Photons carry momentum, and thus their scattering not only modifies light propagation but at the same time induces forces on particles. Confining mobile scatterers and light in a closed volume thus generates a complex coupled nonlinear dynamics. As a striking example, one finds a phase transition from random order to a crystalline structure if particles within a resonator are illuminated by a sufficiently strong laser. This phase transition can be simply understood as a minimization of the optical potential energy of the particles in concurrence with a maximization of light scattering into the resonator. Here, we generalize the self-ordering dynamics to several illumination colors and cavity modes. In this enlarged model, particles adapt dynamically to current illumination conditions to ensure maximal simultaneous scattering of all frequencies into the resonator as a sort of self-optimizing light collection system with built-in memory. Such adaptive self-ordering dynamics in optical resonators could be implemented in a wide range of systems from cold atoms and molecules to mobile nanoparticles in solution. In the quantum regime, it enables exploration of uncharted regions of multiparticle phases, allowing simulation of Hopfield networks, associative memories, or generalized Hamiltonian mean field models.
© 2014 Optical Society of America
Polarizable particles in an optical resonator which are coherently illuminated from the side at sufficient intensity will undergo a phase transition from homogeneous to crystalline order accompanied by super-radiant collective light scattering [1–3]. This transition can be understood from a simultaneous maximization of collective scattering of pump light into the cavity and the depth of the corresponding optical potential created via interference of pump and cavity light [4,5]. In a monochromatic plane wave geometry, the particles form a Bragg-like grating structure, which optimally couples cavity and pump wave so that the intensity of the scattered light grows with the square of the particle number. In a lossy cavity, the ordering process is dissipative and cools the particles, so that in the long time limit light is constantly collectively scattered into the cavity. For varying frequency and geometry of the illumination, the particles continuously tend toward a configuration with maximal scattering, rendering the system an adaptive self-optimizing light collection device.
Here, we study a generalized model of these self-ordering dynamics with several light frequencies applied simultaneously, each of which tends to push the particles toward a competing order. As generic example, we study a spectral composition of laser light with several light frequencies tuned closely to different cavity modes. As the longitudinal cavity modes form an equidistant comb of distinct resonances, a corresponding illumination is implementable with standard comb technology at comparable technical complexity to the single-mode case. For sufficiently distinct pump frequencies, light scattering between different modes can be neglected. Hence, no precise phase locking is required and also the computational complexity of the model grows only linearly with the number of modes. Besides studying the multitude of stationary solutions of the coupled particle-field dynamics, we also numerically compute the time evolution for larger particle ensembles and illumination frequencies, including friction and momentum-diffusion terms. Note that a more complex dynamics arises for cavities with degenerate mode families as, e.g., in a confocal cavity, which already exhibits a very rich structure for a single pump frequency [6,7]. Interestingly, even for particles in free space, collective light scattering can induce some spatial bunching and ordering via optical binding, which has been extensively studied in theory and experiments [8–15], and a rather complex motional dynamics via collective light scattering is also predicted for corresponding multifrequency configurations as, e.g., optical lattices .
After a short presentation of the semiclassical dynamical model in Section 2, we will study the forces and stationary states in generic two- and three-particle configurations in Section 3. Typical scenarios of dynamical evolution trajectories in the particle-field phase space of growing complexity are studied in Section 4, which is followed by long-term studies of many-particle dynamics including memory effects.
We consider identical polarizable point particles of mass representing, e.g., individual atoms, molecules, or nanoparticles trapped within an optical resonator supporting a large number of modes of similar finesse with wave numbers and frequencies (see Fig. 1). The particles are illuminated transversely by lasers with frequencies , each closely tuned to one of the cavity modes but sufficiently detuned from internal optical excitations to ensure linear polarizability and negligible spontaneous emission. Each particle scatters light in and out of the th cavity mode with phase and coupling strength depending on its position . For computational simplicity, we restrict the particles’ motion along the cavity axis, but 2D and 3D traps should lead to essentially similar physics. Following standard adiabatic elimination procedures, the coherent dynamics of the system can be described by the effective Hamiltonian 
Here, is the effective pump strength, is the light shift per photon, and is the detuning between laser and cavity mode frequency. For nanoparticles, is directly proportional to the particle’s polarizability . and denote position and momentum of the th particle, while and are the bosonic annihilation and creation operators of the th cavity mode field. The illumination pattern is specified by choosing a subset of modes with pump amplitudes . Including dissipation via cavity photon loss requires us to solve the master equation
While the full quantum model exhibits intriguing physical behavior as quantum phase transitions even for a single frequency [7,18], the enlarged case of two modes and few particles already touches the limits of current numerical computability . Here, we focus on the essential physics of crystallization and collective light scattering for many particles and modes, which forces us to use simplifications. Hence, we treat particle motion classically and assume coherent states with complex amplitudes for the cavity modes. Fortunately, this approximation works well in related treatments of cavity cooling  and self-ordering. The corresponding semiclassical equations for the coupled particle-mode dynamics can be written as 
The Langevin noise term and the weak frequency dependence of , , and will be mostly neglected in the following. These equations, obtained in similar form previously [6,20], contain the essence of multicolor self-ordering. Generalizations to include spontaneous emission leading to extra momentum diffusion of the particles are straightforward to introduce in principle, but are much harder to analyze in practice.
3. SELF-ORDERED STATES AND LIGHT SCATTERING OF FEW PARTICLES IN A MULTICOLORED FIELD
Before starting a detailed and extensive numerical analysis, let us first examine a few simple but instructive few-particle cases where we look for stationary states. In the bad cavity limit, where cavity losses happen on a shorter timescale compared to particle motion, the field adiabatically follows the particle positions and can be expressed in the form
In the bad cavity limit, the force on the th particle [Eq. (3b)] thus is effectively a function of the particle positions. Hence, when we determine the points in configuration space where all ’s are vanishing, we obtain the equilibrium points of the system. To determine their stability in a strongly damped case, where , a stability criterion which only involves checking the sign of the real part of the eigenvalues of the Jacobian of the vector containing the ’s is employed.
A. Two Particles
As a first nontrivial example, we consider two particles subject to transverse red-detuned multicolor pumping, where we first choose low-order cavity modes to graphically better display the physics. Depending on their positions, they scatter a different fraction of the various pump fields into the cavity. For the known case of a single pump frequency as shown in Fig. 2 on the left, we find a periodic pattern of strong scattering configurations when both particles sit on field antinodes with the same phase. Adding extra pump lasers close to other longitudinal modes only slightly shifts the stable equilibrium points with respect to the single-pump frequency configuration and creates some extra equilibria (see Fig. 2 on the right). However, the total amount of scattered light at each stable point now strongly varies, which implies different local trap depths. In a statistical equilibrium distribution, one will then find the particles more likely in deeper potential wells. As these are naturally associated with stronger light scattering, the system adapts toward optimal light collection. Note that strong scattering more likely occurs along the diagonal, where both particles occupy the same well and scatter with exactly the same phase.
Clearly, the optimum scattering positions for the different frequencies do not coincide, as demonstrated in Fig. 3, where we color-code three different scattering intensities by different colors. For most particle positions one or two colors dominate, but there are a few spots with nearly equal intensity scattering, yielding “white” scattered light.
B. Three Particles
The more complex case of three particles and several frequencies invokes a configuration space of a cube of length . The distribution of the stable equilibrium points for two different illumination conditions is depicted in Fig. 4 by small spheres, where size and color of the spheres represent the total amount of scattered light at the corresponding stable point. Again, the points with maximum scattering are on the diagonal (i.e., all three particles are in the same well). Note that in the chosen example, the number of stable points is much larger than for two particles; it strongly increases with particle number as the number of possible distributions of particles among the optical potential minina grows. Note, however, that the optical potential has to be self-consistently derived from the particle distribution, so that we were not able to find an explicit corresponding scaling law with particle number.
4. ADAPTIVE DYNAMICS OF THE COUPLED PARTICLE FIELD SYSTEM
So far, we have studied light fields and forces at fixed particle positions, where points of vanishing force give equilibrium positions. Equations (3), however, also describe dynamical properties of the system. As known from the single-mode case, the delayed field response during the self-ordering for negative detuning induces friction (cavity cooling) so that the system reaches a steady state . To mimic cavity cooling as well as other possible damping, e.g., by a background gas in the system, in the following numerical simulations we simply add an effective linear friction force to Eq. 3(b). This is only an approximation to the actual full dynamics, but will guide the system toward a stationary equilibrium state as desired.
A. Dynamics for Strongly Damped Particle Motion
Let us study the coupled particle-field dynamics in a simplified form first, where the friction force is large so that the particles quickly relax to a stationary velocity for a given light force (over damped case). The light fields in turn will continuously adapt to the current particle positions and the coupled system evolves to a close-by equilibrium configuration exhibiting a local light scattering maximum similar to the case of a self-consistent optical lattice . When the illumination condition changes, the particles will evolve toward a new equilibrium point, which better adapts to the momentary pump field configuration. We visualize these dynamics by periodically repeating a series of different pump light patterns. In this case, after some initial position changes, the particles find a suitable closed loop in the configuration space and periodically follow the illumination sequence. In each step, they quickly arrange to a local optimum configuration to maximize combined light scattering. Due to the complexity of the optical potential landscape, for different initial conditions a multitude of such loops are attained for the same illumination sequence. These can be distinguished by a characteristic corresponding output intensity sequence. Typical paths in the configuration space of three particles for a specific illumination sequence with different initial positions are shown in Fig. 5, nicely demonstrating this effect.
B. Dynamics with Noise Forces on the Particles
In any real system, damping is accompanied by noise forces on the particles, which in a simple form are modeled by random momentum kicks on individual particles as used for Brownian motion. Their effective strength relates to a temperature parameter. In the full dynamics [Eq. (1)], we also get white noise term representing field fluctuations as derived in Ref. , which in the optical domain at room temperature just represent vacuum fluctuations of the cavity field. While in a quantitatively correct description, these and even more noise sources would have to be carefully modeled and scaled, we restrict ourselves here to their qualitatively most important effect, which is that they render the force-free stationary configurations of the particles metastable. Driven by fluctuations, the system will eventually leave any such equilibrium configuration and evolve toward a new metastable state. In Fig. 6, we show a typical simulated trajectory for two scatterers over an extended time. While the particles rapidly evolve in the dark areas between different equilibrium points, the trajectory concentrates close to bright areas of the background picture, denoting strong light scattering. The system thus explores a large volume of configuration space but preferentially stays at points of strong scattering, where trapping is efficient.
This effect of localization near scattering maxima is displayed more quantitatively in Fig. 7. On the left we see that often both particles stay in the same well, with hopping between 0.5 and 1.5. These configurations correspond to points of maximal scattering as shown in the right picture. Because available phase space is rather limited for two particles in 1D, they will eventually meet each other at the same position where collective scattering is strongest. As the trapping potential is deeper at such points, the time it takes the particles to diffuse out of the corresponding optical well is much larger than for shallow minima with low light scattering. Hence, adding noise finally helps the particles to find more stable equilibria with enhanced scattering rates, which is the basis of the system-inherent scattering optimization mechanism. This dynamic adaptation and optimization mechanism certainly requires and deserves a more in-depth and quantitatively accurate study. However, due to the large number of parameters and possible mode choices, a systematic investigation goes beyond the scope of this work and we will concentrate on a different aspect, namely the memory properties of the dynamics in the following.
C. Time Evolution of Larger Ensembles with Randomly Varying Pump Light
While the previous examples nicely illustrate the basic physical mechanism of multicolor self-ordering, more interesting scenarios appear for many higher-order modes and larger particle numbers with smaller individual couplings. Since in the corresponding configuration space, a huge number of stationary states corresponding to local energy minima exist, it is far less likely for the system to come close to a global optimum, and many intermediate configurations appear. In the following, we consider many particles and tens of modes to study the extent to which we still get enhanced collective light scattering into several modes. Fortunately, in the classical regime of particle and field dynamics, this only requires a moderate increase in computing power. As particle phase space cannot be graphically depicted here, we show only important collective quantities as the total light intensity scattered into all modes, and the sum of order parameters for all modes
Of course, for large particle and mode numbers with numerous parameter choices, we can only discuss a few typical cases. As before, we assume high field-seeking particles and sufficiently red-detuned cavity pumping to avoid heating and nonlinear instabilities. Since we are interested in finding stationary configurations not so much in the precise time evolution toward them, we assume sufficiently strong friction again with negligible noise. We choose five illumination patterns consisting of about 50 distinct frequencies of equal pump strength randomly chosen out of the set , with and .
As in Section 4, we switch between the chosen illumination patterns after a prescribed time, when a stationary configuration is reached and the particles do not move anymore. In contrast to the periodic repetition considered previously, here we randomly choose one of the five patterns at every switch so that no closed loops are attained and the system will evolve nontrivially on a longer timescale.
One realization of these dynamics is shown in Fig. 8, where we plot the total scattered light intensity and the order parameter sum . We observe that and is roughly monotonically increasing, demonstrating that the system continuously improves its adaptation to the changing illumination. Due to the vast amount of possible configurations, this is a rather slow process, continuing at least to 8000 switches. After a sufficiently long time, the particles have found areas of configuration space with well-adapted positions to all five illumination patterns. Both the time needed to reach such a state and the resulting light intensity strongly vary for different realizations (i.e., different sequences of illumination patterns), but the qualitative behavior is generally similar. Interestingly we observed that if one particular mode is pumped in all of the applied illumination patterns, i.e., it is kept on continuously, the scatterers often preferentially adapt to scatter into this particular mode with much less light scattered to the others. Such a trapping gets more likely the higher the order of the mode is, as such a mode possesses more local minima. Overall, such a dominance of a particular mode results in a lower total scattering intensity summed over all modes. We also observe that the number of particles in clusters (i.e., with zero distance) grows with increasing . This mimics a lower effective particle number with higher individual coupling. In a more realistic model in 2D or 3D, such clusters can be expected to form much slower, and momentum diffusion will eventually split existing clusters and prevent lumping of the system.
Note that one can also interpret the adaptive ordering as acquisition of a memory of past illumination conditions. We see that when we apply any of the five illumination patterns after a long time evolution, more light will be scattered for each configuration than for the random particle distribution in the beginning. Hence, the system remembers that this illumination has been applied before, with the information stored in the order of the atoms.
5. CONCLUSIONS AND OUTLOOK
We have seen that the coupled particle-field evolution of mobile scatterers with multifrequency illumination in an optical resonator exhibits a wealth of intriguing phenomena beyond simple regular self-ordering. For a proper choice of the detunings and pump powers, the particles evolve toward a multitude of different spatial configurations, locally minimizing their optical potential energy and at the same time maximizing total light scattering into the cavity. For time-varying illumination conditions, the system continuously optimizes its light-scattering properties and acquires a memory of past conditions. This speeds up adaptation to a new equilibrium when similar conditions reappear, which increases overall scattering efficiency with time. Adding noise and diffusion allows the system to explore larger volumes of configuration space, which results in a configuration diffusion dynamics toward close to optimum scattering conditions for many light frequencies simultaneously. This includes concurrent super-radiant scattering into several cavity modes. Hence, we can consider the system an adaptive and self-learning light collection system with built-in memory. While implementations with a cold gas in a high- cavity would give straightforward possibilities to experimentally study such dynamics, alternative setups using mobile nanoparticles in solutions provide equally interesting experimental platforms .
Austrian science fund (FWF) (F4013-Phy).
We thank Wolfgang Niedenzu for stimulating discussions.
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