Optimal cooling of multiple levitated particles: Theory of far-field wavefront shaping

The opportunity to manipulate small-scale objects pushes us to the limits of our understanding of physics. Particularly promising in this regard is the interdisciplinary field of levitation, in which light fields can be harnessed to isolate nanoparticles from their environment by levitating them optically. When cooled towards their motional quantum ground state, levitated systems offer the tantalizing prospect of displaying mesoscopic quantum properties. While the interest in levitation has so far been focused mainly on manipulating individual objects with simple shapes, the field is currently moving towards the control of more complex structures, such as those featuring multiple particles or different degrees of freedom. Unfortunately, current cooling techniques are mostly designed for single objects and thus cannot easily be multiplexed to address such coupled many-body systems. Here we present an approach based on the spatial modulation of light in the far field to cool multiple nano-objects in parallel. Our procedure is based on the experimentally measurable scattering matrix and on its changes with time. We demonstrate how to compose from these ingredients a linear energy-shift operator, whose eigenstates are identified as the incoming wavefronts that implement the most efficient cooling of complex moving ensembles of levitated particles. Submitted in parallel with Hüpfl et al. [Phys. Rev. Lett. 130, 083203 (2023)], this article provides a theoretical and numerical study of the expected cooling performance as well as of the robustness of the method against environmental parameters.

Figure 1

(a) Simplistic illustration of the cooling concept. A many-body system is composed of nanoparticles (blue beads) in random motion (black arrows). The system is opened on two sides, through which spatially modulated light fields (“incident wave,” blue to yellow patterns) get injected. The wavefronts (orange wave lines) are designed to counteract the motion of the nanoparticles. (b) In our numerical simulations, this cooling approach is implemented in a 2D system, such as a thin waveguide (gray structure, small extension in the $y$ direction), in which nanoparticles (blue cylinders) experience a 2D motion in the ($x,z$) plane. The shaped wavefronts, which are used to cool the motion of the nanoparticles, get injected from both sides. Inside the waveguide, the field is composed of propagating transverse electric modes (the intensity distribution of the first two modes on each side are indicated by orange shapes). The shaped incident and outgoing wavefronts are expressed as a linear combination of such modes through the coefficient vectors ${\vec{c}}^{\text{in}}, {\vec{c}}^{\text{out}}$. These input and output vectors are connected through the scattering matrix, $\mathit{S}\left(t\right)$, which evolves through time $t$ due to particles' motion.

Figure 2

(a), (b) Optimal cooling states applied to a single circular nanobead with radius 750 nm in (a) and 75 nm in (b), respectively. The waveguide in which the particles move has a width of $W=5.5 \mu \mathrm{m}$, corresponding to $M=10$ propagating waveguide modes. The instantaneous velocities of the beads are marked by white arrows, while the optimal cooling states create scattering patterns that produce forces marked by red arrows. The quantity ${\rho }_w^{\text{trans}}$ below the panels characterizes the correlation between the velocity and the force vectors (limited by the globally optimum value of ${\rho }_w^{\text{trans}}=-1$). (c) Optimal cooling states applied to a rectangular particle experiencing translation and rotation. The waveguide dimensions are identical to (a) and (b), while the longer edge of the rectangle is set to $\approx 1.5 \mu \mathrm{m}$. The rigid body translation and rotation of the particle are marked by the straight and curved white arrows, respectively. The optimal cooling state induces a force on the center of mass and a torque that are marked by the straight and curved red arrows. ${\rho }_w$ is the total correlation of center of mass and angular velocity against the optical force and torque.

Figure 3

(a) Results of the cooling procedure applied to 10 triangles, 10 squares, and 10 pentagons, respectively, moving in a waveguide without friction (all particles have the same outer radius of $r=150$ nm). Blue, orange, and green indicate the particle geometry (see inset). Dotted and solid curves display the evolution of the translational, $E_{\text{kin}}^{\text{trans}}$, and rotational, $E_{\text{kin}}^{\text{rot}}$, kinetic energy throughout the process. The sampling time $\mathrm{\Delta }{t}_{\text{cool}}=1 \mu \mathrm{s}$, and the light field is characterized by a power $P=20 \mu \mathrm{W}$, a wavelength $\lambda =532$ nm, and $M=10$ transverse waveguide modes. Compared to translational degrees of freedom, rotational degrees of freedom are addressed at characteristic times dependent of the particles' shapes. (b) Evolution of the correlation between force velocity (${\rho }_w^{\text{trans}}$, blue) and between torque-angular velocity (${\rho }_w^{\text{rot}}$, orange) during the cooling of the 10 squares; see orange curves in (a) (data include a running average over 10 time steps). At early times, only the translation degrees of freedom are cooled and ${\rho }_w^{\text{trans}}$ settles to high negative values while ${\rho }_w^{\text{rot}}$ remains low. The situation is reversed when the rotational degrees of freedom start to be cooled. A snapshot of the optical field (yellow to blue pattern) produced by an optimal cooling state is provided in inset. (c) Simulation results for $N_{\text{scat}}=10$ beads of radius 75 nm, subject now to a stochastic motion as in Eq. (5) that is characterized by a damping rate $\gamma =1$ kHz and a surrounding temperature $T_{\text{env}}=30\mathrm{K}$. All the other parameters are identical to the ones used in (a) and (b). In phase (i), the velocities of the beads start out thermally distributed according to a (low) initial temperature of $T=0.01·T_{\text{env}}$. Thermalization to the bath temperature at $T_{\text{env}}=30\mathrm{K}$ is observed in the first $\approx 2$ ms (the horizontal blue dotted line indicates the prediction from the equipartition theorem). Over the next $\approx 1$ ms in the constant state phase (ii), a field constant over time ($P=20 \mu \mathrm{W}$) gets injected from both waveguide leads and produces an increase in energy. Afterwards, the field is turned off and in phase (iii), from 3 ms to 5 ms, the particles thermalize back towards $T_{\text{env}}$. Finally, in the cooling phase (iv), our method is applied using a power of 20 $\mu \mathrm{W}$, 2 $\mu \mathrm{W}$ or 200 nW (solid orange, dashed green, and dotted red curves, respectively), leading to a significant drop in energy.

Figure 4

(a) $N_{\text{scat}}=5$ circular nanobeads ($r=75$ nm, white circles) are trapped by the standing-wave pattern of a laser (blue-to-yellow surface). This trapping laser (${\lambda }_{\text{trap}}=1.5 \mu \mathrm{m}, P_{\text{trap}}=200 \mu \mathrm{W}$) gets injected from both sides of the waveguide along its fundamental transverse mode. The stochastic motion of the beads is described by Eq. (5) ($\gamma =6$ Hz, $T_{\mathrm{env}}=30\mathrm{K}$) with the additional influence of gravity ($g$, red arrow) and of the trapping field. The cooling light field (not shown here) is characterized by a wavelength $\lambda =532$ nm and $M=10$ transverse waveguide modes. (b) Evolution of $E_{\text{tot}}$ throughout the cooling process. In the first 500 $\mu \mathrm{s}$ (light blue region), no cooling procedure is applied, and the beads stay thermalized at $T_{\text{env}}$ (blue dashed line). The procedure is launched at $t=500 \mu \mathrm{s}$ with a cooling power of 200 nW (yellow region) and is observed to lower $E_{\text{tot}}$ to ${10}^{-25}J$ in $\approx 2$ ms. (c) Power spectral density, $|S_{zz}|$, of the spatial component $z$ for the rightmost of the five particles in (a) (similar behaviors are found for all the particles). The orange curve indicates the spectrum measured before cooling, which is characterized by a main resonance at ${\omega }_z$ = 40 kHz and extra harmonics (e.g., ${\omega }_z/2, 3/2{\omega }_z, 2{\omega }_z$). The green (blue) curve displays $|S_{zz}|$ obtained after cooling the system with a cooling power of $P=20$ nW (200 nW), while the white dashed (yellow solid) line describes its fitting by a Lorentzian model that is used to extract the system's temperature. (d) The procedure of (a) and (b) is systematically applied to different sets of $N_{\text{scat}}=5$ trapped nanobeads, while varying the cooling power $P$. The total energy at the start (end) of the simulation $E_{\text{tot}}\left(t_0\right)$ ($E_{\text{cool}}^{\text{sim}}$) is marked by blue squares (orange circles). The dotted green (dashed red) line depicts the estimator of the cooled energy $E_{\text{cool}}$ of Eq. (8) [the optimal cooled energy $E_{\text{cool}}^{\text{opt}}$ of Eq. (10) for ${\overline{\rho }}_w=-1$].

Figure 5

(a)–(c) Total energies at the outset of the cooling procedure [$E_{\text{tot}}\left(t_0\right)$, blue squares] and at its end ($E_{\text{cool}}^{\text{sim}}$, orange circles), resulting from the simulations of the systematic cooling of random initial configurations of circular nanobeads ($r=75$ nm). The cooling light field is characterized by a power $P=2 \mu \mathrm{W}$ and a wavelength $\lambda =532$ nm. The dotted green (respectively dashed red) line depicts the estimator of the cooled energy $E_{\text{cool}}$ of Eq. (8) [respectively the optimal cooled energy $E_{\text{cool}}^{\text{opt}}$ of Eq. (10)]. In (a), the number of particles, $N_{\text{scat}}$, is varied while the number of transverse waveguide modes $M=10$ is fixed and the purple crosses ($E_{\text{cool}}^{\text{opt},\text{sim}}$) report cooling performed under the optimal conditions of Eq. (10). In (b), $M$ is varied while we set $N_{\text{scat}}=10$. In (c), $N_{\text{scat}}=10$ and $M=10$, while we vary the number of transverse waveguide modes $M_{\text{used}}$ used to assemble the energy-shift operator ${\mathit{Q}}_{\text{ES}}$. (d) Efficiency of the cooling of $N_{\text{scat}}=10$ nanobeads, while varying the damping rate, $\gamma$, and the laser power $P$. The green dots indicate when the cooled energy is at most half the initial energy (red crosses show when this is not the case). The blue solid and orange dashed lines indicate the bounds provided by Eq. (11).

Figure 6

The blue shapes depict two rigid dielectrics of cylindrical shape at time $t=t_1$, which are tagged with a permittivity ${\epsilon }_i(.,t_1)$ and whose center of mass is labeled $z_i\left(t_1\right)$. The lighter shaded shape depicts the position of the particles at a time $t=t_0$ (permittivity ${\epsilon }_i(.,t_0)$ and center of mass $z_i\left(t_0\right)$). Throughout the motion of the particles, a location $x$ inside the first dielectric at $t=t_0$ gets displaced to $r_i(x,t)$ at time $t$.

Figure 7

(a), (b) Initial total energy (blue squares, $E_{\text{tot}}\left(t_0\right)$) and end total energy (orange circles, $E_{\text{cool}}^{\text{sim}}$), recorded when systematically cooling ($P=2 \mu \mathrm{W}$) a random configuration of circular nanobeads ($r=75$ nm and $M=10$). In (a) the sampling time, $\mathrm{\Delta }{t}_{\text{cool}}$, is varied while the damping rate is fixed to $\gamma =6$ Hz. In (b) the damping rate is varied while the sampling time is fixed to $\mathrm{\Delta }{t}_{\text{cool}}=1 \mu \mathrm{s}$. The green dotted (red dashed) line marks the prediction given by Eq. (E15) [Eq. (E17)], labeled as $E_{\text{cool}}$ ($E_{\text{cool}}^{\text{opt}}$).

Figure 8

(a) A system containing $N_{\text{scat}}=10$ ciruclar nanobeads with various radii $r$ is cooled using different sampling times, $\mathrm{\Delta }{t}_{\text{cool}}$. The green dots (red crosses) indicate when the cooled energy is (is not) at most half the initial thermal equilibrium energy. The blue line describes the boundary provided in Eq. (F4). (b) Systems composed of $N_{\text{scat}}=10$ nanobeads are subject to different surrounding temperatures, $T_{\text{env}}$, and cooled using different sampling times, $\mathrm{\Delta }{t}_{\text{cool}}$. The green dots (red crosses) indicate when the cooled energy is (is not) at most half the initial energy. The blue solid line describes the boundary provided in Eq. (F4), while the orange dashed line corresponds to the averaged lower boundary of Eq. (F2). In both cases good agreements between numerics and analytical estimates are found.

Figure 9

Time evolution of the total energy of a system composed of 10 nanobeads ($r=75$ nm), which are cooled in a waveguide of M=20 modes and using a power of 2 $\mu \mathrm{W}$, while noise deteriorates the measurement of $\mathit{S}$. The continuous red, dash-dotted green, dashed orange, and dotted blue curves correspond to noise levels characterized by signal-to-noise ratios of 50 000, 500, 50, and 5, respectively.

Figure 10

$N=4$ circular beads with radius $r=750$ nm evolving in a waveguide are cooled with a sampling rate of $\mathrm{\Delta }{t}_{\text{cool}}=1$ ms, while using a cooling laser of wavelength $\lambda =1500$ nm allowing for four propagating transverse modes and a power $P=40$ nW. The black solid line depicts the kinetic energy of the system, while the orange dotted (blue dashed) line indicates the stationary energy of the (un)cooled system.