ n + 1 /
2. We quantify the distance to the GS via the fidelity F ( ρ, σ ) = Tr (cid:20)q √ σρ √ σ (cid:21) = h | ρ | i = 2 Z d µ Z d νW ( µ, ν ) e − µ e − ν . (21) V. DERIVATION OF THE BAYESIAN UPDATE RULE WITH IMPERFECT READOUT
In this section we present the derivation of the Bayesian update rule in Eq. (3) of the main text, which accountsfor imperfect readout fidelities. We define the readout fidelity as the probability of correctly detecting a given TLS0state, i.e. we do not account for errors in state preparation. Furthermore we assume that the readout fidelity is equalfor both states. This can be expressed by the conditional probabilities: p (1 | ↑ ) = f, p (0 | ↑ ) = 1 − f, (22) p (0 | ↓ ) = f, p (1 | ↓ ) = 1 − f. (23)Where p (1 | ↑ ) ( p (0 | ↓ )) is the conditional probability that the TLS-up (down) state is detected if the TLS is in theup (down) state. The spatial probability distribution after up state detection is given by P ( z |
1) = X s = {↑ , ↓} P ( z, s |
1) = X s = {↑ , ↓} P ( z | s, p ( s |
1) = X s = {↑ , ↓} p (1 | s ) p ( s ) p (1) P ( z | s, . (24)Furthermore, we know that P ( z | s, ≡ P ( z | s ) and that the conditional spatial distribution for spin state s = {↑ , ↓} is given by P ( z | s,
1) = 1 p ( s ) |h s | U ( z ) |↓i| P ( z ) , (25)as derived in Sec. I, and p ( s ) = Z ∞−∞ d z |h s | U ( z ) |↓i| P ( z ) . (26)Here and in the following P ( z ) is the spatial probability distribution of the initial state of the oscillator. PluggingEq. (25) into Eq. (24) yields the spatial probability distribution given the measurement result 1 P ( z |