Towards quantum simulation with circular Rydberg atoms

Towards quantum simulation with circular Rydberg atoms

T.L. Nguyen    J.M. Raimond    C. Sayrin    R. Cortiñas    T. Cantat-Moltrecht    F. Assemat    I. Dotsenko    S. Gleyzes    S. Haroche Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL Research University, UPMC-Sorbonne Universités, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France    G. Roux    Th. Jolicoeur LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France    M. Brune brune@lkb.ens.fr Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL Research University, UPMC-Sorbonne Universités, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France
July 5, 2019
Abstract

The main objective of quantum simulation is an in-depth understanding of many-body physics. It is important for fundamental issues (quantum phase transitions, transport, …) and for the development of innovative materials. Analytic approaches to many-body systems are limited and the huge size of their Hilbert space makes numerical simulations on classical computers intractable. A quantum simulator avoids these limitations by transcribing the system of interest into another, with the same dynamics but with interaction parameters under control and with experimental access to all relevant observables. Quantum simulation of spin systems is being explored with trapped ions, neutral atoms and superconducting devices. We propose here a new paradigm for quantum simulation of spin- arrays providing unprecedented flexibility and allowing one to explore domains beyond the reach of other platforms. It is based on laser-trapped circular Rydberg atoms. Their long intrinsic lifetimes combined with the inhibition of their microwave spontaneous emission and their low sensitivity to collisions and photoionization make trapping lifetimes in the minute range realistic with state-of-the-art techniques. Ultra-cold defect-free circular atom chains can be prepared by a variant of the evaporative cooling method. This method also leads to the individual detection of arbitrary spin observables. The proposed simulator realizes an XXZ spin- Hamiltonian with nearest-neighbor couplings ranging from a few to tens of kiloHertz. All the model parameters can be tuned at will, making a large range of simulations accessible. The system evolution can be followed over times in the range of seconds, long enough to be relevant for ground-state adiabatic preparation and for the study of thermalization, disorder or Floquet time crystals. This platform presents unrivaled features for quantum simulation.

I Introduction

Understanding strongly-coupled many-body quantum systems is a problem of paramount importance. They present fascinating properties, such as quantum phase transitions Sachdev (2007), topological phases Bernevig and Hughes (2013), quantum magnetism Schollwöck et al. (2008), quantum transport Dittrich et al. (1998) or many-body localization Nandkishore and Huse (2015). Exploring this complex physics is essential for fundamental issues, such as fractional quantum Hall states Cage et al. (2012) or high-temperature superconductivity Phillips (2012). It may also lead to solutions to high-energy physics problems such as relativistic quantum field theories Cirac et al. (2010). Finally, it bears the promise of applications based on materials with engineered properties.

The quantum many-body problem is all the more challenging that explicit analytical solutions are only available in a limited set of cases. Solid state experiments have to face the lack of access to some relevant quantities (entanglement properties for instance). Brute-force numerical exact diagonalization techniques face the exponential growth of the Hilbert space. In the restricted set of problems without the so-called sign problem Troyer and Wiese (2005) there are successful algorithms from the quantum Monte-Carlo family that allow for numerically exact solutions Suzuki (1993). However, many interesting physical problems are outside of this class. In one-dimensional physics problems, the DMRG algorithm White (1992, 1993); Schollwöck (2005) is very successful but requires specific entanglement properties.

The ideal tool to address many-body physics would be a ‘quantum simulator’ Feynman (1982); Lloyd (1996); Georgescu et al. (2014), transcribing the dynamics of the system of interest into another one that is under complete experimental control. Its parameters can be tuned nearly at will, all its observables can be measured. In principle, a general purpose quantum computer could be turned into a ‘digital’ quantum simulator at the expense of an embarrassingly high amount of resources Lloyd (1996); Raeisi et al. (2012). A more realistic approach is the ‘analog’ quantum simulator Manousakis (2002), with the same complexity (number of spins for instance) as the system of interest. An analog simulator made up of a few tens of spins would already surpass any classical machine Buluta and Nori (2009). Analog quantum simulation is one of the most promising domains of quantum information science.

This paper proposes a new paradigm for analog quantum simulation of spin arrays, based on laser-trapped circular Rydberg atoms, protected from spontaneous emission decay Kleppner (1981) and reaching extremely long lifetimes in the minute range. It combines a deterministic preparation and read-out of defect-free chains containing a few tens of atoms. The strong dipole-dipole interaction between the giant atomic dipoles emulates a fully tunable spin- XXZ chain Hamiltonian Baxter (1982). The chain dynamics can be followed over one second for a chain containing a few tens of atoms, corresponding to elementary exchange times. This analog simulator could supersede other platforms, even though they have already achieved impressive performance.

i.1 State of the art

Trapped ions Wineland (2013) are excellent tools for digital simulation Schindler et al. (2013), since they combine long coherence times, high-fidelity gates and individual unit-efficiency state-selective detection. The digital simulation of a QED process is a remarkable achievement Martinez et al. (2016). Ions are also well-suited for analog quantum simulation of spin arrays. The spin-spin interaction is simulated by a laser-induced coupling of the ions’ internal states with their motional modes. This interaction can be tuned between a long-range regime (independent upon the distance between the ions) and a mid-range one (decreasing as the cube of the distance) Kim et al. (2009); Islam et al. (2013). Recent experiments demonstrated quantum random walks of excitations in spin- or spin-1 chains Jurcevic et al. (2014); Senko et al. (2015), spectroscopy of spin waves Jurcevic et al. (2015), many body localization Smith et al. (2016) and thermalization Clos et al. (2016). First 2-D simulations of spin-squeezing with long-range interactions Bohnet et al. (2016) have been reported. Engineered interactions in the nearest-neighbor regime of great interest are not available yet.

Superconducting circuits are thriving, with qubits interacting directly or via their common coupling to cavities Devoret and Schoelkopf (2013); Wallraff et al. (2004). They are adapted to digital Barends et al. (2015, 2016) or analog Eichler et al. (2015); Neill et al. (2016) simulations. The experiments involved so far either only a few high-quality qubits Salathé et al. (2015); Roushan et al. (2017), a moderate number of damped systems Fitzpatrick et al. (2017) or even a large number of strongly damped ones Boixo et al. (2014), for which quantum speed-up is an open question Heim et al. (2015).

Cold atoms in optical potentials are a remarkable platform for quantum simulation Lewenstein et al. (2007); Bloch et al. (2008, 2012). They can emulate the quantized conductance of a mesoscopic channel Krinner et al. (2015). Their joint coupling to an optical Fabry-Perot cavity implements the Dicke phase transition Baumann et al. (2010), more perspectives being offered by photonic band-gap cavities Douglas et al. (2015); Tiecke et al. (2014); Manzoni et al. (2017). Many experiments use optical lattices, with unit filling in the Mott-insulator regime Greiner et al. (2002) and individual site imaging Kuhr et al. (2003); Sherson et al. (2010); Haller et al. (2015); Parsons et al. (2016). Inter-site tunneling and on-site interactions implement a Bose-Hubbard Baier et al. (2016) or Fermi-Hubbard Cheuk et al. (2016) Hamiltonian, on which complex entanglement properties can be measured Kaufman et al. (2016); Parsons et al. (2016). Controlled disorder created by a speckle pattern Sanchez-Palencia and Lewenstein (2010) leads to explorations of many-body localization Schreiber et al. (2015). Experiments reach now domains beyond the grasp of theoretical methods and classical computations Choi et al. (2016). Lattice dynamical manipulations Lignier et al. (2007); Bloch et al. (2012) or multi-level atoms Gerbier and Dalibard (2010) open the way to the simulation of gauge fields and topological phases Stuhl et al. (2015); Mancini et al. (2015); Eric Tai et al. (2016). However, following long term dynamics, such as that of spin glasses, is challenging, since it requires very long lattice lifetimes. Alternative solutions with smaller lattice spacings and higher tunneling rates have been proposed Romero-Isart et al. (2013); Gonzalez-Tudela et al. (2015) but not realized yet. Polar molecules DeMille (2002); Yan et al. (2013) or magnetic atoms Lahaye et al. (2009); Baier et al. (2016) can also be used to enhance the interactions.

Rydberg atoms Gallagher (1994) experience giant dipole-dipole interactions. The van der Waals potential Raimond et al. (1981) is in the MHz range for inter-atomic distances of a few microns. These interactions lead to the dipole blockade mechanism Lukin et al. (2001): a resonant laser can excite only one Rydberg atom out of a micron-sized volume, since the first excited atom detunes all the others from laser resonance Dudin and Kuzmich (2012); Barredo et al. (2014). This leads to non-classical excitation statistics Amthor et al. (2010); Malossi et al. (2014); Ebert et al. (2014); Weber et al. (2015); Urvoy et al. (2015); Teixeira et al. (2015), to quantum gates Saffman et al. (2010); Wilk et al. (2010); Isenhower et al. (2010); Ravets et al. (2014), to self-organization of Rydberg excitations Schausz et al. (2012), and to giant optical non-linearities Stanojevic et al. (2013); Paredes-Barato and Adams (2014); Tiarks et al. (2014); Maghrebi et al. (2015); Tresp et al. (2016); Thompson et al. (2017). These features are promising for quantum simulation Weimer et al. (2011); Lesanovsky (2012); Schönleber et al. (2015). Coherent excitation transport Barredo et al. (2015); Labuhn et al. (2016) and synthetic spin arrays based on ground-state dressing with a Rydberg level Zeiher et al. (2016) have been demonstrated. However, the experiments have to face the finite lifetime of the laser-accessible Rydberg levels (few hundred of s) and the blackbody-induced state transfers Goldschmidt et al. (2016). Moreover, in all experiments so far, the Rydberg atoms are not trapped. The strong van der Waals forces between the atoms cause then a rapid explosion or collapse of the atomic ensemble Teixeira et al. (2015), limiting further its useful lifetime. Replacing the actual excitation to a Rydberg level by a ground-state laser dressing solves the problem only in part Glaetzle et al. (2015). Simulations of slow processes over long times are, for the time being, beyond the reach of low-angular-momentum Rydberg atom simulators.

i.2 Principle of the proposed simulator

Figure 1: Pictorial scheme of the proposed circular state quantum simulator.

We propose here a circular-state quantum simulator, schematized in Fig. 1, which combines the best features of the other platforms and avoids some of their bottlenecks. Rydberg atoms in circular states, i.e., states with maximum angular momentum, are trapped in the ponderomotive potential induced by laser fields Avan et al. (1976); Dutta et al. (2000). These low-field seekers are radially confined on the axis (axis assignment in Fig. 1) by a Laguerre-Gauss ‘hollow beam’ at a 1 m wavelength. They are longitudinally confined in a one-dimensional adjustable lattice produced by two 1 m-wavelength beams, propagating in the plane at small angles with respect to the axis. In the following, we will consider for the sake of definiteness two lattices with inter-site spacings m and m, corresponding to a strong or moderate dipole-dipole interaction, respectively. The main decay channel of circular levels (spontaneous emission on the microwave transition towards the next lower circular level) is efficiently inhibited Hulet et al. (1985) by placing the atoms in a plane-parallel capacitor, which also provides a static electric field defining the quantization axis (the plane of the circular orbit is thus parallel to the capacitor plates). A method based on a van der Waals variant of evaporative cooling Masuhara et al. (1988) prepares deterministically long chains of atoms. It also leads to an efficient detection of individual atomic states.

The spin-up and spin-down states of the simulator are encoded in the circular levels with principal quantum numbers 50 and 48, respectively, connected by a two-photon transition. The dipole-dipole interaction provides a general spin- XXZ chain Hamiltonian Baxter (1982) with nearest-neighbor interactions. Its parameters can be adjusted at will over a short time scale by tuning the static electric field and a near-resonant microwave dressing. This complete freedom in the choice of the model Hamiltonian is a unique feature of the circular state quantum simulator.

The dynamics of a chain with a few tens of spins can be followed over up to about spin-coupling times. The final state of each spin can be individually measured. Adiabatic evolutions through quantum phase transitions, sudden quenches and fast modulations of the interaction parameters are within reach. This proposal thus opens promising perspectives for the simulation of spin systems in a thermodynamically relevant limit, beyond the grasp of classical computing methods.

In Section II, we recall the main properties of circular Rydberg atoms and discuss their dipole-dipole interaction. Additional details are given in Appendix A. Section III is devoted to the interaction Hamiltonian of an atom chain and to the rich phase diagram of the corresponding spin system, with details on the associated numerical simulations in Appendix B. Section IV is devoted to the laser trapping of circular atoms and to their protection from loss mechanisms, with technical details in Appendices C and D. Section V is devoted to the deterministic preparation of a Rydberg atom lattice with unit filling (see also Appendix E). Section VI presents the results of state-of-the-art numerical simulations showing that the simulator reaches a thermodynamically relevant regime. We examine the most interesting perspectives in the concluding Section VII.

Ii Circular Rydberg atoms and van derWaals interaction

The circular states have a large principal quantum number and maximum orbital and magnetic quantum numbers:  Gallagher (1994). They are the states closest to the circular orbit of the Bohr model, with a radius (: Bohr radius). Their wavefunction is a torus, with a small radius , centered on this orbit. This anisotropic orbit is stable only in a directing electric field , normal to the orbit, defining the quantization axis and isolating the circular state from the hydrogenic manifold Gross and Liang (1986) (Appendix A). The circular states cannot be excited directly from the ground state. Their preparation relies on a complex but efficient and fast process, combining laser and radio-frequency photons absorption Signoles et al. (2014). These states have long radiative lifetimes, scaling as (25 ms for ). The microwave transitions between neighboring circular states are strongly coupled to the electromagnetic field. These remarkable properties make them ideal tools for experiments on fundamental quantum processes in cavity quantum electrodynamics experiments Haroche and Raimond (2006); Haroche (2013).

The large dipole matrix elements between circular levels make them particularly sensitive to the dipole-dipole interaction. Two atoms in the same circular state experience a van der Waals, second-order interaction proportional to (: interatomic distance), repulsive in the proposed geometry (the interatomic axis, , is perpendicular to the quantization axis , see Fig. 1). For atoms in different circular states, and , this interaction competes with the resonant Förster-like transfer of energy (‘spin exchange’) from one atom to the other: . This exchange process is at first order in the dipole-dipole interaction when . Scaling as , it then overwhelms the repulsive interaction, realizing a spin model in which the spin exchange is by far the dominant interaction. With , the exchange is negligible compared to the van der Waals interaction. We chose here a more flexible simulator. With , the van der Waals and exchange interactions are of the same order of magnitude, scaling both as . Their competition opens, as we show below, a wide range of possibilities to engineer interatomic potentials.

The dipole-dipole interaction mixes the circular states with neighboring elliptical states (Appendix A), since it breaks the cylindrical symmetry of the Stark effect. These elliptical states have decay channels that are not inhibited by the capacitor (Appendix C). This deleterious mixing effect can be reduced by using a large enough directing electric field and a magnetic field parallel to it.

A careful optimization led us to choose the and states to represent the ‘spin-up’ and ‘spin-down’ states. With the field values  Gauss and V/cm, the intrinsic lifetime of interacting atoms exceeds 90 s for the smallest m interatomic distance. Lower principal quantum numbers would lead to an annoyingly small inhibition capacitor spacing. Higher principal quantum numbers would lead to larger spacings and dipole-dipole couplings. However, the transition frequencies between adjacent Rydberg manifolds is reduced and the lifetime reduction due to increased blackbody-induced transfer rates (Appendix C) is not compensated by the increase in couplings.

The interaction Hamiltonian for a pair of atoms reads, in terms of the atomic pseudo-spin operators (Appendix A)

(1)

The positive exchange term, , is nearly independent of the directing electric field . It is proportional to , strong ( kHz) for m or moderate (2.3 kHz) for m. The frequency shift , of the order of , also proportional to , exhibits a slow field dependency (Appendix A). A unique feature of the circular state interaction is that varies significantly, from negative to positive values, with the electric field amplitude. The sign of can thus be controlled and the ratio (independent on ) can be tuned over a large range by adjusting the control fields, as illustrated on Fig. 2. Over this complete range, the atomic lifetimes remain extremely long ( s).

Figure 2: Variation of , in units of , with the directing electric field amplitude . Dots result from the numerical diagonalization of the complete atomic Hamiltonian for and  Gauss (magenta, black, blue, green, red, cyan and purple dots respectively). The colored lines are a guide to the eye. The horizontal solid line and the dotted lines correspond to the pure XY spin- exchange model and to the isotropic models, respectively. The shaded background and the light gray lines give a qualitative estimation of the lifetime of a pair of interacting atoms at a m distance. This estimation is based on explicit lifetime calculations for the and values corresponding to the plotted dots.

Iii The emulated XXZ model

iii.1 Spin Chain Hamiltonian

We now turn to a chain of interacting atoms at a constant spacing . The Hamiltonian reads

(2)

where is the atomic transition energy ( GHz for the two-photon transition). We have here assumed that the pairwise dipole-dipole interactions are additive and we have neglected the next-nearest-neighbor interaction (64 times smaller than the nearest-neighbor one). Note that the atoms at the ends of the chain ( and ) have a single neighbor and thus an energy shift (), which is half that of the atoms in the bulk (). The generalization of this Hamiltonian to arrays with higher dimensions is straightforward.

In this Hamiltonian, the atomic frequency is, by many orders of magnitude, the largest, making the ground state and the dynamics trivial. The situation is more interesting when driving the atoms by a -polarized classical field at a frequency , close to resonance with the atomic two-photon transition (). The interaction with this field is, within an irrelevant phase choice for the classical driving field, represented by the effective two-level Hamiltonian

(3)

where (considered as positive without loss of generality) is the effective Rabi frequency on the two-photon transition. Adding this term to the chain Hamiltonian, switching to an interaction representation defined by the unitary operator and using the rotating wave approximation, we get the final dressed-chain Hamiltonian

(4)

where and . We recognize here a spin- XXZ chain Hamiltonian Des Cloizeaux and Gaudin (1966); Yang and Yang (1966a, b, c); Baxter (1982), in which and describe the Ising coupling and spin-flip exchange, respectively. The detuning plays the role of an effective longitudinal magnetic field, while is an effective transverse field.

The field-independent term defines the fundamental exchange time scale for this Hamiltonian, s at m and 108s at m. A unique feature of the simulator is that all other parameters of the Hamiltonian are under experimental control. The and parameters are determined by the classical microwave source dressing the atomic transition and is controlled by the directing fields and (Fig. 2). All the Hamiltonian parameters can thus be changed or modulated over a nanosecond time scale, infinitely short as compared to . This is a unique feature of this simulator.

iii.2 Phase diagram

Figure 3: Sketch of the phase diagram of Hamiltonian (5) based on the results of Fig. 4 and Ref. Dmitriev et al., 2002a.
Figure 4: Numerical phase diagram of the XXZ in a transverse field. (a): MPS results for the order parameters , , and von Neumann entropy (from left to right) for the Hamiltonian (5) on an open chain with spins.(b): same data for a open spin chain. The order parameters defined in (7) are computed with for and for . Red regions represent ferromagnetic ordering while blue ones represent antiferromagnetic (Néel) ordering. The gray lines are guides to the eyes for the quantum phase transition lines, inferred from symmetry arguments (horizontal lines) and from the von Neumann entropy plot for . They have been used to delineate the phases in Fig. 3

The case already provides a rich ground-state phase diagram, spanning a variety of key many-body problems. In this Section, we review this diagram in the thermodynamic limit for the bulk of the simulator, putting aside the edge effects. Setting in (4), the generic Hamiltonian reads

(5)

which boils down to the XXZ model in a transverse field Kurmann et al. (1982); Müller and Shrock (1985); Mori et al. (1995); Hieida et al. (2001); Dmitriev et al. (2002a, b); Dutta and Sen (2003). This model is relevant, in particular, in the interesting physics of the CsCoCl Kenzelmann et al. (2002); Breunig et al. (2013) or BaCoVO Grenier et al. (2015) quantum magnets. Its phase diagram is sketched on Fig. 3 and exhibits interesting quantum phase transitions.

The main four phases are associated with different symmetry breakings of the generic Ising symmetries ( and ). The competition between these phases is driven by the sign and strength of the parameter and by the magnitude of the transverse field . At large , the field polarizes all spins close to the -direction. This phase is gapped, does not break any symmetry and has a non-degenerate ground state. Using the terminology of the Ising model in a transverse field Dutta et al. (2015), we call it the “paramagnetic phase” (although it is ferromagnetically ordered along the -direction) and denote it by P. This phase is separated from the others by Ising transition lines (red lines in Fig. 3).

The three symmetry-breaking phases stem from the line corresponding to the pure XXZ model. This model has three phases: a gapped ferromagnetic phase, F, for , a gapless (critical) Luttinger liquid phase Haldane (1981); Giamarchi (2004) for and a gapped Néel phase, N, along the -direction, for . The F and N phases have doubly degenerate ground states and break the symmetry, with an additional breaking of translational symmetry for the Néel phase. When a transverse magnetic field is applied (), the two gapped F and N phases are stable until the gap closes at the Ising transition line, at which the system enters the P phase. For the Luttinger liquid phase, a non-zero transverse field immediately opens a gap. The associated broken symmetry is , corresponding to a Néel ordering in the -direction (N phase). This order is eventually destroyed by the transverse field through an Ising transition toward the P phase.

The boundaries between the three phases, F, N and N (green horizontal lines in Fig. 3), with broken symmetries emerge from the Heisenberg points . Along these lines, the gapless system presents additional symmetries. Indeed, the Heisenberg points correspond to a SU(2) symmetry, which, under the application of the transverse field (), is reduced to U(1). The upper line corresponds to the Heisenberg model under an external field Yang and Yang (1966a, b), for which a Luttinger liquid phase survives up to the critical field , at which a commensurate-incommensurate transition occurs Pokrovsky and Talapov (1979); Schulz (1980). On the opposite Heisenberg point , the transformation maps the model onto the ferromagnetic Heisenberg chain. It has, as the other Heisenberg point, a SU(2) symmetry, lowered to U(1) when the transverse field is applied. Thus another straight critical line emerges from this Heisenberg point, separating N from F. Due to the model mapping transformation, this coexistence line ends with a lower critical field than the one Alcaraz and Malvezzi (1995).

This spin- model presents other remarkable features. The integrability of the model is an essential concept to discuss relaxation and thermalization. The model is integrable by the Bethe ansatz when and on the critical lines emerging from the Heisenberg points. In particular, corresponds to the XY model that maps onto free fermions Barouch and McCoy (1971). In the limit, the model maps onto the (anti)ferromagnetic Ising model in a transverse field, which also maps onto free fermions Pfeuty (1970), and is thus integrable. Away from these limits, the model is non-integrable.

The qualitative plot of Fig. 3 is supported by numerical results based on matrix-product state (MPS) simulations White (1992, 1993); Schollwöck (2005, 2011); ITensor () (Appendix B). We define the average magnetization along the axis () as

(6)

For symmetry reasons, must be zero on non-degenerate finite-size ground state. Therefore, the ordering of the spins is better captured by order parameters defined from correlations as

(7)

where , , and where is “large enough”, to be specified for a given .

We plot in Fig. 4 the magnetization and order parameters along the three spin axes as a function of and for and open spin chains. The first column shows that, as expected, the magnetization increases steadily with . The region with a large value corresponds to the P phase. Along the line and for , we observe magnetization plateaus, corresponding to a succession of ground states with fixed total magnetization along . These finite-size effects are gradually smoothed out away from this line Müller and Shrock (1985); Dmitriev et al. (2002a).

The order parameters and show the strength of Néel and ferromagnetic ordering across the phase diagram. While most phase transitions are rather steep, the N N transition at is much smoother due to strong finite-size effects. In this region, the gaps are indeed the smallest (the Luttinger liquid to N transition is of the Berezinskii-Kosterlitz-Thouless type Berezinskii (1971); Kosterlitz and Thouless (1973); Kosterlitz (1974)).

The features of the phase diagram and its finite-size effects are also conspicuous when plotting the von Neumann entropy where is the reduced density-matrix of the first spins in the chain. Along the critical lines, one expects Calabrese and Cardy (2004, 2009) a logarithmic divergence of the entropy (for open boundary conditions) with for Luttinger liquid phases and for Ising transitions. In the gapped phases, the entropy remains finite, and decreases when the gap increases. It displays plateaus along the line reminiscent of the magnetization plateaus. The rapid variation of the entropy when increasing within the N phase is due to fact that the MPS variational state breaks the symmetry (see Appendix B).

Figure 4 shows that the chain Hamiltonian exhibits a wide variety of interesting behaviors. It also shows that, in most regions, finite size effects are not too large. A good approximation of the thermodynamical limit can be reached with 40 atoms only, setting realistic goals for the circular state quantum simulator.

Iv Preservation and trapping of circular Rydberg atoms

iv.1 Circular atoms lifetime

These remarkable features of the spin-chain Hamiltonian are only relevant if the circular atoms can be preserved and trapped for times much longer than , even much longer than their natural lifetime ( ms for ). They should thus be protected from spontaneous emission and from other loss mechanisms. We show in this Section that this ambitious goal can be achieved with state-of-the-art techniques.

D. Kleppner pointed-out Kleppner (1981) and experimentally demonstrated Hulet et al. (1985) that spontaneous emission can be inhibited by placing atoms in a structure with no field mode close to resonance with the atomic transition. The unique spontaneous decay channel for the circular states in a zero temperature environment is a -polarized transition towards the next lower circular state. It is inhibited in the plane-parallel capacitor providing when its plates are separated by a distance smaller than half the radiated wavelength,  mm for the transition. In an ideal, infinite capacitor, the inhibition is complete and the circular level lifetime is infinite.

A more realistic calculation should take into account the finite size and conductivity of the capacitor. We have numerically computed the residual spontaneous emission rate, , for a capacitor with square plates (made up of gold cooled below 1 K) of side , using the CST-studio software suite (Appendix C). Figure 5 shows the ratio as a function of and . The inhibition is large as soon as is larger than 10 mm. We choose for the following discussion an operating point with  mm and  mm, corresponding to a 50 dB inhibition rate, i.e., to a  s lifetime for . Note that the spontaneous emission inhibition for is even stronger, since the emission wavelength is larger. The opening between the capacitor plates is large enough to provide convenient optical access to the trapping region.

The capacitor also inhibits the -polarized dressing microwave required, in particular, to engineer the chain Hamiltonian . However, due to the sensitivity of Rydberg atoms to microwave fields, this drive requires only a low power. It can thus be applied on the atoms in an evanescent mode to which a powerful enough source is coupled, for instance through tiny ( mm diameter) irises pierced in the capacitor plates. According to simulations, these irises do not significantly affect the spontaneous emission inhibition.

Figure 5: Spontaneous inhibition ratio, (log scale) as a function of the capacitor spacing and size . The dashed vertical line corresponds to . The open red triangle shows the chosen operation point  mm and  mm with a 50 dB inhibition.

Spurious effects conspire to reduce the lifetime (Appendix C). Blackbody photons induce a -polarized transition from the circular state towards elliptical states in a higher manifold. The transition rate for this polarization is enhanced by a factor in the capacitor. Cryogenic temperatures are thus required to limit this effect. We assume  K, a typical base temperature for He refrigerators. The effect of the collisions with the background gas is small for a background pressure in the torr range, accessible in a cryogenic environment Gabrielse et al. (1990); Diederich et al. (1998). We must also include in the loss mechanisms the contamination by elliptical states due to the dipole-dipole interaction, photoionization, which turns out to be quite negligible for circular states, or the elastic diffusion of trapping-lasers photons.

We finally find (Appendix C) that the levels lifetimes, including all foreseeable loss mechanisms, exceeds 47 s in the useful range of values (even longer lifetimes can be reached by increasing further and (at the expense of a reduced tunability of in the latter case). A 40-atom chain is thus expected to have a useful lifetime of at least 1.1 s, corresponding to spin exchange periods at m. That one can follow the dynamics of a spin chain over such long times is a unique feature of the circular state quantum simulator.

iv.2 Circular atom trapping

The circular atoms must obviously be trapped in order to take benefit of these long lifetimes. Trapping them through the Stark or Zeeman effects has been proposed Hyafil et al. (2004); Mozley et al. (2005) or realized Anderson et al. (2013). These techniques, however, do not lead to flexible trap architectures. We consider instead, following Dutta et al. (2000), an optical laser trap.

The nearly free valence electron of the circular atom experiences a positive ponderomotive energy Avan et al. (1976) proportional to the laser intensity ,

(8)

where and are the electron’s charge and mass, respectively, and where is the laser angular frequency (much larger than the electron’s orbital frequency). The electron is thus attracted towards low intensity regions. The ponderomotive energy is 14.8 MHz (about 1 mK) in the 10 m waist of a 1 W, 1 m-wavelength laser. It is about ten times larger than the potential experienced by a ground-state rubidium atom in the same conditions.

The electronic attraction towards intensity minima is transmitted to the Rydberg atom as a whole (note that the ponderomotive energy of the ionic core is quite negligible due to its large mass). We propose to radially trap the rubidium atoms along the axis (Fig. 1) by a 0.5 W, 1 m-wavelength hollow beam in a (0,1) Laguerre-Gauss (LG) mode focused to a 7 m waist Carrat et al. (2014). The transverse trapping frequencies are then  kHz. At the edges of the inhibition capacitor, the LG beam diameter is 0.6 mm. The laser power hitting the plates (60 nW) and dissipated in the cold environment is thus much less than the cooling power of the He refrigerator.

The longitudinal lattice (along ) should have an inter-site spacing adjustable at least between 5 and 7m and provide a tight confinement to reduce the variations of the dipole-dipole interactions due to the residual atomic motion. Note that the residual motion along the transverse axes is much less worrisome, acting only at the second order on the interatomic distance. In order to get a simply adjustable spacing, we suggest to use the interference at a small angle between two 1 m-wavelength laser Gaussian beams, offset in frequency by a few tens of MHz with respect to the LG beam to avoid interferences with the transverse trap beam. They propagate in the plane at an angle with respect to the axis. For m, ( for m). Their waist is 7 m along and 200 m along , so as to cover the whole length of the chain. With a power of 1.45 W in each beam for m (2.8W for m), we get  kHz and a longitudinal trap depth of nearly 4 MHz, i.e. 200 K (Appendix D). The power hitting the capacitor is also negligible for these beams. Figure 6 presents the total ponderomotive potential for m. The deep traps are regularly spaced along the axis. The extent of the atomic motion ground-state in these nearly harmonic traps is  nm.

Note that, for a position-dependent laser intensity, the potential acting on the atom is the average of the ponderomotive energy over the atomic orbital Dutta et al. (2000). We show (Appendix D) that this effect plays no role when the atom remains in the harmonic region close to the bottom of the trap. We also estimate the decoherence due to the atomic motion in the residual trap anharmonicity. The coherence time (0.2 s) corrresponds to at m.

Figure 6: Cut in the plane of the ponderomotive potential produced by a Laguerre-Gauss and two interfering gaussian beams at a 1 m-wavelength. The potential values are given in frequency units by the color map on the right.

We have suggested here a set of operating parameters adapted to the conservation of a strongly interacting long chain over extended times. Other compromises can be made, according to the experimental goals. Smaller couplings can be obtained with a larger inter-site spacing , limiting the impact of the residual atomic motion of the chain dynamics (Section VI). Much tighter traps can be obtained with higher laser powers, at the expense of a reduced lifetime. Longer lifetimes can be reached in very high electric and magnetic fields, at the expense of a reduced tunability of the Hamiltonian parameters.

V Deterministic preparation and detection of circular atom chains

The -atom chain must be prepared deterministically. Techniques based on the Mott transition Greiner et al. (2002) achieve a unit filling of a ground-state atom lattice. They are not easily applicable to the large lattice spacings envisioned here. Real-time feedback allows one to prepare regular arrays of independent dipole traps with unit filling Barredo et al. (2016); Endres et al. (2016). However, the preparation of circular levels from the ground state has a finite efficiency, leading to gaps in the final Rydberg chain. The dipole blockade mechanism could lead to nearly regular Rydberg atoms arrangements after the excitation of a BEC or of a lattice Pohl et al. (2010); Schausz et al. (2012) but, according to our simulations Nguyen (), interatomic distance variations are large and lead to an excessive atomic motion in the final traps.

We thus discuss in this Section an innovative chain preparation method based on a variant of evaporative cooling Masuhara et al. (1988). Its principle is to start with an irregular chain and a large random number of atoms trapped in a laser tube and to progressively compress and ‘evaporate’ this chain until the required interatomic spacing and atom number are reached. The evaporation provides cooling nearly down to the ground state of the trap, leading to very small motional effects and dephasing. We show that the chain evaporation technique also leads to an efficient state-selective individual detection of each atom.

Figure 7: Sketch of the proposed chain preparation and detection sequence. : atom chip. : science capacitor. : field-ionization detection region. Note the axes orientation in the lower left.

Figure 7 presents a conceptual scheme of the experiment. The sequence (detailed in Appendix E) begins with the preparation near a superconducting atom chip of an elongated Viteau et al. (2011) Rb atom thermal cloud cooled below 1 Nirrengarten et al. (2006); Roux et al. (2008), near quantum degeneracy. This sample is trapped in a red-detuned focused laser beam and brought inside the ‘science’ capacitor . We suppress the ground state trap and laser-excite a low angular momentum Rydberg state in the dipole blockade regime, leading to a random Rydberg atom chain ( atoms) with inter-atomic spacings of the order of 9 Teixeira et al. (2015); Hermann-Avigliano et al. (2014). We get rid of the residual ground-state atoms with a resonant pushing laser pulse and transfer the Rydberg atoms into using a -polarized evanescent rf field. During the few microseconds required for this sequence, atomic motion is negligible.

The Laguerre-Gauss radial confinement beam is then switched on. We also switch on two 1 m-wavelength ‘plug’ Gaussian beams parallel to . They create two energy barriers on the axis, centered at . The ‘right’ plug () is lower than the ‘left’ one. We then slowly compress the trap by reducing . We increase accordingly the van der Waals repulsive interaction up to a point where the energy of the right-end atom compares to that of the weak plug. Further compression ejects atoms, one at a time, above the weak plug. The ‘evaporation’ of an atom removes a part of the global energy, providing a cooling mechanism reminiscent of the evaporative cooling Masuhara et al. (1988). The final atom number, , is determined by the height of the weak plug and by the final value of .

Figure 8: Number of atoms left as a function of the distance between the two plug beams. The thick curve gives the average over 100 realizations of the evaporation process. The atom number variance is indicated by the blue-shaded area.

Numerical simulations of the classical atomic dynamics reveal the efficiency of this process. Figure 8 presents the average and the variance over 100 realizations of the evaporation sequence of the number of remaining atoms as a function of the final value. For atom numbers lower than 45, we observe clear steps in the evolution of . The zoom around (inset) shows that the atom number variance cancels for optimal values. Stopping the evaporation process at such trap lengths deterministically prepares a string with a prescribed atom number. The interatomic spacing is finely tuned through a final adjustment of . The lattice is then adiabatically turned on, trapping the atoms in their respective sites (the plugs remain on with an adjusted power to compensate the repulsion of the end atoms by their single neighbor).

The complete preparation sequence simulated here lasts 1.3 s (Appendix E). In order to avoid atomic decay during this relatively long time interval, the electric field can be raised to a large value, leading to an individual atom lifetime  s. The final longitudinal position dispersion with respect to the lattice sites is  nm for =14 atoms, corresponding to only oscillation quantum (110 nm for , i.e. quanta). A full quantum model would be clearly required. It is out of the scope of this paper, and will be used in the next Section for an order of magnitude estimate of the influence of the atomic motion. We have checked with 3-D simulations of the dynamics that the transverse position dispersions are of the same order of magnitude as .

The evaporation procedure can also be used for an efficient detection of the spin states. At the end of the spin-chain evolution, the exchange interaction can be halted by casting with a ‘hard’ microwave -pulse onto . The exchange interaction is in the mHz range. The energy states of the spins are thus frozen from then on. The repulsive van der Waals interactions being nearly unchanged, the evaporation process can be resumed. The lattice is switched off, the right plug is lowered, and is slowly decreased, expelling atoms one at a time. The atoms escape along the axis, guided by the LG beam at a velocity determined by the height of the weak plug. They fly towards the field-ionization region ( on Fig. 7). The levels and are selectively detected there with near-unit detection efficiency. This simple scheme reads out the spin states in the up/down basis. Adding a hard microwave pulse before freezing the interaction, we can rotate the equivalent spin at will and thus detect any spin observable (the same for all atoms) and its correlation functions along the chain. Microwave pulses acting on individual atoms on their way from to make it possible in principle to measure arbitrary quantum observables of the spin chain.

The ability to measure, as a function of time, the states of the individual spins opens a wealth of possibilities. It is instrumental to access complex correlation functions and entanglement properties in the spin chain.

Vi Numerical simulation of adiabatic evolution through a quantum phase transition

In this Section, we discuss the observation of quantum phase transitions using this setup. In particular, we investigate the influence of the residual atomic motion around the lattice sites. We include the effect of the classical atomic motion in the spin-chain Hamiltonian discussed in Section III and in the numerical simulations of the system dynamics. This effect is quite dependent upon the relative values of the exchange frequency and of the trap oscillation frequency . We thus explore numerically the two cases, m and m, corresponding respectively to and to .

vi.1 Hamiltonian with a classical atomic motion

We treat the atomic motion as classical and independent from the spin dynamics. We use the results of the numerical simulations of the evaporation process (Section V and Appendix E) as an input for the atomic trajectories and perform averages over the outcomes of many (100) realizations of the evaporative chain preparation. The Hamiltonian including the atomic motion can be written as

(9)

We have introduced

(10)

where is the position of atom . A regular lattice ( within a constant offset) corresponds to . An important remark is that, even though the absolute strengths of the coupling coefficients fluctuate with position and time, the ratios of these couplings are constant, in time and along the chain. The motion induces thus a highly correlated noise on the couplings.

Figure 9: Simulation of an adiabatic preparation of the ground state for the ferro-para transition at and with spins. (a), (b) and (c) total magnetizations , and fidelity for  kHz. (d), (e) and (f) same curves for  kHz [note that the time scale and vertical scale for the fidelity differ from those of frames (a-c)] The insets in frames (a) and (d) depict the optimized ramp . For each quantity , the black curve gives its average over 100 realizations of the classical atomic trajectories. The shaded area represents the corresponding standard deviation of the distribution over trajectories. The red curves corresponds to a situation with atoms fixed at the lattice sites. The blue curves correspond to the exact ground state of the Hamiltonian for motionless atoms. In each frame, the vertical/horizontal dotted lines correspond to the expected quantum phase transition point.

The term adds a random longitudinal magnetic field along the -direction. We chose the dressing frequency so as to cancel the average value of this field: , where and the over-line denotes the average over many realizations of the atomic trajectories. Still, a residual magnetic field, , remains on the two edge sites . This field breaks locally the symmetry and polarizes the edge spins in the -direction. It is an asset or a drawback depending on the purpose of the quantum simulator. It is, for instance, an asset while entering a ferromagnetic phase. It creates a perturbation that naturally triggers the build-up of the order parameter. Note that, for large enough chains and in gapped phases, these edge effects are relevant only over the correlation length scale. The physics of the model can still be captured anyway in the bulk of the chain.

vi.2 Adiabatic evolution through a quantum phase transition

We now investigate the evolution of the system in an adiabatic evolution through a quantum phase transition line. We perform simulations of the full system dynamics under the Hamiltonian (9) for up to atoms using exact diagonalization. We infer, from Fig. 4, that a favorable situation to probe a quantum phase transition is the ferro-para transition FP. It has little finite-size effects and a strong ordering in the ferromagnetic phase. We take thus , which corresponds to V/cm and Gauss in Fig. 2 and leads to .

The edge fields are then negative and favor the spin-up ferromagnetic state . This state is actually the ground state of (9) for and and can be straightforwardly prepared experimentally. We note that, in the opposite limit, , the polarized state , where become the ground state. Starting from an exact ground state is an ideal situation for an adiabatic preparation protocol. We choose thus to start from , to vary from to and then to decrease by reversing the function. This protocol has two goals. First, we follow the behavior of the observables along the path in order to probe the transition and, second, this cycle allows us to probe the deviations from adiabaticity through the comparison between the observables in the direct and return ways.

Adiabatic theory suggests Roland and Cerf (2002); Kim et al. (2011) to use non-linear ramps for , with a velocity proportional to the square of the gap to the first excited states. In the presence of motion, we phenomenologically found good non-linear ramps with a velocity inversely proportional to the derivative calculated in the ground state (very low velocity values are replaced by a constant lower bound). In order to save computing time, we first optimize the ramps using simulations for spins and reuse them for the largest calculation ().

We plot on Fig. 9, for a spins chain, the average values and the standard deviation (over 100 atomic motion realizations) of and . We also plot the fidelity of the time-evolving state, , with respect to the ideal ground state for a given : . Frames (a-c) correspond to  kHz, frames (d-f) to  kHz. The optimized ramps are given in the insets of frames (a) and (d). The realistic averaged curves (black lines) are compared to the ground state (blue lines) and to the time evolutions obtained with the same time-dependent protocol operating on atoms at fixed positions (red lines).

In the thermodynamic limit, the transition (indicated by the vertical dashed lines in Fig. 9) would be signaled by a vanishing of at the critical point and a discontinuity in the slope of , both with critical exponents belonging the Ising universality class. On a finite chain, the transitions are smoothed out. The data of frames (a-c) in Fig. 9 clearly exhibit, for  kHz, the expected behavior of the magnetization observables around the phase transition points. However, imperfections are conspicuously revealed by the intermediate oscillations in and the reduced final value of (which, in principle, should return to its initial value, 1). The protocol generates “heating”, mostly close to the transition points, and the fidelity accordingly sharply drops at the transition.

Part of these imperfections are due to the atomic motion, as shown by the differences between the black and red curves. These motion-induced imperfections increase rapidly when the sweep time is increased. We are thus driven to use a rather fast ramp (the total duration  ms of the sequence corresponds to only). Accordingly, part of the imperfections are due to the breaking of the adiabaticity criterion, as illustrated by the difference between the red and blue curves.

A lower value (2.3 kHz) leads to a considerably improved situation, as shown in frames (d-f) of Fig. 9. The atomic motion is effectively decoupled from the spin dynamics. This decoupling allows us to use a much slower ramp. The total duration  ms corresponds now to . The differences of the observables in the three situations are then negligible. The final fidelity of the 14-spin state reaches an outstanding value of 0.99.

These preliminary results show that it is fairly easy to achieve operating conditions such that the residual classical atomic motion has a quite negligible influence on the spin dynamics. The long lifetime of the spin chain is instrumental to realize slow evolutions fulfilling the adiabaticity criterion. This would allow us to explore properly the complete phase diagram and the quantum phase transition phenomenon. Obviously, further studies could lead to further optimizations of the ramps making it possible to operate at larger couplings over a reduced time scale and to the exploration of the other transitions in the phase diagram.

Vii Conclusion

We have shown that state-of-the-art techniques make it possible to build a spin-chain quantum simulator based on laser-trapped circular Rydberg atoms. This simulator combines the flexibility of atomic lattices, the individual atomic observables read-out typical of ion trap together with the strong dipole-dipole interactions of Rydberg atoms. Defect-free atomic chains can be prepared by an evaporative cooling method, which leaves the atoms finally near their vibrational ground state. Evaporation also provides us with a unit efficiency individual spin detection. A proper microwave dressing leads to a fully tunable spin- XXZ chain Hamiltonian. Its parameters are under direct experimental control, a unique feature of this simulator. The long lifetime of the laser-trapped circular atoms, protected from spontaneous emission, makes it possible to follow the dynamics over unprecedented time intervals, in the range of times the spin flip-flop period. Moreover, the individual detection of all spin observables makes it possible to access a wealth of interesting properties, such as entanglement properties and local entropies.

A circular state simulator with about 40 atoms could address important problems of many-body quantum physics. We have shown that slow variations of the Hamiltonian parameters make it possible to explore precisely the quantum phases of the XXZ model, generating the ground states with a high fidelity. This bears two complementary interests. On the one hand, the precise determination of non-trivial observables in the ground state and the comparison with state-of-the-art numerical approaches will assess the quality of the simulator. On the other hand, variations of the protocol from its optimal implementation will lead to the generation of defects. We could thus explore the adiabaticity limits, a particularly important topic in the context of adiabatic quantum computation Chandra et al. (2010), quantum annealing Boixo et al. (2014); Heim et al. (2015) and Kibble-Zurek mechanism Zurek et al. (2005).

An essential perspective for such ground-state physics is to explore the spin-one Haldane phase Haldane (1983, 1983) using the ladder geometry. Separately prepared parallel chains could be brought in interaction (by moving their Laguerre-Gauss transverse trapping beams), leading to a square ladder geometry. Using the anisotropy of the dipole-dipole interaction, the signs of the coupling between legs (along ) and rungs (along ) of a properly oriented ladder can be different. This leads to two antiferromagnetic chains that are ferromagnetically coupled. This model, in part of its phase diagram, realizes the Haldane phase Narushima et al. (1995); White (1996); Hijii et al. (2005). This phase possesses a non-trivial topological order den Nijs and Rommelse (1989), which can be straightforwardly measured in this context, and fractional spin-1/2 edge states with the open boundary conditions typical of our simulator Hagiwara et al. (1990). The exploration of this non-trivial physics is one of the first main incentives to build a circular state simulator.

Another interesting low-energy physics problem is that of a disordered XXZ chain Ma et al. (1979); Dasgupta and Ma (1980); Doty and Fisher (1992); Fisher (1994); Iglói and Monthus (2005). Adding a laser speckle field to the optical lattice, it is fairly easy to produce random shifts of the atoms with respect to their equilibrium positions, randomly modulating the dipole-dipole interactions. In the regime, this model displays the paradigmatic competition between localization and interactions, opening the way for Bose-glass physics Giamarchi and Schulz (1987, 1988); Giamarchi (2004). Another striking feature of this model is the emergence of random singlet phases Ma et al. (1979); Dasgupta and Ma (1980); Doty and Fisher (1992); Fisher (1994); Iglói and Monthus (2005), with their unusual long range correlations and entanglement properties Refael and Moore (2004); Laflorencie (2005) in disordered systems. Remarkably, the random singlet phase of the Heisenberg point would be accessible thanks to the possibility to tune on all bonds.

The ability to modulate rapidly the Hamiltonian parameters also opens a vast realm of possibilities Silveri et al. (2017). Periodic modulations could be used to realize spectroscopic investigations of the elementary excitations of the system. They bear a particular interest at the critical point of the Ising transition (the one studied in Section VI), as shown by its remarkable integrable features Zamolodchikov (1989); Kjäll et al. (2011), recently investigated in condensed matter experiments Coldea et al. (2010); Morris et al. (2014). The long lifetime of the circular simulator would be instrumental in studying low-frequency excitations, not easily accessed in other contexts.

Floquet engineering corresponds to periodic variations of the couplings much faster than . It allows one to design effective Hamiltonians that are not accessible with the usual control parameters Lignier et al. (2007); Kolovsky (2011); Goldman and Dalibard (2014). This is a particularly interesting perspective to enlarge the field of applications of the circular state quantum simulator, since all parameters of can be easily modulated at high frequencies. In the same spirit, the proposed Rydberg set-up notably makes it possible to study Floquet time crystals Else et al. (2016); Sacha and Zakrzewski (2017).

Instantaneous quenches can be realized by a sudden variation of the Hamiltonian. There is a whole range of questions on quenches that would benefit from long observation times. Whether an isolated quantum system displays equilibration and thermalization is a fundamental issue of statistical physics Dziarmaga (2010); Polkovnikov et al. (2011); D’Alessio et al. (2016); Borgonovi et al. (2016); Neill et al. (2016). As the spin-chain Hamiltonian has integrable points, one could investigate the interplay between thermalization and integrability Kinoshita et al. (2006). The intermediate relaxation time regime contains information on the propagation of correlations at the origin of the relaxation process Cheneau et al. (2012). Another remarkable scenario is the pre-thermalization Gring et al. (2012a). Some observables reach rapidly a metastable steady-state, while the system is not yet in its thermal equilibrium. Only few experiments have been carried out in this regime Gring et al. (2012b). Finally, the dephasing time of a sub-system could be directly measured Barthel and Schollwöck (2008). Combining quench protocols with disordered Hamiltonians offer a way to address the issues related to many-body localization Basko et al. (2006); Pal and Huse (2010); Nandkishore and Huse (2015); Lüschen et al. (2016). In particular, the long simulation times would allow one to follow the logarithmic increase of the entropy that signals the many-body localization transition Žnidarič et al. (2008).

Beyond the spin arrays physics, the circular state simulator could explore a new regime of spin-boson interaction Leggett et al. (1987); Le Hur (2008); Porras et al. (2008). Shallow optical lattices lead to a situation in which the spin exchange is strongly coupled to the atomic motion Manzoni et al. (2017). The joint motion of the atoms would then entangle with the spins, leading to a situation, in which numerical simulations are far out of reach even for moderate spin numbers. In particular, the common coupling of the spin ensemble to the same bath could mimic correlated errors, which are one of the key problems for quantum error correction in quantum information protocols.

Finally, extensions of the evaporative chain preparation to full 2-D or even 3-D geometries can also be envisioned. This extension of the circular state quantum simulator capability would allow it to address a domain where understanding the nature of the ground state is even more challenging.

Acknowledgements.
We acknowledge funding by the EU under the FET project ‘RYSQ’ (ID: 640378) and by the ANR under the project ‘TRYAQS’ (ANR-16-CE30-0026). We are grateful to B. Douçot, Th. Giamarchi, Ph. Lecheminant, D. Papoular and P. Zoller for fruitful discussions.

Appendix A Circular states and their van der Waals interaction

For Rydberg levels with a high angular momentum, the quantum defects are negligible and the hydrogenic model is an excellent approximation. In vanishing electric and magnetic fields, the circular state with principal quantum number , , is degenerate with the enormous hydrogenic manifold. Any perturbation admixes it with other high- ‘elliptical’ states Gross and Liang (1986). In a static electric field, the manifold degeneracy is partially lifted Gallagher (1994). The eigenstates of the Stark Hamiltonian in an electric field along can be sorted out by their magnetic quantum number ( is no longer a good quantum number since the spherical symmetry of the Hydrogen atom is broken). The energy spectrum of the manifold arranges as a triangle whose tip is the circular level , as shown in Fig. 10, isolated from the nearest elliptical states . A magnetic field , also along , lifts the near-degeneracy of with . The circular state is now stable against stray field perturbations. The circular level experiences a negative second-order Stark shift, scaling as ,  MHz/(V/cm) for . The differential Stark shift on a transition between two circular states is much lower [ kHz/(V/cm) on the two-photon transition].

Figure 10: Diagram of the Stark levels with the highest magnetic quantum numbers. The circular state is at the tip of the triangle of Stark levels sorted according to .

Due to their high angular momentum, circular states cannot be reached directly by laser excitation of the ground state. Their preparation relies on the laser excitation of a low- Rydberg state, followed by a series of -polarized radio-frequency transitions between Stark levels, performed in an adiabatic rapid passage sequence Nussenzveig et al. (1993). A good control of the radio-frequency field polarization leads to an efficient ( % efficiency and purity) and rapid (few s) transfer into the circular state Signoles et al. (2014). Field-ionization provides a state-selective detection with near unit efficiency Maioli et al. (2005).

For a pair of interacting Rydberg atoms at a distance along , perpendicular to the quantization axis , the dipole-dipole interaction reads

(11)

where and are the distances of the two Rydberg electrons to their respective cores and where the are the spherical harmonics for the two electron positions.

We encode the spin-up and spin-down states of the simulator on the and circular states, connected by a two-photon transition at frequency  GHz. In the basis , , , , the dipole-dipole interaction reads, in a second order perturbative approximation,

(12)

In terms of the Pauli operators for the two atoms, , this interaction can be rewritten as

(13)

where

(14)
(15)
(16)
(17)

Note that the sign of the exchange term is irrelevant since it can be changed by a mere redefinition of the absolute phase of the basis levels. We thus chose it to be positive. The term is a mere redefinition of the energy origin, that will no longer be explicitly included in our discussions. The term results from the differential van der Waals shift between the two atomic levels and plays the role of a longitudinal field in the spin model. The and terms describe the longitudinal and transverse (exchange) spin-spin interactions respectively.

In order to determine precisely these coefficients, we perform an explicit numerical diagonalization of the pair Hamiltonian, including the Zeeman and Stark perturbations (note that the dipole-dipole interaction breaks the cylindrical symmetry of the Stark levels in the proposed geometry, preventing us from using approximate analytical solutions). We have to restrict the total Hilbert space in order to perform the computation. We limit its basis to levels whose principal quantum numbers differ by from 48 or 50 (the coupling matrix elements decrease rapidly when increases). We also select values differing by at most from those of the levels or interest. Most of the computations are performed with a basis of 361 pair states. For a few values of the fields, we have checked that the interaction changes by only % when using a three times larger basis.

We have first computed the interaction between two atoms in as a function of the interatomic distance, for  Gauss and  V/cm. The uncoupled pair state is found to be mainly contaminated by the symmetric pair state. The energy variation of the levels is in excellent agreement with a dependence for  m. For smaller distances, the interaction is too large to agree with the perturbative van der Waals dependence.

For m, we find  GHz m, a value independent (within ) of the electric and magnetic fields in the relevant range. The other coefficients have a marked dependency on and , varying by 10 to 20% for  V/cm and  Gauss. Their values for  V/cm and  Gauss are  GHz m,  GHz m and  GHz m.

Figure 11: Variations of as a function of the electric field for and  Gauss (magenta, black, blue, green, red, cyan and purple dots respectively). The colored lines are a guide to the eye.

Accordingly, in terms of the spin model,  kHz at m (2.3 kHz at 7 m) is independent of the fields, whereas and vary over large ranges. Figure 2 shows the variations of as a function of and . Fig. 11 shows the corresponding variations of . Note that and do not depend upon . We observe that the dependence flattens when increases. On the other hand, a larger value reduces the mixing of the circular states and the elliptical states, and accordingly increases the levels lifetime (Appendix C). We thus chose the largest value for which the spin chain can be tuned over the complete phase diagram,  Gauss.

Appendix B Details on numerical simulations

Figure 12: (a): Gaps to first and second eigenstates for a chain with periodic boundary conditions and spins. (b): Ferro () and antiferro () order parameters from MPS calculations.

Numerical simulations of the spin Hamiltonian are conducted using the ED (Exact Diagonalization) and MPS (Matrix Product States) techniques.

ED has been mostly used on small systems and to treat quasi-exactly the time-evolution in the presence of atomic motion. It has been used to compute the excitation gaps to the first and second excited states with periodic boundary conditions, shown on Fig. 12. For symmetry breaking phases, the gap to the first excited state must vanish in the thermodynamical limit, while the gap to the second eigenstate must vanish only on critical lines. This expected behavior is qualitatively well reproduced numerically in spite of finite-size effects around the line, reminiscent of the magnetization plateaus.

The MPS calculations are performed using the ITensor library ITensor (). We use typically up to 1200 kept states for spins with open boundary conditions. In many regions of the phase diagram, there are almost classical low-lying excited states, in which the algorithm gets easily trapped, even on small systems. To help circumvent this issue, we include noise in the reduced density-matrix White (2005) for the first sweeps of the algorithm. Furthermore, deep in the symmetry broken phase, the ground state is almost degenerate on large systems (all eigenstates are eigenvectors of the symmetries). The MPS algorithm converges thus towards a superposition of these finite-size ground states that effectively breaks the symmetries, and that have a lower entanglement entropy. This is illustrated on Fig. 12, where the local ferromagnetic and Néel order parameters are computed from local magnetization. The algorithm randomly converges towards one of the two symmetry breaking states. The Néel order along never shows up on local observables simply because the algorithm works with real states (the Hamiltonian is purely real).

Appendix C Loss mechanisms

The spontaneous emission rate inhibition results from the reduction of the classical electromagnetic field mode density at the atomic emission frequency. It can thus be computed with a classical approach Haroche (1992); Hinds (1991). For an atom in the middle of an ideal, infinite plane parallel capacitor (plate separation along the axis), the spontaneous emission rate modification factors and for - and -polarized transitions (w.r.t. ) at wavelength respectively read

(18)

where the square brackets in the summation limits stand for the integer part. For , . The inhibition is perfect in a capacitor below cut-off and a polarization parallel to the plates.

In order to get a more realistic value, we use a numerical approach taking into account the finite size and conductivity of the capacitor. We compare the total power radiated in free space by a -polarized tiny antenna at the 61.41 GHz frequency of the transition to that radiated by the same antenna placed in the capacitor. This computation is performed using the CST Microwave Studio software suite. We have first tested the method with a very large capacitor made up of an ideal conductor. The results are in excellent agreement with the predictions of Eqs. (C). We have then computed the spontaneous emission in a finite capacitor with electrodes made of gold cooled below 1 K (conductivity m Lide (1996)). Note that superconducting electrodes cannot be used in this context, since they are incompatible with the directing magnetic field . The results of this calculation are presented on Fig. 5. We choose the operating point  mm and  mm, providing a 50 dB inhibition.

The lifetime of isolated circular atoms is also limited by the absorption of -polarized residual blackbody photons. The dominant processes are the transitions from to the elliptical states . Transitions to higher manifolds are negligible, since the matrix elements and the blackbody number of photons per mode drop rapidly with the upper principal quantum number. The capacitor-induced rate enhancement for these transitions (a factor 1.8) is computed from Eqs. (C). At  K, a typical base temperature for a He refrigerator, we find the excitation rates of and to be 1/630 s and 1/360 s, respectively.

For interacting atoms, the circular states get mixed with elliptical states, which can emit or absorb -polarized or high-frequency photons. These processes are not inhibited by the capacitor. The numerical diagonalization of the full pair Hamiltonian provides the expansion of the coupled states on the spherical basis. Using these results, we compute the total decay rate of the coupled levels, including spontaneous decay and blackbody-induced transfers modified by the capacitor.

Figure 13: (a) Lifetime of an atom in interacting with another atom in the same state at a m distance, as a function of the electric field and of the magnetic field .

Figure 13 presents a color plot of the lifetimes, computed in an ideal capacitor, of two atoms at a m distance, as a function of the electric field and of the magnetic field . Similar results are found for . The lifetime increases with and , due to the decrease of the circular state contamination when the directing fields increase (for an isolated atom, the lifetime depends on , but is found to be nearly independent of for  V/cm). Ideally, we should thus aim for the largest field values. However, the tunability of decreases rapidly when increases (Fig. 2). To get a flexible simulator, we are thus limited to  Gauss, and hence to an individual atom lifetime between 88 s for  V/cm and 145 s for  V/cm. Note that can be raised during the chain preparation and detection phases, making radiative losses negligible during these lengthy procedures.

We have also estimated the dipolar relaxation mechanism Boesten et al. (1996), involving a transition from a pair of atoms in towards a pair of atoms in . This process releases an energy much larger than the trap depth. The two elliptical atoms would thus escape at a high velocity. The matrix element between the initial trapped state and the final high energy plane wave is very small, making the process negligible.

Microwave superradiance Gross and Haroche (1982) does not contribute to a lifetime reduction. First, spontaneous emission and, hence, superradiance on the two-photon transition from to is totally negligible. Superradiance on the one-photon transitions towards the or states could be a concern. However, all atoms are in the upper state of the transition. We thus consider only the emission of the first photon in a superradiant cascade, which occurs at a rate times larger than for a single atom, a trivial statistical factor. We have already taken into account this effect when stating that the useful chain lifetime is times that of an individual atom.

Collisions with the background gas also limit the lifetime. The state-changing cross-sections for the circular state colliding with Helium gas at room temperature have been calculated for quite a few final states in de Prunelé (1985). Comparable estimates are given in Yoshizawa and Matsuzawa (1984). Reference de Prunelé (1986) shows that these cross-sections are nearly independent of the electric field, up to 0.2 times the ionization threshold (i.e. up to 20 V/cm for ).

Extrapolating to all final states the cross-sections given in de Prunelé (1985), we estimate the total cross-section to be of the order of 2000 atomic units for . Intuitively, it should scale as the surface of the circular orbital, a torus with main radius and minor radius . We infer an order of magnitude estimate for , about 10 times the geometric cross-section. The collision lifetime is thus 400 s at a gas density  m, corresponding to  mbar at 1 K. Such vacuum conditions can be met easily in a cryogenic environment Gabrielse et al. (1990); Diederich et al. (1998), due to the intense cryopumping by all surfaces around the atoms.

Laser trapping competes with photoionization. For low angular momentum Rydberg states, photoionization is fast, with a lifetime in the s range for realistic traps Saffman and Walker (2005). The situation is radically different for the circular levels Dutta et al. (2000). They are nearly impervious to photoionization. In simple terms, the Rydberg electron absorbs an optical photon with a high momentum only when coming close to the core, a situation which never happens for circular states.

The hydrogenic photoionization cross-section at frequency for the state is computed for isotropic and unpolarized radiation in Beterov et al. (2007). It can be used for an order of magnitude estimate in a polarized laser beam:

(19)

where all quantities are expressed in atomic units and where is the modified Bessel function of the second kind. For large values and a laser field at a m wavelength, the argument of the Bessel functions is large (170 for ). We can thus use the asymptotic expansion of to lowest order. We get, in SI units:

(20)

with , and being, respectively, the Rydberg and fine structure constants. The cross-section decreases exponentially with , down to about  m for . A simple estimate based on the wavefunctions in -representation confirms this order of magnitude. Note also that the photoionization rates have been measured as a function of up to  Saffman and Walker (2005). The exponential decrease with is conspicuous on these data. The extrapolation to the circular states confirms that photoionization is indeed negligible.

Another loss channel is the elastic diffusion of the trapping laser by the nearly-free Rydberg electron. This Compton-like process is different from photoionization. The electron receives a momentum kick corresponding to a rather large recoil energy (300 MHz), of the order of the Stark levels separation [ MHz/(V/cm)]. A diffusion may thus cause a transition towards an elliptical state. The diffusion cross-section can be evaluated with the classical Thompson diffusion model. Averaging the laser intensity on the atomic motion in the actual trap (peak-to-peak amplitude 70 nm) and on the electronic motion around the core ( nm), we find that the average time between diffusions is 180 s. This is a worst case estimate of the contribution to the circular state lifetime, since not all diffusion events are expected to change the atomic state.

Cause Lifetime (s)
Residual spontaneous emission 2500
Blackbody induced processes 630
Level mixing 88
Dipolar relaxation
Photoionization
Collisions with background gas at torr 400
Compton elastic diffusion in trap
Predicted lifetime 47
Table 1: Decay channels for a pair atoms at m, =6 V/cm and =13 Gaus. The corresponding lifetimes are given for a single atom in seconds.

Adding all relevant sources of losses, summarized in Table 1, we find an individual atomic lifetime of 47 s, leading to a 1.2 s lifetime for a 40-atom chain.

Appendix D Ponderomotive trap

The trap is formed by the combination of a standing wave produced by the interference at a small angle between two elongated Gaussian 1m-wavelength laser beams (1.45 W each for m and 2.8 W each for m) together with a 1m-wavelength Laguerre-Gauss beam of order and waist m (0.5 W power). The intensity of the Laguerre-Gauss beam at a distance from the symmetry axis in its focal plane reads

(21)

providing a quadratic trapping potential for small motion. The total depth of the transverse trap is then 6 MHz (300 K), while that of the longitudinal lattice is 4 MHz (200 K). Near the trap center, the ponderomotive potential is harmonic with trap frequencies  kHz and  kHz.

The ponderomotive potentials estimated above assume that the electron has a fixed position in the trap. In fact, it orbits around the core. As shown in Dutta et al. (2000), the ponderomotive energy must be averaged over the electron probability density in the circular state . This average can of course be performed numerically.

An excellent analytical approximation is obtained by assuming that the electron is on the Bohr orbit with radius , in the plane. Using the harmonic approximation to the ponderomotive potential, it is easy to show that the averaging results in a simple offset on the trapping potential, , where and are the trap frequencies for a motion in the plane of the circular orbit. This offset amounts to  kHz for . We have checked that this simple model differs from the numerical integration over the electron’s probability density by less than 4%.

Such an offset does not change the trap characteristics. The offsets experienced by and differ by 1.7 kHz, resulting in a constant shift of the atomic transition frequency. We thus expect that, to first order, the motion of the atoms in the trap does not contribute to any dephasing of the spin states.

In order to estimate the residual motional dephasing, we must include the anharmonicity of the trapping potential. The dominant effect corresponds to the motion along . Using the numerical potential averaging, we find that the atomic transition frequency varies quadratically with , being shifted by 12 Hz for  nm (this shift can be interpreted as a relative difference in the trapping frequencies for the two levels). For a motion in the trap with a 65 nm amplitude (prediction of the numerical simulations of the evaporation process for small chains), this corresponds to a 160 ms coherence lifetime, much larger than the spin exchange time .

The flip-flops of the spins in the chain evolution slightly change the interatomic van der Waals forces and thus the equilibrium atomic positions. If this modification was large, this would lead to an entanglement between the spin-chain dynamics and its motional excitations (phonons). This would be a rich and complex situation, the exploration of which is an interesting perspective for the future of this simulator Manzoni et al. (2017). Nevertheless, we first aim at minimizing this effect and thus choose a tight enough trap.

It is easy to estimate an order of magnitude of the atomic displacement from the center of the trap, , in units of the ground-state extension

(22)

where plays the role of the Lamb-Dicke parameter of ion traps. Here, for m, and . The atomic displacement being much smaller than the ground-state extension, the entanglement with the motion is negligible. We indeed predict from an explicit analytical model in the simple case of two atoms that the exchange between the spins is not appreciably modified. Note that the situation would be much worse when using a dipole-allowed transition to encode the spins. For instance, for the one-photon transition, the exchange coupling is of the order of 12 MHz. The resulting forces are strong enough to expel the atoms from the trap!

Appendix E Evaporation process

We have performed a detailed simulation of the deterministic preparation of a atom chain. We start from a thermal cloud cooled near quantum degeneracy in an elongated dipole trap formed by a 780 nm-wavelength focused laser beam, displaced adiabatically from the atom chip to the science capacitor. We assume a state-of-the-art Viteau et al. (2011) cloud of about 2000 atoms with a mm length.

We turn off the dipole trap and apply a s-long laser pulse to bring the atoms into the Rydberg state in the dipole blockade regime. We use 780 nm- and 480 nm-wavelength lasers Hermann-Avigliano et al. (2014), tuned on resonance with the two-photon transition from to , and away from resonance with the intermediate state. The final positions of the excited Rydberg atoms are simulated using a Monte-Carlo rate equation model including the laser line-width (250 kHz) and the van der Waals interactions Nguyen (). About 100 Rydberg atoms are excited, separated by m. For such separations, the van der Waals interaction between the atoms is weak, comparable to the laser line-width.

This excitation stage is immediately followed by the transfer into the circular state in an adiabatic rapid passage sequence, lasting a few microseconds Signoles et al. (2014). The transfer is induced by a -polarized radio-frequency field produced by the four electrodes on the side of . We finally apply a short pulse of a resonant  nm laser to push out the remaining ground-state atoms. The motion of the atoms is negligible during the preparation stage, lasting s. The final atomic velocities are randomly chosen, with a thermal distribution at a 1K temperature.

Figure 14: Deterministic chain preparation sequence as a function of time (in milliseconds). (a) distance between the plug beams. (b) height of the plug beam barriers (red and green lines for the right and left plugs respectively) and plug beam waists (black line). (c) Longitudinal trap depth (solid line) and oscillation frequency (dashed line). (d) atomic trajectories. The trajectories of ejected atoms are interrupted after a short time for clarity. The inset exhibits the small final residual longitudinal motion. (e) Atomic kinetic energy (red line) and van der Waals potential energy (blue line) averaged over 100 realization of the evaporation sequence.

Figure 14(a-c) presents the timing (total duration 1.3 s) of the optimized chain preparation sequence for starting from this initial configuration as well as the results of a numerical simulation of the 1-D atomic trajectories (we have also performed some 3-D simulations to estimate the transverse atomic motion). This sequence should be performed with the largest possible and values to limit the radiative losses (Appendix C). The sequence is divided in four successive phases:

  1. Switch-on of the Laguerre-Gauss transverse trap, in combination with two ‘plug’ beams (100 ms). The m-wavelength Gaussian plug beams have a m waist (this large value results in a smoother evaporation in phase II). They are initially separated by  mm. The height of the associated barriers is smoothly raised from zero to 4 MHz (left beam) or 3 MHz (right beam) – panel (b). We simultaneously quickly compress the chain by reducing the distance from 1 mm down to 0.5 mm – panel (a). This fast compression saves time without significantly modifying the preparation efficiency.

  2. Actual evaporation until the required atom number is reached (1000 ms). The distance between the two plug beams is slowly reduced. The atomic chain is compressed, building up the repulsive van der Waals forces. The last atom on the weak plug side is expelled out of the trap as soon as its energy exceeds the height of the barrier. This phase stops here at m to reach the target value .

  3. Final adjustment of the chain (100 ms). The weak plug barrier is raised to 4 MHz, preventing further evaporation. In the meantime the waists of the plug beams are reduced – panel (b) – to provide a finer control of the atomic positions (this stage, experimentally complex, could be replaced by the adiabatic switching-off of the 30 m-waist plug beams and the simultaneous adiabatic switching-on of 10 m-waist beams). The length is slightly adjusted [inset in Fig. 14(a)] to provide a final m interatomic distance

  4. Adiabatic installation of the longitudinal lattice (100 ms) – panel (c). The amplitude of the residual motion in the traps is accordingly reduced.

The 1-D classical dynamics simulation is complex, since the motion of these coupled atoms is chaotic. The exponential sensitivity to the initial conditions makes it necessary to compute statistics over many realizations. Many numerical methods do not conserve the total energy, resulting in artificial excitation or damping of the system. We thus use a symplectic integrator with a sixth-order Runge-Kutta-Nyström method Blanes and Moan (2002).

Figure 14(d) presents the atomic trajectories in one of these simulations. The four phases are clearly apparent. In the first one, during the installation of the plug beams and the fast compression, rapid atomic escapes occurs from both sides while the plugs are still weak. This initial evaporation stops after  ms. The chain is then compressed more slowly. Evaporation above the weak plug resumes at the beginning of phase II, atoms escaping in the positive direction. The evaporation events seem to reduce the residual motion of the remaining atoms.

This qualitative insight is confirmed in Fig. 14(e), which presents the kinetic and potential van der Waals energies per atom averaged over 100 realizations of the evaporation sequence. During the evaporation stage II, the kinetic energy clearly decreases. The evaporation above the plug barriers provides a cooling reminiscent of the evaporative cooling Masuhara et al. (1988). The final motion of the 40-atom chain after stage IV has a typical extension  nm (see inset in Fig. 14(e)). The 3-D simulations indicate that the transverse motion extensions and are of the same order of magnitude as . Note that the transverse motion does not appreciably modify the interatomic distance.

Running 100 times the simulation, continued for 1.4 s to the end of the evaporation stage II, when the final chain only contains one atom, we obtain the average number of atoms and its standard deviation as a function of the final length presented in Fig. 8.

During evaporation, the atoms, particularly those close to the end of the final chain, transiently experience rather large trap laser intensities. We have estimated the associated loss rate due to Compton diffusion events in a full 3-D simulation. It is small, less than 3% for the atoms at the extremities of the chain, about 1% for the bulk atoms. Selective microwave transitions from the circular states towards a lower manifold and field-ionization of the remaining atoms could be used for a final purification of the chain before switching on the longitudinal lattice.

The atomic detection stage simply resumes the evaporation stage II after removing the longitudinal lattice and lowering the right plug beam. This process is clearly less critical, the only requirement being to keep the order of the atoms. The velocity of the ejected atoms in the guiding LG beam is determined by the height of the weak plug,  m/s for 3 MHz. The atoms thus reach the detection region, about 2 cm away, after a 125 ms delay, short as compared to their lifetime.

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