Abstract

Optically trapped nanoparticles have recently emerged as exciting candidates for tests of quantum mechanics at the macroscale and as versatile platforms for ultrasensitive metrology. Recent experiments have demonstrated parametric feedback cooling, nonequilibrium physics, and temperature detection, all in the classical regime. Here we provide the first quantum model for trapped nanoparticle cooling and force sensing. In contrast to existing theories, our work indicates that the nanomechanical ground state may be prepared without using an optical resonator; that the cooling mechanism corresponds to nonlinear friction; and that the energy loss during cooling is nonexponential in time. Our results show excellent agreement with experimental data in the classical limit, and constitute an underlying theoretical framework for experiments aiming at ground state preparation. Our theory also addresses the optimization of, and the fundamental quantum limit to, force sensing, thus providing theoretical direction to ongoing searches for ultraweak forces using levitated nanoparticles.

© 2016 Optical Society of America

1. INTRODUCTION

Optically trapped nanoparticles can support explorations of macroscopic quantum mechanics as well as ultrasensitive metrology very well, since they can be isolated from the environment in a trap, cooled, and detected—all using a single laser beam without the need for an optical cavity [15]. Experiments with optically trapped harmonically oscillating subwavelength dielectric particles [610] (see Fig. 1) have recently realized feedback cooling [1,11], nonlinear dynamics [2], nonequilibrium physics [4], coupling to spin degrees of freedom [12], and thermometry [3]. All experiments thus far have been carried out in the classical regime.

 

Fig. 1. (a) Image of the trapped nanoparticle and (b) schematic of the experiment modeled in this article. The schematic shows the electro-optic modulator (EOM), polarizing beam splitters (PBS1 and PBS2), high numerical-aperture lenses (L1 and L2), the detector (D), the beam dump (BD), and the feedback circuit gain (G).

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While several groups are currently exploring ways to access the nonclassical regime of such systems, we present here the first (to our knowledge) quantum theory of trapped nanoparticle optical feedback cooling and force sensing. The impetus for investigating cooling comes from the fact that although levitated particles have been successfully cooled in optical resonators (to 10 K [13] and 64 K [14]), cavityless cooling has been able to reach much lower temperatures (50 mK [11]) starting from the same initial (room) temperature. It is, however, an open theoretical question as to whether the quantum ground state can be prepared without using a cavity. Our model addresses this important question and reveals a viable route to nanoparticle ground state preparation for ongoing cavityless experiments. It also identifies the dissipative mechanism underlying experimental cooling as being nonlinear in nature, in contrast to standard experimental techniques, which depend on linear damping, and standard theory, which relies on linear response analysis [1420]. Lastly, the model yields analytical results for the ensuing nonexponential decay of the phonon number, which shows excellent agreement with our experimental data in the classical regime. The motivation for investigating force detection is provided by the use of levitated cavityless nanoparticles in several ongoing searches for various ultraweak forces [5,2123]. Again, it is an open question as to whether and to what extent force detection is limited by the effects of quantum backaction in those systems. To clarify this issue, we derive in this article expressions for the force sensitivity and the standard quantum limit of force detection. We expect our new results on cooling and force sensing in the quantum limit will be invaluable as this still nascent field matures.

A schematic of the physical system under consideration, similar to the setup in Ref. [11], is shown in Fig. 1. A subwavelength polarizable dielectric sphere is confined at the focus of a Gaussian trapping beam, and its motion is detected using a probe beam, polarized orthogonal to the trap. The detected signal is processed and fed back to the trap beam to cool the particle. We analyze this configuration by dividing it into a “system” and a “bath.” The system consists of the nanomechanical oscillator and the optical probe and trap. The bath consists of the optical modes into which light is scattered by the nanosphere, and the background thermal gas present in the experiment. We proceed to derive a quantum model by identifying the electromagnetic modes relevant to the problem, constructing the system and bath Hamiltonians, and deriving the master equation for the system [24,25]. All conclusions stated in this article follow from this master equation.

2. MODEL

The configuration Hamiltonian can be written as

H=Hm+Hf+Hint.
In Eq. (1), the first term on the right-hand side represents the mechanical kinetic energy Hm=|p|2/2m, where p is the three-dimensional momentum of the nanoparticle and m its mass. The second term in Eq. (1) is the field energy Hf=ϵ0|E(r)|2d3r, where E(r) is the sum of the trap Et, probe Ep, and background Eb electric fields. We model the trap and probe modes as Gaussian beams, and the background using a plane wave expansion. We find, after dropping a constant term,
Hf=ωpaa+μd3kωkaμ(k)aμ(k),
which is simply the sum of the probe and background field energies with a and aμ(k) representing the corresponding standard bosonic field operators. Finally, in Eq. (1) the interaction Hamiltonian is given by Hint=VP(r)·E(r)d3r/2=αpV|E(r)|2d3r/2, where we have assumed that the dielectric has a volume V, and that it has a linear polarizability density αp, i.e., the polarization density is P(r)=αpE(r). Using the expressions for the electric fields from Supplement 1, we can evaluate Hint for small particle displacements r, and rewrite Eq. (1) as
H=HS+HB+HSB,
where this system Hamiltonian is
HS=ωpaa+jωjbjbjjgjaa(bj+bj),
with mechanical trapping frequencies ωj, optomechanical coupling constants gj, and mechanical operators that obey the standard commutation relations [bj,bj]=1 (j={x,y,z}). In Eq. (3), the bath Hamiltonian is HB=μd3kωkaμ(k)aμ(k) and the system–bath interaction Hamiltonian is HSB=ϵcϵ0Vd3r[Et(r)+Ep(r)]·Eb(r), which represents the scattering of the trap and probe fields into the background (see Supplement 1 for details).

We now trace over the bath modes, applying the standard Born and Markov approximations, since the system–bath coupling is weak and the bath correlations decay quickly [17,24]. We also trace over the x and y degrees of particle motion, since the dynamics along the three axes are independent of each other, and it suffices to analyze a single direction [11]. The net result of our calculation is a master equation for the density matrix ρ(t) describing the optical probe and the z motion of the nanoparticle,

ρ˙(t)=1i[HS,ρ]At2D[Q]ρ+Lscρ,
where the first term on the right-hand side represents unitary evolution of the system with HS=ωpaa+ωzbzbzgzaa(bz+bz). The second term corresponds to the positional decoherence of the nanoparticle due to the scattering of trap photons, with the Lindblad superoperator D[Qz]ρQzQzρ+ρQzQz2QzρQz, where Qz=bz+bz, and At is the heating rate due to trap beam scattering as defined in Supplement 1. The third superoperator describes the loss of photons from the probe, also due to scattering by the nanoparticle, Lsc[ρ(t)]=B(D[a]+(7ωp2z2/5c2)D[aQz])ρ, where z is the oscillator length.

The nanoparticle also experiences collisions with background gas particles at the ambient temperature T. This effect may be accounted for by adding to the right hand of Eq. (5) the superoperator [26]

B[ρ(t)]=Dp2D[Qz]ρDq2D[Pz]ρiηf4m[Qz,{Pz,ρ}],
where Pz=i(bzbz) and curly braces denote an anticommutator. The first term on the right-hand side corresponds to momentum diffusion, and Dp=2ηfkBTz2/2, where kB is Boltzmann’s constant. The second term describes position diffusion, with Dq=ηf2/(24kBTm2z2). The third term accounts for friction, and by Stokes’ law we have ηf=6πμrd, where rd is the radius of the nanoparticle and μ is the dynamic viscosity of the background gas. As shown earlier, internal and center-of-mass heating of the nanoparticle due to optical absorption and blackbody radiation are negligible in systems such as ours, as are particle size and shape effects, as well as trap beam shot noise [11,27].

We now characterize the measurement of the oscillator displacement using input–output theory from quantum optics [28] applied to the nanoparticle. Specifically, the incoming probe field ain interacts with the nanoparticle, and the outgoing probe field aout carries a signature of this interaction (as shown in Supplement 1):

aout=ain+αχ2Qz(t),
where χ=4gzΔt is the scaled optomechanical coupling, with integration time Δt (determined by the detection bandwidth), and we have written the probe beam as a coherent state a=iα+v, with α being a classical number and v a bosonic annihilation operator. A homodyne measurement on the output field yields a current [28],
Ih=χ2ΦQz(t)+χ2Φξ(t),
where Φ=α2Δω is the average detected flux of probe photons and ξ(t) is a stochastic variable with mean ξ(t)=0 and correlation ξ(t)ξ(t)=δ(tt).

In the experiment, the detected current Ih is frequency doubled, phase shifted, and fed back to modulate the power of the trapping beam [11]. This results in a feedback Hamiltonian, HfbIh(t)Ih(tδt)Qz2, where ωzδt is the phase delay introduced by the feedback circuit. When the phase is chosen such that Ih(tδt)Pz, the feedback Hamiltonian can be represented by Hfb=GIfbQz3, where G is the dimensionless feedback gain related to the trap intensity modulation [5,29],

MΔItItGχ2Φbzbzωz,
and the feedback current is Ifb=χ2ΦPz+2χ2Φξ(t), where ξ(t) has the same properties as ξ(t) (see Supplement 1 for details). This form of the Hamiltonian implies a feedback force, Ffb=Hfb/qz, which is equivalent to that used in experiments in the classical regime [5]. Taking the Markovian limit where the feedback occurs faster than any system time scale, and applying quantum feedback theory for homodyne detection [30], we find that the following superoperator must be added to Eq. (5):
F[ρ(t)]=iχ2ΦG[Qz3,{Pz,ρ}]χ2ΦG2D[Qz3]ρ,
where the first term on the right-hand side represents the desired cooling effect of the feedback, and the second term the accompanying backaction. We emphasize that in contrast to standard optomechanics, the feedback and backaction terms are highly nonlinear in the oscillator variables. The presence of this nonlinearity distinguishes our system from conventional cavity optomechanics and results in qualitatively different dynamics, as we show below.

The full master equation, assembled from Eqs. (5), (6), and (10), is then

ρ˙(t)=1i[H˜S,ρ(t)](At+Ap)D[Qz]ρ(t)/2BD[a]ρ(t)+B[ρ(t)]+F[ρ(t)],
where the new system Hamiltonian, H˜S=ωpvv+ωzbzbziαgz(vv)(bz+bz), accounts for the linearization of the probe implemented above, and the master equation now includes a mechanical decoherence term due to scattering from the probe, in addition to the trap beam, with heating rate Ap.

A sample experimental data set of the measured position of the nanoparticle along the y axis is shown in Fig. 2. At atmospheric pressures, Brownian effects, as given by Eq. (6), dominate, and the position of the particle follows a diffusive evolution, as can be seen in Fig. 2(a). At lower pressures the particle’s evolution becomes increasingly ballistic. The ensuing harmonic motion is shown in Fig. 2(b), both in the absence and in the presence of feedback [Eq. (10)]. The decrease in amplitude of the harmonic motion is due to the presence of parametric feedback cooling.

 

Fig. 2. (a) Diffusive evolution of the trapped nanoparticle’s position at atmospheric pressure, (b) harmonic motion of the nanoparticle at a lower pressure of 4×103 mbar. The reduction in the amplitude of the harmonic motion corresponds to the turning on of feedback, i.e., cooling.

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3. PHONON DYNAMICS

Employing the master equation [Eq. (11)] to consider the question of ground state occupation, tracing out the optical probe field, and using the resulting reduced master equation for the nanoparticle only, we find the equation for the dynamics of the phonon number (Nbzbz), N˙=JN2KN+L, where J=12[G9G2]χ2Φ, K=ηf/m+J, L=DJ/2, the dot denotes a time derivative, and D=Dp+Dq, with Dp=Dp+At+Ap accounting for positional decoherence. We assume that the nanoparticle is described by a thermal state [24,3134], for which N2=2N2+N [35], a relation that simplifies the phonon dynamics to

N˙=2JN2(J+K)N+L.
The total effect of parametric feedback on the phonon dynamics is contained in the parameter J, which is determined by the difference between the feedback cooling and backaction heating. In the experiments J0 [1,11], making the phonon dynamics of cooling nonlinear and the oscillator energy loss nonexponential, as shown below. We stress that this behavior is qualitatively different from standard quantum cavity optomechanical theory, which characterizes cooling as a linear damping process resulting in an exponential decay of energy (see, e.g., [31,32] and Eq. (82) in [18]). We note that G=Gopt=1/18 maximizes N˙ in Eq. (12), with the maximum nonlinear cooling rate Jmax=χ2Φ/3.

Assuming the initial condition N(0)N0=Dpm/ηfkBTeff/ωz, where Teff is the effective temperature of a bath due to gas and optical scattering combined, the analytical solution to Eq. (12) is

N(t)=(J+K)4J+12Jτtanh(tτ+θ),
where θ=tanh1[(2JN0+J+K)τ] and the cooling timescale τ=2[(J+K)2+8JL]1/2. From Eq. (13), the steady state phonon number is
NsslimtN(t)=12Jτ(J+K)4J
ηf2mN02J=Dp+At+Ap2J,
where the approximation is valid for N01. To reach the ground state, we need to maximize the feedback cooling J, which can be done by setting G=Gopt. We also need to minimize gas heating, which can be accomplished by going to low pressures and cryogenic temperatures, such that Dp is negligible in Eq. (14). Below we discuss situations involving realistic experimental parameters.

Two plots of the nonlinear phonon dynamics are shown in Fig. 3(a) for the z and y motion at 103 mbar, along with experimental values measured by us (circles). The solid curve represents N(t) as given in Eq. (13), while the dotted curve gives the corresponding equation for motion along one of the transverse directions y (which is nearly degenerate with x, i.e., ωyωx); see Fig. 1(b). In Fig. 3(b) we show three plots of the steady-state phonon number as the vacuum pressure is tuned. The solid and dotted curves represent Nss [Eq. (14a)] at 300 K for z and y motion, respectively, while the circles are experimental data. As can be seen, in all cases there is very good agreement between theory and experiment. The dashed curve in Fig. 3(b) predicts the steady-state phonon number for an identical configuration, but placed in a cryostat at 4 K. The ground state can be prepared if we start at high pressures, the particle is cooled while continuously increasing the feedback gain, as in [11], and we keep the trap modulation M=10%. Proceeding in this manner, we find that below 105mbar optimal feedback J=Jmax can be achieved, and the ground state occupied.

 

Fig. 3. (a) y and z phonon cooling dynamics [Eq. (13)], (b) steady-state phonon number versus pressure [Eq. (14a)]. The circles represent the experimental data, and the solid curve is our theoretical model for a fused silica sphere (ϵr=2.1 and density=2200kg/m3) of radius rd=50nm, 1064 nm trap (100 mW) and probe (10 mW) beams, mechanical frequency ωz/2π=38kHz, χ107, and trap intensity modulation M0.1%. The dotted lines represent the equivalent curves for one of the transverse degrees of freedom (ωy/2π=138kHz). The dashed curve in (a) represents the prediction of our theory for a setup placed in a cryostat with the feedback chosen optimally, keeping M10%.

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We note that practical cooling to lower phonon numbers is currently limited by a number of factors. These include high pressures enforced by nanosphere loading technologies, classical errors from the electronic feedback loop and laser noise, measurement uncertainties due to detector bandwidth limitations, and collection inefficiencies of the scattered light [3,5,11,22]. However, these problems are technical rather than fundamental, and efforts are under way to overcome these limitations [36], strongly suggesting the possibility of cooling to the ground state using the parameters presented in this article.

4. FORCE SENSING

We now consider force sensing using the nanoparticle model given by the master equation of Eq. (11). Since the state of the nanoparticle is continuously monitored, the master equation can be unraveled in terms of a set of Langevin equations describing the evolution of the quadratures Qz and Pz plus a stochastic force due to the measurement backaction, which gives [37,38]

Q˙z=L0#[Qz]=ωzPz,P˙z=L0#[Pz]+F/mωzz=ωzQzΓPz+F/mωzz,
where L0# is the Liouvillian superoperator (dual to the superoperator L0 appearing in the master equation ρ˙=L0[ρ]), defined by Tr(ρL0#[A])Tr(AL0[ρ]) for any arbitrary operator A [39]. The parameter Γ=Γ0+δΓ, where Γ0 is the gas damping and δΓ12χ2ΦG(N+1/2) is the nonlinear feedback damping [2,5,11]. Finally, F=FT+FF is the sum of the (independent) stochastic forces due to thermal and feedback backaction heating, respectively, with zero mean and correlations FT(t)FT(t)=STδ(tt) and FF(t)FF(t)=SFδ(tt), with
ST=2mΓ0kBTeff,SF=54mωzχ2ΦG2(2N2+2N+1).
The presence of the N-dependent factor in Eq. (16) implies that the feedback noise is dependent on the system state, and is therefore nonadditive. Furthermore, the dependence is nonlinear in N. Both of these features are fundamentally different from the typical additive feedback noise in standard cavity optomechanics, which is independent of the state of the system [33].

We convert Eq. (15) into the second-order differential equation for the position q¨z+Γq˙z+ωz2qz=F/m, and take its Fourier transform to find the position spectrum q˜z(ω)=χm(ω)F˜(ω), where

χm(ω)={m[(ωz2ω2)iωΓ]}1
is the optomechanical susceptibility of our oscillator. Finally, the positional power spectral density (PSD) noise spectrum is given by
|q˜z(ω)|2=|χm|2(ST+SF)+z2χ2Φ,
where the last term in the equation comes from the shot noise of the measured signal [Eq. (8)]. A typical example data set of the positional PSDs at moderate vacuum is shown in Fig. 4, along with fits to the theoretical expression of Eq. (18).

 

Fig. 4. Experimentally measured positional PSDs for all three degrees of freedom, with the dark lines representing the theoretical fits to the data [Eq. (18)]. Data were taken at a moderate vacuum pressure of 10 mbars and clearly show the Lorentzian shape of the resonance. These fits were used to extract the values of ωj, Γ, and Nss.

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In view of the fact that trapped nanoparticles offer the possibility of ultrasensitive force measurements [5,2123], we express our measurement noise spectrum [Eq. (18)] in terms of the estimator q˜(ω)/χm in order to investigate the fundamental limits of such measurements. The sensitivity of force estimation is set by the force noise PSD,

|F˜(ω)|2=ST+SF+SS(ω),
where SS(ω)=SS(0)[(1(ω/ωz)2)2+(ωΓ/ωz2)2] and SS(0)=(mzωz2)2/χ2Φ. Only the last term carries a ω dependence in Eq. (19). A plot of SS(ω) is shown in Fig. 5 in the high as well as low total damping Γ regimes, both of which are experimentally accessible [11,12]. The minimum value of SS(ω), and therefore the optimal force sensitivity, occurs at the response frequency ωopt=ωz2Γ2/2.

 

Fig. 5. (a) Plots of the shot noise force PSD [the last term of Eq. (19)] versus the normalized mechanical frequency ω/ωz for low and high total damping Γ. The minimum occurs for ωopt=ωz2Γ2/2. (b) Plot of the force sensitivity as a function of the normalized optical power Φ/ΦSQL at high vacuum. The standard quantum limit is reached when the shot noise balances the recoil and backaction noise [Eq. (19)].

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The first two terms in Eq. (19) scale linearly with the optical power, while the shot noise scales inversely (i.e., ST+SFΦ and SS1/Φ). Therefore, there is a power that minimizes the total noise, representing the standard quantum limit for our system. Assuming that the feedback is optimal (i.e., J=Jmax), the standard quantum limit is reached when χ2ΦSQLΓ/8, and equals

|F˜|2SQL=2mΓ0kBT+4mωz(At+2Γ).
The first term in Eq. (20) represents a thermal contribution from the background gas; the second term is due to the scattering of photons from the trapping beam; and the third term contains the effects of light scattering and shot noise from the probe, as well as the feedback backaction. At the low vacuum pressures currently available (i.e., 107mbar), the gas contribution is negligible, implying a minimum force sensitivity of |F˜|2SQL1021N/Hz=zN/Hz and optimal probe power of ωpΦSQL2mW, where the remaining system parameters have been taken from the caption of Fig. 3. Even at this limit the system can be readily used to test for violations of Newtonian gravity (1018N) [22] with moderate measurement bandwidths. However, backaction effects will impose long interrogation times on experiments searching for new small-scale (1021N) [21] and Casimir forces (1024N) [5,21]. Conversely, for short measurement times, our calculations show that backaction effects, which are of interest in their own right in optomechanics [40], can be observed at moderate laser powers and readily attainable vacuum pressures.

5. CONCLUSIONS

To conclude, we have presented a quantum model that describes the cooling and force sensing characteristics of an optically trapped subwavelength dielectric particle. We have shown that the predictions of this model for cooling are in very good agreement with experimentally measured occupation values in the classical regime. Further, we have demonstrated that quantum ground state preparation is challenging but achievable in anticipated experiments. Finally, we have derived the standard quantum limit to force sensing, indicating experiments where quantum backaction may play an important role. The model we present opens the door to the characterization of the quantum behavior of a system, important for macroscopic quantum mechanics [1,11], optical tweezing [2], ultrasensitive metrology [3], and nonequilibrium physics [4]. With the proper identifications, our theory is also applicable to electromechanical systems with parametric feedback [41].

Funding

Office of Naval Research (ONR) (N00014-14-1-0442, N00014-14-1-0803); Institute of Optics; University of Rochester.

Acknowledgment

We are grateful to C. Stroud, A. Aiello, B. Zwickl, and S. Agarwal for useful discussions. LPN acknowledges support from a University of Rochester Messersmith fellowship. ANV thanks the Institute of Optics for support.

 

See Supplement 1 for supporting content.

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35. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004).

36. P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015). [CrossRef]  

37. N. Gisin and I. C. Percival, “The quantum-state diffusion model applied to open systems,” J. Phys. A. 25, 5677–5691 (1992). [CrossRef]  

38. J. Halliwell and A. Zoupas, “Quantum state diffusion, density matrix diagonalization, and decoherent histories: a model,” Phys. Rev. D 52, 7294–7307 (1995). [CrossRef]  

39. K. Hornberger, Entanglement and Decoherence, Vol. 768 of Lecture Notes in Physics (Springer, 2009).

40. T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013). [CrossRef]  

41. L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011). [CrossRef]  

References

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  1. L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
    [Crossref]
  2. J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
    [Crossref]
  3. J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
    [Crossref]
  4. J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
    [Crossref]
  5. L. P. Neukirch and A. N. Vamivakas, “Nano-optomechanics with optically levitated nanoparticles,” Contemp. Phys. 56, 48–62 (2015).
  6. T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
    [Crossref]
  7. Z. Q. Yin, A. A. Geraci, and T. C. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
    [Crossref]
  8. J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).
  9. Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
    [Crossref]
  10. M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
    [Crossref]
  11. J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
    [Crossref]
  12. L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).
  13. J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
    [Crossref]
  14. N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
    [Crossref]
  15. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
    [Crossref]
  16. F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
    [Crossref]
  17. O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
    [Crossref]
  18. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
    [Crossref]
  19. P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. Lpz. 525, 215–233 (2013).
    [Crossref]
  20. P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).
  21. A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105, 101101 (2010).
    [Crossref]
  22. G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
    [Crossref]
  23. D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113, 251801 (2014).
    [Crossref]
  24. A. C. Pflanzer, O. Romero-Isart, and J. I. Cirac, “Master-equation approach to optomechanics with arbitrary dielectrics,” Phys. Rev. A 86, 013802 (2012).
    [Crossref]
  25. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 2002).
  26. L. Diósi, “Quantum master equation of a particle in a gas environment,” Europhys. Lett. 30, 63–68 (1995).
    [Crossref]
  27. D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
    [Crossref]
  28. C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd ed. (Springer, 2004).
  29. S. Mancini, D. Vitali, and P. Tombesi, “Optomechanical cooling of a macroscopic oscillator by homodyne feedback,” Phys. Rev. Lett. 80, 688–691 (1998).
    [Crossref]
  30. H. M. Wiseman and G. J. Milburn, “Quantum theory of optical feedback via homodyne detection,” Phys. Rev. Lett. 70, 548–551 (1993).
    [Crossref]
  31. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
    [Crossref]
  32. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
    [Crossref]
  33. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
    [Crossref]
  34. O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
    [Crossref]
  35. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004).
  36. P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
    [Crossref]
  37. N. Gisin and I. C. Percival, “The quantum-state diffusion model applied to open systems,” J. Phys. A. 25, 5677–5691 (1992).
    [Crossref]
  38. J. Halliwell and A. Zoupas, “Quantum state diffusion, density matrix diagonalization, and decoherent histories: a model,” Phys. Rev. D 52, 7294–7307 (1995).
    [Crossref]
  39. K. Hornberger, Entanglement and Decoherence, Vol. 768 of Lecture Notes in Physics (Springer, 2009).
  40. T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
    [Crossref]
  41. L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
    [Crossref]

2015 (6)

J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
[Crossref]

L. P. Neukirch and A. N. Vamivakas, “Nano-optomechanics with optically levitated nanoparticles,” Contemp. Phys. 56, 48–62 (2015).

L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref]

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

2014 (5)

D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113, 251801 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
[Crossref]

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).

2013 (8)

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref]

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref]

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
[Crossref]

Z. Q. Yin, A. A. Geraci, and T. C. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. Lpz. 525, 215–233 (2013).
[Crossref]

P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
[Crossref]

2012 (2)

A. C. Pflanzer, O. Romero-Isart, and J. I. Cirac, “Master-equation approach to optomechanics with arbitrary dielectrics,” Phys. Rev. A 86, 013802 (2012).
[Crossref]

J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
[Crossref]

2011 (4)

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
[Crossref]

2010 (2)

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105, 101101 (2010).
[Crossref]

2009 (1)

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

2008 (1)

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

2007 (3)

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
[Crossref]

1998 (1)

S. Mancini, D. Vitali, and P. Tombesi, “Optomechanical cooling of a macroscopic oscillator by homodyne feedback,” Phys. Rev. Lett. 80, 688–691 (1998).
[Crossref]

1995 (2)

L. Diósi, “Quantum master equation of a particle in a gas environment,” Europhys. Lett. 30, 63–68 (1995).
[Crossref]

J. Halliwell and A. Zoupas, “Quantum state diffusion, density matrix diagonalization, and decoherent histories: a model,” Phys. Rev. D 52, 7294–7307 (1995).
[Crossref]

1993 (1)

H. M. Wiseman and G. J. Milburn, “Quantum theory of optical feedback via homodyne detection,” Phys. Rev. Lett. 70, 548–551 (1993).
[Crossref]

1992 (1)

N. Gisin and I. C. Percival, “The quantum-state diffusion model applied to open systems,” J. Phys. A. 25, 5677–5691 (1992).
[Crossref]

Anders, J.

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

Arita, Y.

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref]

Arndt, M.

P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

Asenbaum, P.

P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Atherton, D. P.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

Barker, P.

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

Barker, P. F.

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref]

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref]

Bateman, J.

J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).

Berthelot, J.

P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

Blaser, F.

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

Bose, S.

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref]

Carmichael, H. J.

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 2002).

Chang, D. E.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

Chen, J. P.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

Cirac, J. I.

A. C. Pflanzer, O. Romero-Isart, and J. I. Cirac, “Master-equation approach to optomechanics with arbitrary dielectrics,” Phys. Rev. A 86, 013802 (2012).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

Clerk, A. A.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

Cross, M. C.

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
[Crossref]

Cunningham, M.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

Deesuwan, T.

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

Delic, U.

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

Dellago, C.

J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
[Crossref]

Deutsch, B.

J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
[Crossref]

Dholakia, K.

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref]

Diósi, L.

L. Diósi, “Quantum master equation of a particle in a gas environment,” Europhys. Lett. 30, 63–68 (1995).
[Crossref]

Fonseca, P. Z. G.

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref]

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd ed. (Springer, 2004).

Genes, C.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Geraci, A. A.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

Z. Q. Yin, A. A. Geraci, and T. C. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

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O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
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A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105, 101101 (2010).
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P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

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Z. Q. Yin, A. A. Geraci, and T. C. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
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S. Mancini, D. Vitali, and P. Tombesi, “Optomechanical cooling of a macroscopic oscillator by homodyne feedback,” Phys. Rev. Lett. 80, 688–691 (1998).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
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Matheny, M. H.

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
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P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. Lpz. 525, 215–233 (2013).
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H. M. Wiseman and G. J. Milburn, “Quantum theory of optical feedback via homodyne detection,” Phys. Rev. Lett. 70, 548–551 (1993).
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D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113, 251801 (2014).
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J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
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M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
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L. P. Neukirch and A. N. Vamivakas, “Nano-optomechanics with optically levitated nanoparticles,” Contemp. Phys. 56, 48–62 (2015).

L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).

L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
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L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).

L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
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J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).

P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

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I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
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P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
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J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
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J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
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L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
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J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
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D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
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D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105, 101101 (2010).
[Crossref]

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N. Gisin and I. C. Percival, “The quantum-state diffusion model applied to open systems,” J. Phys. A. 25, 5677–5691 (1992).
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A. C. Pflanzer, O. Romero-Isart, and J. I. Cirac, “Master-equation approach to optomechanics with arbitrary dielectrics,” Phys. Rev. A 86, 013802 (2012).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
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T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
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P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
[Crossref]

L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
[Crossref]

J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
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T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
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G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
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T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
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D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

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D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113, 251801 (2014).
[Crossref]

Romero-Isart, O.

A. C. Pflanzer, O. Romero-Isart, and J. I. Cirac, “Master-equation approach to optomechanics with arbitrary dielectrics,” Phys. Rev. A 86, 013802 (2012).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

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L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).

Roukes, M. L.

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
[Crossref]

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M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref]

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P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

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P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
[Crossref]

Stutz, J. H.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

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C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

S. Mancini, D. Vitali, and P. Tombesi, “Optomechanical cooling of a macroscopic oscillator by homodyne feedback,” Phys. Rev. Lett. 80, 688–691 (1998).
[Crossref]

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J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).

Vahala, K. J.

Vamivakas, A. N.

L. P. Neukirch and A. N. Vamivakas, “Nano-optomechanics with optically levitated nanoparticles,” Contemp. Phys. 56, 48–62 (2015).

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L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
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C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
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L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).

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D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
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H. M. Wiseman and G. J. Milburn, “Quantum theory of optical feedback via homodyne detection,” Phys. Rev. Lett. 70, 548–551 (1993).
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D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
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D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
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I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
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P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. Lpz. 525, 215–233 (2013).
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P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
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L. P. Neukirch and A. N. Vamivakas, “Nano-optomechanics with optically levitated nanoparticles,” Contemp. Phys. 56, 48–62 (2015).

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Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. (a) Image of the trapped nanoparticle and (b) schematic of the experiment modeled in this article. The schematic shows the electro-optic modulator (EOM), polarizing beam splitters (PBS1 and PBS2), high numerical-aperture lenses (L1 and L2), the detector (D), the beam dump (BD), and the feedback circuit gain (G).
Fig. 2.
Fig. 2. (a) Diffusive evolution of the trapped nanoparticle’s position at atmospheric pressure, (b) harmonic motion of the nanoparticle at a lower pressure of 4 × 10 3 mbar. The reduction in the amplitude of the harmonic motion corresponds to the turning on of feedback, i.e., cooling.
Fig. 3.
Fig. 3. (a)  y and z phonon cooling dynamics [Eq. (13)], (b) steady-state phonon number versus pressure [Eq. (14a)]. The circles represent the experimental data, and the solid curve is our theoretical model for a fused silica sphere ( ϵ r = 2.1 and density = 2200 kg / m 3 ) of radius r d = 50 nm , 1064 nm trap (100 mW) and probe (10 mW) beams, mechanical frequency ω z / 2 π = 38 kHz , χ 10 7 , and trap intensity modulation M 0.1 % . The dotted lines represent the equivalent curves for one of the transverse degrees of freedom ( ω y / 2 π = 138 kHz ). The dashed curve in (a) represents the prediction of our theory for a setup placed in a cryostat with the feedback chosen optimally, keeping M 10 % .
Fig. 4.
Fig. 4. Experimentally measured positional PSDs for all three degrees of freedom, with the dark lines representing the theoretical fits to the data [Eq. (18)]. Data were taken at a moderate vacuum pressure of 10 mbars and clearly show the Lorentzian shape of the resonance. These fits were used to extract the values of ω j , Γ , and N ss .
Fig. 5.
Fig. 5. (a) Plots of the shot noise force PSD [the last term of Eq. (19)] versus the normalized mechanical frequency ω / ω z for low and high total damping Γ . The minimum occurs for ω opt = ω z 2 Γ 2 / 2 . (b) Plot of the force sensitivity as a function of the normalized optical power Φ / Φ SQL at high vacuum. The standard quantum limit is reached when the shot noise balances the recoil and backaction noise [Eq. (19)].

Equations (21)

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H = H m + H f + H int .
H f = ω p a a + μ d 3 k ω k a μ ( k ) a μ ( k ) ,
H = H S + H B + H S B ,
H S = ω p a a + j ω j b j b j j g j a a ( b j + b j ) ,
ρ ˙ ( t ) = 1 i [ H S , ρ ] A t 2 D [ Q ] ρ + L sc ρ ,
B [ ρ ( t ) ] = D p 2 D [ Q z ] ρ D q 2 D [ P z ] ρ i η f 4 m [ Q z , { P z , ρ } ] ,
a out = a in + α χ 2 Q z ( t ) ,
I h = χ 2 Φ Q z ( t ) + χ 2 Φ ξ ( t ) ,
M Δ I t I t G χ 2 Φ b z b z ω z ,
F [ ρ ( t ) ] = i χ 2 Φ G [ Q z 3 , { P z , ρ } ] χ 2 Φ G 2 D [ Q z 3 ] ρ ,
ρ ˙ ( t ) = 1 i [ H ˜ S , ρ ( t ) ] ( A t + A p ) D [ Q z ] ρ ( t ) / 2 B D [ a ] ρ ( t ) + B [ ρ ( t ) ] + F [ ρ ( t ) ] ,
N ˙ = 2 J N 2 ( J + K ) N + L .
N ( t ) = ( J + K ) 4 J + 1 2 J τ tanh ( t τ + θ ) ,
N ss lim t N ( t ) = 1 2 J τ ( J + K ) 4 J
η f 2 m N 0 2 J = D p + A t + A p 2 J ,
Q ˙ z = L 0 # [ Q z ] = ω z P z , P ˙ z = L 0 # [ P z ] + F / m ω z z = ω z Q z Γ P z + F / m ω z z ,
S T = 2 m Γ 0 k B T eff , S F = 54 m ω z χ 2 Φ G 2 ( 2 N 2 + 2 N + 1 ) .
χ m ( ω ) = { m [ ( ω z 2 ω 2 ) i ω Γ ] } 1
| q ˜ z ( ω ) | 2 = | χ m | 2 ( S T + S F ) + z 2 χ 2 Φ ,
| F ˜ ( ω ) | 2 = S T + S F + S S ( ω ) ,
| F ˜ | 2 SQL = 2 m Γ 0 k B T + 4 m ω z ( A t + 2 Γ ) .

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