Many-body chiral edge currents and sliding phases of atomic spinwaves in momentum-space lattice

Yongqiang Li, 2 Han Cai, Da-wei Wang, Lin Li, Jianmin Yuan, and Weibin Li Department of Physics, National University of Defense Technology, Changsha 410073, P. R. China Department of Physics, Graduate School of China Academy of Engineering Physics, Beijing 100193, P. R. China Interdisciplanery Center for Quantum Information and State Key Laboratory of Modern Optical Instrumentation, Zhejiang Province Key Laboratory of Quantum Technology and Device and Department of Physics, Zhejiang University, Hangzhou 310027, China MOE Key Laboratory of Fundamental Physical Quantities Measurement, Hubei Key Laboratory of Gravitation and Quantum Physics, PGMF and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China School of Physics and Astronomy, and Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems, The University of Nottingham, Nottingham NG7 2RD, United Kingdom (Dated: March 18, 2020)

A key element to build many-body correlations is strong two-body interactions. Atoms excited to electronically high-lying (Rydberg) states provides an ideal platform due to their strong and long-range van der Waals interactions (e.g. a few MHz at several µm separation [34]). Benefiting directly from this interaction and configurable spatial arrangement [35][36][37], Rydberg atoms have been used to emulate topological dynamics [38,39] in a time scale typically shorter than Rydberg lifetimes (typically 10 ∼ 100µs). On the other hand, Rydberg dressed states, i.e. electronic ground states weakly mixed with Rydberg states [40][41][42][43][44], have longer coherence time (hundreds of ms) and relatively strong long-range interactions [45]. Rydberg dressing has been demonstrated experimentally in magneto-optical traps [46], optical tweezers [47] and lattices [48]. A remaining open question is to identify routes to create chiral states by making use of Rydberg dressed atoms.
In this work, we propose a new lattice setup to explore chiral edge currents of an interacting many-body system via hybridizing long-lived atomic hyperfine states with electronically excited Rydberg states. By mapping to momentum space [49], collective excitations (atomic spinwaves) of ultracold gases form effective sites of an extended two-leg Bose-Hubbard model with competing laser-induced complex hopping and Rydberg dressed interactions. With this new setup, previously untouched many-body phases and correlated chiral dynamics can be realized in momentum-space lattice (MSL). Through dynamical mean-field calculations, we identify novel chiral edge currents in the many-body ground state. The strong dressed interaction suppresses edge currents, leading to a blockade in momentum space. By incorporating the Rydberg dressing to the two-legs, we find exotic translational-symmetry-broken sliding insulating and superfuid/supersolid phases in momentum-space lattice.  Fig. 1(a)] are created by an off-resonant and a resonant standing wave lasers along the x-axis [with wave vector k = k cx (x is the unit vector) and wave length λ = 2π/k c ]. For small number of excitations, the spinwaves are described by free bosons [49]. By projecting the spinwave to momentum space with a characteristic momentum k c [49], we obtain a ladder of A-leg and B-leg for the |a and |b states [ Fig. 1(b)], respectively. The jth site of A-leg (B-leg) represents a collective state with wave vector 2jk c [(2j − 1)k c ]. When the standing wave lasers are phase mismatched [ Fig. 1(a)], a synthetic magnetic field is generated [50]. This gives a nearest-neighbor hopping with complex amplitudes along and between the ladders, described by the Hamiltonian where φ, h o and h r are the flux, intra-and inter-leg hopping amplitudes. In our setting, state |a (|b ) is coupled to a Rydberg state |R a (|R b ) by an off-resonant laser. This induces a state-dependent two-body soft-core shape interaction V σ (x) ≡ C σ /(r 6 σ + |x| 6 ), where C σ and r σ (σ = a, b) are the dispersion coefficient and characteristic distance of the interaction potential, respectively. Their values can be engineered by tuning parameters of the dressing laser [42][43][44][45]. The Hamiltonian for the interactions of the dressed state is, is the Fourier transformation of the corresponding interaction.Ṽ σ (k) decays rapidly with increasing k as typically r c λ [ Fig. 1(c)]. Finally the two-leg Bose-Hubbard Hamiltonian is given by H = H c + H d − i,σ µ σ n iσ with n iσ = σ † i σ i and µ σ to be the atomic density at site (momentum) i and chemical potential in state |σ . Details of the Hamiltonian can be found in the Supplemental Material (SM). In the following, we choose V ≡Ṽ b (0) as the unit of the energy.
Symmetry and ground state phases in the interaction free case-The coupling Hamiltonian H c possesses where τ , C and T are the time-reversal, chiral and translational symmetry operators. The chiral symmetry shows that currents after swapping the two states will remain the same if the flux is shifted simultaneously by π, i.e. φ → φ + π. The system does not preserve the time-reversal symmetry in general except when φ = π/2, where the ground state energy exhibits double degeneracy [50], already leading to rich chiral phases (see examples in SM).
Solving H c exactly we obtain three types of band minima, i.e. a single minimum at real-space coordinate x = 0 or = 0, and two minima, as show in Fig. 1(d), analogous to spin-orbit coupling setup in real space [12] (the states |a and |b treated as spin | ↑ and | ↓ , respectively). The corresponding chiral phases are characterized by current J σσ ≡ −2Im(h i,j σ † i σ j ) for the bond j → i and the leg σ → σ , similar to the definition in real-space lattices [52,53]. As shown in Fig. 2, the ground state prefers a number of chiral superfluid phases (CSFs). When the inter-leg coupling h r is strong, currents along the ladders (edges) have opposite directions, i.e. J AA × J BB < 0 and J AB = 0 [ Fig. 2(c)], denoted by CSF m phase (condensed into band minimum at x = 0). When h r and h o are comparable, we have a different chiral superfluid phase (CSF v condensed into band minimum at x = 0) where currents on the ladders and rungs satisfy J AA × J BB < 0, and J AB = 0. The CSF m and CSF v phases are analogues of the Meissner and vortex phases of real-space ladder systems [16][17][18][19][54][55][56][57][58][59][60][61][62][63][64][65][66].
When h r is further decreased, two band minima become degenerate [see Fig. 1(d)]. At the minima, currents on both ladders flow in the same direction, i.e. J AA × J BB > 0. This leads to a particularly interesting anti-chiral edge current (denoted by CSF p ) in the system. Compared to real-space lattices [18,62], a fundamental difference of our scheme is the staggered magnetic flux, i.e. the flux of a plateau in the zigzag lad- There are three quantum phases with different types of edge currents, including the CSFm and CSFv phases with chiral edge currents, and the CSFp phase with anti-chiral edge current. Here, the filling Ntot/L lat = 0.125 with Ntot being the total number of atoms in the lattice and L lat the lattice size [51], and the dashed lines denote the non-interacting system. Interaction effects on phase transitions for a non-interacting (c) and interacting (d)(e) systems. Note that in the absence of two-body interactions, the currents for φ = 5π/4 can be obtained from (c) by using the chiral symmetry (i.e. swapping the state |a and |b ). The other parameters ho = 0.1 and V = 0 (c); ho = 0.1 and V = 1 (d); ho = 0.1 and V = 1 (e). (f) Interplay between flux and interaction for a fixed hopping amplitude ratio. Inset: Interaction-induced CSFv-CSFm phase transition for a fixed flux φ = 0.6π (g). Periodic boundary condition is used in the calculation.
der is φ and its neighbor π − φ. The system then hosts asymmetric band structures in the regime h o /h r 1, as shown in Fig. 1(d), with atoms in A-(B-) leg favoring the left (right) minimum [ Fig. S2(b)]. As a result, the staggered flux induced asymmetric condensation of particles in each ladder changes directions of edge currents (a perturbative derivation for h o /h r 1 can be found in SM). As far as we know, this new quantum phase has not been studied in real-space lattice. Since the CSF p phase emerges in the limit h o /h r 1, one could create this phase through an adiabatic manner. Starting with large h r and h o , one can adiabatically reduce (increase) h r (h o ) to enter the CSF p phase region while keeping h o (h r ) fixed. The anti-chiral edge current will be induced and saturate to a finite value even though h r /h o → 0.
Stable anti-chiral currents and excitation blockade by single Rydberg dressing. -In the single Rydberg dressing the B-leg is coupled to a Rydberg state (i.e. V a (x) = 0 and V b (x) = 0). We employ a bosonic dynamical meanfield calculation that captures both quantum fluctuations and strong correlations in a unified framework [52,53,67] (see SM for details). The reliability of this approach has been confirmed by a comparison with an unbiased quantum Monte Carlo simulation [68].
Intuitively, one would expect currents are suppressed by the two-body interaction. On the contrary, the CSFs phases are stable against the dressed interaction [see The two-body interaction plays a vital role in the dynamics. This is illustrated with a situation where initially two excitations at the first site of the A-leg (i.e. n  Turning on the interaction, the excitation probability in the B-leg is reduced apparently. A closer examination shows that the double occupation probability P 2 = j | ψ(t)|(b † j ) 2 / √ 2|ψ(0) | 2 in the B-leg is significantly suppressed by the strong interaction. |ψ(t) = exp(−iHt)|ψ(t = 0) is the many-body state at time t. This excitation suppression could be regarded as an interaction blockade in momentum space, analogue to the Rydberg blockade effect in real space [69,70].
The dynamically generated currents show features of the ground state phases, though their strengths are suppressed by the interaction. For example, when the parameters are in the CSF v phase, for example, the currents counter-propagate on the two legs at the beginning of the evolution [ Fig. 3(c)-(e)]. Surprisingly, the currents for φ = 5π/4 along the legs become co-propagating when two-body interactions are important [ Fig. 3(f)]. This behavior is similar to the currents found in the CSF p phase.
Sliding phases when both chains are laser dressed to Rydberg states.-A double Rydberg dressing is realized when the two legs are coupled to two different Rydberg states. For energetically close Rydberg states, this leads to similar dressed potentials for the two legs. Without loss of generality, we focus on the case withṼ a (k) =Ṽ b (k) in the following.
Without hopping (h r = h o = 0), the ground state is an unusual insulating state with a high degree of degen- is the binomial coefficient, N a and N b denote the number of particles, and L a and L b the lattice number in A-and B-legs, respectively. Its depends on the total number of particles but not their distributions in the momentum-space ladder, i.e. changing their locations in the corresponding leg costs no energy. In Fig. 4(a), we illustrate this phase with an example of five bosons along each leg. We attribute this phase as a sliding insulating phase (SI), analogous to long-sought sliding superfluid phase [71][72][73][74][75][76]. The underlying physics of the emergent SI studied here is a result of momentumspace Rydberg coupling (all sites equally coupled) from the long-range real-space Rydberg interaction. Varying the chemical potential, the ground state forms Devil's staircase structures with non-integer filling [ Fig. 4(b)], which breaks the lattice translational symmetry.
Turning on the hopping, the sliding insulating phase is stable in the low hopping regime (h o , h r V ), as shown in the phase diagram Fig. 4(c) and (d) (detailed discussions for finite hopping shown in SM). For larger hopping amplitudes, density fluctuations are stronger and a sliding superfluity phase (SSF) appears. The SI-SSF phase transition can be determined by examining the ground state energy [Fig. 4(e)]. Note that the SSF breaks both lattice translational and gauge symmetries, and therefore can be considered as a sliding supersolid phase. In the weakly interacting regime (h o , h r V ), kinetic energy dominates and normal superfluid (NSF) appears [ Fig. 4 In the presence of external magnetic flux, the chiral symmetry is broken, i.e. nonzero local currents are found in the sliding phases. In the weakly interacting limit, normal chiral superfluid phases emerge in the ground state.
Conclusion-A momentum-space lattice model suitable for investigating topological physics and correlated many-body dynamics is proposed. Using single and double Rydberg dressing, the ground state exhibits a series of exotic phases, including anti-chiral edge currents, sliding insulating and superfluid states. Strongly correlated dynamics, such as the state dependent excitation blockade, is found due to the competition between the chirality and strong Rydberg interactions. Compared to schemes based on superradiant Dicke states characterized by steady states [49,50,77], our setting permits to explore correlated chiral phenomena both in and out of equilibrium coherently. Compared to real-space magnetic ladder system, our system is essentially a spin-orbit coupling setup in momentum space, and supports three distinctive chiral broken phases.
Our study paves new routes towards the study of chirality with interacting spinwaves in higher dimensions and external driving. In frustrated lattices (e.g. honeycomb lattices), emergent quantum topological dynamics can be investigated in a Hamiltonian system, e.g., by quenching the system from a trivial to topological state. When dissipation is introduced, this opens opportunities to uncover stability of edge modes, as well as to explore competing dynamics between atomic interactions and dissipation in an open quantum system (example of quantum Zeno effect is discussed in SM.).

EXTENDED BOSE-HUBBARD MODEL IN MOMENTUM SPACE The Hamiltonian in momentum space
Here we consider N three-level atoms, i.e. ground state g(r i ), another ground state a(r i ) and Rydberg dressed state b(r i ), where r i is the position of the ith atom with random distribution. The atoms are initially prepared in the ground state |G ≡ |g 1 ...g N . A standing wave laser couples the atomic a and b states with vectors k 1 = −k 2 = k c . In the rotating wave approximation, the Hamiltonian reads where C b and r b is the dispersion coefficient and characteristic distance of the soft-core shape interaction, respectively [S42-S45]. Collective atomic excitation operators in momentum space are introduced as We transform the Hamiltonian from position space to momentum space via The total Hamiltonian in momentum space can be written as , and the transformation is valid for many excitations if the excitation number is much less than the atom number [S49].
Here, if we switch on another far-tuned standing wave lasers and couple the |a state to another Rydberg state, then an extra interaction term − j 2h o cos(2k c r j + φ)(|a j a j | − |b j b j |) appears due to the AC Stark shifts [S50]. The total Hamiltonian is given by: To investigate ground states of bosonic gases loaded into momentum-space lattices, described by Eq. (S7), we establish a bosonic version of dynamical mean-field theory (BDMFT) on the ladder system with z = 4, where z is the number of neighbors connected by hopping terms. As in fermionic dynamical mean field theory, the main idea of the BDMFT approach is to map the quantum lattice problem with many degrees of freedom onto a single site -"impurity site" -coupled self-consistently to a noninteracting bath [S78]. The dynamics at the impurity site can thus be thought of as the interaction (hybridization) of this site with the bath. Note here that this method is exact for infinite dimensions, and is a reasonable approximation for neighbors z ≥ 4. In the noninteracting limit, the problem is trivially solvable in all dimensions, all correlation functions factorize and the method becomes exactly [S79].

BDMFT equations
In deriving the effective action, we consider the limit of a high but finite dimensional optical lattice, and use the cavity method [S78] to derive self-consistency equations within BDMFT. In the following, we use the notation h ij for the hopping amplitude between sites i and j, and define creation field operator d † for the state |σ [σ = a(b)] to shorten Ham. (S7). And then the effective action of the impurity site up to subleading order in 1/z is then expressed in the standard way [S78, S79], which is described by: with Weiss Green's function and superfluid order parameter Here, we have defined the the diagonal and off-diagonal parts of the connected Green's functions where . . . 0 denotes the expectation value in the cavity system (without the impurity site).
To find a solver for the effective action, we return back to the Hamiltonian representation and find that the local Hamiltonian is given by a bosonic Anderson impurity model.
where the chemical potential and interaction term are directly inherited from the Hubbard Hamiltonian. The bath of condensed bosons is represented by the Gutzwiller term with superfluid order parameters Φ (0) . The bath of normal bosons is described by a finite number of orbitals with creation operatorsâ † l and energies l , where these orbitals are coupled to the impurity via normal-hopping amplitudes V l and anomalous-hopping amplitudes W l . The anomalous hopping terms are needed to generate the off-diagonal elements of the hybridization function.
The Anderson Hamiltonian can be implemented in the Fock basis, and the corresponding solution can be obtained by exact diagonalization of BDMFT [S78]. After diagonalization, the local Green's function, which includes all the information about the bath, can be obtained from the eigenstates and eigenenergies in the Lehmann-representation Integrating out the orbitals leads to the same effective action as in Eq. (S8), if the following identification is made where l +iωn , and means summation only over the nearest neighbors of the "impurity site".
In next step, we make the approximation that the lattice self-energy Σ i,lat coincides with the impurity self-energy Σ i,imp , which is obtained from the local Dyson equation The real-space Dyson equation takes the following form: Here, the self-consistency loop is closed by Eq. (S14)-(S18), and this self-consistency loop is repeated until the desired accuracy for values of parameters l , V l and W l and superfluid order parameter Φ is obtained.
FIG. S1. (Color online) Phase diagram of the quantum system with interacting spinwave states in a momentum-space lattice with the flux φ = π/2 and filling factor Ntot/L lat = 0.125.
Without two-body interactions, or with single Rydberg dressing, there are two special case for the flux φ = 0 and φ = π/2. For the case φ = 0, the phenomena are trivial, and the system does not support edge currents in the absence (presence) of Rydberg long-range interactions. For the case φ = π/2, the band structure of the lattice system is actually a double-valley well with two degenerate band minima connected by time-reversal symmetry, indicating that more ground states appear in the case. Here we choose the parameters: the filling factor N tot /L lat = 0.125 (L lat being the lattice size) and φ = π/2. We observe there are four stable phases in the diagram with different types of ground state edge currents, including CSF 2 and CSF 3 with currents on the rung but with a suppressed global current on both ladders withJ AA = i J i AA /L lat ≈ 0 andJ BB = i J i BB /L lat ≈ 0, CSF m with currents only on the ladders, and CSF v with currents on both ladders and rungs. The physical reason of the suppressed edge currents of the CSF 2 and CSF 3 phases is that, J i AA ≈ 2h o n i sin(φ − 2kx A ) ≈ 0 and J i BB ≈ −2h o n i sin(φ − 2kx B ) ≈ 0 with φ = π/2, n i being the filling at site i, and condensing at 2kx A ≈ π/2 and 2kx B ≈ −π/2, respectively.

DENSITY DISTRIBUTION IN REAL SPACE AND MOMENTUM SPACE WITH SINGLE RYDBERG DRESSING
In the main text, we identified three phases in the case of single Rydberg dressing. The density distributions in momentum and real space are vastly different in the three phases. Here we will show both real-space and momentumspace density distributions in these phases, which might be observed directly through time-of-flight experiment. In the Meissner phase, the size of the vortex is infinite and the density is uniform in the momentum-space lattice. On the other hand, in the CSF p and CSF v phases, the system exhibits vortex structures, where the densities are distributed inhomogeneously and the vortexes are separated into different regions (the averaged inter-leg current i J i BB /L lat ≈ 0 and = 0 for the CSF p and CSF v phases, respectively).
Actually, we have observed that there are three different kinds of band structures in the noninteracting system, as shown in Fig. 1(d), where band minima are localized at zero (CSF m ), finite-value (CSF v ), and doubly degenerate points (CSF p ) in real space [S50]. The interacting system also supports three types of condensation in real space, and the resulting phenomena are that, in the CSF p phase, the maximal density for the |a and |b states, localized at x A and x B , respectively, separates by about 2kx Fig. S2(b) for x A and x B , and more discussions on the anti-chiral current are shown in the next section], and in the CSF v phase, peak positions of the two states are identical in real space condensing at nonzero value. In the CSF m phase, however, the maximal value of the density in the two states is centered at 2kx = 0 in real space. This indicates that we can directly identify the CSF p phase through the real-space distribution.

CSFp PHASE WITHOUT TWO-BODY INTERACTIONS
When the two-body interaction is vanishing, the Hamiltonian of an ensemble of atoms in real space is given by where j is the index of atoms. The eigenvalues of the Hamiltonian are E ± = ± √ 2 × h 2 0 [1 + cos(4kx j − 2φ)] + h 2 r [1 + cos(2kx j )]. In the limit h r → 0 the two eigenvalues as a function of kx j will cross at kx j = φ/2 + π/4. We can also find that when k 1 x j = φ/2 (for state |a ) and k 2 x j = φ/2 + π/2 (for state |b ), the eigenenergies are local minimal. The current j are zero in this limit. Now focusing on the case φ = π/4 and turning on the coupling h r between the two states, the two local minimal points are coupled. When h r h 0 , the two local minimal points are still nearly degenerate. The minimal points are shifted slightly with respect to k 1 and k 2 . We can expand the lower branch E − of the eigenenergy around h r ∼ 0 up to second order and find the shifts, ∆k 1 x j ≈ −h 2 r sin φ/4h 2 0 and ∆k 2 x j ≈ −h 2 r sin φ/4h 2 0 . The current > 0. Similarly, we find thatJ BB > 0. HereJ AA >J BB because the state |b is weakly occupied in the ground state, due to small h r . This explains the results shown in the main text. The unique properties of the sliding phase can be understood by considering a single chain. In the following we will consider the A-leg with N excitation from the atomic hyperfine ground state as an example. In the momentum space, all the excitations are coupled by the infinite long-range Rydberg interactions. The interaction energy E int = V N (N −1)/2 regardless of the particle distribution. The energy gap by adding one particle ∆E + N = E N +1 −E N = N V and removing one particle ∆E − N = E N − E N −1 = (N − 1)V depends on particle numbers linearly, where E N denotes the ground state energy of the system. This is approximately valid even in the presence of a weak hopping. To verify this, we have carried out exact diagonalization (ED) calculations for unit filling N = L a . To calculate the gap, we have considered system size up to N = 10 and L a = 9, whose Hilbert space is 43758. Our numerical results show that the energy gap is indeed proportional to particle numbers (see Fig. S3). On the other hand, the ratio of the two gaps ∆E + N /∆E − N = N/(N − 1) when the hopping vanishes. The numerical simulation shows that the ratio is approximately correct (Fig. S3). The deviation in the numerical calculation is due to the finite size effect.

SLIDING PHASES IN THE STRONGLY INTERACTING REGIME
Moreover, we can understand the existence of the sliding phase from the relative bandwidth of the system, i.e. the ratio between the hopping and interaction energy. First, the maximal value of the hopping energy is h[ N (N + 1) + (N − 1)(N + 2)]/2, corresponding to a configuration where (N −1)/2 and (N +1)/2 atoms locate at two neighboring sites. In the thermodynamic limit, the relative bandwidth is estimated as This power decay indicates that the relative bandwidth vanishes for the Rydberg-dressed system in the thermodynamic limit. As a result, a sliding phase emerges in the strongly interacting limit. As a comparison, we note that the relative bandwidth in the Mott insulator state of the Bose-Hubbard model, described by H = −h i,j b † b+V /2 i n i (n i −1), is ∆ D ∝ 1/n(n − 1) with filling n (i.e. the mean number of atoms in a site). The corresponding bandwidth is constant in the thermodynamic limit.
It becomes difficult to carry out ED calculation for larger systems, such as two coupled chains with larger L lat . Here we calculate energy distributions for different numerical simulations via the dynamical mean-field theory. The ground state energies, with different density distributions, localize in a small region (compared to hopping amplitudes), indicating the sliding nature of the quantum many-body states at finite hopping amplitudes, as shown Fig. S4. The broadening of the spectra is largely caused by the kinetic energy of different configurations of the finite-size system [see Fig. S4(b), (c), (e) and (f)]. As examples, we also show density distributions in momentum space from two different numerical simulations [ Fig. S4(b) and (c) in SI, and Fig. S4(e) and (f) in SSF region] but with the same parameters. In different simulations, the particles distribute randomly in each ladder, as shown in Fig. S4(b)(c) and (e)(f). Experimental observation of such distribution could be a direct evidence of the sliding phase. When the long-lived hyperfine states are replaced with low-lying decay states, we realize a superradiance lattice [S49], or incorporate an effective decay in the present setup. To highlight the collective decay, we study dissipative dynamics of the A-leg solely and turn off the dressing laser. It could be realized by coupling the level |a with a strong classical field off-resonantly. Spontaneous Raman process happens in a rare probability, which depends on the detuning. It's a reversal process of the state initialization as we mentioned above.
Without two-body interactions, we will study a single excitation in the A-leg, whose dynamics is now described by a master equationρ where H A = − i (h 0 e iφ a † i a i+1 + H.c). Γ j are decay rate of the j-th site. Propagation of the spinwave depends on hopping h 0 and decay rate Γ j . The latter is site (momentum) dependent for a superradiance lattice. With this consideration, Γ j = 0 only for phase matched spinwave components [S80], i.e. at sites with index j = 0 and 1 [see Fig. 1 We consider a single excitation initially occupies the state |n (a) −4 = 1 and propagates to the middle of the lattice. One typically expects that the propagation is coherent away from the central sites, and becomes dissipative once approaching to the decaying sites. This is true when the decay rate Γ j is small, where a large fraction of the population will be lost [ Fig. S5(a)]. The dynamics changes qualitatively when Γ j is large, where the population is largely reflected at the zeroth site [ Fig. S5(b)(c)].
We attribute the reflection to the quantum Zeno effect [S81]. Stronger decay behaves similar to a frequent measurement of occupations in the zeroth site. It prohibits the occupation of its neighboring sites from hopping to the initially not occupied center sites. As a result, the loss occurs at an effective, smaller rate h o /Γ 2 0 [S82]. To illustrate this, we consider the simplest model which contains only the two sites with index j = −1 and j = 0. The initial state is |ψ(0) = |n −1 , n 0 with n −1 = 1 and n 0 = 0 (the subscript indicates the site index.). This state couples to the state |φ = |0, 1 , which will decay at a rate Γ The initial state evolves according to |ψ(t) = exp(−iE (2) 0 t)|ψ(0) . Hence the remaining probability of the initial state is The initial state decays at an effective rate Γ eff = 4h 2 0 /Γ 0 , which decreases with increasing Γ 0 . We numerically calculate the remaining population as a function of Γ 0 at time τ = h 0 t = 3.8, when the reflection occurs. As shown in Fig. S5(d), the effective decay rate is linearly proportional to ∼ h o /Γ 2 0 , confirming the analytical prediction.

EXPERIMENTAL PROPOSAL FOR REALIZING STRONGLY CORRELATED PHENOMENA
Chiral edge states can be realized in momentum space, either using mechanical momentum states of cold atoms [S83-S87], or collective excitation formed by collective excitations of electronic states [S88]. A unique advantage of the latter is that thermal resistant edge states can be probed, since the momentum-space lattice [S49, S77] of collective atomic excitations is immune to the motional entropy of atoms. The first proof-of-principle experiment has demonstrated chiral edge currents at the room temperature recently [S50]. However, electronic excited states suffer from fast spontaneous decay, inducing a steady state in the pump-dissipative system and destroying the quantum nature of the system. A clean quantum system in the momentum-space lattice in the presence of strong interactions is required to simulate strongly correlated phenomena.

Realization for the Hamiltonian (1) without dissipation
We can get rid of the radiative dissipation by selecting three hyperfine spin states in ground levels of 87 Rb with |g being |5 2 S 1/2 , F = 1, m = 1 , |a being 5 2 S 1/2 , F = 1, m = −1 , and |b being |5 2 S 1/2 , F = 2, m = −2 . The atomic level scheme and the configuration of the coupling beams are plotted in Fig. S6(a)(b). The spin states are split by a bias magnetic field. The inter-leg coupling h r in purple is realized by Raman interaction which composed by the two fields around D1 line (in purple), while the one in blue is realized by a 6.8GHz microwave field. Since microwave field transfers negligible momentum, the MSL is slightly tilted in momentum space, as shown in Fig. S6(c). The intra-leg coupling h 0 is implemented by the standing waves around D2 line, whose frequency is set in the middle point between a and b levels to introduce the π phase shift between the intra-leg couplings of A-and B-legs. The phase φ in Ham.
(1) can be controlled by manipulating the phase of the microwave and optical driving fields. Note here that, the small difference between the wave vectors of the standing wave is negligible. In the low-excitation regime, these atomic spinwave states are described by bosons [S89].
Here we choose Rydberg 60S state as an example, whose lifetime is 102 µs. With the detuning ∆ d = 8 MHz and Rabi frequency Ω R = 3 MHz of the dressing laser, the effective lifetime in the Rydberg dressed state is 2.9 ms. We obtain r c = 4.54 µm and V = 158.2 kHz. This large soft-core radius r c leads to a short-range interaction in momentum space, as it is far larger than the wavelength of the standing wave laser (∼ 780 nm). The inter-and intra-leg coupling strengths h r and h o can vary in a large parameter regimes. For example, we choose the parameters ∆ zeeman = 50 MHz being the Zeeman splitting between |g and |a , ∆ r = ∆ a = ∆ b = 3.4 GHz being the detunings [see details in Fig. S6(a)], Ω s = 40 MHz being the Rabi frequencies of optical fields with s = 0, 1, 2, and Ω mw = 500 kHz being the effective Rabi frequency of the microwave field. The corresponding h r and h 0 are ∼ 50 kHz ( E r ≈ 4 kHz, with E r being the recoil energy), which are larger enough to observe coherent dynamics in microsecond timescale. 5 2 S 1/2 5 2 P 1/2 (d) 6.8GHz

Loading the excitations and measurement
To observe the dynamics in MSL, we need to initialize excitations in the A-leg with a post-selection process. The BEC is prepared in ground state before turning on the Ham. (S7). Then we apply a strong classical field (thick) and a single photon (thin) in Fig. S6(d), forming a Raman coupling. When the single photon is not observed by a detector on its incident direction, we know one excitation is loaded in level a [S88]. Since the coherent time of level a is long enough, we can repeat the process twice to prepare the two-excitation state in zeroth site in A-leg. For the Zeno dynamics, we need to prepare a single excitation on the nth site in A-leg and introduce an effective decay to the zeroth one (see more details in the next subsection). After the single excitation is loaded into the zeroth site, we can apply two π-pulses of blue and purple h r couplings in a sequence. Such a pulse pair transport the excitation from the zeroth site to the first site in the A-leg. We can repeat the process for n times to finish the initialization.[S90].
We can also pump the excitation to the MSL when the Hamiltonian is on. By tuning the frequency of the pumping microwave field to the energy of the ground state in MSL, we can excite a specific state with high fidelity since the state width is very narrow. To prepare the total excitations N tot N BEC , the pulse area of the pumping microwave field is roughly estimated as √ N BEC Ω p t = πN tot , where Ω p is the effective Rabi frequency of the pumping microwave field, √ N BEC is the collective enhancement of N BEC atoms, and t is the pulse duration. By measuring the probability distribution of the a-and b-level atoms in momentum space via time of flight imaging, we can obtain the distribution of the excitation in MSL, which is expected to show the strongly correlated phenomena, e.g. ground state chiral current [S18], excitation blockade, and Zeno dynamics.