Evaporative cooling to a Rydberg crystal close to its ground state

We theoretically show how to obtain a long one-dimensional crystal near its quantum ground state. We rely on an evaporative cooling scheme applicable to many-body systems with nonzero-ranged interactions. Despite the absence of periodic potentials, the final state is a crystal which exhibits long-range spatial order. We describe the scheme thermodynamically, applying the truncated Boltzmann distribution to the collective excitations of the chain, and show that it leads to a novel quasi-equilibrium many-body state. For longer chains, comprising about 1000 atoms, we emphasize the quasi-universality of the evaporation curve. Such exceptionally long 1D crystals are only accessible deep in the quantum regime. We perform our analysis on the example of an initially thermal chain of circular Rydberg atoms confined to a one-dimensional (1D) geometry. Our scheme may be applied to other quantum systems with long-ranged interactions such as polar molecules.

We theoretically show how to obtain a long one-dimensional crystal near its quantum ground state. We rely on an evaporative cooling scheme applicable to many-body systems with nonzeroranged interactions. Despite the absence of periodic potentials, the final state is a crystal which exhibits long-range spatial order. We describe the scheme thermodynamically, applying the truncated Boltzmann distribution to the collective excitations of the chain, and show that it leads to a novel quasi-equilibrium many-body state. For longer chains, comprising about 1000 atoms, we emphasize the quasi-universality of the evaporation curve. Such exceptionally long 1D crystals are only accessible deep in the quantum regime. We perform our analysis on the example of an initially thermal chain of circular Rydberg atoms confined to a one-dimensional (1D) geometry. Our scheme may be applied to other quantum systems with long-ranged interactions such as polar molecules.
Up to now, the investigation of 1D quantum crystals has been hindered by the difficulty of obtaining large crystals in this geometry, where thermal and quantum fluctuations both destroy long-range order in macroscopic systems [1]. Nevertheless, crystallization does occur in finite-sized systems [15]. It has been unambiguously observed in the absence of any external periodic potential only in small systems of up to fifty ions [16][17][18][19] or ten electrons [20,21]. The realization of larger 1D crystals requires going deep into the quantum regime. There, thermal fluctuations are suppressed, and long-range order is only limited by quantum fluctuations, which are less stringent [1]. The realization of large 1D crystals will pave the way towards the investigation of 1D quantum crystals, where one may look for e.g. giant plasticity through the tunneling of defects [22,23].
We focus on one way of obtaining spatial order which relies on strong nonzero-range dipole interactions between Rydberg atoms [24]. Rydberg atoms are ideally suited for quantum information processing [25,26] and quantum simulation [2,28]. Nontrivial many-body states [29][30][31] of up to 50 atoms manipulated with optical tweezers have been prepared through resonant coupling to Rydberg states [32][33][34][35]. Rydberg states may be weakly admixed to the atomic ground state [36][37][38] or resonantly excited [39] so as to study the interplay between anisotropic interactions and disorder or frustration [40]. Quantum gases resonantly coupled to Rydberg states have been predicted to exhibit a quantum phase transition to a Rydberg crystal [41], leading to a univer- The potential maxima VL and VR satisfy VL < VR, so that atoms are expelled from the left edge of the trap.
sal scaling behavior observed in the critical region [42]. In all those cases, low-angular-momentum Rydberg states were considered, leading to a strong limitation on the lifetime (100 µs per atom, a few µs for many atoms), limiting the size of the system. Circular Rydberg atoms [43][44][45][46], whose excited electron has maximal orbital and magnetic quantum numbers, overcome this limitation and offer a very promising platform for the quantum simulation of many-body problems [2]. Using spontaneous emission inhibition [47,48], their already long lifetime (30 ms) is expected to be extended to more than 1 min. This timescale allows for implementing an evaporative cooling scheme applicable to Rydberg atoms [2], whose classical analysis shows great promise for reaching extremely low temperatures.
In this Letter, we show that large 1D Rydberg crystals may be prepared very close to their quantum ground state in realistic experimental conditions [2,49] through this evaporative cooling scheme. Despite the absence of any spatially periodic potential, these crystals exhibit long-range spatial order. This is in stark contrast to the classical analysis of 1D systems, which would predict the absence of long-range order [1]. We introduce a quantum thermodynamic model, applying the truncated Boltzmann distribution to the collective excitations of the chain. We show that it leads to a novel quasi-equilibrium regime which differs from the truncated Bose-Einstein distribution applicable to quantumdegenerate gases [50]. In contrast to dilute systems where the evaporation is driven by two-body collisions [4], the mechanism we describe here hinges on manybody physics, whereby the phonons present in the chain lead to the expulsion of a single atom. Hence, it is related to the quantum evaporation of liquid helium [52][53][54], also predicted to affect cold bosonic atoms [55].
We first consider a fixed number N of Rydberg atoms confined in a 1D trap of fixed size L (see Fig. 1). We illustrate our model using the parameters of Ref. [2]. The atoms are confined radially using the ponderomotive potential [56] induced by a Laguerre-Gaussian laser beam [57, chap. 2]. They are trapped axially between two optical plugs yielding the potential The barrier width and heights are, respectively, w = 30 µm, V L /h = 3 MHz, and V R /h = 4 MHz. The trap size L = x R − x L is slowly decreased from its initial value so as to induce successive atomic expulsions, providing the evaporative cooling. Unlike for gases, the barrier heights remain constant during the whole process. The atoms interact via the strongly repulsive van der Waals interaction V (x i , x j ) = C 6 /|x i − x j | 6 with C 6 /h = 3 GHz µm 6 , corresponding to 87 Rb atoms with the principal quantum number n = 50. The equilibrium positions x 0 1 , . . . x 0 N are evenly spaced in the bulk of the chain, but not on the edges, due to the finite spatial extent of the barriers. Two neighboring atoms are distant by l ≈ 5 µm, leading to interaction energies C 6 /l 6 ≈ h · 200 kHz. We describe the atomic vibrations in terms of a quadratic Hamiltonian: In Eq. (1), the N vibrational modes {ũ k } have the frequencies ω 1 < . . . < ω N , and the {p k } are their conjugate momenta. They are related to the atomic displacements {u n = x n − x 0 n } through the orthogonal matrix R. The applicability of Eq. (1) only requires local order [58, Sec. I]: the averages (u n+1 − u n ) 2 1/2 involving two neighboring atoms should remain small compared to l = L/N . For a thermal chain at the temperature T , this requires k B T < 2C 6 /l 6 , and is well satisfied for up to 1000 atoms with l ∼ 5 µm and k B T < ∼ h 100 kHz ≈ k B 5 µK. Classical thermodynamics. For a given configuration characterized by the phonon mode energies {ǫ k } 1≤k≤N and phases {φ k } 1≤k≤N , the position of the leftmost atom at time t is u 1 (t) = k R 1k (2ǫ k /(mω 2 k )) 1/2 cos(ω k t+ φ k ). It remains trapped as long as |u 1 (t)| < u M , where u M = x 0 1 − x L . We consider the time-averaged meansquare displacement u 2 1 = u 2 M N k=1 ǫ k /E Mk , where the quantities E Mk = mω 2 k u 2 M /R 2 1k increase with k. Hence, for a given α, the lowest-energy configurations for which  ) is set by ω 1 . Furthermore, numerical simulations of the classical (cl) dynamics of the atom chain [59] have shown the atomic motion to be chaotic. Hence, exploiting ergodicity, the trapped configurations are those with E < E cl M . We describe the quasi-equilibrium thermodynamics of the chain using a Boltzmann distribution truncated at the energy E cl M , whose partition function reads: In Eq. (2), β = 1/(k B T ) is the inverse temperature, E = H({p k ,ũ k }) and P (a, z) = γ(a, z)/Γ(N ) is the normalized lower incomplete gamma function [3]. The mean (quadratic) energy U cl (L, T ) associated with the Hamiltonian H and the entropy S cl (L, T ) follow from The function P (a, z) also appears in the thermodynamics of the evaporation of a gas (a = 3 for a harmonic trap) [4]. Here, a = N ranges from 40 to 1000, so that the role of truncation is strongly enhanced with respect to gases of ground-state atoms [58, Sec. III]. It is important for k B T > ∼ E cl M /N . For larger T , all trapped configurations are equally populated. The probability density for a configuration to have the energy E is Both U cl and S cl reach finite maxima U N max (L) and S N max (L) (see Fig. 2), where: For fixed N , both maxima increase with L, because less  the energy and entropy reflect the non-truncated thermodynamics of a harmonic oscillator chain. They overlap with U quant , S quant for a range of values of T , yielding the full quantum thermodynamic functions (see Fig. 2). Evaporation. We now describe the evaporation process. Initially, the chain comprises N = N I atoms in a trap of size L (N ) = L I , with the energy U (N ) = U I . For all considered parameters, U I ≫ E ZP , signalling the classical regime, and U I ≪ E cl MI , so that it is described by non-truncated thermodynamics. Thus, U I /N I = k B T I is the initial temperature. We adiabatically compress the chain by slowly decreasing L (see Fig. 3). Hence, the entropy S (N ) remains constant. Expelling an atom is irreversible, therefore N also remains constant. However, T and U (N ) increase, whereas U f ). At this point, the leftmost atom is expelled from the trap, its kinetic energy being the barrier height V L . The (N − 1) remaining atoms thermalize to the new initial energy U Here, V The complete evaporation curve consists of a repeated sequence of these two steps. Figure 4 compares our classical (dark red) and quantum (red) predictions, down to the trap size The result of our classical model closely matches the classical-dynamics simulations reported in Ref. [2] (Fig. 14, phase II). Our quantum approach predicts that, starting from N I = 100 atoms, the final state with N F = 40 atoms obeys a Bose-Einstein distribution with U F /(N F h) = 7.0 kHz, slightly above the zero-point energy E ZP /(N F h) = 5.9 kHz. The shown average energies account for the uncertainty ∆U I = U I / √ N I = h·6.5 kHz on U I , which washes out their jaggedness due to the expulsions (Fig. 3) . Smaller values of ω ⊥ will lead to quasi-1D chains exhibiting the 'zigzag' transition observed with ion chains [23,63] and in electronic systems [64].
Quasi-universality for longer chains. The final state of such a long chain is a crystal exhibiting true long-range order, with all spatial correlators C nm = (u n − u m ) 2 ≪ l 2 [58, Fig. S1]. This is only possible deep in the quantum regime, where thermal fluctuations are suppressed [1]. The crystalline order may be fully characterized experimentally through microwave spectroscopy, revealing the regularity and fluctuations of the lattice parameter, combined with spatially-resolved ground state imaging [33,65].
We have introduced a quantum thermodynamic model for the evaporative cooling of 1D Rydberg atom chains [2]. Unlike the evaporative cooling of ground-state atoms, the final temperatures accessible with our scheme are not of the order of the barrier heights. Instead, they are determined by the maximum energy u max (l) compatible with the trap. This reflects the many-body character of the evaporation scheme and leads to final temperatures that are radically lower than the barrier heights by three orders of magnitude. We have shown that, under realistic experimental conditions, this scheme yields large near-ground-state Rydberg crystal. The long-range spatial order of these 1D structures is a feature of the deep quantum regime. Our scheme will also apply to other interacting 1D systems such as polar molecules [66,67]. There, the nonzero-ranged interaction between the particles is provided by the dipole-dipole interaction, which scales with 1/r 3 and may be made purely repulsive in low-dimensional geometries [68].
Outlook. The following directions warrant further investigation. (i) For higher initial temperatures or mean atom spacings, the initial state is a liquid and Eq. (1) does not hold, but our scheme will still drive the system towards its crystalline ground state. (ii) For longer chains, a prolonged evaporation going beyond the regime of Fig. 5 leads to E quant M < ∼ E ZP +hω N , in which case the calculation of the quantum thermodynamic functions is more involved. (iii) The timescale ensuring adiabaticity is set by the anharmonic processes neglected in Eq. (1). (iv) Our scheme is also applicable in 2D, where the expected ground state is a hexagonal crystal which we shall investigate both theoretically and experimentally.

Supplemental material
This document provides complementary information on the following topics: I. the applicability of the quadratic Hamiltonian; II. the anharmonic terms and their twofold role; III. the partition function and its numerical evaluation; IV. the observability of the adiabatic plateaux with constant atom numbers; V. the quasiuniversal description for long chains and its limits.

THE QUADRATIC HAMILTONIAN
The Hamiltonian describing the harmonic vibrations of the atoms about their equilibrium positions {x 0 n } (Eq. 1 in the main text) is applicable as soon as the chain exhibits local order. Indeed, in the chain bulk, the trapping potential is negligible and, within the nearest-neighbor approximation, the interaction energy of atom n is E I n = Here, x n = x 0 n + u n is the position of atom n. Expanding E I n to second order in the displacements {u n }, and exploiting the neartranslational invariance, we find that the harmonic approximation is valid if η n = 21 (u n+1 − u n ) 2 /l 2 < 1, where l = L/N is the mean interatomic distance and the average (u n+1 − u n ) 2 is the spatial correlator between two neighboring atoms. For a thermal distribution, this condition reduces to k B T < 2C 6 /l 6 . Accounting for the trap and the truncated thermodynamics, we find this criterion to be well satisfied all along the evaporation for the long chain of Fig. 5 in the main text (see Fig. S1(a)).
The present criterion is less stringent than asking for the chain to be in a crystalline phase. This is especially true in 1D where thermal fluctuations quickly rule out long-range order [S1]. For example, the long chain of Fig. 5 exhibits no long-range spatial correlations in its initial state (N i = 1000, l i = 5.5 µm, k B T i /h = 65 kHz). This can be seen on Fig. S1(b): the correlator (u n − u m ) 2 /l 2 > 1 for distant atoms. However, our scheme brings the chain close to its quantum ground state, which does exhibit long-range correlations ( (u n − u m ) 2 /l 2 ≪ 1 for all n and m, see Fig. S1(c)).

ANHARMONIC EFFECTS
The leading anharmonic contribution to the Hamiltonian follow from the third-and fourth-order terms in the displacements {u n }. For gases, they yield two-body collisions which are essentially instantaneous. By contrast, for Rydberg chains, they generate many-body correlations over the characteristic time τ propag for propagation along the chain, set by the sound velocity. They are mostly due to interactions and occur in the chain bulk, where their probability does not depend on position (see Fig. S2). They are much less probable near the edges, where the trapping potential leads to larger distances between the static equilibrium positions of the atoms.
The role of these anharmonic processes is twofold. First, they are responsible for thermalization and ergodicity on a timescale involving τ propag . Second, they set the (longer) timescale ensuring the adiabaticity of the compression between two atomic expulsions. The classical-dynamics simulations reported in Ref. [S2] have shown that, for the shorter chain of Fig. 3 in the main text (N i = 100), compression rates of the order of 40 µm/ms are adequate. The optimal compression rate will be investigated elsewhere.
For gases, anharmonic processes directly drive the atomic expulsions, which immediately follow two-atom collisions during which one atom has acquired enough energy. Their relation to expulsions is more involved for Rydberg chains. If the trap size is such that an expulsion is expected (T → ∞), ergodicity causes the system to explore various configurations until the leftmost atom is expelled with the energy V L . If no expulsion is expected (T finite), the compression of the trap causes an increase in energy due to the atoms on the edges of the chain being set in motion towards the bulk. Expelling the leftmost atom before thermalization has taken place (i.e. with an energy > V L ) is likely to involve a two-atom collision at the open end of the trap. There, anharmonic terms are strongly suppressed (see Fig. S2), so that these higherenergy expulsions are rare. Instead, the energy increase is most often mediated, through harmonic vibrations, to the chain bulk where thermalization occurs. The rare cases in which the leftmost atom is expelled are not captured by our thermodynamic model. However, they are not a hindrance as long as their rate remains small: instead, they speed up the evaporation process with respect to our thermodynamic prediction. The presence of a single open end (the left end on Fig. 1 of the main text) is favorable for two reasons: (i) it leads to longer propagation times and, hence, more efficient thermalization; (ii) it helps reduce the rate of non-thermalized expulsions.

THE PARTITION FUNCTION
Normalized lower incomplete Gamma function -The thermodynamics of the (classical or quantum) truncated Boltzmann distribution involve the normalized lower incomplete Gamma function P (a, z), defined as [S3]: For given values of the trap size L and atom number N , the classical partition function Z cl is proportional to P (N, βE M )/β N . Hence, a is of the order of N , whereas z = βE M is the ratio of the threshold energy to the temperature. For a given a, the function P (a, z) resembles a step function (see Fig. S3(a)) which is equal to 0 for  small z (representing the truncation for large T ) and to 1 for large z (truncation plays no role for small T ). The smooth transition occurs for z ≈ a, so that truncation plays a role for k B T /E M > ∼ 1/a. The parameter a = 3 for a gas in a truncated 3D harmonic trap [S4], whereas for Rydberg chains a ≈ N ranges from 40 to 1000. Hence, Rydberg chains are affected by the truncation starting from much lower temperatures than gases are. . We go beyond the quasiclassical integral expression and include the leading-order quantum correction, proportional tō h 2 [S5, §33]. Hence, we write Z quant = Z cl (1 + h 2 χ 2 ), where the correction h 2 χ 2 is expressed in terms of the moments x 2 k cl , p 2 k cl and x 2 k p 2 k cl of Z cl . We find: Numerical evaluation -The evaluation of U (L, T ) and S(L, T ) involves calculating P (a, z) for 40 ≤ a ≤ 1000. In order to capture the steep variation of these functions for z ∼ a, we resort to arbitrary-precision numerics using the Boost.Multiprecision C++ library [S6].

CONSTANT ATOM NUMBER PLATEAUX
Between two atomic expulsions, the chain undergoes an adiabatic compression during which N remains constant (see Fig. 3   text and k B T I /h = 65 kHz, these plateaux are visible when the remaining trapped atom number N < ∼ 45 (see Fig. S4), in agreement with the classical-dynamics results of Ref. [S2]. The plateaux are resolved earlier on for lower initial temperatures and later on for higher ones.
Hence, starting from the first atomic expulsion, u remains close to the universal curve u = u max (l), within small deviations which decrease like 1/N . Furthermore, the entropies s (N ) = s max (l Fluctuations -The quasi-universality of the evaporation constrains the fluctuations ∆u and ∆s on the energy and entropy per particle to follow those on the atomic distance, ∆l. Neglecting the small deviations from the universal curves u = u max (l) and s = s max (l), they satisfy ∆u/∆l = u ′ max (l) and ∆s/∆l = s ′ max (l) (see Fig. S5). The constraint on ∆u/∆l has an important consequence. As l decreases, u max (l) tends towards e ZP (l) (see Fig. 5(b) in the main text). Hence, the derivative u ′ max (l) goes to zero. The fluctuations ∆u do not vanish, therefore ∆l increases and so does ∆n = (n/l)∆l (see Fig. S5(a)). Thus, as long as the quasi-universal regime holds, the constant-N plateaux will be poorly resolved. If the evaporation proceeds further, it will eventually drive the system out of the universal regime. Then, we expect to recover the short-chain behavior described in Sec. . For the chain considered in Fig. S5, this occurs beyond the validity range of our assumption E quant M ≫ E ZP +hω N , and will be investigated elsewhere. Non-universality of N/N I -The entropy per particle s(l, u) may be seen as a function of l and u. The derivative ∂s/∂u| l = 1/T goes to zero on the curve u = u max (l), which is reached for T → ∞. However, our numerical results show that ∂s/∂l| u diverges along the curve u = u max (l) (see Fig. S6). Therefore, s(l, u) may not be linearized near this curve, and the entropy difference   f ) goes to zero slower than 1/N . This rules out any exact universal behavior for the atom number fraction n = N/N I . However, the deviation from universality is small. For a given N I , we consider two initial energies u I1 < u I2 , and compare the curves n uI1 (l) and n uI2 (l) for l < l 1 , where l 1 is the mean atom spacing leading to the first expulsion for u I1 . Our numerical results show that these two curves nearly satisfy the scaling relation which would have been exact had ∂s/∂l| u not been divergent, namely n uI2 (l 1 )n uI1 (l) ≈ n uI2 (l) (see Fig. S6(a), whose inset highlights the breakdown of this scaling behavior).
The divergence of ∂s/∂l| u = p/T along the curve u = u max (l) signals that the pressure p goes to infinity faster than T does. This starkly contrasts with the behavior of the ideal gas, where p/T = nk B is finite, its constant value being set by the particle density n.