Quantum Brownian Motion of a particle from Casimir-Polder Interactions

We study the fluctuation-induced dissipative dynamics of the quantized center of mass motion of a polarizable dielectric particle trapped near a surface. The particle's center of mass is treated as an open quantum system coupled to the electromagnetic field acting as its environment, with the resulting system dynamics described by a quantum Brownian motion master equation. The dissipation and decoherence of the particle's center of mass are characterized by the modified spectral density of the electromagnetic field that depends on surface losses and the strength of the classical trap field. Our results are relevant to experiments with levitated dielectric particles near surfaces, illustrating potential ways of mitigating fluctuation-induced decoherence while preparing such systems in macroscopic quantum states.


I. INTRODUCTION
Creating macroscopic superpositions of massive systems as a means to understand the quantum-to-classical transition is a task of foundational importance [1]. Among promising experimental platforms for realizing large superpositions of massive objects, levitated optomechanical systems bring together the advantages of optical trapping and cooling methods in terms of control, while being well-isolated from an environment in the absence of mechanical clamping, thus minimizing decoherence [2][3][4][5][6]. There has been astonishing experimental progress in terms of control and manipulation of levitated dielectric nanoparticles -ranging from recent demonstrations of cooling particles down to micro-and milli-kelvins, [7][8][9], to observation of rotational frequencies as large as MHz-GHz with remarkable stabilities [10][11][12].
Interfacing such precisely controlled mesoscopic quantum systems with waveguides further allows for better manipulation and probing mechanisms of the system of interest, as guided photonic modes can couple efficiently to particles in the near-field regime [13][14][15]. Near-field levitated nanophotonics can therefore allow for a strong optomechanical couplings of mesoscopic systems with well-controlled fields, as has been demonstrated in [16].
However, when preparing a system in a macroscopic quantum state near surfaces, one needs to consider that the quantum (and thermal) fluctuations of the electromagnetic (EM) field are enhanced due to the presence of the surface degrees of freedom [17]. The increased density of EM field modes can therefore cause the system of interest to decohere faster in the vicinity of a surface, as has been shown both theoretically and experimentally with regard to the internal degrees of freedom of particles near surfaces [18][19][20][21][22][23][24]. It is similarly imperative to analyze the fluctuation-induced decoherence for the ex- * kanu@umd.edu † ysubasi@lanl.gov ternal degrees of freedom in near-field nanomechanical experiments [25].
In this paper we study the decoherence and dissipation of the quantized center of mass (COM) motion of a neutral dielectric particle trapped near a surface. We show that the open system dynamics of the particle can be described in terms of the quantum Brownian motion (QBM) master equation [26][27][28], and the surfacemodified dissipation and decoherence can be expressed in terms of a modified spectral density of the electromagnetic field. We further draw a correspondence between the surface-induced decoherence of the particle's COM and the collisional model of decoherence [29,30].
The paper is organized as follows. In section II we develop a theoretical model of a polarizable dielectric particle interacting with the EM field in the presence of a surface, deriving the fluctuation-and drive-induced potentials. In section III we derive the QBM master equation for the quantized COM motion, and analyze the resulting decoherence and dissipation of the particle for different surface properties in section IV. We summarize our findings in section V.

II. MODEL
Let us consider a dielectric particle of mass M and polarizabilityᾱ (ω) placed at a distance z from a planar half-space medium of permittivity S (ω), as depicted in Fig. 1 (a). We consider a classical driving field that is incident normally on the surface of the medium and reflected to form a standing wave potential. We assume that the particle is trapped near the first intensity maxima of the standing wave potential near the surface [16].
The Hamiltonian for the total system can be written asĤ where the first term corresponds to the kinetic energy of arXiv:1910.01560v1 [quant-ph] 3 Oct 2019 1. (a) Schematic representation of a dielectric particle of mass M , radius R and polarizabilityᾱ (ω) near a planar halfspace with permittivity S (ω), interacting with the quantum fluctuations of the EM field. We assume a classical trap field incident normally on the surface which creates a standing wave potential perpendicular to the surface. The particle is trapped in the first intensity maxima of the standing wave potential, at a distance z from the surface. (b) The four contributions to the total interaction energy as a result of the interaction between the total electric field and the induced polarization of the particle -(I) corresponds to the classical trap potentialĤ Tr due to the interaction of the trap field with the clasically induced polarization (Eq. (8)). (II) stands for the Casimir-Polder interactionĤ CP arising from the interaction between the fluctuations of the EM field with the fluctuating dipole moment of the particle (Eq. (9)). (III) and (IV) taken together lead to the drive-induced Casimir-Polder HamiltonianĤ DCP (Eq. (10)). (III) arises from the interaction between the classical field and the dipole moment fluctuations and (IV) corresponds to the interaction between the classically induced polarization and the EM field fluctuations.
the particle withp z as the quantized COM momentum along the z-axis. In the absence of a strong transverse confinement we ignore the quantized motion in the xyplane.Ĥ F corresponds to the Hamiltonian of the quantized field in the presence of the medium (see Eq. (A1)). The interaction HamiltonianĤ int represents the electricdipole interaction between the polarizable particle and the total electric field over the volume V of the particle given by [31] whereP (r ) refers to the polarization of the dielectric, E(r ) is the electric field, with r being a point in the volume of the particle. The total electric field at a position r can be expressed asÊ (r ) = E 0 (r , t) +Ê f (r ), where E 0 (r ) is the classical trap field, andÊ f (r ) refers to the fluctuations of the field (quantum and thermal). We assume that the trap field is given as with ω 0 as the frequency, and E 0 (r ) as the amplitude of the electric field at position r which takes the incident and reflected fields into consideration. We further assume that the field is polarized along the xy-plane.
Using the macroscopic QED formalism [32,33], the electric field fluctuations in the presence of a surface are given as (see (A4)) whereḠ λ (r 1 , r 2 , ω) stands for the propagator of a field excitation between points r 1 and r 2 , as described by Eq. (A7)-(A10) [34]. Assuming that the particle has a linear, homogeneous and isotropic polarizabilityᾱ (r , ω) ≡ α (ω) 1, we can write the induced polarization of the particle asP (r ) = P 0 (r , t) +P f (r ) [31], where is the polarization induced by the classical field and corresponds to the polarization induced due to the fluctuations of the EM field. We now assume that the dielectric particle is pointlike, with the COM position of the particler M =1r+ẑe z such that r corresponds to the classical COM coordinates andẑ represents the quantum fluctuations of the COM motion along the z-axis about the classical trap position.
Using Eq. (3)-(6), one can rewrite the interaction Hamiltonian Eq. (2) aŝ whereĤ corresponds to the trap Hamiltonian, stands for the Casimir-Polder (CP) interaction Hamiltonian, and is the driven Casimir-Polder (DCP) interaction. The first term in the above stands for the fluctuating dipole interacting with the classical trap field, and the second term corresponds to the classically driven dipole interacting with the EM field fluctuations at the position of the particle, as depicted by the processes (III) and (IV) in Fig. 1 (b). We study each of these contributions in detail in the following.

A. Classical Trap
The classical trap potential to zeroth order in the COM fluctuations is given as where we have taken a time-average over the electric field. We note that the factor of 1/2 is introduced to avoid the double sum of the energy associated with the interaction of the induced polarization and electric field [31]. We have further assumed here that the dielectric particle has negligible internal loss with a real polarizability such that α (ω) = α * (ω). ExpandingĤ Tr to second order in the COM fluctuationsẑ around the classical equilibrium position r 0 , and ignoring constant energy shifts, one obtains the trap potential asV corresponds to the frequency of the trap due to the classical field. We have assumed here that the electric field amplitude for the standing wave formed by the classical trap field goes as E 0 (r 0 ) ∼ e ik0z as a function of z.

B. Casimir-Polder Interaction
Considering the interaction between field and polarization fluctuations to zeroth order in COM fluctuationsẑ, such thatĤ (0) CP ≡ −P f (r) ·Ê f (r), one can obtain the Casimir-Polder potential as U CP (r) ≡ 1 2 Tr F ρ FĤ CP . This can be evaluated in first order perturbation theory as [33] where we have assumed that the field density matrixρ F corresponds to a thermal state with temperature T and n th (ω) = f † λ (r, ω) ·f λ (r, ω) = 1 e ω/(k B T ) −1 is the average number of thermal photons in the mode ω. All surface properties enter into consideration through the scattering Green's tensorḠ sc (r, r, iξ) (see Eq.(A10)) corresponding to the propagation of a virtual photon from the position (r) of the particle to the surface and back. Given that imaginary frequencies are associated with virtual interactions, the above potential can be physically understood as coming from the interaction between the fluctuations of the dipole and those of the vacuum EM field, summed over all frequencies of virtual photons exchanged between the particle and the surface. The second term corresponds to the scattering and reabsorption of thermal fluctuations of the EM field by the particle off the surface.

C. Drive-induced Casimir-Polder Interaction
The linearized part of the interaction Hamiltonian with respect to the classical field gives a drive-induced contribution to zeroth order in the COM fluctuations given bŷ One can derive a corresponding drive-induced Casimir-Polder potential in second-order perturbation theory as [35] (see Appendix C for details) The above shift is analogous to the resonant Casimir-Polder shift for the excited state of a two-level atom [33,35,36]. This can be understood as coming from a process wherein a classically-induced dipole scatters a photon off of the surface and reabsorbs it.

D. Total Potential
The total potential for the classical coordinate of the particle can be written as the sum of Eq. (11), (12) and (14) as U Tot (r) = U Tr (r) + U CP (r) + U DCP (r), which yields the classical equilibrium position r 0 of the particle such that ∂ z U Tot (r) | r0 = 0.
Expanding the CP and DCP Hamiltonians to second order inẑ around r 0 would lead to additional corrections to the trap frequency. Assuming that the potential associated with the CP and DCP contributions arê 2 , the total free Hamiltonian for the particle is the sum of its kinetic energy term and the harmonic trap potential given bŷ where the total trap frequency is defined as Ω ≡ Ω 2 Tr + Ω 2 CP + Ω 2 DCP .

III. QBM FOR THE PARTICLE IN THE PRESENCE OF A SURFACE
We now study the open system dynamics of the quantized COM motion of the particle as the system of interest, interacting with the fluctuations of the EM field as the bath. It can be seen that in the presence of an external trapping field, to the lowest order in field fluctutations, the coupling between the quantized COM motion and the EM field fluctuations arises due to the driveinduced Casimir-Polder HamiltonianĤ DCP . Expandinĝ H DCP to first order in the COM motion fluctuations, we obtain an interaction Hamiltonian between the quantized COM motion and the fluctuations of the field aŝ whereB (t) is the bath operator defined aŝ We remark that while the last two terms in the above with ∂ z E 0 (r) and ∂ z P 0 (r) vanish at the field intensity maxima, the equilibrium position r 0 is shifted from that point due to the presence of the CP and DCP potentials.
Moving to a rotating frame of reference with respect to the total free Hamiltonian, we can write the interaction Hamiltonian (Eq. (16)) in the interaction picture as In the interaction picture we can thus describe the dynamics of the COM in terms of a Born-Markov master equation as [27] whereρ M refers to the density matrix for the quantized COM motion of the particle. Performing a trace over the EM field in the above one obtains the following QBM master equation where first term corresponds to trap frequency renormalization, the second term corresponds to dissipation or friction, the third term represents decoherence in the position basis, and the last term corresponds to momentum diffusion. The dissipation and noise kernels in the above master equation are given as (see Appendix D for details of the derivation) where we have defined the bath operator in the inter- The kernels D (τ ) and N (τ ) correspond to the standard QBM dissipation and noise kernels, respectively. Here J (ω, r 0 ) ≡ J free (ω, r 0 ) + J sc (ω, r 0 ) is the effective spectral density of the EM field in the presence of the surface, with the free space and scattering contributions given by where we have defined To physically interpret the spectral density obtained above, we note the following features from Eq. (22) • The spectral density scales as the square of the induced dipole of the dielectric particle. The part of total spectral density that depends on (α (ω 0 )) 2 arises due to the classically-induced dipole, corresponding to the emission and reabsorption of a photon by the classical dipole. The part of the spectral density that depends on (α (ω)) 2 arises from the interactions of the fluctuating dipole with its image via the classical trap field. Terms that go as ∼ α(ω 0 )α(ω) can be understood as coming from processes where a classical dipole scatters a photon, inducing a fluctuating dipole in the medium, which in turn interacts with the classical dipole via the trap field. This can be seen from the derivation of the dissipation and noise kernels in Appendix D.
• Given that the imaginary part of the surface scattering Green's tensor ImḠ sc corresponds to the surface loss, the density of modes increases near a lossy surface. This indicates that a lossy surface with a large number of fluctuating degrees of freedom leads to a larger dissipation and decoherence for the quantized COM dynamics, as a result of the fluctuation-dissipation theorem. Previously it has also been shown that surface loss leads to additional dissipation and decoherence for the internal degrees of the particle [18][19][20][21][22][23][24].
• In addition to the surface-induced modifications, there is also dissipation and decoherence due to the interaction of the particle with the free space EM field modes, as given by the free space Green's tensor contribution. This can be understood as arising from scattering of the classical drive photons by the particle into free space modes.
We can now define the dissipation and decoherence coefficients as This shows that the dissipation and decoherence depend only on the effective spectral density evaluated at the mechanical sideband frequencies of the driving field. It can be seen that the above expressions are analogous to those for optomechanical damping and radiation pressure induced noise [37]. This allows us to simplify the master equation Eq.
where we have defined the renormalized free Hamiltonian for the center of mass as H M which includes the frequency renormalization due to the first term in Eq. (19), and ignored the momentum diffusion term [27].

IV. DECOHERENCE AND QUANTUM FRICTION FOR A DIELECTRIC NANOSPHERE
As a concrete example, we now evaluate the decoherence and dissipation for a dielectric nanosphere near a planar half-space. The parameter values corresponding to the particle and surface, and simplifying assumptions are given as follows.

A. Parameter values and Assumptions
• We consider a dielectric nanosphere made of silica with a radius R = 72 nm, and density ρ ≈ 2000 kg/m 3 [16].
• We assume that the particle is trapped in a harmonic potential along the z axis, with a trap frequency Ω ≈ 3 MHz.
• Considering that the classical drive frequency is much larger than that for the mechanical trap ω 0 Ω, and ω 0 k B T one can simplify Eq. (24) and (25) to obtain the following simple expressions for the dissipation and noise • For the purpose of estimation we consider in the following that the equilibrium position is roughly given by the classical trap field intensity maximum, such that we can approximate the spectral density in Eq. (22) as where we have assumed that ∂ z E 0 (r 0 ) ≈ 0 and defined the free-space and scattering recoil Green's tensor as where we have assumed the trap field to be polarized along the x-axis.

B. Surface properties
We calculate the influence of the free space and surface scattered EM field given by the the imaginary part of the Green's tensor as follows.

Free space
Observing that the free space recoil Green's tensor is G free (ω, r 0 ) = ω 3 15πc 3 , using Eq. (31) the contribution to the spectral density due to the free space EM field modes is given by where we have defined analogous to the dissipation rate of a dipole of strength d ≡ α(ω0)+α(ω) 2 |E 0 | interacting with the vacuum EM field. It can be seen from Eq. (30) that this yields a decoherence rate of which corresponds to the position decoherence of the particle arising from the scattering of drive photons into free space modes.

Perfect conductor
For a perfectly conducting planar surface (with the Fresnel coefficients r p = 1 and r s = −1) the scattering part of the recoil Green's tensor is given as (see Eq. (B6)), wherez ≡ k 0 z is the dimensionless distance of the particle from the surface. In the near-field limitz 1, we find that G NR sc,pc (ω, r 0 ) ≈ ω 3 15πc 3 , where NR stands for nonretarded regime. This yields the density of modes in the subwavelength limit as Comparing with Eq. (33), we see that the density of modes is twice that of the free space, which can be understood as a sum of the field radiated by the dipole and its image. The corresponding localization parameter is given as We note that in the absence of surface losses the decoherence of the particle in the near-field regime is independent of its distance from the surface, as can be seen from Fig. 2 (b).

Metal
For a metal surface we assume the permittivity function to be given by the Drude model The scattering recoil Green's tensor is given as (see Eq. (B6)) where the Fresnel coefficients are as given by Eq. (B2). In the near-field limit the scattering recoil Green's tensor can be simplified to G NR sc,met (ω, r 0 ) ≈ 3ω 3 , yielding a density of modes near a metal surface as One can thus write the decoherence of the particle near a metal half-space as where γ sc (r 0 ) ≈ 3 8z 3 Im S (ω0)−1 S (ω0)+1 γ 0 (ω 0 ) is surfacemodified photon scattering rate (see Eq. (C16)). As seen from Fig. 2 (b), the decoherence of the particle's COM in the near field regime is well-approximated by the above expression.

C. Correspondence to collisional model of decoherence
We note that the deocherence term in the master equation Eq. (26) is of the position localization decoherence (PLD) form [30]. To understand this, we note that a similar form of decoherence term can also be obtained from a collisional model of decoherence, wherein the system in consideration is bombarded by individual scatterers from the environment. As each scattering bath particle interacts with the system via a local interaction and gets correlated, it acquires some information about the system's position as a result. Thus, upon tracing out the bath, the system exhibits decoherence in the position basis. Particularly in the limit where the scatterer has a much longer de Broglie wavelength compared to the coherence length scale of the system one obtains a decoherence term as in Eq. (26) [29,30]. This correspondence in the decoherence dynamics from two different models suggests that the decoherence of a particle near a surface arises due to scattering of virtual photons off of the surface.
We note that the decoherence rate due to scattering of photons in free space goes as Λ free ∼ k 2 eff γ eff (see Eq. (35)), where k eff refers to an effective wavevector for the scattered photon and γ eff is the rate of scattering.
Considering that Λ NR met ∼ γ sc (r 0 ) /z 2 , we deduce from Eq. (42) that the virtual photons inducing decoherence have an effective de Broglie wavelength ∼ k −1 eff ∼ z that scales as the distance of the particle from the surface. We remark that a similar effective de Broglie wavelength was previously also derived in [36] in the context of recoil heating of a driven atom near a surface.

V. DISCUSSION
To summarize, we have derived a quantum Brownian motion master equation for the quantized center of mass motion of a dielectric particle trapped near a surface. Considering the particle to be trapped with an external classical field, we find that there are three different contributions to the total potential seen by the particlea classical trap potential, Casimir-Polder potential and driven CP potential, as illustrated in Fig. 1(b). Taken together, these potentials lead the particle to be trapped close to the first intensity maxima of the standing wave potential formed by the classical field (see Section II D). The interaction between the quantized COM motion and the fluctuations of the EM field to the lowest order is described by a linear expansion of the driven CP potential (Eq. (10)) about the equilibrium position. Tracing out the EM field fluctutations, we arrive at a second-order Born-Markov master equation describing the dissipative dynamics of the particle's COM as given by Eq. (19). The resulting dynamics is governed by a quantum Brownian motion master equation, with an effective spectral density that is determined by the polarizability of the particle, properties of the surface, and the strength of the external trapping field (Eq. (22)). The dissipation and decoherence that arise as a result can be understood as coming from the classically induced dipole scattering field fluctuations, and the fluctuating dipole scattering the drive photons. We then estimate the decoherence and dissipation for a dielectric nanosphere near different surfaces, and find that the quantized COM decoherence and dissipation increase in the presence of a lossy medium (Fig. 2). We further illustrate a correspondence between the resulting decoherence and that from the collisional model in the long-wavelength limit [29,30].
Comparing the resulting decoherence due to surface fluctuations with that arising from other sources as a benchmark, we observe that surface-induced decoherence can potentially pose a fundamental limit for preparing a dielectric particle in macroscopic COM quantum states. It can be seen from Appendix E that the decoherence due to background gas scattering and blackbody radiation can be reduced significantly by going to lower pressures and temperatures, respectively. In the present analysis we have derived the spectral density that governs the surface-modifications to fluctuation phenomena for the quantized COM of a particle. This could allow one to systematically modify the surface properties and drive strength in order to mitigate the surface-induced dissipation and decoherence. As quantum optical systems are being increasingly miniaturized, and mesoscopic quantum components being regularly interfaced with surfaces and waveguides at nanoscales, our results provide new insights into tailoring fluctuation phenomena in these regimes [41][42][43][44].

Appendix A: Medium-assisted EM field
Using the macroscopic QED formalism [32,33], the Hamiltonian for the vacuum EM field in the presence of the surface can be written as withf † λ (r, ω) andf λ (r, ω) as the bosonic creation and annihilation operators respectively that take into account the presence of the media. Physically these can be understood as the ladder operators corresponding to the noise polarization (λ = e) and magnetization (λ = m) in the medium-assisted EM field, at frequency ω, created or annihilated at position r. The medium-assisted bosonic operators obey the canonical commutation relations The electric and magnetic field operators evaluated at the position of the particle are given as The coefficientsḠ λ (r 1 , r 2 , ω) are defined as with (r, ω) and µ(r, ω) as the space-dependent permittivity and permeability, andḠ (r, r , ω) as the Green's tensor for a point dipole near a surface [32,33]. The Green's tensor is defined as the solution to the Helmholtz equation in the presence of the boundary conditions The total Green's tensor can be expressed as whereḠ free (r 1 , r 2 , ω) andḠ sc (r 1 , r 2 , ω) refer to the free space and scattering components of the total Green's tensor.

Appendix B: Scattering Green's tensor near a planar surface
For a point dipole located at the position r 1 near an infinite planar half-space, one can write the scattering Green's tensor as [32] G sc (r 1 , r 2 , ω) = 1 8π with |r 1 − r 2 | = r, (r 1 + r 2 ) · e z = (z 1 + z 2 ), and we have defined the relative coordinate vector between the points r 1 and r 2 as r1−r2 |r1−r2| ≡ x12 r , 0, z1−z2 r T . Here r s,p are the Fresnel reflection coefficients for the s and p polarizations reflecting off the surface, and κ 2 ⊥ = −k 2 + k 2 , where k = ω/c. Assuming that the medium can be treated as homogeneous and isotropic, and can be well-described in terms of its bulk optical properties at the length scales of the particle-surface separations, we can consider that all the information about the surface material is accounted for in the following Fresnel reflection coefficients In the non-retarded limit (z 1), one can expand the Fresnel coefficients in Eq. (B2) to lowest order in For coincident points (r 1 = r 2 = r 0 ), one can write the scattering Green's tensor as [32] G sc (r 0 , r 0 , ω) = 1 8π The double z-derivative of the xx component of the recoil Green's tensor (see Eq. (32)) is then given as The free space Green's tensor between the points r 1 and r 2 is given as

Appendix C: Derivation of the driven Casimir-Polder potential
Using time-dependent second-order perturbation theory, we can define the energy correction and the modification to the dissipation rate of the system arising due to the driven CP HamiltonianĤ where we have defined the electric field and polarization fluctuation operators in the interaction picture as [45]. The average is taken over the thermal state of the field. We note that the shifts and decay rates are consistent with those derived via the second-order Born-Markov master equation [27].
We further divide the above into 4 separate terms as such that χ = (I) + (II) + (III) + (IV). Let us consider the first term as follows where in the second step we take an average over the field density matrixρ F =ρ th , such that Physically this is equivalent to saying that the virtual excitations of the EM field emitted and absorbed by the particle occur at the same frequency, position and space coordinate, and have an average number expectation value of n th (ω). We can further simplify (I) as where in the first step we have made the rotating wave approximation and used the fluctuation-dissipation rela-tion [32] λ=e,m In the second step while performing the integral over τ , we note that ∞ 0 dτ e iτ x = πδ (x) + iP 1 x . To evaluate the principal value term, we make a contour integral over the first and the second quadrants of the upper half complex plane.
We can similarly simplify the remaining terms to find that (I) = (II) = (III) = (IV).
This yields the total potential as This is the driven CP potential as given in Eq. (14), which is in agreement with the result in [35] at zero temperature. We can similarly also find the surface-modified scattering rate as γ sc (r) = µ 0 ω 2 0 (α (ω 0 )) 2 (2n th (ω 0 ) + 1) For a particle in the near-field limit of a surface with permittivity S (ω), for n th (ω 0 ) 1 the above can be approximated as

Appendix D: Derivation of the dissipation and noise kernels
Let us consider the two-time correlation functions of the bath operators as follows, We further divide the correlator above into sixteen parts as follows such that B (t)B (t − τ ) = 16 j=1 C j (τ ). We now consider the first term closely as follows = µ 0 2π cos (ω 0 τ ) dω ω 2 (α (ω 0 )) 2 E 0 (r 0 ) · ∂ z ImḠ (r 0 , r 0 , ω) ∂ z · E * 0 (r 0 ) n th (ω) e iωτ + {n th (ω) + 1} e −iωτ , (D20) where we have averaged over the thermal state of the field and used the fluctuation-dissipation relation for the Green's tensor in obtaining the third step.