A Supplemental Material

Mobile bound states of Rydberg excitations in a lattice

Abstract

Spin lattice models play central role in the studies of quantum magnetism and non-equilibrium dynamics of spin excitations – magnons. We show that a spin lattice with strong nearest-neighbor interactions and tunable long-range hopping of excitations can be realized by a regular array of laser driven atoms, with an excited Rydberg state representing the spin-up state and a Rydberg-dressed ground state corresponding to the spin-down state. We find exotic interaction-bound states of magnons that propagate in the lattice via the combination of resonant two-site hopping and non-resonant second-order hopping processes. Arrays of trapped Rydberg-dressed atoms can thus serve as a flexible platform to simulate and study fundamental few-body dynamics in spin lattices.

\normalem

Introduction. Interacting many-body quantum systems are notoriously difficult to simulate on classical computers, due to the exponentially large Hilbert space and quantum correlations between the constituents. It was therefore suggested to simulate quantum physics with quantum computers Feynman (1982), or universal quantum simulators consisting of spin lattices with tunable interactions between the spins Lloyd (1996). Dynamically controlled spin lattices can realize digital and analog quantum simulations. Quantum field theories not amenable to perturbative treatments are often discretized and mapped onto the lattice models for numerical calculations. Spin lattices are fundamental to the studies of many solid state systems, where the competition between the interaction and kinetic energies determines such phenomena as magnetism and superconductivity.

Realizing tunable spin lattices in the quantum regime is challenging. Several systems are being explored to this end, including trapped ions Blatt and Roos (2012); Zhang et al. (2017), superconducting circuits Houck et al. (2012); Neill et al. (2017), quantum dots Hensgens et al. (2017) and other solid state systems. Cold atoms in optical lattice potentials are accurately described by the Hubbard model, representing perhaps the most versatile and scalable platform to realize various lattice models Gross and Bloch (2017). The Hubbard model for two-state fermions or strongly interacting bosons at half filling can implement the lattice spin-1/2 model Kuklov and Svistunov (2003); Duan et al. (2003). The spin-exchange interaction then stems from the second-order tunneling (superexchange) process Trotzky et al. (2008); Chen et al. (2011) and the interspin Ising interaction can exist for atoms or molecules with static magnetic or electric dipole moments Lahaye et al. (2009); Moses et al. (2017). These interactions are, however, weak (tens of Hz or less), which makes the system vulnerable to thermal effects even at ultra-low temperatures of nK Greif et al. (2013); Boll et al. (2016); Mazurenko et al. (2017).

Here we propose a practical realization of a tunable spin lattice model with an array of trapped atoms Barredo et al. (2016); Endres et al. (2016). The atomic ground state dressed by a non-resonant laser with a Rydberg state Bouchoule and Mølmer (2002); Johnson and Rolston (2010); Macrì and Pohl (2014) represents the spin down state, while another Rydberg state corresponds to the spin-up state (see Fig. 1). Controllable spin-exchange interactions are then mediated by the dressing laser and resonant dipole-dipole exchange interaction (scaling with distance as ) between the atoms on the Rydberg transition. van der Waals interactions between the excited-state atoms (scaling as ) serve as Ising-type interaction between the spins Schauß et al. (2012, 2015); Labuhn et al. (2016); Lienhard et al. (2017); Bernien et al. (2017). Due to long lifetimes of the Rydberg states and large energy scales of their interactions, this system is essentially at zero temperature. This permits observation of coherent quantum dynamics of spin-excitations – magnons.

We study the dynamics of magnons in the spin-lattice with long-range spin-excitation hopping and nearest neighbor interactions. Apart form scattering states, we find exotic interaction-bound states of magnons Fukuhara et al. (2013). The bound pairs of magnons can propagate in the lattice via resonant two-site spin exchange and non-resonant second-order exchange interactions [see Fig. 1(a)]. We note that the spin lattice model can be mapped onto the extended Hubbard model with spinless fermions or hard-core bosons: In the extended Hubbard model with low filling, particle tunneling from site to site and the attractive or repulsive interactions between the particles at the neighboring sites correspond, in the spin-lattice model, to the excitation hopping via spin-exchange and to the Ising interspin interaction, respectively. The bound states of magnons are then equivalent to interaction bound states of particles in the (extended) Hubbard model Winkler et al. (2006); Fukuhara et al. (2013). But our solution goes beyond the bound-state solutions of the Hubbard model Piil and Mølmer (2007); Petrosyan et al. (2007); Valiente and Petrosyan (2008, 2009); Valiente (2010) and it can be easily generalized to arbitrary-range hopping and interactions. We find that longer-range hopping of individual magnons leads to the increased, and tunable, mobility of the bound pairs of magnons.

Figure 1: (a) Spectrum of two spin (Rydberg) excitations in a lattice versus the center of mass quasi-momentum . The scattering states form a continuum spectrum (black), Eq. (5). The bound states for strong (red lines) and weak (blue lines) repulsive interactions are obtained from the spin-lattice Hamiltonian (dashed lines), Eq. (7), and from exact diagonalization of the Hamiltonian for the system sketched in (b) (solid lines). In the simulations, we used a lattice of size and periodic boundary conditions, with the spin model parameters , for the red and blue dashed lines, respectively. Inset illustrates the motion of the bound pair via resonant two-site hopping and second order hopping . (b) Level scheme of atoms to realize a spin-lattice model. Atoms in Rydberg states and undergo dipole-dipole exchange interaction with rate . The atomic ground state is dressed with the Rydberg state by a non-resonant laser with Rabi frequency and detuning . The spin-up and spin-down states correspond to and . Interactions between the atoms in state lead to formation of mobile bound states of Rydberg excitations. The parameters in numerical simulations shown in (a) correspond to , , , and for the red and blue solid lines, respectively.

Interacting spin excitations in a lattice. We consider a spin lattice model described by Hamiltonian ()

(1)

where and are the raising and lowering operators for the spin at position , and is the projector onto the spin-up state. In Eq. (1), the first term is responsible for the spin transport via the exchange interaction , while the second term describes the interaction between the spins in state with strength . Both and have finite range and depend only on the distance between the spins at positions and .

Hamiltonian (1) preserves the number of spin excitations. For a single excitation, the interaction does not play a role, and the Hamiltonian reduces to , where denotes the state with the spin-up at position in a lattice of spins (we assume periodic boundary conditions), and is the range of the exchange interaction. The transformation diagonalizes the Hamiltonian, , which indicates that the plane waves with the lattice quasi-momenta () are the eigenstates of with the eigenenergies .

Consider now two spin excitations. We denote by the state with one spin-up at position and the second spin-up at . With this notation, the transport and interaction terms of the Hamiltonian (1) are given by

(2)
(3)

We introduce the center of mass and relative coordinates. Making the transformation (), we obtain the total Hamiltonian that is diagonal in the basis of the center of mass quasi-momentum : , where

(4)

with SM (). The two-body wavefunction can be cast as , where the relative coordinate wavefunction depends on the quasi-momentum as a parameter via the effective hopping rates in . There are two kinds of solutions of the eigenvalue problem for , corresponding to scattering states of asymptotically free magnons and to the interaction-bound states.

The wavefunction for the scattering states has the standard form containing the incoming and scattered plane waves , where is the (finite) range of the interaction potential , and the phase shift depends on . The energies of the scattering states are simply given by the sum of energies of two free magnons,

(5)

where and are the center of mass and relative quasi-momenta. In Fig. 1(a) we show the spectrum of the scattering states, assuming the range of the spin-exchange interaction with , while . Note that due to the longer range hopping , the spectrum at does not reduce to a single point as in Valiente and Petrosyan (2008, 2009), but has a finite width , see also Piil and Mølmer (2007).

The bound state solutions correspond to a normalizable relative coordinate wavefunction, . We assume nearest-neighbor interaction, and in Eq. (4). We set and , with some constant, and make the ansatz

(6)

The physical intuition behind this recurrence relation is that every (discrete) position can be reached from positions and with the amplitudes and . We then obtain SM () , , and the energy of the bound state

(7)

The first term on the right-hand-side of this equation does not depend on the interaction and it describes two-site resonant hopping of the excitation over the other excitation, , with rate . This process is resonant because the relative distance , and thereby the interaction energy, are conserved during this two-excitation “somersault”. The second and third terms are contributions from the second-order () and third-order () hopping processes. The last term is the energy shift due to interaction .

Figure 2: (a) Diagram of values of , for the existence of bound states (light-blue shaded region). (b) Wavefunction versus the relative distance for several values of the center of mass quasi-momentum . The parameters are the same as in Fig. 1, with (left graph and red line in (a)), and (right graph and blue line in (a)) where the bound state does not exist in the vicinity of .

The above solution is valid under the conditions that bound-state wavefunction is normalizable. Inserting into Eq. (6), we obtain that the wavefunction exponentially decays with distance , and therefore is normalizable, when . In Fig. 2(a) we show the values of and , forming a triangular region, for which there exists an exponentially localized bound state. With only nearest-neighbor hopping (), we recover the condition of Refs. Valiente and Petrosyan (2008, 2009). For a given set of parameters , the bound state may not exists for all values of the center of mass quasi-momentum , since both and depend on . In general, the closer is the point () to the boundary of the shaded region in Fig. 2(a), the less localized is the bound state wavefunction, as we illustrate with two examples in Fig. 2(b). In Fig. 3 we show the diagrams of and versus for the existence of the bound states. Clearly, for certain sets of parameters, the bound states do not exist at all, or exist only within a certain interval of values of .

Figure 3: Diagram of values of , for fixed (upper panel), and , for fixed (lower panel), versus , for the existence (white regions) and absence (black regions) of the bound states.

Rydberg dressed atoms in a lattice. The spin lattice model of Eq. (1) might be realized with a regular array of atoms in Rydberg states and . We could excite one or more atoms to state and prepare all the remaining atoms in state . Assuming the transition is dipole allowed, resonant dipole-dipole interaction between the atoms separated by lattice sites would lead to transfer of excitations via the exchange interaction with rate , where is the interaction coefficient and is the lattice constant Barredo et al. (2015); de Léséleuc et al. (2017). Atoms in the Rydberg states also interact via the van der Waals interactions , which would map onto the interactions between the spin excitations Schauß et al. (2012, 2015); Labuhn et al. (2016); Lienhard et al. (2017); Bernien et al. (2017), provided differs from the interaction between the and state atoms.

Typically, however, the resonant dipole-dipole interaction is orders of magnitude stronger than the van der Waals interactions , since the latter originate from non-resonant dipole-dipole interactions, , with large Förster defects, Saffman et al. (2010). Small interactions will preclude the interplay between the spin transport and spin-spin interactions. To mitigate this problem, we propose to dress trapped ground state atoms with the Rydberg state . The dressing laser would then mediate hopping of the Rydberg excitation to nearby atoms in the dressed ground state with rates which can be made comparable to, or even weaker than, the effective interaction between the excitations. Rydberg dressing of ground-state atoms Bouchoule and Mølmer (2002); Henkel et al. (2010); Johnson and Rolston (2010); Pupillo et al. (2010); Wüster et al. (2011); Macrì and Pohl (2014) is a versatile tool for tuning interatomic interactions to simulate various lattice models Schempp et al. (2015); Glaetzle et al. (2015); van Bijnen and Pohl (2015); Buchmann et al. (2017); Zeiher et al. (2016, 2017).

We consider an array of single atoms with the level scheme shown in Fig. 1(b). The ground state of each atom is coupled to the Rydberg state by a laser with Rabi frequency and large detuning . We assume that is much larger than the resonant dipole-dipole interactions between the and state atoms separated by lattice sites. The van der Waals interactions are assumed to be still weaker, so the hierarchy of the energy scales is .

The laser instills a small admixture of the Rydberg state to the ground state SM (). We then identify the dressed ground state with the spin-down state, , while the spin-up state is . Neglecting the interactions between the dressed ground state atoms, we adiabatically eliminate the nonresonant state and obtain effective excitation hopping rates between the atoms separated by lattice sites. Since , we can truncate to range . More careful considerations show that the hopping rates for a Rydberg excitation are slightly altered when another Rydberg excitation is in a close proximity SM (). We assume that the lifetime of the Rydberg state is longer than the timescale for the system dynamics and neglect dissipation. The number of atoms prepared in state is then conserved. Decay via the non-resonant state is suppressed by the factor of .

For the effective interaction potential between the excitations we obtain , where both terms scale with distance as . We assume that is dominated by the nearest-neighbor van der Waals interaction between the atoms in Rydberg states . Corrections to the level shift of Rydberg dressed atoms in the vicinity of the Rydberg excited atom lead to small contribution to and weak longer range interaction SM (). Despite these small variations of and with distance between Rydberg excitations, the spin-lattice model approximates well the properties of interacting Rydberg excitations, including the two-excitation bound states shown in Fig. 1(a).

The dynamics of Rydberg excitations in a lattice and their bound states can be prepared and observed with the presently available experimental techniques. We envisage an array of single atoms confined in a chain of microtraps Barredo et al. (2016); Endres et al. (2016). Using focused laser beams, selected atoms can be resonantly excited from the ground state to the Rydberg state , while the dressing laser is turned off, . Next, turning on the dressing laser, will lead to the admixture of the Rydberg state to the ground state atoms, which will induce the excitation hopping between the atoms due the dipole-dipole exchange interaction. With realistic experimental parameters SM (), hopping rates kHz and can be achieved. This will allow observation of non-trivial dynamics of the excitations on the timescale of Rydberg state lifetimes s. With a proper choice of state , we can ensure appropriate interaction strength , which will result in the formation of tightly bound Rydberg excitations that are still mobile as they propagate with rate . Free Rydberg excitations and their scattering states can be discriminated from the interaction-bound states spectroscopically or by the fast and slow dynamics, respectively. Turning off the dressing laser would freeze the dynamics and individual Rydberg excitations can be detected with high efficiency and single-site resolution Labuhn et al. (2016); Bernien et al. (2017); Lienhard et al. (2017).

Conclusions. We have shown that spin lattice models with controllable long range hopping and interactions between the spin excitations can be realized with Rydberg dressed atoms in a lattice. We have found mobile bound states of spin excitations which are quantum lattice solitons. It would be interesting to consider bound aggregates of more than two magnons which may form mobile clusters that can propagate via resonant long-range hopping process. In turn, multiple clusters can form a lattice liquid Mattioli et al. (2013); Dalmonte et al. (2015), while including controllable dephasing and disorder Schempp et al. (2015); Schönleber et al. (2015) may change the transport of (bound) Rydberg excitations from ballistic to diffusive or localized. Hence, this system can be used to simulate and study few- and many-body quantum dynamics in spin lattices.

Acknowledgements.
We thank Michael Fleischhauer and Manuel Valiente for valuable advice and discussions. F.L. is supported by a fellowship through the Excellence Initiative MAINZ (DFG/GSC 266) and by DFG through SFB/TR49. D.P. is supported in part by the EU H2020 FET Proactive project RySQ. We are grateful to the Alexander von Humboldt Foundation for travel support via the Research Group Linkage Programme.

Appendix A Supplemental Material

a.1 Details of derivation of the two-excitation wavefunction in a spin lattice

Consider two spin excitations in a lattice. The transport and interaction Hamiltonians are given by Eqs. (Interacting spin excitations in a lattice.) and (3) in the main text, namely

(8)

and

(9)

where denotes the state with the excited spins at positions and .

We introduce the center of mass and relative coordinates: takes discrete values, and takes values. In terms of these coordinates, the transport Hamiltonian reads

(10)

Similarly to the single excitation case, we can diagonalize the center of mass part of by the transformation , where () is the center of mass quasi-momentum:

(11)

where . The interaction Hamiltonian remains diagonal in these coordinates,

(12)

and the total Hamiltonian can be cast as

We have thus reduced the two-body problem for

(13)

to a one-body problem for the relative coordinate wavefunction , which depends on the center of mass quasi-momentum as a parameter via the effective hopping rates in .

Our aim is to solve the eigenvalue problem

(14)

for the relative coordinate wavefunction . The scattering solutions are expressed via the plane waves as given in the main text. We present here the details of derivation of the bound solutions corresponding to a normalizable [localized] relative coordinate wavefunction, [with ].

We assume range (nearest-neighbor) interaction, and in Eq. (12), leading to the Hamiltonian

(15)

This results in the equation

(16)

We set and , with some constant to be determined by the normalization. We make an ansatz for the wavefunction,

(17)

The physical meaning of this recurrence relation is that every site can be reached from the previous two sites and with the amplitudes and . Starting from position , the wavefunction at any can then be written as

(18)

where is the floor function, and the binomial coefficients count the weights for different path from site to . For instance, we can reach from by three one-site hoppings , or by two-site hopping followed by one-site hopping , or vice versa, . Using the ansatz (18) in Eqs. (A.1) for we obtain a set of three equations,

(19a)
(19b)
(19c)

for the unknowns . Solving these equations, we obtain

(20)

while the energy of the bound state is

(21)

The physical meanings of the various terms of this equation are discussed in the main text.

We finally discuss the conditions of validity of the above solution under which the bound-state wavefunction is normalizable, . Assuming and inserting into Eq. (17), we obtain the quadratic equation with the solutions

We can now write the wavefunction as

(22)

and determine the coefficients from and , leading to . This is of course the same wavefunction as in Eq. (18). More important, however, is that we have found that exponentially decays with distance , and therefore is normalizable, when both .

Truncation of interaction range

Figure 4: Scattering (black) and bound-state (red solid line) spectra obtained by exact numerical diagonalization of Hamiltonian with long range interactions and (). The bound state energy of Eq. (21) (dashed blue line), obtained with truncated interactions ( and ), is nearly indistinguishable from the exact result.

Our formalism to obtain the bound states of excitations in a lattice can be easily extended to longer range hopping and interaction . We are, however, mainly concerned with the typical case of resonant dipole-dipole exchange interaction, leading to , and van der Waals repulsive or attractive interaction, leading to . We have therefore truncated to range and to range . In Fig. 4 we show the spectra for the scattering and bound states obtained without the truncation. This figure clearly demonstrates that the above approximations are well justified for the power-law decay of the strengths of and with distance.

a.2 Derivation of the effective excitation hopping rate and interaction strength for Rydberg dressed atoms in a lattice

Consider an ensemble of atoms in a lattice with period , with one atom per site. A spatially uniform laser field of frequency couples the ground state of each atom to the Rydberg state with the Rabi frequency and detuning , see Fig. 1(b) of the main text. Resonant dipole-dipole interaction between the atoms at positions and leads to the exchange interaction with rate , where is the interaction coefficient. Including also the van der Waals interactions between the Rydberg states, the Hamiltonian in the rotating frame reads ()

(23)

where are the atomic operators.

We take the detuning of the laser field to be much larger than the Rabi frequency as well as the resonant dipole-dipole interactions between the Rydberg-state atoms separated by lattice sites. The van der Waals interactions are assumed to be still weaker, .

Rydberg dressing

For a single (isolated) atom, the dipole-dipole and van der Waals interactions are irrelevant, and the Hamiltonian reduces to that for a two level system,

(24)

[We set the energy of the ground state to zero and work in a rotating frame in which the energy of state is also zero]. The eigenstates and corresponding eigenvalues of this Hamiltonian are

(25)

For , the eigenstate , with shifted energy (ac Stark shift), corresponds to the ground state with a small admixture of the Rydberg state . We identify this Rydberg dressed ground state with the spin-down state, , while the spin-up state is .

A pair of dressed ground-state atoms would interact with each other via the Rydberg state components. Each atom is in state with probability and therefore the two-atom interaction strength is Buchmann et al. (2017). We neglect this weak interaction and instead focus below on the interatomic interactions that are up to second order in . Hence, with atoms in a lattice, all in the dressed ground state, the total energy shift is

(26)

This constant energy shift can be disregarded by redefining the zero-point energy, e.g., by absorbing the ac Stark shift into the laser detuning, .

Single excitation

Assume now that one atom is excited to state while the rest of the atoms are in the dressed ground state. Our aim is to derive the effective hopping rate of the single Rydberg excitation in the lattice and the modification of the ac Stark shifts of the ground state atoms in the vicinity of the excited one. We are interested in the interatomic interactions that are up to second order in , which thus involve no more that one (virtual) excitation. It is therefore sufficient to consider the two atom state

(27)

and the corresponding Hamiltonian

(28)

where and denote the positions of the two atoms, and we defined . The equations for the amplitudes of the state vector are then

(29a)
(29b)
(29c)
(29d)

We adiabatically eliminate states containing the highly detuned Rydberg state . To that end, we set and and solve the last two equations for and . Inserting the solution into the first two equations, we obtain

(30a)
(30b)

We can interpret these equations as follows: The dressed state atom at position acquires an energy shift

(31)

which depends on the position of the excitation. Besides, states and are coupled via exchange interaction

(32)

This effective excitation hopping rate depends on the relative distance .

Hence, the total energy of atoms in a lattice with a single excitation is