Proposal for Anderson localization in transverse spatial degrees of freedom of photons

TL;DR

This paper proposes an experimental setup to observe Anderson localization of light in the transverse spatial degrees of freedom of photons using a spatial light modulator to create a weak random potential, with numerical simulations confirming feasibility.

Contribution

It introduces a novel experimental configuration that separates dispersion and potential effects, enabling new studies of Anderson localization in photon transverse profiles.

Findings

Numerical simulations confirm the feasibility of the proposed setup.

The setup allows observation of Anderson localization in transverse photon profiles.

The method separates dispersion and potential effects in the evolution of light.

Abstract

We propose an experimental setup for studying the Anderson localization of light in the continuous transverse spatial degrees of freedom of the photons. This physical phenomenon can be observed in the transverse profile of a paraxial and quasi-monochromatic beam of light using a spatial light modulator. The light modulator acts in the laser beam as a weak random potential. Here, differently from the standard models studied in the literature, our setup splits the dispersion and potential terms along the beam evolution. By numerical simulations we confirm the feasibility of our experimental proposal.

Equations

$$\frac{i}{k}\frac{\partial }{\partial z}A(x,y,z)=-\frac{1}{2k^2}\left(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}\right)A(x,y,z).$$

$$\frac{i}{k}\frac{\partial }{\partial z}\varphi (x,z)=-\frac{1}{2k^2}\frac{{\partial }^2}{\partial x^2}\varphi (x,z),$$

$$U(q_x)=e^{-iz{q}_x^2/2k}.$$

$$U^{\prime }=e^{-id{q}_x^2/2k}{e}^{if(x)}.$$

$$U^{\prime \prime }=\prod _{j=1}^N{e}^{-id{q}_x^2/2k}{e}^{if(x)}.$$

$$i\hslash \frac{\partial \varphi }{\partial t}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\varphi }{\partial x^2}+V(x)\varphi ,$$

$$U=\exp \{-i[p^2/2m]\Delta t/\hslash \}\exp \{-i[V(x)]\Delta t/\hslash \},$$

$$V(x)=V_0\sum _{j=1}^{S}v(x-x_j),$$

$$v(x)={\left(\sigma \sqrt{\pi }\right)}^{-1}\exp \left(-\frac{x^2}{{\sigma }^2}\right),$$

Full text

I Introduction

Anderson localization - the phenomenon of transport suppression of wave due to a destructive interference of the many paths associated with coherent multiple scattering from the modulations of a disordered potential - continues fascinating researchers since the appearance of Anderson’s 1958 paper (Anderson, 1958). This effect is a characteristic of wave physics and occurs when the disordered potential presents a weak amplitude, making the localized state with exponentially decaying tails and absence of diffusion. The Anderson localization has been experimentally demonstrated in many scenarios such as Bose-Einstein condensate (BEC) of atoms (Wiersma et al., 1997; Störzer et al., 2006), in 2D (Billy et al., 2008) and 1D (Roati et al., 2008) disordered photonic lattices. From the theoretical viewpoint, the Anderson localization has been studied in synthetic photonic lattice with random coupling (Pankov et al., 2019), in biological nanostructures of native silk (Choi et al., 2018), on the surface plasmon polariton (Petráček and Kuzmiak, 2018), and in Bose–Einstein condensate with a weakly positive nonlinearity under the influence of chaotic potentials (Cardoso et al., 2012, 2016).

Potential applications ranging from solar cells to endoscopic fibers have also motivated the investigation of the Anderson localization of light that arisen by the understanding that the localization is ubiquitous to all wave systems, in particular it should be present in optics, where there is the remarkable analogy between the paraxial equation for electromagnetic waves and the Schrödinger equation ruling the quantum phenomena (Soukoulis, 2012; Wiersma, 2013). It is worth to mention that light in optical domain furnishes an ideal system to investigate localization effects since we have preserved coherence and non-interacting bosons (photons) in order to satisfy the two required assumptions of Anderson’s model: time-invariant potential and absence of interaction (Segev et al., 2013).

In this context, transversal Anderson localization of light - firstly proposed in (Nurligareev et al., 2005) and rediscovered a decade later in (De Raedt et al., 1989) - has started a promisor experimental scenario for studying localization effects. It occurred due mainly to the experimental observation of discrete transversal solitons (Fleischer et al., 2003) that has opened the way to investigate localization effects in paraxial disordered photonic systems, culminating with the remarkable experimental observations of the transverse Anderson localization of light in 3D random media (Schwartz et al., 2006; Lahini et al., 2008). In addition, the experimental achievement of (Segev et al., 2013) in a two-dimensional photonic lattice with random refractive index fluctuations induced on a photorefractive crystal using an optical interference pattern, together large refractive index available in optical fiber, becomes possible to glimpse transverse Anderson localization as an effective waveguiding mechanism for the light in random optical fiber (Karbasi et al., 2012).

On the other hand, the transverse spatial continuous variables associated with paraxial beam profile provide a robust and fertile testing scenario for investigations of fundamentals of quantum mechanics related to the EPR paradox (Walborn et al., 2011), quantum entanglement (Tasca et al., 2008; Di Lorenzo Pires et al., 2009), entanglement beyond Gaussian states of light (Gomes et al., 2009a; Tasca et al., 2013), production of purely nonlocal optical vortex (Gomes et al., 2009b, 2011), quantum information (Walborn et al., 2006, 2008; Almeida et al., 2005), quantum computation (Tasca et al., 2011), and simulation of chaotic (Lemos et al., 2012) and relativistic system such as Dirac equation (Silva et al., 2019). Here, taking advantageus of the facilities available in linear paraxial optics scenario, we propose the experimental realization of the quantum Anderson Localization dynamics in the spatial degrees of freedom of the photons in a monochromatic paraxial light. To this end, a spatial light modulator (SLM) is used to implement a potential in the transverse profile of the field propagating freely, resulting to Anderson Localization in the transverse position of the photon.

The paper is structured as follows. In Sec. II we discuss the transversal continuous variables. The experimental setup is proposed and detailed in Sec. III. In Sec. IV, we present the results of a numerical simulation, showing the feasibility of the proposal. Finally, we present our conclusions and final remarks in Sec. V.

II Transverse Continuous Variables

We propose an optical physical system to implement Anderson localization in the transverse profile of a laser beam. The complex amplitude of the electric component of a electromagnetic field can be represented by $\vec{E}(x,y,z)=A(x,y,z)\vec{\epsilon}e^{-ikz}$, where $\vec{\epsilon}$ is the polarization vector, $k=2\pi/\lambda$ is the wavenumber and $A(x,y,z)$ is the complex envelope. Consider $A(x,y,z)$ a function that varies slowly on the neighborhood of wavelength $\lambda$, such that the complex envelope of a monochromatic laser beam satisfies the paraxial Helmholtz equation (Saleh and Teich, 1991), given by

[TABLE]

The above equation behaves like a bidimensional, alias (2+1), Schrödinger equation for the complex monochromatic field $A(x,y,z)$ in free space, with the propagation variable $z$ of the field being analogous to the time in the standard Schrödinger equation while $x$ and $y$ are the transverse positions of the field. It is known that Eq. (1) admits a Gaussian solution representing the transversal profile of the laser beam. Without loss of generality, one can study the system in only one spatial transverse dimension, and from Eq. (1) obtains the form

[TABLE]

where $\phi(x,z)$ is the component of the transversal profile in $x$ direction. Note that, if one considers the variable $z$ as a time variable and replace the wavenumber $k$ to $1/\hbar$, the Eq. (2) becomes completely analogous to the standard one-dimensional Schrödinger equation for the field $\phi(x,z)$ in free space for a particle of unitary mass. By this analogy, David Stoler introduced the isomorphism between the Hilbert space in the transverse variables of single photons with the Hilbert space of the nonrelativistic quantum state of single point particle. The operators $\boldsymbol{x}$ and $\boldsymbol{\theta}_{x}$ are defined as canonical conjugate operators that obey the commutation relations $[\boldsymbol{x},\boldsymbol{\theta}_{x}]=i/k=i\hbar$ or $[\boldsymbol{x},\boldsymbol{q}_{x}]=1$, where the dimensional transverse momentum is $q_{x}=\frac{1}{i}\partial/\partial x=\frac{1}{i}\theta_{x}k$ (Stoler, 2008; Nienhuis and Allen, 1993).

III Experimental setup

In our experimental proposal, we consider the use of transverse profile of a quasi-monochromatic and paraxial light beam and two SLMs properly positioned so that the beam suffers multiple reflections, each one of them with the same random phase impression. This experimental setup shown in Fig. 1 allows us to observe the Anderson localization in the transversal profile of the photons.

In the free propagation path the amplitude of the light beam is well described by the Eq. (2). Indeed, in Fresnel approximation the evolution operator can be represented by the transfer function (Saleh and Teich, 1991)

[TABLE]

Following, the interaction of the light beam with the SLM leads to the field suffers a phase changing according to the transformation $e^{if(x)}$. In order to observe the Anderson localization of the photons, one need to use a random profile function $f(x)$. Then, the transformation in the complex envelope of the field after a reflection in the SLM and the propagation in free space by a distance $d$, before the next reflection, it is represented by the evolution operator

[TABLE]

Consequently, after $N$ reflections in the SLM’s, as represented by the setup in Fig. 1, the resulting transfer function is given by

[TABLE]

On the other hand, the wavefunction $\phi$ of a massive point particle subject to the potential $V(x)$ is well described by the Schrödinger equation

[TABLE]

whose evolution operator is given by $U=\exp\{-i[p^{2}/2m+V(x)]t/\hbar\}$. In numerical methods, the split-step method it is commonly employed to solve the Eq. (6). The main idea of this method is to split the evolution operator $U$ into the time interval $\Delta t$ in the form

[TABLE]

by using the well known Baker-Campbell-Hausdorff formula. The precision of this approximation depends on the value of $\Delta t$. In fact, the smaller the time interval $\Delta t$ is greater the accuracy of the approximation is. We emphasize that the result obtained in Eq. (7) is similar to that of Eq. (4). Then, since the Eq. (6) admits the Anderson localization to appropriate random potentials $V(x)$, we expect a similar behavior to the transverse profile of the light beam in the experimental setup depicted in Fig. 1.

IV Numerical results

In the experimental setup shown in Fig. 1, the spatial light modulators SLM1 and SLM2 are responsible to create a random profile for the phase print $V(x)\equiv f(x)/dt$, acting as a potential in Schrödinger-like equation (2), with $dt$ being a small parameter. As an example, in Fig. 2(a) we display a typical potential profile obtained via a superposition of 300 speckles generated in random positions. A detailed description, including a algorithm for an experimental generation of speckle patterns of light can be found in Ref. (Huntley, 2009).

To implement the speckle patterns similar to that of Ref. (Huntley, 2009) in the SLM, we construct an algorithm based on the Ref. (Cheng and Adhikari, 2010). To this end, the phase printing function $V(x)$ can be modeled by a set of $S$ identical spikes randomly distributed along the $x$ axis (Sanchez-Palencia et al., 2008)

[TABLE]

with $V_{0}$ being the strength of the spike, and $v(x-x_{j})$ are the profile associated to a single spike at random position $x_{j}$. We assume Gaussian spikes given by (Sanchez-Palencia et al., 2008)

[TABLE]

with a specific width $\sigma$. The statistical average of the disordered potential (8) and the auto-correlation function for the phase printing function $V(x)$ are given by $\langle V(x)\rangle\equiv\int_{-L}^{L}V(x)dx/2L=V_{0}/D$ and $C(d)=\langle V(x)V(x+d)\rangle-\langle V(x)\rangle^{2}$, respectively, with $D$ being the average spacing between spikes in the spatial extension $2L$. Also, the average spike height $V_{S}$ is defined by $V_{S}=\left[\int_{-L}^{L}dx\left(V(x)-\langle V(x)\rangle\right)^{2}/2L\right]^{1/2}$. As a typical example we set $S=300$, $L=30$, $V_{0}=1$ and a small width $\sigma=0.1$.

The numerical simulations were performed by splitting the dispersion term due to the free propagation of the field and the phase printing generated as effect of the spatial light modulators under this field. Indeed, this is truly physical mechanism involved in the present protocol, once these effects does not occurs simultaneously.

In Fig. 2(b) we display the variance of the the localized state $\sqrt{\langle x^{2}\rangle-\langle x\rangle^{2}}$ versus the evolution time, alias the longitudinal propagation coordinate $z$ of the light beam. The parameters used in our simulations were in a such way that the time interval $\Delta t=1$ corresponds to 100 reflections of the light beam by the SLMs. As a result, the absence of variance growth implies the state localization. In order to evidence the result shown in Fig. 2(b), we display in Fig. 2(c) the light beam profile ($|\phi|^{2}$) by considering a Gaussian input state in the real time evolution up to a generic time $t=100$, corresponding to $10^{4}$ interactions of the beam with the SLMs.

Finally, in Fig. 2(d) we consider the light beam profile of a typical configuration in log scale obtained numerically from the Eq. (5) at $t=100$. One can observe from this plot that the tails of the output state (solid line) decay much more smoothly (almost linear on average) than that for the Gaussian state, shown by the dashed curve in Fig. 2(d). This is the main evidence of the Anderson localization (Wiersma et al., 1997; Störzer et al., 2006; Schwartz et al., 2006; Lahini et al., 2008; Billy et al., 2008; Roati et al., 2008).

V Conclusion and Final Remarks

We propose here a new experimental platform to study the Anderson localization in the transverse degrees of freedom of photons, employing SLM’s to print a random phase in the transversal profile of a laser beam. We confirmed the presence of Anderson localization in the transverse degree of freedom of photons by direct numerical simulations. The simplicity of the experimental configuration proposed here to control the disordered potential and to detect the transversal intensity of the beam is a prime advantage of the work. The tuning of the disordered potential continues to be one of the prime challenges of the experimental observation of the Anderson Localization in others scenarios. On the other hand, the spatial light modulator absorbs 2% of the beam intensity at each reflection, attenuating the beam according to the number of reflections. However, this can be contoured using avalanche photodetectors (Gomes et al., 2009a). Furthermore, the increasing technological development of more efficient SLM’s will enable to observe the resulting beam of light in CCD cameras. Finally, we hope that this work opens new possibilities to study the Anderson localization phenomenon in quantum optics.

Acknowledgments

We acknowledge acknowledge financial support from the Brazilian agencies CNPq (#304073/2016-4, #425718/2018-2 & #479956/2013-8), CAPES, and FAPEG (PRONEM #201710267000540, PRONEX #201710267000503). This work was performed as part of the Brazilian National Institute of Science and Technology for Quantum Information (INCT-IQ).