Arbitrary synthetic dimensions via multi-boson dynamics on a one-dimensional lattice

The synthetic dimension, a research topic of both fundamental significance and practical applications, is attracting increasing attention in recent years. In this paper, we propose a theoretical framework to construct arbitrary synthetic dimensions, or N-boson synthetic lattices, using multiple bosons on one-dimensional lattices. We show that a one-dimensional lattice hosting N indistinguishable bosons can be mapped to a single boson on a N-dimensional lattice with high symmetry. Band structure analyses on this N-dimensional lattice can then be mathematically performed to predict the existence of exotic eigenstates and the motion of N-boson wavepackets. As illustrative examples, we demonstrate the edge states in two-boson Su-Schrieffer-Heeger synthetic lattices without interactions, interface states in two-boson Su-Schrieffer-Heeger synthetic lattices with interactions, and weakly-bound triplon states in three-boson tight-binding synthetic lattices with interactions. The interface states and weakly-bound triplon states have not been thoroughly understood in previous literatures. Our proposed theoretical framework hence provides a novel perspective to explore the multi-boson dynamics on lattices with boson-boson interactions, and opens up a future avenue in the fields of multi-boson manipulation in quantum engineering.


Introduction
Dimensionality is one of the most important concepts in modern physics. In the condensed-matter physics, systems with different dimensionalities exhibit vast different behaviors. Notable examples include topological insulators with protected surface states [1][2][3][4] and superconducting electronic gases confined in quantum structures [5][6][7][8][9], which all exhibit significant dependence of system dimensionality.
To explore phenomena unique to high-dimensional physics, the artificial synthesis of extra dimensions in low-dimensional platforms has attracted great interest in recent years [10][11][12][13][14], partly because experimental platforms, including dielectrics [15], plasmons [16,17], atoms [11,18,19], and magnetons [20], feature relatively convenient fabrication and manipulation in low dimensions. By connecting internal degrees of freedom of particles to form apparent artificial lattices [13], one can successfully construct synthetic dimensions which, together with spatial dimensions, make the dimensionality of a physical system beyond that of real space. This focus on synthetic dimensions is of fundamental significance in physics, and also holds promising applications in the field of optical communication and quantum information processing [21][22][23][24][25].
Quantum many-body physics, as an important research subject in the quantum physics, receives intensive studies, with its counterparts also greatly explored by quantum simulations in optical platforms [26][27][28]. As an outstanding example, repulsively bound boson pairs, also called doublons, have been unveiled as a result of Bose-Hubbard Hamiltonian [29]. Following this important discovery, quantum problems of two-particle states with one-dimensional (1D) interactive Hamiltonians, such as the tight-binding Bose-Hubbard model [30], the tight-binding Bose-Hubbard model with a parabolic potential [31], with a small impurity potential [32], and with nonlocal interactions [33][34][35], Su-Schrieffer-Heeger (SSH) Bose-Hubbard model with nonlocal interactions [36][37][38][39][40], as well as twoparticle problems with Bloch oscillation in an external electrical field [41][42][43], have been investigated with very broad interest in both condensed-matter and optical societies. In particular, when two correlated indistinguishable boson dynamics are under exploration, the 1D-2D mapping approach [30,32,33,35,39,40,43] has been used to convert the two-boson dynamics on 1D lattices into the singleparticle dynamics on two-dimensional (2D) lattices. This mapping method significantly facilitates the theoretical analysis and deepens our understanding of two-boson interacting dynamics in the quantum regime.
Inspired by the works mentioned above, in this paper, we propose to construct arbitrary synthetic dimensions with multiple bosons in a 1D array, i.e., an N-boson synthetic lattice, by generalizing the 1D-2D mapping method to a 1D-N-dimensional counterpart. After properly choosing the symmetry restriction of the wavefunction, this N-dimensional synthetic space with N indistinguishable bosons features the periodicity along each of its synthetic dimensions, making it possible to predict the N-boson dynamics by utilizing the well-established band structure approach. Related to, but distinct from previous works [44][45][46][47], which have constructed extra dimensions beyond the original system by resorting to the photon number on each available lattice site, our theoretical framework harnesses the 4 lattice site number that each individual boson resides on to achieve the dimension augmentation. As for demonstrations, we provide examples exploring multi-particle physics with extended nonlinearity including the two-boson case supporting the topological interface states, as well as the three-boson case predicting the boson blockade effects in three dimensions. Our work hence points out a fundamentally different perspective for studying many-body dynamics with nonlinearity from a physical picture with synthetic dimensions, which also shows potential applications in quantum simulations and quantum information processing with synthetic dimensions in optical systems.
The remaining parts of this paper are organized as follows. Section 2 describes the fundamental theoretical framework of the N-boson synthetic lattice that will be adopted throughout this paper. Sections 3,4 and 5 give examples addressing the richness of various systems where our theoretical framework is applicable. Section 6 discusses some possible experimental implementations followed by a brief conclusion.

Theoretical framework
We start to establish our theoretical framework by examining two indistinguishable bosons on an infinite 1D tight-binding lattice, and then move on to the N-boson case. The two-boson case has been addressed in several previous researches as the 1D-2D mapping approach [30,32,33,35,39,40,43]. Here we first re-iterate this approach in a more explicit and systematic way. The Hamiltonian of the system is: where g, a real number, is the coupling strength between neighboring sites, and k b ( † k b ) is the annihilation (creation) operator of bosons on the k-th lattice site as shown in Fig. 1(a). We take 1 = throughout the paper for simplicity. In Schrödinger's picture, the two-boson state is: where 0 is the vacuum state. By substituting Eqs. (1) and (2) Here, the conventionally standard restriction on mn v is to follow the exchange symmetry, i.e., we let Also, the probability amplitude that one boson is observed on the m-th lattice site and the other on the which shows that the dynamics of mn v is mathematically equivalent to the single-particle dynamics on a 2D tight-binding lattice. This lattice, referred to as a two-boson synthetic lattice, is illustrated in Fig.   1(b), and the process of constructing such two-boson synthetic lattice can be denoted as 1D-2D mapping.
The procedure for the construction of such synthetic lattice can be generalized for N-boson states.
We consider the tight-binding Hamiltonian of Eq. (1) operating on an N-boson state From the Schrödinger equation, we can calculate that ( ) and the square of the modulus of 1 ,, N v  is the N-th-order correlation function: Because of the exchange symmetry, Eq. (9) can finally be simplified to ( ) ( ) Eq. (13) is mathematically identical to the coupled-mode equations [48] Since the coupling strengths are uniform in this synthetic lattice along all of the N dimensions ( 1 ,, N  ), we can define an N-boson band structure analytically: where   T 1 ,, N kk = k and i k is the wavevector along the i  direction in Eq. (14). The group velocity of the particle wavepacket in the synthetic lattice associated with a specific N-boson state can be calculated using the band structure by computing the derivative of the energy spectrum with respect to the wavevectors. One notices that the band structure for a 1D tight-binding lattice is ( )  , , , , , , , , , the temporal profile of the boson excitation, takes the form of a modulated Gaussian envelope where 0  is the normalization coefficient, 0 t and  are the center and the width of the Gaussian pulse respectively, and   is the excitation energy. Finally, the multi-boson excitation probabilities can be calculated according to Eq. (12), and the average boson number on the k-th lattice site of the hosting 1D tight-binding chain ( Fig. 1(a)) can be determined by Some typical simulation results of the two-boson dynamics with Hamiltonian Eq. (1) are displayed in Fig. S2 [49].
Before ending this section, we note that, for illustration purposes, the synthetic lattice described in this section is rather simple with only real-valued, nearest-neighbor coupling and no interaction.
However, more complex behaviors including non-trivial topological and interacting features can be created in the N-boson synthetic lattice based on the same theoretical framework as introduced here. To highlight these more complex features, in the following, we discuss two bosons on an SSH lattice without/with interactions (Sections 3 and 4), and two/three bosons on a tight-binding extended Bose-Hubbard lattice (Supplemental Notes 2 [49] and Section 5).

Two bosons on an SSH lattice without interactions
As the first illustration of interesting topological behaviors in the N-boson synthetic lattice, here we examine the two-boson dynamics on a 1D SSH lattice. The 1D SSH model is perhaps one of the simplest models that exhibit a nontrivial energy band topology [52], and has been widely studied [53][54][55][56][57][58]. Here we show that dynamics of two non-interacting bosons on a 1D SSH lattice can be mapped to a 2D SSH model. A 1D SSH lattice, as shown in Fig. 2(a), features alternating coupling strengths of 1 g and 2 g , and consequently there are two types of lattice sites on the chain which are denoted as k C and k D in the k-th unit cell, respectively. 1 g is referred to as intracell coupling and 2 g as intercell coupling.
The corresponding Hamiltonian and the two-boson state under consideration are expressed by , we obtain the differential equations: Now we apply the exchange symmetry restriction of the wavefunction ( Again, this exchange symmetry is not a physically necessary restriction on the wavefunction, yet it is mathematically essential in obtaining Eq. (21). Eq. (21) gives the two-boson synthetic lattice as illustrated in Fig. 2(b), which corresponds to a standard 2D SSH model [59]. Mathematically, this lattice features translational symmetry along both m and n axes with period 2, and its band structure can be analytically obtained as:  22) can also be expressed as where ( )

because no boson-boson interaction is considered in
Hamiltonian Eq. (18) [61]. A 2D SSH lattice exhibits topological features since edge modes exist in the nontrivial phase ( 12 gg  ) because of nonzero Berry connection and Zak phase [60]. Here we examine an SSH stripe which is infinite along the m axis but has finite width along the n axis ( 15 15 n −   ). (Strictly speaking, the trivial/nontrivial phase transition point is not exactly at 12 gg = , depending on the stripe width, but takes 12 gg = in the limit of infinite stripe width [60].) One notes that such an SSH stripe is not physically implementable by our platform of two bosons on a 1D lattice, since the wavefunction exchange symmetry requires that the lattice should have reflection symmetry along mn = axis; yet this SSH stripe is still a useful mathematical construct for analyzing edge modes. In Figs  Although the band structure analysis shown in Fig. 3 is based on an infinite SSH stripe, the edge modes still preserve in a finite two-boson synthetic lattice (i.e. a square-shape 2D SSH panel as in Fig.   2(b)), which we demonstrate with numerical simulations in Fig. 4. In simulations, we consider a 1D SSH lattice with 62 lattice sites (31 unit cells), i.e., 15 15 k −   in Fig. 2(a). It gives a two-boson synthetic lattice with 62 62  sites in two dimensions. For the convenience of plotting in Fig. 4, lattice site k C in Fig. 2(a) is labeled as the 2k-th site (i.e.,     Fig. 4(b). One clearly sees the edge modes in the twoboson synthetic lattice in Fig. 4(b), which features bi-directional propagation because the corresponding mode A (with a positive slope) and A' (with a negative slope) in Fig. 3(b) are both excited by the source.
The corresponding evolution of the average boson numbers in the 1D lattice in Fig. 4(a)  In this section, we utilize a simple example to show that a perspective from a synthetic lattice helps one to understand the N-boson dynamics in a 1D lattice. This capability will be much more important when the interaction between bosons is introduced, which we will show in the next two sections.

Two bosons on an SSH lattice with interactions
An interacting quantum system certainly show a far richer physical behavior as compared with a non-interacting systems. The study of the N-boson synthetic lattice provides a unique perspective to explore multi-boson problems with interactions. As an illustration, we consider two bosons on the same 1D SSH lattice as that in Section 3, but with boson-boson interactions of the extended Bose-Hubbard type. The corresponding Hamiltonian is described by where U is the interaction strength and R is the interaction range.    g Eq. (29) therefore again can be mapped to a synthetic lattice in two dimensions as shown in Fig. 5(a).
Here U appears as the onsite potential on the diagonal with a width of 2R + 1 in this two-boson synthetic lattice, as a result of the nonlinear interaction terms in Eq. (27), as indicated by the green-colored lattice sites in Fig. 5(a). These lattice sites with nonzero onsite potentials compose the 'nonlinear lattice region', while the sites without onsite potentials compose the 'linear lattice region'.  6(d) and 6(e)). It is shown that, of the two degenerate bands in each group, one band is symmetric with respect to the line mn = (Figs. 6(c) and 6(e)) and the other one is anti-symmetric (Figs. 6(b) and 6(d)).
However, only the symmetric modes can be physically excited because of our exchange symmetry restriction of the wavefunction. The pair of anti-symmetric interface modes (bands 1, 3) is an artifact produced from the 1D-2D mapping method, and does not physically exist.  Fig. 7(a), one can see that the interface mode is excited and preserved throughout the simulation, as the distance between the two bosons keeps stationary. The boson scattering in Fig. 7(a) is negligible because the bands of interface modes are relatively flat, so the group velocity of the wavepacket is small. In Fig. 7(b), we take the case that 0 C and 6 D are excited but the lattice is in the trivial, nonlinear phase ( 1 3 gg = , 2 gg = , 2 Ug = ). There is no interface mode observed. In Fig. 7(c), the lattice is in the nontrivial, nonlinear phase, but two bosons are excited simultaneously on site 0 C , and thus bulk modes are excited instead. Finally, in Fig. 7(d), 0 C and 6 D are excited and the lattice is in the nontrivial phase but 0 U = , and no interface mode is found as expected. The simulations here prove that, for the Hamiltonian in Eq. (27), the nonzero onsite potential U, the topologically nontrivial phase ( 21 gg  ), and an initial excitation on the linear-nonlinear interface are all necessary conditions to successfully excite an interface mode in Fig. 5(c).
The successful demonstration of the interface modes exhibits the capability of our approach, with the two-boson synthetic lattice and the band structure analysis, to predict new physical phenomena. We also note that several previous studies used the modified Bethe ansatz and gave explicit solutions to some specific Hamiltonians with short-range interactions [30,33,[35][36][37]40], yet these methods still face challenges when the interactions are arbitrarily nonlocal ( 2 R  ) or the total boson number is large ( 3 N  ). Our theoretical approach proposed here is more general and, as we show in Sections 4 and 5, is able to deal with multi-boson dynamics with long-range interactions and discover exotic states in these systems.

Three bosons on a tight-binding lattice with interactions
We have discussed the 2D synthetic SSH lattice induced by the two-boson physics in Sections 3 and 4. In this section, we demonstrate a three-dimensional (3D), three-boson synthetic lattice, which is constructed by three indistinguishable bosons on a 1D tight-binding extended Bose-Hubbard Hamiltonian, and show that it possesses exotic trimer states.
We consider a Hamiltonian which goes beyond previous studies [62][63][64] with nonlocal interactions: and the three-boson state For this three-boson case, the wavefunction exchange symmetry is defined as , and then the permutation sum in Eq. (32) can be removed: The first six terms in Eq. (33) correspond to a 3D tight-binding lattice, which spans the full 3D

( )
,, m n p space with equal coupling strengths g, as is shown by the cubic structure in Fig. 8 which can be shown by the energy isosurfaces in Fig. 8 Fig. 9(a). One can see that there are four separate quasi-continuum bands. We find that these bands correspond to different multi-boson states when we plot the eigenstates of these four quasicontinuum bands in the 1 l -2 l plane, as shown by the diagram in Fig. 9 categories, which are dominated by eigenstates from different bands. In the blue regions as shown in Fig. 9(b), we find that they correspond to the band with lowest energy in Fig. 9(a) Fig. 9(d)), residing in the second-lowest energy band in Fig. 9(a). Moreover, the red region denotes the tightlybound triplon states (see the eigenstate distribution at Fig. 9(f)) with the strongest interaction (consequently the highest energy) in Fig. 9(a) Fig. 9(e)) in the orange regions. Such an exotic state in Fig. 9(e) has not been thoroughly understood previously in the model without the nonlocal interaction [62]. In this state, compared to the tightly-bound trimer in the red region, the relative distances between each two of these three particles cannot be all smaller than R, while compared to the dimer-monomer 20 states in the cyan regions, none of the relative distances between each two of these three particles is allowed to be larger than 2R, albeit there is no artificial discontinuity at 2R in the Hamiltonian in Eq. (30). This 'virtue potential wall' at 2R is entirely attributed to the boson-boson blockade effect.
Therefore, in this fascinating state in Fig. 9(e), three bosons are loosely localized together, neither too close to nor too far from one another.

Discussions and conclusions
In this paper, we propose a theoretical framework to treat the multi-boson dynamics in a onedimensional lattice as an N-boson synthetic lattice by applying exchange symmetry restrictions to the wavefunction. In such an artificially constructed N-boson synthetic lattice, one can mathematically perform band structure analysis and the resulting N-dimensional band structure provides a novel perspective to analyze the multi-boson dynamics on 1D lattices. We show that, for complicated Hamiltonians with boson-boson interactions, projected band structures could be understood from the full N-dimensional band structure, and nontrivial multi-boson states (such as interface modes in Section 4 and weakly-bound triplon states in Section 5) can be successfully predicted. In addition, the 1D lattice required in this paper to hold these multiple bosons, can be in either real or synthetic dimensions per se.
If this 1D lattice itself is in synthetic dimensions (e.g. formed by exploiting boson frequencies, orbital angular momenta, etc. [13]), our theory suggests that this multi-boson approach might enable the construction of arbitrarily multi-dimensional systems on an otherwise zero-dimensional platform, We also briefly suggest several possible experimental platforms for exploring N-boson dynamics in the 1D lattices where our proposed approach can be conducted. For lattices in the real space, coupled photonic waveguides and cavities [42,43,[65][66][67][68][69][70][71], cold bosonic atoms in optical lattices [72][73][74][75], and superconducting circuits [76][77][78][79] are state-of-the-art technologies for implementations. For example, the coupled waveguide array fabricated to study the two-photon quantum walks in a 1D SSH lattice [71] gives an experimental demonstration of our discussions in Section 3. As for the synthetic space, 1D tight-binding model and SSH Hamiltonians without interactions are readily accessible via synthetic frequency dimensions in few ring resonators with electro-optic modulation [15,57,[80][81][82]. Artificial boundaries of the lattices along frequency dimensions could be achieved by incorporating group velocity dispersions in ring resonators or introducing lossy absorbers at particular frequencies [70,83], and interactive terms in Hamiltonians can be constructed from four-wave-mixing processes with the nonlinear effect [84]. Additionally, arbitrarily long-range coupling in synthetic frequency dimensions is also possible by special designs of the electro-optic modulation profile [85].
In conclusion, we systematically develop a theoretical framework on the creation of the N-boson synthetic lattice that enlightens insights into the studies of both synthetic dimensions and the physics of multiple interactive indistinguishable bosons. The connection between multiple bosons and multiple dimensions is highlighted. Through band structure analysis, novel dynamics of multi-boson states are unveiled, and can be confirmed by numerical simulations. Our study hence points out a new way towards the studies of boson-boson interactions and multi-boson dynamics on lattices, and also holds potentials for exploring important multi-boson manipulations together with nonlinearity, nontrivial topology, and/or boson entanglements with possible applications in fields of quantum computations, quantum simulations, and quantum information processing.