Binding lab

In this lab, we study optical binding (light-induced dipole-dipole) interactions and probe the collective dynamics of several optically trapped nanoparticles. Our experiment offers time-dependent control and independent readout of particle motion, allowing us to reconstruct and analyze the collective states.

The experiment and our latest results are described in detail below.

THE APPARATUS

Trap creation

NON-HERMITIAN DYNAMICS

Linear motion

Hopf bifurcation

TEAM

Postdocs

Dr. Murad Abuzarli

PhD students

Livia Egyed

Clemens Jakubec

Manuel Reisenbauer

Master's students

Ludovico Ricciardelli

THE APPARATUS

Trap creation

To create a tunable tweezer array, we use the acousto-optical deflectors (AOD). They make use of the acousto-optical effect, diffracting light through the interaction with a sound wave that is sent through the material using a piezo-electric transducer. The angle of the first diffraction order θ of light at wavelength λ on a lattice depends on the lattice constant d as sin(θ)=λ/2d. In an AOD, the lattice constant d is given by the wavelength of our radio-frequency (RF) tone as the sound wave modifies the AOD crystal density.

We use two perpendicularly aligned AODs to create the two-dimensional trap array. Each AOD is run with n RF tones, creating n first diffraction orders. The first diffraction orders of the first horizontally aligned AOD are aligned along the horizontal axis, with the distance between them proportional to the frequency difference between the RF tones. We then reimage them into the second, vertically aligned AOD, such that each beam gets diffracted n times again, but now along the vertical axis, creating an n x n array of first diffraction orders. The RF driving frequency and phase are imprinted onto the light diffracted by the AODs. By running both AODs with the same n RF tones, the frequencies of the resulting trap array will be equal along its diagonal, as these are the traps created by getting diffracted exactly once by each of the n RF tones (see Figure 1).

Figure 1: The setup to create and control the laser beams. We create a 1D tweezer array from the laser beams on the diagonal of the 2D array, where the optical frequencies can be made degenerate. We tilt a Dove prism to rotate the tweezer array into the horizontal plane.

The resulting laser beam array is sent through a 7:1 telescope, where we select only the diagonal traps with a mechanical slit in the focal plane. The transmitted laser beams are rotated 45 degrees into the horizontal plane with a Dove prism. At the position of the trapping lens, the two traps spatially overlap again, allowing us to focus them both down close to the diffraction limit using an aspheric lens with a numerical aperture of NA = 0.77 and trapping the particles in the tweezers. The distance between the two focal spots now depends on the angle at which the light gets diffracted upon the AODs. For example, we can tune the distance by changing the driving tones of both AODs in opposite directions, while keeping the optical frequencies the same (see Figure 2).

Figure 2: Tuning the driving frequencies changes the distance between the traps while leaving the frequency of the diagonal traps unchanged.

Detection

To detect the particles' motion along the optical axis, we perform an interferometric measurement. We collect the light scattered off our particles in the backward direction as the motion is imprinted as a phase shift φj ≈ 2kzj. To ensure the independent detection of two particles, we reimage the trapping plane onto a mirror prism to reflect the two signals in opposite directions. Finally, we direct them towards single-mode 50:50 fiber beamsplitters aligned in a 4f configuration with the prism. This is a fiber-based adaptation of a confocal detection scheme, which allows us to increase the information content of the collected light over the detection background. A sketch of our detection scheme can be seen in Figure 3 (for clarity, beamsplitters are shown here as being in free space). The collected back-scattered light is overlapped with a local oscillator with variable frequency ωLO on the beamsplitter to form a balanced homodyne or heterodyne detection.

Figure 3: Detection of the backscattered light. We reimage the trapping plane onto a right-angled prism in order to spatially separate the light scattered by the two particles. We then mode-match the dipole radiation pattern with the fiber mode at the fiber tip using a confocal alignment to maximize the signal-to-noise ratio.

Distance calibration

As explained in the section Trap creation, the distance between the particles is proportional to the frequency difference of the drive tones. To convert this frequency difference into a physical distance, we use the distance-dependent interference between the light scattered off the two particles. A camera along the particle connection axis records the light scattered off both particles along the same axis (see Figure 4). As we change the inter-particle distance d, we observe a change in spot-brightness according to

where Δφ corresponds to the phase difference between the tweezers. By fitting this dependence, we can find a conversion factor between the change in the RF drive frequencies and the interparticle distance.

Figure 4: Distance calibration scheme. We change the distance between the particles while recording their interference pattern on a camera. The periodicity of the pattern is used to calculate the conversion between frequency difference and moved distance.

OPTICAL BINDING

The interaction between particles arises through resonant light scattering. Each particle can be viewed as a point dipole that scatters light toward the other particle according to Erj = k3αEj eikd cos θ/dε0, where θ is the axis of polarization of the incoming light, α is the particle polarizability, and Ej is the incoming electric field of particle j. The scattered field thus has the same frequency and phase as the incident field, decays linearly with the distance d, and acquires a traveling phase kd. The light scattered from one particle can interfere with the trapping field of the second particle if the two trapping fields are of equal optical frequency. This interference effect leads to a position-dependent force along the optical axis. For small particle motion along the z-axis (z1, z2 ≪ λ), we can approximate this force as [1,2]:

with the reciprocal coupling rate gr = cos(kd) cos(Δφ)/kd and anti-reciprocal coupling rate ga= sin(kd) cos(Δφ)/kd.

The forces F1 and F2 are, in general, non-reciprocal, meaning that F1 ≠ -F2 if ga ≠ 0. This apparent breaking of Newton's third law results from the phase-dependent interference at the two-particle sites (see Figure 5). It leads to non-Hermitian particle dynamics, which prohibits us from defining a general, joint system Hamiltonian when tracing out the light fields.

Equations of motion

In the linearized regime, we can write the equations of motion for the slow-changing amplitudes a1 and a2 of the particle oscillations [2]

with the non-Hermitian dynamical matrix HNH

Here, ΔΩ = Ω2-Ω1 is the mechanical frequency difference of the uncoupled oscillators and γ is the mechanical damping rate. Depending on the interparticle distance d and the optical phase difference Δφ, the interaction can be tuned from reciprocal to unidirectional to anti-reciprocal. In general, the matrix HNH is non-Hermitian for any anti-reciprocal coupling rate ga ≠ 0.

[1] Rieser et al., Science 377, 987 (2022)

[2] Rudolph et al., Phys. Rev. Lett. 133, 233603 (2024)

Figure 5: Schematic example of anti-reciprocal interaction. The two tweezers have a relative optical phase difference of Δφ=π/2. Due to the distance between them, light traveling from one particle to the other acquires an additional phase shift of π/2. This results in a destructive interference of the scattered and trapping light at the position of the first particle and a constructive interference at the position of the second particle. The resulting forces are maximally nonreciprocal as F1 = F2.

Figure 6: Phases of quantum optical binding interactions depending on the interparticle distance d and the optical phase difference Δφ: I. fully reciprocal, II. unidirectional, III. anti-reciprocal, IV. no classical binding but forces due to shot noise. Figure adapted from Phys. Rev. Lett. 133, 233603.

INTERACTION TYPES

Depending on the interparticle distance and the relative optical phase, we can engineer various interaction types: reciprocal, anti-reciprocal, and any combination of the two with varying weights. We show the two "extreme" cases and the case of the unidirectional interactions below.

Reciprocal

The coupling between the particles is fully reciprocal only for the special case of ga = 0, where we retrieve conservative system dynamics and a joint system Hamiltonian exists. Due to the coupling, the particles' motion hybridizes into normal modes that can be tuned by changing the detuning of mechanical frequencies via the relative trap power. At the zero detuning, i.e., when the eigenfrequencies of the uncoupled system are degenerate, the normal mode eigenfrequencies split if gr > γ/2, where γ corresponds to the intrinsic damping rate. Here, the normal modes correspond to the center-of-mass (CoM, "in-phase") mode z+=(z1+z2)/2, and the breathing ("out-of-phase") mode z−=(z1-z2)/2. While the frequency of the CoM mode remains equal to that of the uncoupled system, the frequency of the breathing mode at the position of the avoided crossing shifts away by 2gr. Figure 7 shows how the particles' motion hybridize as we bring their frequencies closer together by increasing the trap stiffness of the first particle (above) and decreasing it for the second particle (below).

We can engineer the reciprocal interaction by, e.g., setting the distance between the particles to a multiple of the wavelength (d = nλ, with n∈N). Modifying the optical phase difference between the tweezers is used to tune from maximum positive coupling (Δφ = 0) over no coupling (Δφ = π/2) to maximum negative coupling (Δφ = π). Flipping the sign of the coupling rate is observed as a change in the sign of the frequency shift of the breathing mode [1].

[1] Rieser et al., Science 377, 987 (2022)

Figure 7: Spectrograms of the motion of the first (above) and second particle (below) for changing relative trap power/mechanical frequency detuning. The trap stiffness increases (decreases) for the first (second) particle. For equal trap power, we observe an avoided crossing, and each particle moves as a combination of two normal modes with equal weights.

Unidirectional

In the case of setting the distance and phase difference such that kd = ±Δφ = ± (π/4, 3π/4), the reciprocal and anti-reciprocal coupling rates are equal in magnitude ga = ±gr. This realizes a unidirectional coupling in our system, such that, e.g., the motion of one particle creates a force that acts on the other particle, but the force in the other direction is zero. We observe this effect in the spectrograms of the particles' motion, as shown in Figure 8, where gr = ga such that the first particle doesn't act on the second particle. The first particle's spectrogram shows a contribution of the second particle's motion, while the first particle's motion is not seen in the spectrogram of the second particle. At the cross of the two lines, we see an exceptional point (EP) where the system's eigenvectors coalesce. For all other mechanical detunings, there are two normal modes with real eigenfrequencies, forming a "pseudo-Hermitian" system.

Figure 8: Spectrograms of the motion of two particles for changing relative trap power. The spectrogram of particle 1 features a stronger peak at its intrinsic frequency and a weaker peak that shows the motion of particle 2. The motion of particle 2 remains unaffected by particle 1, as there is no peak corresponding to the particle 1 motion.

Anti-reciprocal

The case of fully anti-reciprocal interaction can be obtained in analogy to the reciprocal case by now setting gr = 0, which can be done by, e.g., setting kd = ±π/2. The force acting on the two particles is now of equal sign (Figure 5), as their coupling rates are equal in magnitude but opposite in sign. This amplifies oscillation amplitudes, as one particle pushes the other while simultaneously getting pulled by the latter. This interaction breaks the time-reversal (PT) symmetry, as the reverse process cannot happen.

Figure 9 shows the coalescence of the eigenfrequencies of the new eigenmodes at two exceptional points near zero mechanical detuning. In contrast to the unidirectional case, a pair of EPs now exists between which the eigenfrequencies are degenerate. Still, their damping rates become non-degenerate, as shown later in Figure 10.

Figure 9: Spectrograms in the case of anti-reciprocal interaction. In a certain region close to the zero mechanical detuning, the eigenfrequencies become degenerate, giving rise to the PT-symmetry broken system dynamics.

NON-HERMITIAN DYNAMICS

Most of the works in nonreciprocal dynamics explore the region between reciprocal and unidirectional dynamics. Therefore, the anti-reciprocal interaction between trapped nanoparticles is quite unique. In our latest work [3], we focused on this interaction regime and probed the collective dynamics of two nanoparticles in the linear and nonlinear regime. We show a summary of our results below.

[3] Reisenbauer, Rudolph, Egyed et al., Nature Physics 20, 1629 (2024)

Linear motion

The complex eigenvalues of the non-Hermitian dynamical matrix HNH due to an anti-reciprocal interaction are given by:

where γ is the mechanical damping rate (in our system, dominated by gas collisions) and ΔΩ = Ω2-Ω1 is the mechanical frequency difference of the uncoupled oscillators. The eigenfrequencies and damping rates of the coupled system correspond to the real and imaginary parts of these eigenmodes Ω± = Ω0 + Re(λ±) with Ω0=(Ω1+Ω2)/2 and γ± = -Im(λ±), respectively. For some values of ΔΩ, the resulting eigenfrequencies become degenerate. In contrast, their damping rates become non-degenerate. The effect can be represented by an interference of particles' Stokes and anti-Stokes sidebands, which yields two normal modes with amplification ("gain") and suppression ("loss"). The region is bounded by two exceptional points (EPs), for which eigenfrequencies and damping rates are the same as the system eigenvectors coalesce (Figure 10).

Figure 10c shows the coalescence of the eigenfrequencies of the new eigenmodes at exceptional points for small detunings ΔΩ. For detunings between the EPs, we can reconstruct the PSD of the new normal modes at the maximum splitting of the damping rates, as seen in Figure 10b. Furthermore, we observe a stable phase delay between the two particles as their motions become strongly correlated in the region between the EPs, the PT-symmetry broken regime.

Figure 10: a. Due to the anti-reciprocal interaction, the interference of the Stokes and anti-Stokes sidebands results in two normal modes with different damping rates. b. We reconstruct the normal modes from raw time traces of the particles' position (blue and orange spectra). Two peaks with different damping rates show the amplified (red) and the suppressed mode (green). c. The frequency sweep shows the region of degenerate eigenfrequencies and non-degenerate damping rates.

Hopf bifurcation

In our experiment, we can further tune the interaction strength by changing the polarization of the tweezer fields, which modifies the intensity of the scattered light along the particle connecting axis. Our linearized theory breaks down for strong enough coupling rates ga > γ/2 as the effective damping of the amplified normal mode becomes negative, leading, in theory, to infinite oscillation amplitudes. In practice, nonlinear terms in the optical forces stabilize the increasing amplitudes, forming a limit cycle in the position-velocity phase space with an amplitude Ā that is comparable in size to the wavelength (see Figure 11). The nonlinear equations of motion for Ā (in units of λ/2π) and the phase delay ψ:

The equations of motion can be solved to show three different regions:

Region I: This is the PT-symmetric region, where the eigenfrequencies are real and non-degenerate. The motion is linear, and some correlation is visible.

Region IIa: This is the part of the PT symmetry-broken region where the particle motion is still linear, but the eigenfrequencies are degenerate, and the damping rates are non-degenerate.

Region IIb: This is the limit cycle phase where the motion is nonlinear. The motion is strongly correlated.

We observe these phases clearly from the measurements of the limit cycle radius Ā (Figure 11b) and the phase delay ψ (Figure 11c).

Figure 11: a. Spectrogram of the system eigenfrequencies as a function of the mechanical detuning shows the PT symmetry-broken regime where the frequencies are degenerate. b. In this regime, a limit cycle phase arises where the particles' dynamics become nonlinear. c. Particles move collectively due to the interaction, with the largest correlation in the limit cycle phase.

Figure 12: Phases of non-Hermitian dynamics of two nanoparticles. a. Theory solutions to the equations of motion show three distinct regions. b. The measured limit cycle amplitude matches the theoretical expectation well. Right: Position-velocity phase spaces for particle 1: (i) for almost no interaction and far-detuned oscillators, the distribution is purely thermal; (ii) and (iii) above the Hopf bifurcation, the distribution becomes a limit cycle with an increasing amplitude Ā as the coupling rate is increased; (iv) at the edge of the PT symmetry-broken regime, the thermal distribution broadens due to the proximity to the exceptional point.

The full phase diagram obtained from the solution to the equations of motion is shown in Figure 12a. We characterize the transition between the linear and nonlinear dynamics by measuring the displacement amplitude Ā of the particles's motion, which has a discrete step at the Hopf bifurcation line. In Figure 12b, we show the measured displacement amplitude for different system parameters, which agrees with the theoretical prediction [3]. Insets (i) - (iv) show characteristic probability distributions in the position-velocity distributions of one particle, with a limit cycle clearly observed in (ii) and (iii).

[3] Reisenbauer, Rudolph, Egyed et al., Nature Physics 20, 1629 (2024)

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