Quantum boomerang effect of light

Abstract

The quantum boomerang effect is a counterintuitive phenomenon in which a wave packet launched with finite momentum in a disordered medium returns to its origin. However, up to now, the experimental exploration of this boomerang effect remains largely unexplored. Here, we report the observation of this effect with light in an on-chip, one-dimensional (1D) disordered waveguide lattice. After benchmarking the system through Anderson localization, we launch a kinetic light beam into the system and track its center of mass (COM): it first moves away from its starting point, arrives at a maximum-valued point, reverses its direction, and returns to its original position over time, revealing the real-space observation of the photonic quantum boomerang effect. We also show two methods to accelerate and control the return: a symmetric gradient loss and time-varying coupling control to effectively increase the return velocity. Both strategies are realized experimentally and captured by our model. These results establish a controllable photonic platform for boomerang physics and open an avenue for future study in nonlinear and many-photon regimes.

Light launched into a disordered photonic chip first drifts, then unexpectedly reverses and returns to its point of origin. This work presents the first real-space observation of this quantum boomerang effect and demonstrates how engineered loss and time-varying couplings can accelerate the return dynamics.

Introduction

Controlling light has been at the forefront of modern optics for decades and is of fundamental interest to the photonic community. Advances in controlling light in photonics have led to the development of photonic crystals^1,2 and metamaterials^3,4, which enable intriguing phenomena such as the inhibition of spontaneous emission within the bandgap and the steering of light with a negative refractive index^5–8. More recently, the introduction of topology into photonics, inspired by topological insulators in condensed matter physics, has opened avenues for light control^9–18. The realizations leveraging artificial gauge fields in gyromagnetic photonic crystals^9,10, helical waveguides¹³, silicon photonics¹⁴ et al., offered the conventional photonic systems one-way topological edge states that are immune to imperfections and disorder. This field has provided possibilities for robust light transport, unidirectional waveguiding, and even the realization of exotic topological phases that have no electronic counterpart, such as lasing^19–22 and quantum photon sources^23,24. The further interplay between the topological phases and non-Hermitian physics^25–27 has led to many frontiers in photonics^28–41. One paradigmatic example is the non-Hermitian skin effect^42–52, where extended bulk states condense at one edge, resulting in the funneling of light⁵³. These advances underscore the potential for photonics that leverage the unique properties of topological and non-Hermitian systems.

On the other hand, Anderson localization⁵⁴ is a phenomenon where waves become trapped in a disordered medium due to interference, preventing them from diffusing through the material. This concept was originally introduced by Anderson in 1958. However, nowadays Anderson localization has been experimentally observed in various systems beyond its original electronic context, including photonic lattices^2,55–65, where light waves are confined by a disordered structure, and cold atom systems, where matter waves are localized^66–69. Building on this concept, the quantum boomerang effect-a counterintuitive dynamical feature beyond Anderson localization – was theoretically proposed^70–76, which disproves the common wisdom that, in a highly disordered lattice, waves can only be localized. In this phenomenon, a wave packet launched with an initial velocity in a disordered medium unexpectedly returns to its starting point over time, rather than dispersing or continuing along its trajectory. This intriguing phenomenon can be attributed to the combination of Anderson localization and parity-time symmetry of the system^74,77. The former leads to a time-independent distribution of the wave packet at long times, and the latter leads to the wave packet’s distribution being independent of the direction of momentum [Supplementary Section I-A]. While this effect has been extensively theoretically studied, its realization was only recently reported in the momentum space with ultracold atoms⁷⁷, limiting direct applications and broader transfer to other physical platforms. Altogether, these advances motivate a natural question: can the quantum boomerang effect be realized with light?

In this work, we experimentally realize the quantum boomerang effect in a 1D disordered photonic waveguide lattice, where the propagation axis of z plays the role of time and the field obeys a Schrödinger-like equation mimicking the quantum behaviors of particles¹³. The schematic of our optical system is shown in Fig. 1. We first benchmark the platform by launching a static beam and observing Anderson localization. We then inject a finite-momentum wave packet and track its COM: it moves away from its initial position, arrives at a maximum value, reverses its direction, and returns to its original position over time—consistent with the boomerang trajectory. To speed and control the return, we introduce a parity-symmetric gradient loss that acts like a restoring friction. Side-by-side tests of loss-free and loss-engineered lattices show a consistently faster COM’s comeback to the launch site. We further enhance the effect by slowly increasing the inter-waveguide coupling during the propagation, which effectively raises the return speed (see details in Supplementary Section I-D & IX). Our results not only deepen our understanding of light-matter interactions in disordered systems but also open doors to controlling light for cloaking⁷⁸ and optical tweezers⁷⁹.

Fig. 1. Schematic for the quantum boomerang effect of light.

The disordered optical lattice consists of equidistant waveguides with a lattice constant of d. The bluescale represents the strength of the random potential. The axis z is the propagation direction, and the axis x indicates the transverse interaction direction in Eq. (2). A wave packet with momentum k₀ is launched into the lattice. The red curve represents the COM trajectory.

Results

Physical model

Our model is realized in an array of evanescently coupled waveguides^13,52. The waveguides that can be directly written by femtosecond lasers are equally spaced with a lattice constant of d = 15 μm. The paraxial propagation of light in this system can be well described by the Schrodinger-type equation¹³(see supplementary information for details):

$$i{\partial }_z\mathrm{\Psi }=-\frac{1}{2k_0}{\nabla }^2\mathrm{\Psi }-\frac{k_0}{n_0}\mathrm{\Delta }n\mathrm{\Psi },$$ 1

where Ψ is the electric field amplitude, the Laplacian ∇² acts on the transverse plane, k₀ = 2πn₀/λ is the wavenumber in the background medium, and λ is the wavelength of light. Hereafter, the ambient medium for our system is the fused silica with refractive index n₀ = 1.45, and Δn(x, y, z) is the effective on-site potential, which is propagation-invariant and random in the transverse plane to construct a disordered photonic lattice. The paraxial propagation of light in this photonic system can be further governed by the tight-binding equation¹³:

$$i{\partial }_z{\psi }_p={\sum }_{⟨pq⟩}{c}_0{\psi }_q+V_p{\psi }_p=H{\psi }_p,$$ 2

where p ∈ [ − L, L] is the lattice index, 〈pq〉 indicates the summation is taken over the neighboring sites, c₀ is the coupling strength and V_p represents on-site potential, which is set to be pseudo-random values⁸⁰ given by the form of $V_p^{\left(s\right)}=\delta \cos \left[\sqrt{5}\pi {\left(p+\left(s-1\right)\times 5\right)}^{\alpha }\right]$, where δ represents the strength of disorder, s represents the phase shift parameter in sth realization of disorder and α is set to be 3 hereafter⁸⁰. We note that the pseudo-random potential used here follows the same form as that theoretically investigated in ref. ⁷² in the context of the quantum boomerang effect. The detuning of the on-site potentials can be introduced by changing the writing velocity of the femtosecond laser (Supplementary Section VIII).

Anderson localization of light

To establish the platform for studying the quantum boomerang of light, we first verify Anderson localization in our disordered photonic lattice. By utilizing the beam-propagation method (BPM), the intensity profiles at the output facet with different propagation lengths are numerically simulated (See details in Supplementary Section VII), and the results are plotted in Fig. 2b with a logscale for the vertical intensity axis. In our simulations, the effective coupling strength is about c₀ = 1.55 cm⁻¹. The strength of the pseudo-random potentials is about δ = 2.2 cm⁻¹. The parameter of α is 3, which indicates that all the states in the system are localized⁸⁰. We can see that the ensemble-averaged intensity distribution stabilizes at about z = 4 cm and remains there over propagation length. The profile of the output beam becomes exponentially localized, which is direct evidence of the Anderson localization. Note that without disorder, the output profile exhibits ballistic transport, and its width increases linearly with the propagation length (Supplementary Section II).

Fig. 2. Anderson localization of light in the 1D disordered optical lattice.

a The experimental characterization setup. (BPF band-pass filter, M mirror, S slit, L lens, BS beam splitter, CCD charge-coupled device.) b, c The logarithmic intensity profiles at propagation lengths of z = 2, 4, 6, 8 and 10 cm. d, e The intensity distributions of light at different propagation lengths. Panels b, d (c, e) show the numerical (experimental) results averaged over 10³ (21) realizations. Source data are provided as a Source Data file.

To experimentally verify Anderson localization in our photonic lattice, a laser beam with a wavelength of 635 nm is initially launched into the central waveguide of the 1D optical array, fabricated via femtosecond-laser direct writing¹³. The optical characterization setup is shown in Fig. 2a. The measured intensity profiles at the output facet are averaged over 21 realizations at propagation lengths of z = 2, 4, 6, 8, and 10 cm, respectively. As shown in Fig. 2d, e, the light beam becomes exponentially localized at about z = 4 cm, and the measured intensity profiles closely match the simulations (Fig. 2b, c). Note that we further characterize the localization behavior by using the inverse participation ratio (IPR), and the IPR in our model finally stabilizes at a finite plateau value (IPR ~ 0.12), indicating that the wavefunction has relaxed into a superposition of exponentially localized eigenmodes. The result is shown in Supplementary Section VIII-D. Together, the numerical and experimental results reveal that the ensemble-averaged intensity patterns undergo a transition from ballistic transport to localization when the disorder is introduced, which confirms that our disordered photonic lattice exhibits Anderson localization of light.

Quantum boomerang effect of light

Having presented Anderson localization of light in our photonic lattice incorporating disorder, we next move forward to the interaction between the disordered lattice and the kinetic input wave packet with nonzero momentum. The kinetic beam can be expressed in a normalized Gaussian form of $\psi \left(x_p,z=0\right)\propto \exp \left(-\frac{x_p^2}{2{\sigma }_0^2}+i{k}_0{x}_p\right)$, where σ₀, k₀ represent the width and momentum of the initial wave packet. Consequently, under the evolution of the photonic system, the COM of a launched wave packet as a function of propagation distance can be calculated by x(z) = ∑_px_p∣ψ(x_p, z)∣², and 〈x〉 indicates the ensemble-averaging of the COM over disorder realizations. A wave packet propagating in the disordered lattice can be divided into two parts: a localized part and a moving part. Initially, the moving part dominates, causing the COM to move away from its original position. As the wave evolves, however, the intensity of the moving part diminishes, and the localized part begins to take over. Consequently, the COM reaches a maximum value, denoted as ${⟨x⟩}_{\max }$, before eventually returning to its initial position over time.

The next step is to select the optimal momentum value that can maximize the turning distance, thereby broadening the detectability of the phenomenon. We address this with simulations that sweep experimentally accessible parameters. In the absence of disorder, the wave packet with a fixed momentum will acquire a group velocity that is governed by the dispersion relationship of the lattice. At the output facet at z = 15 cm, the relation between the momentum of k₀ and the maximum moving distance of $x_{\max }$ is shown in Supplementary Fig. S4a (Supplementary Section III). We can see that the maximum traveling distance occurs at the momentum of $k_{0}d=\frac{\pi }{2}$ corresponding to the dispersion with the maximum group velocity (see inset in Fig. 3a). Intuitively, we envision that this momentum induces the strongest boomerang effect. This is confirmed by numerical simulations, as shown in Fig. 3a. The relationship between the ${⟨x⟩}_{\max }$ and the momentum reveals that the wave packet with the initial momentum of $k_{0}d=\frac{\pi }{2}$ travels the farthest before returning to its starting point. Therefore, we choose $k_{0}d=\frac{\pi }{2}$ hereafter for the observability of the boomerang effect in experiments.

Fig. 3. Quantum boomerang effect of light in the 1D disordered optical lattice.

a The relationship between the momentum and the furthermost distance ${⟨x⟩}_{\max }$. The inset shows the corresponding energy band of a clean lattice. b The COM at different propagation lengths. The blue curve and the shading indicate the averaged COM and the standard error (Err.) of the numerical results (tight-binding simulation, averaged over 10³ realizations). The purple dots indicate the experimental data (averaged over 21 realizations). The parameters are set to be c₀ = 1.55 cm⁻¹, δ = 2.2 cm⁻¹, σ₀ = 3d, and L = 100. Source data are provided as a Source Data file.

The disordered photonic lattices and characterization setup are identical to those in Figs. 1 and 2a. A 635 nm laser is shaped into a broad Gaussian input through an optical slit, producing a beam with width σ₀ = 3d. The momentum of the Gaussian beam can be controlled by a rotated mirror and set to be $k_{0}d=\frac{\pi }{2}$ (Supplementary Section VIII-B). The initial input beam is then launched into the sample, centered at the optical waveguide with the index of p = 0. To reconstruct the wave-packet trajectory, we record the output-facet intensity for photonic chips with propagation lengths of 0, 0.5, 1, 1.5, 2, 3, 4, 5, 7, 10 and 14 cm through a commercial CCD camera.

The measured COM trajectory of the wave packet is shown in Fig. 3b, with the solid line (dots) representing the numerical (experimental) results. We can clearly see that the COM moves rapidly along the x direction over the first z = 1 cm, reaches a maximum value of about 1.8d, then reverses, and drifts back toward the starting position of x = 0, providing a direct real-space observation of the quantum boomerang of light. Note that in Supplementary Section I-B, we show the turning point and return time follow the mean free path and the scattering time predicted by the 1D theory, in agreement with experiment. Due to the finite sample length, the COM does not fully reach the starting position by z = 14 cm.

Enhancement

A natural question arises: can we accelerate the quantum boomerang effect? Generally, loss is considered detrimental to optical experiments, particularly those involving photonic devices and quantum optics. However, surprisingly, we will show in the following that a specific design of loss configuration can facilitate the realization of the quantum boomerang effect by accelerating the return of the wave packet, meanwhile preserving the maximum excursion. The optical loss is configured to follow a symmetric gradient pattern as shown in Fig. 4b, expressed as $-i\gamma \mid p\mid {\psi }_p^{†}{\psi }_p$, where p, γ represent the lattice index and the strength of the gradient loss. The evolution of the COM of the launched wave packet is traced out by numerical simulations as shown in Fig. 4a. The depth of the blue color indicates the gradient strength of 0, 1.5 × 10⁻³c₀, 3.0 × 10⁻³c₀. We surprisingly find that a stronger gradient causes the COM to return to its starting position of x = 0 more quickly, which facilitates the experimental realization of the quantum phenomena in our system. In Fig. 4c, we present the relationship between the strength of gradient loss and the COM at z = 10 cm, which verifies that the introduction of gradient loss can accelerate the return of the wave packet. The experimental results agree well with the numerical ones. A general Ehrenfest analysis shows that uniform loss is inert, an antisymmetric gradient induces a drift that destroys the boomerang, while the symmetric gradient enhances the return, matching simulations and measurements. More details can be found in Supplementary Section I-C. Note that the symmetric gradient loss alone cannot lead to the quantum boomerang effect in the photonic lattice without the disorder (Supplementary Section II).

Fig. 4. Introducing symmetric gradient loss.

a The COM trajectories under different gradient strengths. The comeback is accelerated, and the maximum excursion is preserved. b The profile of the gradient loss with γ = 3.0 × 10⁻³c₀. The inset displays the waveguide with breaks. The strength of loss is proportional to the breaking length. c The relationship between the COM and the gradient strength at a fixed propagation length of z = 10 cm (see purple dashed line in panel a). The depth of the blue color represents the strength of the gradient. All the numerical (experimental) results are averaged over 10³ (50) realizations, and the parameters are the same as those in Fig. 3. Source data are provided as a Source Data file.

Following the same philosophy as previously, we now fabricate disordered photonic lattices with an engineered gradient loss and repeat the measurements. The gradient loss is implemented by inserting periodic breakpoints in the optical waveguides (Fig. 4b, inset, Supplementary Section VIII)⁵². The gradient strength is set to be γ = 3.0 × 10⁻³c₀. The measured COM trajectory is shown in Fig. 5a. The solid curve represents the numerical result from tight-binding simulations, and the purple dots represent the measured results. By comparing this plot with that in Fig. 3b, we can see that the COM trajectory is similar, but the wave packet returns to the starting position significantly faster due to the presence of gradient loss. BPM simulations (Supplementary Section VIII-E) agree with both the tight-binding results and the measurements. In Fig. 5b–c, we compare intensity profiles at propagation lengths of z = 1 and 10 cm from BPM and experiment. At z = 1 cm, the COM of the wave packet is displaced to the right by about 1.8d; at the propagation length of z = 10 cm, the COM approaches its original position of x = 0. Together, these data demonstrate the quantum boomerang of light and show that a designed gradient loss can substantially accelerate the return of the wave packet.

Fig. 5. Observation of the enhanced quantum boomerang effect of light.

a The COM trajectory with gradient strength of γ = 3.0 × 10⁻³c₀. The blue curve (purple dots) represents the numerical (experimental) result. b, c The light intensity distribution at propagation length of z = 1 and 10 cm. The patterns are generated with the data from BPM simulations and experiments. All the numerical (experimental) results are averaged over 10³ (50) realizations, and the parameters are the same as those in Fig. 3. Source data are provided as a Source Data file.

Discussion

In conclusion, we have realized the quantum boomerang of light by tracking the center of mass of a finite-momentum wave packet in a disordered on-chip waveguide lattice. After validating the platform through Anderson localization, we observe a clear drift-turn-return trajectory in real space, manifesting the quantum boomerang effect, which is a key signature of the underlying interplay between quantum coherence and disorder. We then achieve deterministic control of this return with a symmetric gradient loss that acts as a restoring friction, accelerating the comeback while preserving the maximum excursion. Furthermore, we find that continuously increasing the coupling strength can accelerate the COM’s return to its origin, thereby strengthening the quantum boomerang effect (See more details in Supplementary Section IX). These results establish a compact, tunable photonic setting for boomerang dynamics and for exploring how disorder and dissipation shape wave transport^81,82. Many intriguing questions arise, for example, the effect of nonlinearity^58,83 on the quantum boomerang effect of light, the behavior of entangled photons^84,85 in this system, or the interaction with quantum many-body physics. Thus, our results not only provide insights into light transport in disordered media but also open avenues for future research in photonics and quantum physics.

Methods

Sample fabrication

The photonic lattices of optical waveguides are fabricated on the fused silica glass (Corning 7980) with various lengths (5, 10, 15, 20, 30, 40, 50, 70, 100, and 140 mm). This is achieved by using the femtosecond laser writing method (repetition rate 600 kHz, wavelength 515 nm, pulse duration 217 fs). The laser pulses are focused onto the sample through an objective lens (×20, NA = 0.45), and induce the refractive index contrast of up to Δn₀ = 7.5 × 10⁻⁴. The distance between the two nearest waveguides is set to be d = 15 μm corresponding to the effective coupling strength of about c = 1.55 cm⁻¹ at the characterization wavelength of 635 nm. The incommensurate on-site potential energy and the gradient loss of the waveguide array are realized by writing with different speeds and introducing different break lengths into the waveguides, as illustrated in Supplementary Fig. S8.

Characterization setup

The experimental setup is illustrated in Supplementary Fig. S9b. The laser beam is initially shaped by the slit of S₁, then focused on the rotated mirror of M₂ by the objective lens of L₁ (×4, NA = 0.1). The angle of θ between the center axis of M₂ and the light path is initially set to be π/4. Next, the obtained elliptical Gaussian beam is sent to the input facet of the sample through a 4-f system (L₂ and L₃). The mirror of M₂ and the input facet of the sample are on the focal point of L₂ and L₃.

To control the momentum k_x of the input Gaussian beam, we rotate the mirror of M₂ by about 0.2°. Consequently, the tilted Gaussian beam has a momentum of k_xd = π/2. In the clean lattice, we observe that the COM of the beam travels farthest away from the incident position by fixing the same propagation distance. The relationship between the COM and the angle of M₂ is depicted in Supplementary Fig. S9c. The error bars represent the experimental data (averaged over 21 realizations), and the curve represents the fitting result.

Author contributions

X.H. and F.W. fabricated the sample and conducted the measurements. Z.W. and X.H. carried out the theoretical modeling and numerical simulations. Z.Y. conceived and supervised this project. Z.Y., Z.W., and X.H. wrote the manuscript with input from B.Y. and S.Z. All authors discussed the results and contributed to the manuscript.

Peer review

Peer review information

Nature Communications thanks the anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The data supporting the findings of this study are presented within the article and Supplementary Information. Source data are provided with this paper.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-68293-8.