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
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 [1 –5]. Experiments with optically trapped harmonically oscillating subwavelength dielectric particles [6 –10] (see Fig. 1) have recently realized feedback cooling [1,11], nonlinear dynamics , nonequilibrium physics , coupling to spin degrees of freedom , and thermometry . All experiments thus far have been carried out in the classical regime.
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  and 64 K ), cavityless cooling has been able to reach much lower temperatures (50 mK ) 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 [14 –20]. 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 [5,21 –23]. 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. , 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.
The configuration Hamiltonian can be written as1), the first term on the right-hand side represents the mechanical kinetic energy , where is the three-dimensional momentum of the nanoparticle and its mass. The second term in Eq. (1) is the field energy , where is the sum of the trap , probe , and background 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, 1) the interaction Hamiltonian is given by , where we have assumed that the dielectric has a volume , and that it has a linear polarizability density , i.e., the polarization density is . Using the expressions for the electric fields from Supplement 1, we can evaluate for small particle displacements , and rewrite Eq. (1) as 3), the bath Hamiltonian is and the system–bath interaction Hamiltonian is , 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 and degrees of particle motion, since the dynamics along the three axes are independent of each other, and it suffices to analyze a single direction . The net result of our calculation is a master equation for the density matrix describing the optical probe and the motion of the nanoparticle,Supplement 1. The third superoperator describes the loss of photons from the probe, also due to scattering by the nanoparticle, , where is the oscillator length.11,27].
We now characterize the measurement of the oscillator displacement using input–output theory from quantum optics  applied to the nanoparticle. Specifically, the incoming probe field interacts with the nanoparticle, and the outgoing probe field carries a signature of this interaction (as shown in Supplement 1):28],
In the experiment, the detected current is frequency doubled, phase shifted, and fed back to modulate the power of the trapping beam . This results in a feedback Hamiltonian, , where is the phase delay introduced by the feedback circuit. When the phase is chosen such that , the feedback Hamiltonian can be represented by , where is the dimensionless feedback gain related to the trap intensity modulation [5,29],Supplement 1 for details). This form of the Hamiltonian implies a feedback force, , which is equivalent to that used in experiments in the classical regime . Taking the Markovian limit where the feedback occurs faster than any system time scale, and applying quantum feedback theory for homodyne detection , we find that the following superoperator must be added to Eq. (5):
A sample experimental data set of the measured position of the nanoparticle along the 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.
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 , , where , , , the dot denotes a time derivative, and , with accounting for positional decoherence. We assume that the nanoparticle is described by a thermal state [24,31 –34], for which , a relation that simplifies the phonon dynamics to1,11], 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 ). We note that maximizes in Eq. (12), with the maximum nonlinear cooling rate .
Assuming the initial condition , where is the effective temperature of a bath due to gas and optical scattering combined, the analytical solution to Eq. (12) is13), the steady state phonon number is 14). Below we discuss situations involving realistic experimental parameters.
Two plots of the nonlinear phonon dynamics are shown in Fig. 3(a) for the and motion at mbar, along with experimental values measured by us (circles). The solid curve represents as given in Eq. (13), while the dotted curve gives the corresponding equation for motion along one of the transverse directions (which is nearly degenerate with , i.e., ); 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 [Eq. (14a)] at 300 K for and 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 , and we keep the trap modulation . Proceeding in this manner, we find that below optimal feedback can be achieved, and the ground state occupied.
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 , 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 and plus a stochastic force due to the measurement backaction, which gives [37,38]39]. The parameter , where is the gas damping and is the nonlinear feedback damping [2,5,11]. Finally, is the sum of the (independent) stochastic forces due to thermal and feedback backaction heating, respectively, with zero mean and correlations and , with 16) implies that the feedback noise is dependent on the system state, and is therefore nonadditive. Furthermore, the dependence is nonlinear in . 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 .
We convert Eq. (15) into the second-order differential equation for the position , and take its Fourier transform to find the position spectrum , where8)]. 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).
In view of the fact that trapped nanoparticles offer the possibility of ultrasensitive force measurements [5,21 –23], we express our measurement noise spectrum [Eq. (18)] in terms of the estimator in order to investigate the fundamental limits of such measurements. The sensitivity of force estimation is set by the force noise PSD,19). A plot of is shown in Fig. 5 in the high as well as low total damping regimes, both of which are experimentally accessible [11,12]. The minimum value of , and therefore the optimal force sensitivity, occurs at the response frequency .
The first two terms in Eq. (19) scale linearly with the optical power, while the shot noise scales inversely (i.e., and ). 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., ), the standard quantum limit is reached when , and equals20) 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., ), the gas contribution is negligible, implying a minimum force sensitivity of and optimal probe power of , 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 ()  with moderate measurement bandwidths. However, backaction effects will impose long interrogation times on experiments searching for new small-scale ()  and Casimir forces () [5,21]. Conversely, for short measurement times, our calculations show that backaction effects, which are of interest in their own right in optomechanics , can be observed at moderate laser powers and readily attainable vacuum pressures.
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 , ultrasensitive metrology , and nonequilibrium physics . With the proper identifications, our theory is also applicable to electromechanical systems with parametric feedback .
Office of Naval Research (ONR) (N00014-14-1-0442, N00014-14-1-0803); Institute of Optics; University of Rochester.
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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