Quantum Model of Cooling and Force Sensing With an Optically Trapped Nanoparticle

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 ultra-weak forces using levitated nanoparticles.

Spectacular advances in experimental science have recently made it possible to observe quantum effects at more macroscopic scales than ever before [1]. These studies investigate, on the one hand, fundamental quantum phenomena such as superposition [2,3], entanglement [4,5], backaction [6] and decoherence [7]. On the other hand, they probe the limits placed by quantum mechanics on practical devices such as displacement [8,9], force [10,11] and phonon noise [12,13]

sensors.
A crucial technique underlying these advances involves the laser cooling of mechanical oscillators [14][15][16][17][18][19]. In these optomechanical systems, the radiation modes that interact with the mechanical oscillator are typically confined in an optical resonator. The operation of such systems in the quantum regime has been achieved experimentally [20], and is well understood theoretically [21][22][23]. However, resonator-based optomechanical systems face some major limitations. For example, they can be addressed only by specific (i.e. resonant) optical wavelengths, offer limited physical access to the mechanical component(s), cannot accommodate large or numerous mechanical elements which lower the cavity finesse by absorption or scattering, and are difficult to scale up.
In response to these challenges, substantial efforts have recently been made to develop cavityless optomechanical systems, using freely propagating radiation modes. A typical configuration consists of a harmonically oscillating dielectric particle, trapped optically and cooled via feedback. Ongoing experiments are aiming to broach the quantum regime [24][25][26][27][28][29][30]. However, to the best of our knowledge, no theoretical model exists which specifies the physical conditions for approaching -or operating in -the Orthogonally polarized Gaussian optical trap (solid line) and probe (dotted line) beams are combined on a polarizing beam splitter (PBS1) and focused on a nanoparticle using lenses L1 and L2. The trap confines the particle, while the probe is used to detect its motion; the foci of the two beams do not coincide, as shown in the inset, resulting in an optomechanical coupling linear in the particle coordinate. A second beamsplitter (PBS2) separates probe and trap beams, the latter being sent to a beam dump (BD). The probe light fluctuations are measured by detector D. The resulting measurement of the nanoparticle position is processed and fed back to the trap beam with strength G to cool the particle via the electro-optic modulator (EOM) [24].
the conditions for mechanical ground state occupation?
The physical system under consideration 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 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 [24]. In the remainder of this Letter, we identify the electromagnetic modes relevant to the problem, construct the system and bath Hamiltonians, derive the master equation for the system [31,32], and deduce and solve exactly the oscillator phonon number dynamics. Finally, we demonstrate agreement of our model with experimental data in the classical regime and also present a viable set of parameters for mechanical quantum ground state preparation.
We write the total electric field of the system as where the classical electric field of the Gaussian trapping beam of frequency ω t is E t = E 0 w 0 π/8G(r, ω t )e iωt(z/c−t) + c.c., where E 0 is is the amplitude of the field, and w 0 is the beam waist [33].
In the paraxial regime, the normalized transverse spatial mode near focus is given by , with r 2 ⊥ = x 2 + y 2 , the beam radius w(z) = w 0 [1 + z/z R 2 ] 1/2 and the Rayleigh range z R = ω t w 2 0 /2c. Further, in Eq. (1), the quantized electric field of the paraxial Gaussian probe beam of frequency ω p , linewidth ∆ω and beam waist w 0 is [34] where ∆r is the shift from the trap focus, e p is the probe polarization chosen orthogonal to that of the trap field, and the annihilation and creation operators for the probe field obey [a, a † ] = 1. Finally, in Eq. (1), the background field into which the photons are scattered by the nanoparticle is given by a superposition of plane waves in all directions µ indexes the polarization, e µ,k is a polarization vector, ω = c|k| and [a µ(k) , a † µ(k ′ ) ] = δ(k − k ′ ). We note that since the plane waves form a complete basis, Eq. (1) involves an overcounting of modes due to the addition of the trap and probe fields. This is justified in optomechanics as the trap and probe modes are of measure zero [33].
The configuration Hamiltonian may now be written, In Eq. (4), the first term on the right hand side represents the mechanical kinetic energy H m = |p| 2 /2m, where p is the three dimensional momentum of the nanoparticle, and m its mass. The second term in Eq. (4) is the field energy H f = ǫ 0 E(r) 2 d 3 r. Using the electric fields supplied above and ignoring the constant trap energy we find which is simply the sum of the probe and background field energies. Finally, the interaction Hamiltonian is given by , where we have assumed that the dielectric has a volume V , that its center-of-mass is located at r, and that it has a linear polarizability α p , i.e. the polarization P(r) = α p E(r). Using Eq. (1) and the expressions for the trap, probe and background fields we can evaluate H int for small particle displacements r, and rewrite Eq. (4) as where the system Hamiltonian is that of standard optomechanics [16] , and ∆x, ∆y, and ∆z are the spatial shifts between the foci of the trap and probe beams along the respective axes. The quantities ℓ j = /(2mω j ) are the oscillator lengths along each Cartesian axis and the mechanical operators obey [b j , b † j ] = 1. In Eq. (6), the bath Hamiltonian is , which represents the scattering of the trap and probe fields into the background, as shown more explicitly in the Supplementary Material.
We now trace over the bath modes, applying the standard Born and Markov approximations, since the systembath coupling is weak and the bath correlations decay quickly [3,31,32]. 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 [24]. 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 nanoparticlė where the first term on the right hand side represents unitary evolution of the system with H The second term corresponds to the positional decoherence of the nanoparticle due to scattering of trap photons, 6 ]. The third superoperator describes the loss of photons from the probe, also due to scattering by the nanoparticle, . 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. (8) the superoperator [35] where P z = i(b † z − b z ), and curly braces denote an anticommutator. The first term on the right hand side corresponds to momentum diffusion and The third term accounts for friction, and by Stokes law we have η f = 6πµr d , where r d is the radius of the nanoparticle and µ is the dynamic viscosity of the background gas. Absorption, heating due to blackbody radiation, and trap beam shot noise are negligible for our system [36].
We now characterize the measurement of the oscillator motion using input-output theory from quantum optics [37] applied to the nanoparticle, as in ion-cooling theory [38]. Specifically, the incoming probe field a in interacts with the nanoparticle, and the outgoing probe field a out carries a signature of this interaction [39] where χ = 4g z ∆t is the scaled optomechanical coupling, with integration time ∆t (ultimately determined by the detection bandwidth), and we have written the probe beam as a displaced vacuum (i.e. coherent state) a = −iα + v, with α a classical number and v a bosonic annihilation operator. A homodyne measurement on the output field yields a current [37] where Φ = α 2 ∆ω is the average detected flux of probe photons, and ξ(t) is a stochastic variable with mean In the experiment, the detected current I h is frequency doubled, phase shifted, and fed back to modulate the power of the trapping beam [24]. This results in a feedback Hamiltonian H fb = GI fb Q 3 z , where G is the dimensionless feedback gain [40,41] and the feedback current is I fb = χΦ P z + √ Φξ ′ (t) , where ξ ′ (t) has the same properties as ξ(t). This form of the Hamiltonian implies a feedback force F fb = −∂H fb /∂Q z which is equivalent to that used in experiments in the classical regime [40]. Taking the Markovian limit where the feedback occurs faster than any system timescale, and applying standard quantum feedback theory for homodyne detection [42], we find that the following superoperator must be added to Eq. (8) where the first term on the right hand side represents the desired effect of the feedback, and the second term the accompanying backaction. The full master equation, assembled from Eqs. (8), (9) and (12) accounts for the linearization of the probe implemented above.
now includes a mechanical decoherence term due to scattering from the probe, with A p = 7ℓ 2 z (ǫ c V ) 2 ω 6 p /(60c 6 π 3 w 2 0 ). Employing this master equation 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 (N the dot denotes a time derivative, and D = D ′ p + D q with D ′ p = D p + A t + A p Φ accounting for positional decoherence. Since the nanoparticle is expected to be in a thermal state [21-23, 31, 33], N 2 = 2 N 2 + N [43], a relation which simplifies the phonon dynamics to Below we will consider the case where J = 0. Before presenting the analytical solution to Eq. (14) for otherwise arbitrary values of J, K and L, we find the optimum value of the feedback, G opt . Optimizing the rate of phonon number decrease in Eq. (14) with respect to G we find Since 0 ≤ N < ∞, the optimal gain is also bounded, i.e. 3χ/4 ≤ G opt ≤ χ. From Fig. 2 (a), seeing that since G opt saturates rather quickly with N , and keeping in mind that feedback cooling is typically initiated at large phonon numbers [24], we pick G opt = χ, assuming the feedback is not changed dynamically. Starting with the initial condition N (0) where T eff is the effective temperature of a thermal bath due to gas and optical scattering combined, the analytical solution to Eq. (14) is where θ = tanh −1 (2JN 0 + J + K)/τ and the cooling timescale τ is given by From Eq. (16) the steady state phonon number is where the approximation is valid for N 0 ≫ 1. In the numerator of the radical, the three terms represent heating from the environment, feedback, and feedback backaction, respectively. If we choose G = G opt , then Eq. (18) becomes N ss ≈ η f N0 24mχ 2 Φ . Key plots using experimental parameters [24,40] are shown in Fig. 2(b) and (c). Figure 2  with experimental data in the classical regime, and also predicts that cooling to the ground state is possible, for optimized but realistic parameters.
To conclude, we have presented a model that describes the quantum optomechanics of an optically trapped subwavelength dielectric particle. We have shown that the predictions of this model are in very good agreement with experimentally measured steady state occupation values in the classical regime. Further, we have demonstrated that quantum ground state preparation lies within existing experimental capabilities. The master equation derived by us provides a general framework by which other aspects can be explored in future research. We are grateful to C. Stroud, A. Aiello, B. Zwickl, and S. Agarwal for useful discussions. This material is based upon work supported by the Office of Naval Research under Award Nos. N00014-14-1-0803 and N00014-14-1-0442. ANV thanks the Institute of Optics for generous support. LPN acknowledges support from a University of

ELECTRIC FIELDS
The total electric field as defined in the main article is written as (S1)

Trapping field
The electric field of the trapping beam of frequency ω t is assumed to be a classical Gaussian beam where w 0 is the beam waist and the mode profile with r ⊥ ≡ x 2 + y 2 , z is the axial propagation direction, E 0 ≡ E(0, 0), w(z) = w 0 1 + z/z R 2 is the beam waist, R(z) = z 1 + z R /z 2 is the radius of field curvature and z R ≡ πw 2 0 /λ = ωw 2 0 /2c is the Rayleigh range. Near focus (z ≪ z R ) the radius of curvature becomes R(z) ≈ z 2 r /z, the Guoy phase becomes arctan(z/z R ) ≈ 0, and therefore our mode profile is approximately (S4)

Probe field
The electric field of the paraxial Gaussian probe beam is a quantized beam with frequency ω p , linewidth ∆ω, and beam waist w 0 and can be written as [S1] where r ′ = r + ∆r, e p is the probe polarization assumed to be orthogonal to that of the trap field, and the annihilation and creation operators for the probe field obey the canonical commutation relation [a, a † ] = 1.

Background field
The background field into which optical scattering occurs is also quantized and can be written in the standard plane wave expansion where µ indexes the polarization, e µ,k is a polarization vector, ω = c|k| and [a µ(k) , a † µ(k ′ ) ] = δ(k − k ′ ).

Our Hamiltonian as given in the main text is
where H m = p 2 /2m is the particle's kinetic energy for the momentum p = p 2 x + p 2 y + p 2 z , and H f and H int are the free field interaction Hamiltonians respectively. The energy of the free field Hamiltonian is Using Eq. (S1) we explicitly write out the contributions as The various terms on the right hand side of Eq. (S9) are dealt with as follows.

Trap beam energy
Equation (S9a) can be neglected as it is purely an offset in the energy due to the classical trap field.
Eq. (S10) becomes which is as expected for the probe energy.

Background field energy
Equation (S9c) corresponds to the energy of the background field and the evaluation can be found in several textbooks [S2], (S13)

Field-background cross-term
The cross term between the trap and background fields [Eq. (S9d)] vanishes as their mutual overlap is very small. This cancellation also represents the avoidance of self-interference and mode overcounting in our model. The cross term between the probe and the background fields vanishes due to the same reason.
where ǫ c = 3(ǫ r − 1)/(ǫ r + 2) is the Clausius-Mossotti relation for the effective relative permittivity of a dielectric due to local field effects, and V denotes integration over the volume of the dielectric particle. Now if when we use the total electric field from Eq. (S1), we get Optical scattering (S17c) Trap potential First we consider the effect of the trap beam given by Eq. (S17a). Since the nanoparticle is smaller than the wavelength of any relevant optical field, the integration can be written as V d 3 r = V δ(q) d 3 r, where q is the center of mass position of the particle. Therefore H int for the trap-particle interaction is given by where q z is the longitudinal coordinate, and q x and q y are the transverse coordinates. We can ignore the overall constant in the above equation and we finally have for our trap Hamiltonian which is a quadratic trap in along all three axes. Using the standard form for a harmonic oscillator, we can write H trap as where ω z = ǫ c ǫ 0 E 2 0 V /(mz 2 R ) and ω x,y = 2ǫ c ǫ 0 E 2 0 V /(mw 2 0 ). Writing our canonical nanoparticle variables as allows us to rewrite Eq. (S20) as

DERIVATION OF THE DETECTED HOMODYNE CURRENT AND FEEDBACK
In order to find the input-output relations for the probe field, we write the system Hamiltonian (for the single degree of freedom q z ) as H s = ω p a † a + ω z b † z b z − ga † aQ z . If we move into the interaction picture for the probe field where a → ae −iωpt , then this becomes H s = ω z b † z b z − ga † aQ z . We assume the probe field is initially a coherent state which can be written as a = −iα + v, where α is a constant and v is a field annihilation operator. In this case the optomechanical coupling in our system Hamiltonian becomes The term proportional to |α| 2 Q z is responsible for simply shifting the mean position of the oscillator and can safely be ignored, leaving us with The Heisenberg equation of motion for v is given bẏ which can be integrated formally to give where the integration is taken over a time ∆t short compared to 1/ω z . If we had picked an initial time t f > t then we would have computed v(t) = v(t f ) − αg z Q z ∆t. By identifying the input state as a in = −iα + v(t 0 ), and the output state as a out = −iα + v(t f ), then we can relate the output and input fields by where we have defined the variable χ ≡ 4g z ∆t. Now the detected photon number operator is given by where the last two terms are negligibly small. The detected homodyne current is found by subtracting off the constant offset, α 2 , and is therefore given as where we have converted from photon numbers to rates by using the identity for the photon flux Φ ≡ a † a ∆ω = α 2 ∆ω, and have introduced the stochastic variable ξ(t) which accounts for the shot noise introduced by the term iα(v in − v † in ). As described in the main text, phase shifting the measured signal (or equivalently adding a short time delay) is equivalent to measuring a different quadrature of motion, i.e. the current that is fed back is I fb = χΦ P z + √ Φξ ′ (t), where ξ ′ (t) has the same properties as ξ(t). Now writing σ ≡ χP z allows us to write the master equation for the feedback in standard notation [S4] where the Liouvillian superoperator K is defined as K[ρ] = [F, ρ]/i and where F is the feedback term that comes from the feedback Hamiltonian H fb = I fb F , and is chosen to be F = GQ 3 z to match the classical physics of the problem as described in the main text. The gain coefficient G may be related to the average relative trap power modulation ∆P/P t in the experiment by using the relation ∆P/P t = GχΦ Q P /ω m ≈ GχΦ N /ω m [S5].