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Abstract
We theoretically and systematically investigate Anderson localization of two bosons with nearest-neighbor interaction in one dimension under short- and long-time scales, two types of disorders, and three types of initial states, which can be directly observed in linear disordered photonic lattices via two experimentally measurable physical quantities, participation ratio and spatial correlation. We find that the behavior of localization characterized by the participation ratio depends on the strength of interaction and the type of disorder and initial condition. Two-boson spatial correlation reveals more novel and unique features. In the ordered case, two types of two-boson bindings and bosonic “fermionization” are shown, which are intimately attributed to the band structure of the system. In the disordered case, the impact of interaction on the two-boson Anderson localization is reexamined and the joint effect of disorder and interaction is addressed. We further demonstrate that the independence of the participation ratio or spatial correlation on the sign of interaction can be eliminated by employing an initial state that breaks one of two specific symmetries. Finally, we elucidate the relevant details of the experimental implementation in a two-dimensional linear photonic lattice.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As one of the celebrated consequences of quantum destructive interference, Anderson localization [1], named after the pioneering work of P. W. Anderson, indicates that disorder has a striking effect on the property of a single quantum particle and can trigger an exponential localization for all the eigenfunctions of system so that any long-distance transport is forbidden, which is in stark contrast to the spatially periodic structure with the Bloch plane wave eigenstates. Inspired by the extremely intriguing physics, multifarious theoretical protocols [2–4], covering dynamical versions [5–7] and non-Hermitian counterparts [8,9], have been investigated. Furthermore, there is a ceaselessly growing passion for the probe and verification of this phenomenon and diverse experimental schemes [10,11], ranging from kicked systems [12–14] to light propagation in spatially random optical media [15,16], noninteracting Bose-Einstein condensates in disordered optical potentials [17,18], microwave cavity fields with randomly distributed scatterers [19], and quantum walk in an integrated array of interferometers [20], have been proposed.
Beyond the single-particle model, quantum many-body system [21] can reveal more abundant and extraordinary physics owing to the introduction of interparticle interactions and the further formation of many-body bound states. Many researchers devote their efforts to the limit of few interacting particles, namely two-body system [22,23], and have made significant progress, for instance, on a theoretical level, the two-body topological states of matter [24–26] and the localization mechanism of two-particle flat band interference instead of disorder [27] have been proposed and investigated, from an experimental perspective, several key experimental activities involving quantum walk of two particles [28] and the interplay between magnetic fields and interacting particles [29,30] have been implemented and accomplished.
On the other hand, understanding the effect of disorder on the localization of quantum many-body system is actually the intention of Anderson [31,32]. Also, compared to the single-particle case, the system may undergo a more complicated process and further exhibit more exotic and profound physics. It turns out that this subject is complex indeed and the results rely on the details of models, the measurement of localization, and the applied numerical technique, which, over the past few decades, has attracted extensive studies in both theoretical [33–35] and experimental [36,37] domains. Lahini et al. predict the quantum correlations in the Anderson localization of two noninteracting particles and find that these two-particle correlations develop unique features which depend on the quantum statistics of particles and on the initial state, such as the dependence of the localization of one particle on the localization of the other and the oscillatory correlations within the localization length [38]. Lee et al. further investigate the effect of on-site interaction on the Anderson localization of two bosons in a one-dimensional (1D) disordered lattice, they explicitly analyze and discuss the interplay between disorder and on-site interaction and clearly show, both quantitatively and qualitatively, the results under different conditions [39]. Specifically, rapid developments and crucial advancements in the engineering and fabrication of photonic lattices have opened up an avenue for the experimental confirmation of these theoretical predictions and the robust, easily constructed, and highly controllable virtues make it an ideal platform to simulate and map these relevant problems [40–44]. However, in a dynamical frame, how other types of interactions affect the two-particle Anderson localization still remains unexplored and it is motivating to address the interplay between disorder and other types of interactions via photonic lattices.
In this paper, we consider that two indistinguishable bosons are subjected to the nearest-neighbor interaction in a 1D disordered lattice. We mainly concentrate on the effect of nearest-neighbor interaction on the two-boson Anderson localization under short- and long-time scales, three types of initial states, and two types of disorders, which is quantified by the participation ratio (PR). Additionally, two-boson spatial correlations are displayed for both ordered and disordered lattices. In the ordered case, we find two types of two-boson bindings and bosonic “fermionization”, two-boson energy spectrum is also shown to explain these relevant phenomena. In the disordered case, we further validate and clarify the effect of nearest-neighbor interaction on the two-boson Anderson localization and the joint effect of disorder and nearest-neighbor interaction is also addressed. By resorting to an initial state that breaks one of two specific symmetries, we also demonstrate that the dependence of the PR or spatial correlation on the sign of interaction can be established. We argue that all these results can be directly observed in a two-dimensional linear disordered photonic lattice.
The rest of this paper is organized as follows. In Sec. 2, the model Hamiltonian and the measurement of localization are introduced and presented. In Sec. 3, the effect of nearest-neighbor interaction on the two-boson Anderson localization under different conditions is analyzed and discussed in detail. Two-boson energy spectrum and anomalous doublon eigenstates found in the scattering continuum are also shown therein to explain the behavior that appear in the spatial correlations. In Sec. 4, we distinguish the repulsive and attractive nearest-neighbor interactions based on a few specific initial states. In Sec. 5, the experimental relevance is expounded in a two-dimensional (2D) linear photonic waveguide array. We give a conclusion in Sec. 6.
2. Model Hamiltonian and methodology
The model under consideration describes two interacting indistinguishable bosons in a 1D lattice composed of $N$ sites, as governed by the following Hamiltonian:
whereHere $a_{i}^{\dagger }$ ($a_{i}$) is the bosonic creation (annihilation) operator at site $i$ and $n_{i}=a_{i}^{\dagger }a_{i}$ is the corresponding particle number operator. The total particle number is conserved since the total particle number operator $P=\sum _{i}n_{i}$ commutes with the above Hamiltonian so that the system will evolve with a fixed particle number. The first and second terms denote the on-site effect with potential energy $W_{i}$ and the nearest-neighbor hopping with tunneling amplitude $T_{i,j}$, respectively. $W_{i}$ and $T_{i,j}$ are site-dependent and restricted to ($\bar {W}-\delta W,\bar {W}+\delta W$) and ($\bar {T}-\delta T,\bar {T}+\delta T$). For the sake of convenience, we refer to $\bar {T}=1$ as the energy unit throughout the paper. While the vanishing $\delta W$ and $\delta T$ signify an ordered lattice, the nonzero $\delta W$ and $\delta T$ imply a disordered one and respectively introduce two types of static disorders, diagonal disorder for the finite $\delta W$ and off-diagonal disorder for the finite $\delta T$. In the disordered condition, both $W_{i}$ and $T_{i,j}$ are uniformly and randomly distributed and assigned to be positive-real within an experimentally realizable parameter regime. The last term stands for the nearest-neighbor interaction between two indistinguishable bosons. $V$ is the interaction strength with attractive interaction $V<0$ and repulsive interaction $V>0$.
The introduction of either diagonal or off-diagonal disorders can trigger the two-boson Anderson localization. What we are concerned about is how the existence of nearest-neighbor interaction affects the two-boson Anderson localization and the emergence of interesting phenomena resulting from the interplay between disorder and interaction. To this end, we resort to two experimentally measurable physical quantities. One is the time-dependent PR, which characterizes the localization length of wave packet $|\Psi (t)\rangle$ of two indistinguishable bosons at time $t$ and can be computed as follows:
where $|\Psi (t)\rangle$ is spanned by the basis $|i,j\rangle$ ($i\leqslant j$) and $i$, $j$ designate the locations of two indistinguishable bosons, respectively. $|\Psi _{i}(t)|^{2}$ is the normalized particle density on the $i$th site, calculated by $|\Psi _{i}(t)|^{2}=\langle \Psi (t)|n_{i}|\Psi (t)\rangle /2$. It is easy to prove that $\mathrm {PR}\sim 1$ and $N$ for the most localized and completely delocalized states, respectively. The other is the time-dependent two-boson spatial correlation defined as which responds to the probability of detecting one boson at site $i$ and the other at site $j$ simultaneously. For various initial state $|\Psi (0)\rangle$ and different interaction strength $V$, a number of novel quantum interference phenomena can be identified and the discrepancies among them can be distinguished via $\Gamma _{i,j}(t)$.3. Two-boson Anderson localization and spatial correlation
In this section, two-boson Anderson localization with nearest-neighbor interaction is investigated by the PR under short- and long-time scales, repulsive and attractive cases, two types of static disorders, and three different initial conditions. Also, two-boson spatial correlation is displayed for both ordered and disordered lattices. For simplicity, each spatial correlation is rescaled by its maximum value. Consequently, $\Gamma _{i,j}(t)/\Gamma _{i,j}^{\mathrm {max}}(t)$ is shown. Furthermore, the periodic boundary condition is imposed.
3.1 $|\Psi (0)\rangle =|0,1\rangle$
Let us start by considering that two indistinguishable bosons are initially positioned at two adjacent sites $|0,1\rangle$. We present the PR as a function of the interaction strength $V$ for a broad scope including both the attractive and repulsive cases, as shown in Fig. 1. It provides a comparison between diagonal and off-diagonal disorders and between short- and long-time scales. The examination of Fig. 1 exposes the following observations:
- (i) In general, long-time evolution is required for the saturation of two-boson localization. Nevertheless, we concentrate on the short-time scale, which is experimentally reasonable, since the behavior of the PR for short-time scale is qualitatively similar to that for extremely long time scale ($t=10^{4}$).
- (ii) Regardless of diagonal or off-diagonal disorders, a symmetrical pattern of the PR with respect to $V=0$ reflects that the effect of nearest-neighbor interaction on the two-boson Anderson localization is independent of the $\mathrm {sgn}(V)$. Moreover, with the increase of $|V|$, two peaks arise nearby the $|V|$ on the order of disorder strength. As a result, appropriate nearest-neighbor interaction suppresses localization and compared to the off-diagonal disordered case, the suppressed degree of localization for diagonal disorder can be greater.
- (iii) Across the two peaks, the further increasing $|V|$ tends to enhance localization. The PR of the diagonal disordered case is sensitive to the further increasing $|V|$ and increases sharply, and the wave packet for diagonal disorder can completely localize on its initial position when $|V|$ becomes larger ($\mathrm {PR}=2$), whereas the PR of the off-diagonal disordered case is insensitive to the further increasing $|V|$ and increases gently, but the wave packet for off-diagonal disorder can also completely localize on its initial position except that stronger nearest-neighbor interaction is needed.
- (iv) While off-diagonal disorder is weaker than diagonal disorder, when the nearest-neighbor interaction is vanishing or moderate, for the same $V$, the PR of the diagonal disordered case is always larger than the off-diagonal disordered case, which means that the wave packet for diagonal disorder is less localized than that for off-diagonal disorder. This phenomenon will collapse as the nearest-neighbor interaction becomes stronger and the wave packet for diagonal disorder is more localized compared to that for off-diagonal disorder.
Fig. 1. Participation ratio for the initial state $|0,1\rangle$ versus the interaction strength $V$ at different times $t$ with the number of sites $N=21$. Diagonal (black solid curve) and off-diagonal (red dotted curve) disorders are chosen as $(\bar {W},\delta W,\bar {T},\delta T)=(2,2,1,0)$ and $(\bar {W},\delta W,\bar {T},\delta T)=(2,0,1,1)$, respectively, averaged over 5000 disorder realizations. A zoomed inset of two peaks at time $t=20$ is shown.
As the PR may merely disclose the effect of the interplay between disorder and nearest-neighbor interaction on the localized degree of wave packet, the investigation on the unique quantum correlation features of two bosons in the presence of disorder and nearest-neighbor interaction is still missing. Figure 2 shows the two-boson spatial correlations for ordered lattice, diagonal disorder, and off-diagonal disorder from top to bottom and with no interaction, weak interaction, and strong interaction from left to right, respectively.
Fig. 2. Two-boson spatial correlations $\Gamma _{i,j}(t)$ for the initial state $|0,1\rangle$. The upper row is for ordered lattice at time $t=4$ with interaction strength: (a) $V=0$, (b) $V=1$, and (c) $V=15$. The middle and lower rows are for diagonal and off-diagonal disorders at time $t=20$ with interaction strength: (d) and (g) $V=0$, (e) and (h) $V=2$, (f) and (i) $V=20$, averaged over 1000 disorder realizations. Disorder strength is the same as Fig. 1. The corresponding normalized particle-density distribution is attached below each spatial correlation. All the spatial correlations are identical to the corresponding attractive cases.
In the ordered case, two noninteracting bosons exhibit spatial bunching behavior and prefer to walk towards one direction [see Fig. 2(a)]. When the nearest-neighbor interaction is introduced, two-boson co-walking probability is promoted, i.e., two minor-diagonal components become pronounced [see Fig. 2(b)], whereas the remaining probabilities decrease as $|V|$ increases [see Fig. 2(c)].
We consider the disordered lattice from now on and for strong disorder, all the normalized particle-density distributions follow the dynamical behavior of single-particle Anderson localization: both sides of each distribution, which correspond to the component of ballistic propagation, have a low particle density and the highest particle density is located at the initial site, which reflects the two-boson localization. Furthermore, some additional information can be extracted from the spatial correlation, e.g., in the noninteracting case, while most of the probabilities are confined around the initial position owing to the effect of disorder, one boson can maintain localization and the other expands ballistically [see Fig. 2(d)]. Moreover, weak nearest-neighbor interaction suppresses certain components, such as the horizontal and vertical components through the initial position [see Fig. 2(e)]. The conservation of particle number demands that these decreased probabilities must be added into other parts of the spatial correlation, which will result in a broader spatial correlation. Therefore, the reduction of these components cuts down the particle density on the two initial sites, by contrast, the particle density on other sites increases due to the enhancement of other components of the spatial correlation. As a consequence, the suppression of localization occurs as the PR forecasts and a more extended particle-density distribution can be found. Figures 2(g) and 2(h) both show more compact spatial correlations and the discrepancies between them are less recognizable since for off-diagonal disorder, both the PRs are smaller and the difference between the PRs is also smaller compared to the diagonal disordered case, but the manifestation of the two spatial correlations is in accordance with the observation for diagonal disorder.
For diagonal disorder, large $|V|$ will thoroughly remove all the remaining probabilities, leaving the two minor-diagonal components alone [see Fig. 2(f)], whereas the PR of the off-diagonal disordered case is lack of the sensitivity for large $|V|$ and possesses a larger value so that two more visible minor-diagonal components emerge [see Fig. 2(i)]. Different from the two minor-diagonal components in Fig. 2(c), we deem the phenomenon to be a joint effect of disorder and nearest-neighbor interaction: disorder to a great extent confines the probability to the initial position [see Figs. 2(d) and 2(g)] and nearest-neighbor interaction further enhances the two minor-diagonal components and decreases the remaining probabilities [see Figs. 2(b) and 2(c)]. Hence, only the two minor-diagonal components with the maximum probability on the initial position survive, which indicates that two bosons are inclined to stay at the initial position all the time rather than to walk together over the lattice like the ordered case. The effect of nearest-neighbor interaction also exists in Figs. 2(e) and 2(h) and the two minor-diagonal components are elongated actually compared to Figs. 2(d) and 2(g).
Without loss of generality, we also consider that two indistinguishable bosons initially occupy the same site $|0,0\rangle$, namely, placed at the center of the lattice. We find that in the ordered case, two noninteracting bosons perform an independent quantum walk over the lattice, which is manifested by the peak per corner of the spatial correlation. The effect of nearest-neighbor interaction is also to bind two bosons and the diagonal component dominates as $|V|$ increases (not reported here). It is worth mentioning that for this initial condition, nearest-neighbor interaction plays the role of Hubbard-type on-site interaction since the spatial correlations solely containing the nearest-neighbor interaction are identical to those in the 1D prototypical Bose-Hubbard model [45,46]. The relevant cause will be clarified later on. Furthermore, for diagonal or off-diagonal disorders, compared to the case of the initial state $|0,1\rangle$, almost qualitatively identical results of both the PR and spatial correlation are obtained.
3.2 $|\Psi (0)\rangle =|-1,1\rangle$
We now turn to another case that two indistinguishable bosons are located at two sites separated by an empty site and the initial state is given by $|-1,1\rangle$. Figure 3 also shows the PR as a function of the interaction strength $V$ for both diagonal and off-diagonal disorders and both short- and long-time scales. Different from the case of the initial state $|0,1\rangle$, the PR increases as $|V|$ increases from the outset. Subsequently, it is nearly saturated as $|V|$ further increases, which means that the enhancement of nearest-neighbor interaction also suppresses localization but the suppressed degree will be almost constant no matter how large $|V|$ becomes. Visibly, for diagonal disorder, there are two dips in Fig. 3 and the slight recovery of localization occurs accordingly. Additionally, for the same $V$, the PR of the diagonal disordered case is always larger than the off-diagonal disordered case so that the wave packet for diagonal disorder is less localized compared to that for off-diagonal disorder all the time. However, the independence of the behavior of the PR on both the time scale and $\mathrm {sgn}(V)$ still persists.
Fig. 3. Participation ratio for the initial state $|-1,1\rangle$ versus the interaction strength $V$ at different times $t$ with the number of sites $N=21$. Diagonal (black solid curve) and off-diagonal (red dotted curve) disorders are chosen as $(\bar {W},\delta W,\bar {T},\delta T)=(2,2,1,0)$ and $(\bar {W},\delta W,\bar {T},\delta T)=(2,0,1,1)$, respectively, averaged over 5000 disorder realizations.
As before, we also present the two-boson spatial correlation in Fig. 4. In the ordered case, the spatial correlation shows a multi-block structure for two noninteracting bosons [see Fig. 4(a)]. The effect of nearest-neighbor interaction forces two bosons to separate to the opposite sides of the lattice so that fermionic antibunching behavior can be observed. Meanwhile, the diagonal and two minor-diagonal components all gradually reduce and eventually vanish as $|V|$ increases [see Figs. 4(b) and 4(c)]. In the noninteracting disordered case, the spatial correlation also shows the horizontal and vertical components through the initial position [see Fig. 4(d)]. The joint effect of disorder and nearest-neighbor interaction still makes two bosons mainly stay at the initial position. However, the increase of $|V|$ reduces the diagonal and two minor-diagonal components [see Fig. 4(e)] and they will never stay at the same site and two adjacent sites when $|V|$ is large enough [see Fig. 4(f)]. Moreover, the suppression of localization occurs, which is mainly caused by the reduction of the diagonal and two minor-diagonal components, and the normalized particle-density distributions for strong nearest-neighbor interaction are almost identical, which means that the suppression of localization is saturated. Compared to the diagonal disordered case, the spatial correlations for off-diagonal disorder are also more compact all the time and the discrepancies among them are also less recognizable, but their manifestation still conforms to the observation for diagonal disorder [see Figs. 4(g)–4(i)].
Fig. 4. Two-boson spatial correlations $\Gamma _{i,j}(t)$ for the initial state $|-1,1\rangle$. The upper row is for ordered lattice at time $t=4$ with interaction strength: (a) $V=0$, (b) $V=15$, and (c) $V=80$. The middle and lower rows are for diagonal and off-diagonal disorders at time $t=20$ with interaction strength: (d) and (g) $V=0$, (e) and (h) $V=20$, (f) and (i) $V=80$, averaged over 1000 disorder realizations. Disorder strength is the same as Fig. 3. The corresponding normalized particle-density distribution is attached below each spatial correlation. All the spatial correlations are identical to the corresponding attractive cases.
3.3 Two-boson energy spectrum and anomalous doublon eigenstates in the scattering continuum
The interesting phenomena emerging from the spatial correlations of the ordered cases are attributed to the band structure of the system and we thus show the energy spectrum of the ordered lattice as a function of the interaction strength $V$ in Fig. 5(a). It is evident that there is a major rectangular band in the middle and scattering eigenstates, which describe two independent bosons with weak spatial correlation [see Fig. 5(b)], make up this continuum. As $|V|$ increases, two branches are separated from this continuum and the gap between the branch and continuum is proportional to $|V|$. Depending on the $\mathrm {sgn}(V)$, one branch exceeds the continuum for $V>0$, whereas the other is lower than the continuum for $V<0$. Both the branches are induced by the nearest-neighbor interaction and consist of bound eigenstates. These bound eigenstates have a large probability of finding two bosons at two adjacent sites [see Fig. 5(c)] and the larger the $|V|$, the more pronounced the tendency. Hence, the initial state $|0,1\rangle$ nearly overlaps with these bound eigenstates so that there is nothing but co-walking throughout evolution in Fig. 2(c).
Fig. 5. (a) Energy spectrum of the ordered lattice with the periodic boundary condition versus the interaction strength $V$. Typical spatial correlations $\Gamma _{i,j}$ of scattering and bound eigenstates are shown in (b) and (c), respectively. (d) Spatial correlation $\Gamma _{i,j}$ for one of anomalous doublon eigenstates in the scattering continuum.
From the above analysis, it seems that the two-boson binding for the initial state $|0,0\rangle$ is impossible since scattering eigenstates cannot yield such a strong spatial correlation and bound eigenstates contribute to the two-boson co-walking only for the initial state $|0,1\rangle$. However, we have examined the spatial correlations of all eigenstates and fortunately found some doublon eigenstates in the continuum when $V\neq 0$, as detailed below, which correspond to the spatial correlation of two bosons mainly occupying the same site [see Fig. 5(d)]. This is novel and the doublon behavior also gradually becomes visible with the increase of $|V|$. The initial state $|0,0\rangle$ thus nearly excites these anomalous doublon eigenstates and two bosons will walk in a binding manner.
When the nearest-neighbor interaction is sufficiently strong, the initial state $|-1,1\rangle$ only incorporates scattering eigenstates, bound and anomalous doublon eigenstates do not participate in its constitution. In consequence, two bosons will never occupy the same site and two adjacent sites. The absence of these anomalous doublon eigenstates in the initial state $|-1,1\rangle$ is equivalent to two bosons being subjected to the hard-core condition and it is well known that the hard-core bosonic system can exhibit the fermionic property, which is due to the map between the hard-core bosonic and fermionic operators achieved by the Jordan-Wigner transformation. As a result, bosonic “fermionization” arises and the initial state $|-1,1\rangle$ for boson generates an identical spatial correlation to that for fermion in the large $|V|$ limit, as shown in Fig. 4(c).
We now provide an analytical solution of anomalous doublon eigenstates found in the scattering continuum. For an ordered lattice, the Hamiltonian of the system with nearest-neighbor interaction is given by
As anomalous doublon eigenstates are found in the continuum, we solve scattering eigenstates whose relative quasimomentums $k$ are real. For boson, $C_{i,j}=C_{j,i}$, which indicates that $\eta (d)=\eta (-d)$, and we have $\eta _{0}=\eta (1)/\cos (k)$. The periodic boundary condition requires $C_{i,j}=C_{i+N,j}=C_{i,j+N}$ so that we can obtain $\eta (d+N)=e^{iKN/2}\eta (d)=(-1)^{n}\eta (d)$ and further have
When $d=\pm 1$, substituting $\eta (d)$ and Eq. (8) into Eq. (7), after some algebraic calculation, one can obtain
Combining Eqs. (9) and (10), one can determine $k$ by solving the following equation:
It is obvious that Eq. (11) is actually an algebraic equation of $e^{ik}$ and is invariant under the transformation $k\rightarrow -k$. Accordingly, we restrict $k\in [0,\pi ]$. Furthermore, the function $\eta (d)$ has an explicit expression:
Hence, we can solve scattering eigenstates by Eq. (12).For a given $V$, there are $N(N-1)/2$ scattering eigenstates in the continuum and we want to verify that whether anomalous doublon eigenstates are in these scattering eigenstates or not. More concretely, for each $K$, the continuum consists of $(N-1)/2$ scattering eigenstates and we find that some of $(N-1)/2$ scattering eigenstates are always anomalous doublon eigenstates. For example, we choose $N=21$, $V=10$, and $K=-20\pi /21$ $(n=-10)$, the normalized $|\eta (d)|^{2}$ for one of anomalous doublon eigenstates is shown in Fig. 6. Obviously, nearly all the probabilities are concentrated on $|\eta _{0}|^{2}$, whereas few probabilities are distributed at the rest of $|\eta (d)|^{2}$, which manifests that two bosons almost only occupy the same site and anomalous doublon eigenstates are in the scattering continuum indeed.
Fig. 6. Normalized $|\eta (d)|^{2}$ for one of anomalous doublon eigenstates with parameters $N=21$, $V=10$, and $K=-20\pi /21$ ($n=-10$).
4. Discrimination between repulsive and attractive interactions
All the PRs and spatial correlations shown so far are independent of whether the nearest-neighbor interaction is repulsive or attractive and the reason behind this result is inherently associated with the symmetries of initial state $|\Psi (0)\rangle$ and observable $M$, namely, $\pi$-boost and time-reversal symmetries [39,47–49]. We now briefly review these two transformations: $\pi$-boost operator $B$ satisfies $B=B^{\dagger }$, $BB^{\dagger }=1$, and $B^{2}=1$, whose effect is appending a site-dependent $\pi$ phase on the operator $a_{j}$ in real space, $Ba_{j}B=e^{-i\pi j}a_{j}$. It is not hard to find that the Hamiltonian with repulsive (attractive) nearest-neighbor interaction under the $\pi$-boost transformation becomes $B(H_{o}+H_{+(-)})B=B(H_{o}-H_{p}+H_{I}^{+(-)})B=-(-H_{o}-H_{p}+H_{I}^{-(+)})=H_{o}-H_{-(+)}$. Time-reversal operator $T$ is antiunitary and can be represented as $KU$, where $U$ is unitary and $K$ takes complex conjugate. The time-reversal transformation converts a state into its complex conjugate, $T|\Psi \rangle =|\Psi ^{\ast }\rangle$, and modifies the time evolution operator, $Te^{-iHt}T^{\dagger }=e^{iHt}$.
In order to understand the crucial role of these two symmetries played in the independence of all the PRs and spatial correlations on the $\mathrm {sgn}(V)$, we first suppose that $|\Psi (0)\rangle$ and $M$ are invariant under the $\pi$-boost transformation and $[H_{o},M]=0$, then the expectation value of $M$ under $H_{o}+H_{+}$ becomes
Next consider the time-reversal invariance of both $|\Psi (0)\rangle$ and $M$ and we have
It thus can be deduced that the equivalence between $\langle M(t)\rangle _{+}$ and $\langle M(t)\rangle _{-}$ can be eliminated if $|\Psi (0)\rangle$ or $M$ are not invariant under the $\pi$-boost or time-reversal transformations. Note that a $|\Psi (0)\rangle$ getting a global $\pi$ phase under the $\pi$-boost or time-reversal transformations, $B|\Psi (0)\rangle =-|\Psi (0)\rangle$ or $T|\Psi (0)\rangle =-|\Psi (0)\rangle$, will not alter the result in Eq. (13) or Eq. (14). All the initial states and observables used in Sec. 3 hold both the $\pi$-boost and time-reversal symmetries so that the $\mathrm {sgn}(V)$ produces no effect on both the PR and spatial correlation.
We now consider the initial state $(|0,0\rangle +|1,2\rangle )/\sqrt {2}$, which breaks the $\pi$-boost symmetry but supports the time-reversal symmetry, to confirm the above analysis. The PRs of both the diagonal and off-diagonal disordered cases as a function of the interaction strength $V$ are shown in Fig. 7. The results for long-time scale are checked to be qualitatively similar to those for short-time scale and for a visualized comparison between the PRs with $\pm V$, we present the results only for short-time scale and show the PR for each $|V|$. Strong disorder diminishes the discrepancies between the repulsive and attractive cases and weaker disorder is thus chosen. One can observe from Fig. 7 that the PRs with $\pm V$ mismatch each other as expected and the PR for attractive interaction is always smaller than that for repulsive interaction. However, the difference between them shrinks as $|V|$ increases. Furthermore, we also plot the two-boson spatial correlations with $V=\pm 2$ for both ordered and disordered lattices in Fig. 8. It can be seen that all the spatial correlations with $+V$ differ from those with $-V$ and the repulsive disordered case [see Figs. 8(c) and 8(e)] exhibits a more broader spatial correlation compared to the attractive disordered case indeed [see Figs. 8(d) and 8(f)].
Fig. 7. Participation ratio for the initial state $(|0,0\rangle +|1,2\rangle )/\sqrt {2}$ versus the interaction strength $|V|$ at time $t=20$ with the number of sites $N=21$. (a) Diagonal and (b) off-diagonal disorders are chosen as $(\bar {W},\delta W,\bar {T},\delta T)=(2,1,1,0)$ and $(\bar {W},\delta W,\bar {T},\delta T)=(2,0,1,0.5)$, respectively, averaged over 1000 disorder realizations. The red dotted and black solid curves represent repulsive and attractive interactions, respectively.
Fig. 8. Two-boson spatial correlations $\Gamma _{i,j}(t)$ for the initial state $(|0,0\rangle +|1,2\rangle )/\sqrt {2}$. The upper row is for ordered lattice at time $t=4$. The middle and lower rows are for diagonal and off-diagonal disorders at time $t=20$, respectively, averaged over 1000 disorder realizations. Disorder strength is the same as Fig. 7. The left and right columns correspond to nearest-neighbor interaction $V=2$ and $V=-2$, respectively.
We next consider the $\pi$-boost symmetrical but time-reversal asymmetrical initial state $(|0,0\rangle +i|1,1\rangle )/\sqrt {2}$. Figure 9 shows the two-boson spatial correlations with $\pm V$ for both ordered and disordered lattices. The preservation of the $\pi$-boost symmetry implies that when we evaluate $\langle M(t)\rangle$ under $H_{+}$, equivalently, $\langle M(t)\rangle$ under $H_{-}$ is also obtained except that time is inverted, i.e., $\langle M(t)\rangle _{+}=\langle M(-t)\rangle _{-}$ in Eq. (13). The inversion of $t$ indicates that the direction of the velocity of the wave packet becomes opposite, in other words, the wave packet propagates over the lattice in an opposite direction. Nevertheless, the time-reversal symmetry guarantees that such a counter-propagation does not work since the velocity composition of the wave packet is symmetrical now, which eventually determines a symmetrical diffusion on the spatial correlation. Therefore, we cannot distinguish $\langle M(-t)\rangle _{-}$ from $\langle M(t)\rangle _{-}$. On the contrary, if the time-reversal symmetry is not satisfied, the velocity composition of the wave packet is no longer symmetrical, the forward and backward propagations will behave dissimilarly and further lead to two specularly symmetrical spatial correlations. This is particularly visible for the ordered case and the spatial correlations with $\pm V$ are reversed with respect to the direction of the center of the initial position [see Figs. 9(a) and 9(b)]. In the disordered case, this phenomenon still persists but is inconspicuous [see Figs. 9(c)–9(f)]. Additionally, two specularly symmetrical spatial correlations yield two inverted normalized particle-density distributions so that the $\mathrm {sgn}(V)$ has no effect on the PR for the $\pi$-boost symmetrical but time-reversal asymmetrical initial state (not reported here).
Fig. 9. Two-boson spatial correlations $\Gamma _{i,j}(t)$ for the initial state $(|0,0\rangle +i|1,1\rangle )/\sqrt {2}$. The upper row is for ordered lattice at time $t=4$ with interaction strength: (a) $V=2$ and (b) $V=-2$. The middle and lower rows are for diagonal and off-diagonal disorders at time $t=20$ with interaction strength: (c) and (e) $V=20$, (d) and (f) $V=-20$, respectively, averaged over 1000 disorder realizations. Disorder strength is the same as Fig. 7. The white dashed line guides the specularly symmetrical axis for eyesight.
5. Experimental feasibility
We now detailedly analyze and discuss some practical issues in relation to the experimental realization of the proposed model. Experimentally, a promising candidate for the simulation of two interacting bosons in a 1D lattice is 2D linear photonic waveguide array [50–52], since the Schrödinger equation of probability amplitude can be formally mapped to the coupled-mode equation of light propagation in a 2D linear photonic waveguide array, which is given by
Fig. 10. Schematic diagram of the two-dimensional linear photonic waveguide array for the off-diagonal disordered case. Each circle represents a waveguide. Red hollow and green-colored circles label waveguides with different refractive indices, respectively, which effectively implements the nearest-neighbor interaction. The two-dimensional geometry is symmetrical upon reflection along the diagonal axis.
6. Conclusions
In conclusion, we have explored the two-boson Anderson localization with nearest-neighbor interaction in a 1D disordered lattice, which is feasible to directly observe in a 2D linear disordered photonic waveguide array. We have detailedly analyzed and discussed the PR for short- and long-time scales, repulsive and attractive cases, diagonal and off-diagonal disorders, and three types of initial states. It turns out that the effect of nearest-neighbor interaction on the Anderson localization depends on the type of disorder and initial condition, whereas it qualitatively has nothing to do with the time scale of evolution and is completely independent of whether the nearest-neighbor interaction is repulsive or attractive. We have also considered and displayed the two-boson spatial correlations under two initial conditions for both ordered and disordered cases. In the ordered case, we have found anomalous doublon eigenstates from the scattering continuum, which, together with bound eigenstates, result in two types of two-boson bindings and bosonic “fermionization”. In the disordered case, the effect of nearest-neighbor interaction on the Anderson localization has been further validated and intuitively elucidated, and the joint effect of disorder and nearest-neighbor interaction has been addressed. Additionally, we have demonstrated that an initial state which is not invariant under the $\pi$-boost or time-reversal transformations can trigger the dependence of the PR and spatial correlation on the sign of interaction so that the repulsive and attractive interactions can be discriminated. We hope that the present work can deepen the understanding of the interplay between disorder and interaction and stimulate more interest in the simulation study of quantum two-body system and Anderson localization based on linear photonic lattices.
Funding
National Natural Science Foundation of China (11874132, 12074330, 61575055, 61822114, 62071412).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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