Creating big time crystals with ultracold atoms

We investigate the size of discrete time crystals in the range s = 10 - 100 (ratio of response period to driving period) that can be created for a Bose-Einstein condensate (BEC) bouncing resonantly on an oscillating mirror. We consider the effects of having a realistic soft Gaussian potential mirror for the bouncing BEC, such as that produced by a repulsive light-sheet, which is found to have a significant effect on the dynamics of the system. Finally, we discuss the choice of atomic system for creating time crystals based on a bouncing BEC and present an experimental protocol for realizing big time crystals. Such a system provides a platform for investigating a broad range of non-trivial condensed matter phenomena in the time domain.


Introduction
In 2012 Frank Wilczek proposed that a quantum many-body system in the lowest state could spontaneously break time-translation symmetry to form a time crystal, in analogy with the formation of a crystal in space [1]. Although such time crystals cannot exist in the lowest state of a quantum system with two-body interactions [2,3] (see [4] for possible time crystals with long-range multi-particle interactions), it was later demonstrated that a periodically driven many-body quantum system can spontaneously break discrete time-translation symmetry to form a discrete time crystal which evolves with a period two-times (s = 2) longer than the driving period [5]. Such a time crystal is predicted to be robust against external perturbations and to persist perpetually in the limit of a large number of particles [6,7]. Similar ideas of discrete time crystals were later proposed for periodically driven spin systems [8][9][10], which in the case of a spin-1/2 system evolve with a period twice as long as the driving period. Experimental evidence of discrete time crystals has since been reported for a range of spin-1/2 systems, including a spin chain of ions [11], nitrogen-vacancy spin impurities in diamond [12] and nuclear spins in organic molecules [13] and ordered crystals [14,15]. In addition, space-time crystals  with periodicity in both space and time  have been reported for a superfluid Bose-Einstein condensate (BEC) of atoms [16,17]. Experiments demonstrating spontaneous emergence of periodic evolution that does not require periodic driving have also been performed in magnon BECs [18,19]. A number of comprehensive reviews on time crystals have recently been published [20][21][22].
In an earlier paper [6] we presented mean-field calculations for a BEC of interacting atoms bouncing resonantly on an oscillating mirror that exhibited dramatic breaking of timetranslation symmetry to form a discrete time crystal. These time crystals can evolve with a period more than an order of magnitude (s ≫ 10) longer than the driving period, thereby creating a large number of available 'lattice sites' in the time domain. Such a system provides a platform for investigating a broad range of nontrivial condensed matter phenomena in the time domain [6,[23][24][25][26][27][28]. In Refs. [29,30] it has been shown that periodically driven systems can also reveal crystalline structures in phase space.
In this paper we investigate the range of sizes (s-values) of discrete time crystals that can be created for a BEC of ultracold atoms bouncing resonantly on an oscillating mirror. We also consider the effects of having a realistic soft Gaussian potential mirror  such as that produced by a repulsive light-sheet  which is found to have a significant effect on the dynamics of the system. Finally, we discuss the choice of atomic system for creating time crystals based on a bouncing BEC and present an experimental protocol for realizing big time crystals.

Single-particle case for a hard-wall mirror
We first consider a single atom bouncing in the vertical direction z on a harmonically oscillating hard-wall mirror in the presence of strong transverse harmonic confinement. Introducing gravitational units 0 = (ħ 2 ( 2 ) ⁄ ) 1 3 ⁄ , 0 = 0 , 0 = ħ ( 0 ) ⁄ and assuming a one-dimensional (1D) approximation, the Hamiltonian of the system in the laboratory frame can be written where the potential ̃( z  0) = ∞ and ̃( z > 0) = 0 for the hard-wall mirror,  and / 2 are the frequency and amplitude of the oscillating mirror in the laboratory frame, respectively, and m and g are the atom mass and gravitational acceleration. Use of gravitational units allows calculations to be performed with dimensionless parameters that are independent of the mass of the atom. Transforming from the laboratory frame to the frame oscillating with the mirror, Eq. (1) becomes [5,31] where  is the amplitude of the time-periodic perturbation in the oscillating frame (hereafter referred to as the amplitude of the oscillating mirror.) The case of a realistic soft Gaussian potential mirror is considered in Section 2.6.
In the classical description, transforming to action-angle variables (I, ) and using the secular approximation, Eq. (2) can be expressed as a single-particle time-independent Hamiltonian [6] where P = I  Is, =  2 3 (3 3 ⁄ ) is the action of the s : 1 resonant orbit, and the effective mass and classical average value of z are given by where s = / and  is the bounce frequency of the unperturbed particle (i.e., for a static mirror). The height of the classical turning point (drop height) of the bouncing atom is then h0 =  2 /(2 2 ). Equation (3) is the effective Hamiltonian of a particle in the frame moving along an s : 1 resonant orbit and indicates that for s ≫ 1 a single resonantly driven atom in the vicinity of a resonant trajectory behaves like an electron moving in a crystalline structure created by ions in a solid state system. That is, eigenvalues of the quantized version of the Hamiltonian (3) form energy bands (see Fig. 1) and the corresponding eigenstates are Bloch waves. These eigenvalues are actually quasi-energies of a periodically driven particle while the eigenstates are Floquet states obtained in the frame moving along the resonant orbit [5,6,23,31]. In the quantum description we will apply the tight-binding approximation by restricting the analysis to the first energy band of the quantized version of the Hamiltonian (3). In the many-body case such an approximation is valid provided the interaction energy per particle is smaller than the energy gap Δ between the first and second energy bands shown in Fig. 1.

Optimal value of the driving strength  for a hard-wall mirror
We estimate the largest  for a hard-wall mirror that is allowed before the dynamics become chaotic, making use of the Chirikov criterion [32]. This criterion estimates the value of  for which two neighbouring resonance islands, s : 1 and (s + 1) : 1, described by Eq. (3), overlap. The distance between the islands (for s ≫ 1) and the half-width of the islands are The resonance islands nearly overlap when which leads to   (/6) 2  0.27. The 2/3 factor in (6) is an empirical correction that allows for the presence of higher-order resonances [32]. This means that for s ≫ 1 the critical value of  is a constant independent of s and . For  = 0.2, we find that, although there is some chaos between the neighbouring s : 1 and (s + 1) : 1 islands, the islands themselves are still not perturbed [6]. For   0.2, the resonance islands become smaller and less suitable for realization of a time crystal. Indeed, quantum states that describe time crystals are located inside the resonance islands and if the islands are too small we need to choose a large value of in order to realize a time crystal. While this is in principle possible, the resulting evolution of ultracold atoms that demonstrates time crystal behaviour becomes very long, see Section 2.4. We conclude that for a hard-wall mirror  = 0.2 is universally good for any s : 1 resonance for which s ≫ 1.

Scaling of parameters with s = /
The quantum secular approximation allows us to obtain the quantum version of the classical Hamiltonian (3) [6]. That is, switching to the oscillating frame by means of the unitary transfomation =̂ω / , the matrix elements of the time-averaged quantum Hamiltonian describing the s : 1 resonance dynamics reads where the |  are eigenstates of the unperturbed part of the Hamiltonian (2), i.e., with  = 0, with the corresponding eigenvalues , and ̂|  = | . Around the resonant value of the quantum number of the unperturbed particle, i.e., for ≈ 0  , the first term on the right hand side of (7) can be approximated by We know how the parameters of Eq. (3) scale with Is. We now investigate how the effective mass meff, obtained from the quantum approach (8), and the matrix element n0s/2|z|n0+s/2, which provides an estimate of the classical average value z, scale with n0 and s. The results, presented in Fig. 2, show that for ≤ 100, the classical scaling is reproduced in the quantum approach if the particle quantum number n0 ≳ 100. This allows us to use the classical analytical expressions to determine the optimal parameters for an experiment. The classical results (4) (black curves) predict 1/|meff|  n0 4/3 and z  n0 2/3 which are observed in the quantum approach for n0 ≳ 100 for any 10  s  100. Right panel: matrix element n0s/2|z|n0+s/2 versus s for n0 = 1000. The classical result (4) (black curve), which predicts z  s 2 for a fixed Is  n0, is observed in the quantum approach (red points).

Number of bounces and energy gap
To demonstrate that a time crystal is created one needs to show that the ultracold atoms evolve periodically with a period -times longer than the driving period = 2 ω ⁄ . If the interactions between atoms are too weak, the subharmonic periodic evolution will be destroyed because the atoms will start to tunnel between lattice sites of the potential in (3), or in other words between wave-packets propagating along the resonant orbit. Therefore, to demonstrate that a time crystal is created, the system needs to have evolved for at least the time period corresponding to the tunnelling time ttunnel of a single atom between neighouring lattice sites. The tunnelling time (for s >> 1) and the bounce period Tbounce of the atom are given by [6] where J is the tunnelling amplitude of the particle between neighbouring sites of the periodic potential in (3),  = [ 2 (3 0 ⁄ )] 1/3 , and Is is denoted here by the quantum number of the unperturbed particle, i.e., Is  n0. We require ttunnel to be as short as possible because then the number of bounces needed to demonstrate that a time crystal is robust against single-particle tunnelling is relatively smalleach bounce off the mirror is a potential source of loss of atoms from the BEC. The number of bounces required to observe quantum tunnelling for noninteracting particles is then Another important parameter that we wish to control is the energy gap E between the first and second bands of the quantum version of the Hamiltonian (3), see Fig. 1. To realize a time crystal, sufficiently strong interactions between atoms need to be be present. However, if the interactions are too strong, the single-band description within the Bose-Hubbard model (see Section 2.5) is no longer valid because higher bands of the Hamiltonian (3) become involved. Thus, E should be as large as possible but this requirement is in contradiction with the requirement of a short tunnelling time ttunnel and we need to find a compromise.
Dividing the classical Hamiltonian (3) by z and rescaling s  , gives which depends only on the single universal parameter The division by   also means that in order to analyse the scaling properties of the system the quasi-energy gap E and the tunnelling amplitude J should be expressed in units of ϵ =   =   2 ⁄ .   (11). The red points are related to the fully quantum secular approach, Eq. (7), for 10 ≤ ≤ 100, 0.007 3 ≤ 0 ≤ 0.06 3 and  chosen so that  is in the range 0  2.5.
With the help of f() = J()/ϵ, we can now express the number of bounces as which indicates that if we choose  (e.g., the optimal value  = 0.2 for a hard-wall mirror) and a value for the energy gap E()/J() (which determines ) the number of bounces Nb is constant and independent of the values of s that we choose. In other words, for fixed  and  we can choose any s : 1 resonance (then from  = constant we obtain n0 and consequently ) and the number of bounces is always the same. This is illustrated in the left panel of Fig. 4 where for different values of  we always choose  so that the gap E/J between the first and second energy bands of the quantum version of the Hamiltonian (3) is about 10 (which for  = 0.2 corresponds to  = 0.456).
The key results of the analysis are presented in the right panel of Fig. 4 where the number of bounces Nb needed for tunnelling of non-interacting atoms between lattice sites of the potential in (3)

Many-body case
We now switch to the case of a BEC of interacting bosonic atoms bouncing on an oscillating mirror in the presence of strong transverse harmonic confinement  and assume the 1D approximation. We restrict the analysis to the Hilbert subspace corresponding to the first energy band of the quantum version of the Hamiltonian (3). This is the resonant subspace where atoms occupy s localized wavepackets ( , ) evolving along the s : 1 resonant orbit with period sT, where T is the driving period [6]. These wave-packets are the time-periodic version of the Wannier states in solid state physics. Within the mean-field approach and restricting to the s-dimensional resonant Hilbert subspace we can expand solutions of the Gross-Pitaevskii equation in terms of localized Wannier-like states ( , ) = ∑ =1 ( , ) and obtain the energy functional (actually the quasi-energy functional) in the form [6] ≈ − with where N is the total number of atoms, g1D = 2as describes the contact interaction between the atoms, as is the s-wave scattering length (in gravitational units) and the Uii and Uij describe the on-site and long-range interaction energies per particle in the Bose-Hubbard model (14). The leading tunnelling amplitudes correspond to the nearest-neighbour hopping J = Ji,i+1 and only this hopping can be kept in the model when we want to describe time crystal dynamics. Such a tight-binding approximation is valid provided the interaction energy per particle is much smaller than the energy gap E between the first and second bands of the effective Hamiltonian (3) which characterizes the overlap of the lowest energy solution  with a single Wannier-like wave-packet wi versus the interaction strength g1DN is presented for different s. The critical values of the interaction strength can be identified in the plots. The figure also shows how strong the interactions need to be in order to be dealing with the lowest energy solution that practically reduces to a single Wannier-like wave-packet wi. This is important information because time crystals in which (z, t)  wi(z, t) can be easily prepared in an experiment, see Section 4. The case s = 40 has been extensively analyzed in [6], where the numerical integration of the full Gross-Pitaevskii equation confirmed the results based on the Bose-Hubbard model. For different s : 1 resonances, the critical interaction strength g1DN differs because the tunnelling amplitudes Jij are slightly different and also because the same g1DN does not necessarily mean the same interaction coefficients Uij in (14). The coefficients Uij depend on the longitudinal width of the atom cloud which, for the optimal values of the parameters, varies from z = 1.3  2.7 for s = 10  100 (Table 2). Thus, a slightly different interaction parameter g1DN and transverse confinement frequency  is required for different s.

Case of a soft Gaussian potential mirror
The calculations in previous sections were based on a simple hard-wall mirror potential. We now consider the case of a realistic soft Gaussian mirror potential, such as that produced by a repulsive light-sheet, where V0 and 0 are the height and width of the Gaussian mirror potential.
For a hard-wall mirror, the trajectories of the bouncing atoms reverse their direction abruptly at the reflection point (i.e., → − ), so that the Fourier transform of the unperturbed periodic trajectories ( ) = ∑ Ω (where Ω is the frequency of the bouncing atom in the absence of mirror oscillations, see (9)) results in amplitudes of the harmonics that decrease with k like ~1 2 ⁄ . In the case of a soft Gaussian potential mirror, the trajectories of the atoms are smoothly reflected, so that the harmonics decrease much faster with k, see left panel of Fig. 6. Because for an s : 1 resonance the amplitude of the potential in (3) is determined by the amplitude of the s-th harmonic, i.e.,    , the Gaussian potential mirror needs to oscillate with an amplitude λ much larger than the oscillation amplitude of a hard-wall mirror in order to have the same effect on the bouncing atoms.
When a particle bounces off a soft Gaussian potential mirror some harmonics of the classical unperturbed orbits are not created at all. The question of which harmonics disappear for a given = 0 / particle ≥ 1 (where particle is the particle's energy) is a complex problem. In the left panel of Fig. 6 we see dramatic drops of certain Fourier components which have a uniform spacing increasing from k = 16  33 for V  1.6  5. We focus here on the 30 : 1 resonance. The right panel of Fig. 6 shows the amplitude 30 of the 30 th harmonic of an unperturbed periodic orbit as a function of = 0 / particle . When approaches one, i.e., V0 = Eparticle, from above there is a series of zeros of 30 . However, for ≳ 10, there is no dramatic drop of  30  and an experiment demonstrating a discrete time crystal can be carried out in this regime. For a given choice of V0 and 0, our primary restriction is to obtain an energy gap of E/J  10. First, we choose a mirror oscillation frequency  and by applying the quantum secular approximation (7) we determine the optimal oscillation amplitude  that leads to E/J  10. We then need to check if the secular approximation is still valid for these parameters by examining the classical phase-space pictures of the action I versus angle  to see if the dynamics is still regular or if it is already chaotic.
We assume here a soft Gaussian mirror potential which corresponds approximately to the repulsive light-sheet mirror used in [33] for the reflection of a 87 Rb BEC dropped from heights up to 300 m. The mirror is formed by a 3W 532 nm laser beam with a waist 0l0 = 10 m and horizontal extension 200 m, which for 39 K atoms corresponds to 0 = 15.5 and Vmax  4.6  10 3 in gravitational units. The hardness of the mirror can be varied by varying the beam waist 0. For a given Gaussian width 0 = 15.5, different mirror potential heights V0/Vmax, and oscillation frequencies around the optimal hard-wall mirror value  = 4.45, we have used the quantum secular approach (7) to determine the amplitude of the mirror oscillations λ required to obtain E/J  10. The results are shown in the left panel of Fig. 7. Now that we have predictions for all parameters, the corresponding classical phase-space pictures have been obtained, examples of which are shown in Fig. 8 for different mirror heights V0/Vmax. For  = 4.45, the phase space around the 30 : 1 resonance islands is regular in all cases except V0/Vmax ≈ 0.2, for which the  needed to obtain E/J ≈ 10 is extremely high ( 100) and the classical motion is no longer regular, and consequently the 30 : 1 resonance for V0/Vmax  0.2 is not suitable for realization of a time crystal. This is because for V0/Vmax ≈ 0.2 (and for a particle energy that fulfills V = V0/Eparticle  5), the 30 th harmonic of the resonant periodic orbit disappears (Fig. 6, left panel). For V0/Vmax  0.2, the drop height h0 does not change too much as a function of V0/Vmax (Fig. 7, right panel) and the optimal number of bounces required for tunnelling to neighbouring wave-packets and the optimal tunnelling amplitude remain nearly constant at ≈ 54 and J  0.0010 for  = 4.45. All of these parameters are very close to the optimal values for the hard-wall mirror.  The mirror oscillation amplitude  is chosen so that the quantum secular approximation (7) predicts a band gap E/J = 10. The phase space around the s = 30 resonance islands, which are located at I  1000  1200 (red horizontal lines), is regular in all cases except V0 ≈ 0.2Vmax, for which the  needed to obtain E/J = 10 is extremely high (cf. Fig. 7, left panel) and the classical motion is no longer regular. Not all islands are visible because some are hidden in the lower, narrow and elongated part of the phase space. For  close to 0 and 2 , i.e., close to the Gaussian mirror, the action I drops to zero because we have used action-angle variables suitable for the hard-wall mirror (which are given analytically) rather than for the Gaussian mirror.
For   4.45 and corresponding to E/J = 10, the classical motion becomes more chaotic than for  = 4.45 and the secular approach is not fully valid, while for   4.45 the number of bounces Nb required for tunnelling to neighbouring wave-packets increases. Thus, the optimal value of  for the soft Gaussian potential mirror is close to the optimal value  = 4.45 for the hard-wall mirror for s = 30.
The relatively large optimal values of  in the case of the soft Gaussian potential mirror, e.g.,  = 2.3 (corresponding to l0/ 2 = 74 nm for 39 K in the laboratory frame) at V0/Vmax = 0.8,  = 4.45, required to obtain a band gap E/J = 10 are more readily accessible experimentally than the optimal hard-wall mirror value  = 0.2 (6.5 nm for 39 K). If still larger mirror oscillation amplitudes  are required in the laboratory, one can choose a mirror potential height closer to V0/Vmax = 0.2 provided the corresponding classical resonance islands are not destroyed by chaos.

Constraints on the maximum number of atoms
To operate in the quasi-1D regime, the transverse standard deviation σ of the atomic density needs to be smaller than the standard deviation σz along the longitudinal direction and the interaction energy per particle should not exceed the excitation energy in the transverse directions (σ and σz are defined as the Gaussian fits to the density of the atomic cloud). The transverse width is determined by the frequency ω of the harmonic potential in the transverse directions, i.e., ⊥ = √ℏ/(2 ω  ). These requirements imply that we need  ⊥ and > | | , where is the atomic s-wave scattering length. When these criteria are fulfilled, a BEC of ultracold atoms is also stable against 'bosenova' collapse that can occur for attractive interactions in three-dimensional space. For atoms bouncing resonantly on an oscillating mirror, the longitudinal width is smallest at the classical turning point and corresponds to z = 1.78. For 39 K atoms, = 30, ω = 4.45 and a Gaussian mirror with 0 max ⁄ = 0.8, 0 0 = 10 m, the minimum longitudinal width is zl0 = 1.15 m; for example, for as 0 = 1.8a0, the maximum number of atoms is Nmax  12,000.
To minimize losses due to three-body recombination, the mean-square atom density  2  at the classical turning point needs to be less than 1/(K3BEC), where K3 is the three-body recombination coefficient and BEC is the lifetime of the BEC. For 39 K 1,+1 atoms and taking K3 = 1.3 (5)  10 29 cm 6 s 1 near the zero-crossing point [34] and BEC  1 s, we obtain n 2 max  10 29 cm 6 ; for example, for σzl0 = σl0 = 1.15 m, the maximum number of atoms is Nmax  17,000.

Choice of Atomic System
In our earlier paper [6], we focussed on the 85 Rb F=2, mF=2 system, which has a broad Feshbach resonance at 155 G with a zero crossing point at 166 G. In Table 1, we compare the Feshbach resonance parameters and time crystal parameters for a hard-wall potential mirror and s = 30 for four atomic systems that have broad Feshbach resonances: 85 Rb F = 2, mF = 2, 39 K 1, +1, 39 K 1, 1 and 7 Li 1, 0. These are also the BEC systems for which bright matterwave solitons have been investigated [34][35][36]. Table 1 Feshbach resonance and time crystal parameters for 87 Rb, 39 K and 7 Li atoms for a hard-wall potential mirror and s = 30.
In an experiment, the s-wave scattering length as needs to be adjusted precisely to zero and to small negative (attractive) values (e.g., asl0 = 1.8 a0). The sensitivity of as to magnetic fields in a Feshbach resonance is determined by the ratio of the background scattering length to the width of the resonance abg/. For 7 Li and 39 K, abg/ is 320 and 75 times smaller, respectively, than for 85 Rb and therefore much less sensitive to stray magnetic fields. 39 K also has the flexibility of having two broad Feshbach resonances, one involving a high fieldseeking 1, +1 state at B0 = 350 G and the other a low field-seeking 1, 1 state at B0 = 505 G. The latter may be useful, for example, if we need to trap the atoms in a magnetic trap prior to producing a BEC in an optical dipole trap.
The gravitational unit of length l0 (which scales as m 2/3 ) is 5.3 and 1.7 times larger for 7 Li and 39 K atoms than for 85 Rb, and hence the mirror oscillation amplitude  and drop height h0 are larger by these amounts. A larger mirror oscillation amplitude is more accessible in an experiment while a larger drop height allows the atom density to be probed with higher spatial resolution during a bounce cycle. On the other hand, the gravitational unit of time t0 (which scales as m 1/3 ) is 2.3 and 1.3 times larger for 7 Li and 39 K atoms than for 85 Rb, and hence the bounce period and the time for tunnelling to neighbouring wave-packets are longer by these amounts, which makes the experiment longer compared to the lifetime of the bouncing BEC. In addition, the rms velocity (which scales as m 1/2 ) for a thermal cloud is 3.5 and 1.5 larger for 7 Li and 39 K atoms than for 85 Rb, which means the atom cloud needs to be 12 and 2.2 times colder than for 85 Rb to have the same velocity spread. A small velocity spread is important for a bouncing thermal atom cloud. Potassium-39 1, +1 and 1, 1 atoms have a much smaller three-body loss rate than 7 Li 1, 0 atoms (Table 1), which allows higher atom densities to be used. 39 K also has certain technological advantages compared with 7 Li: the high-power laser systems at 767 nm (D2) and 770 nm (D1) are readily available commercially; 39 K does not require a Zeeman slower; and it is easier to access the zero crossing point (B0 = 350 G) than for a 7 Li resonance (B0 = 545 G) and to quickly switch or ramp the Feshbach magnetic field.
From the above considerations, we focus here on 39 K as an optimal atomic system. Table 2 summarizes the parameters for 39 K atoms for a hard-wall mirror for the range s = 10  100 and  = 0.2, E/J = 10,  = 0.456, Nb = 54. Ideally, we require a large mirror oscillation amplitude (≫ 10 nm) which is more readily accessible in an experiment, a large drop height ( 100 m) to allow high spatial resolution probing of the atom density during a bounce cycle, and a short tunnelling time ( 1 s) to allow the experiment to be performed in times shorter than the lifetime of the bouncing BEC. The results of Table 2 suggest it should be feasible to create time crystals with sizes in the range s  20  100. = 4.45.
For the soft Gaussian potential mirror and s = 30,  =4.45, we obtain ≈ 54, ℎ 0 ≈ 253 and thus similar parameters to the corresponding hard-wall mirror case. However, the mirror oscillation amplitude for the soft Gaussian potential mirror case is much more accessible experimentally, i.e., 75 nm for 39 K, and Gaussian width 0l0 = 10 m and height V0/Vmax = 0.8, where Vmax  4.6  10 3 . Table 2 Calculated parameters for 39

Experimental Protocol
We present here an experimental protocol to realize a discrete time crystal based on an attractively interacting 39 K BEC bouncing resonantly on an oscillating mirror. As an example, we focus on an s = 30 time crystal and a Gaussian potential mirror with width 0l0 = 10 m and potential height V0/Vmax=0.8, where Vmax  4.6  10 3 . Other time crystals in the range s  10  100 can also be considered, for which estimates of the optimal parameters for a hard-wall mirror are provided in Table 2.
(i) Preparation of 39 K BEC. We create a 39 K BEC using an all-optical technique, similar to that described in [41]. First, a slow beam of 39 K atoms from a 2D MOT is loaded into a 3D MOT where they are cooled to about 5 K in grey optical molasses, using blue-detuned D1 light at 770 nm. The ultracold atoms are then transferred to a large 1064 nm crossed optical dipole trap (CODT) located at the classical turning point  164 m above the atom mirror. The CODT is then compressed and the trap depth slowly reduced to evaporatively cool the dense cloud of atoms to about 100 nK to create a BEC. The trap frequencies are then adjusted to produce a spherical CODT, with trap frequency about 98 Hz, which is chosen so that the longitudinal width (zl0  1.15 m) of the atomic distribution (z,0) 2 matches the width of the Wannier wave-packet w(z,0) 2 at the classical turning point.
(ii) Release of BEC from optical dipole trap. Next, the longitudinal trapping potential is switched off ( = 0) to release the BEC of about 5000 atoms from the CODT to fall on to a 532 nm repulsive light-sheet mirror in the presence of the vertical 1064 nm optical waveguide with confinement frequency 98 Hz. The light-sheet mirror is produced, for example, by an 3W 532 nm laser beam focussed to a waist of 10 m with horizontal extension 200 m [33].
(iii) Setting the mirror frequency and amplitude. Next, the frequency of the oscillating mirror is tuned to the s : 1 resonance:  = ½s[g/(2h0)] 1/2 (e.g., /(2t0) = 2.8 kHz for s = 30 and a drop height 164 m) and the amplitude of the oscillating mirror is set to  75 nm (for a Gaussian mirror height V0/Vmax  0.8) to obtain an energy gap E/J  10 and with the resonance islands not disturbed by chaotic motion. Other s : 1 resonances can be accessed by adjusting the mirror oscillation frequency . The optimal mirror oscillation amplitude  can be varied by adjusting the Gaussian mirror height V0. The maximal interaction energy per particle that we consider here (i.e., Uii/(2J)  2.5) is less than the energy gap E/J  10 between the first and second energy bands. The light-sheet mirror is modulated sinusoidally either mechanically using a piezo crystal on a reflecting optical mirror or an acousto-optic modulator. It may also be possible to modulate the light-sheet mirror by modulating the intensity of the 532 nm laser beam, though this requires further simulations.
(iv) Detection of a discrete time crystal. The density of the atom cloud is determined at fixed positions between the classical turning point and the atom mirror at different moments in time out to typically 2000 mirror oscillations ( 0.7 s), first with the particle interaction turned off when all atoms will have tunnelled out of the initially chosen wave-packet, and then with a suffiently large attractive interaction turned on to break the discrete timetranslation symmetry. In the latter case, atoms occupy an initial wavepacket and evolve along the 30 : 1 resonant orbit without any signature of tunnelling to other wave-packets. For g1DN = 0.07, 0.10, 0.23, the maximal squared-overlap of the lowest energy meanfield solution ψ that describes the discrete time crystal (cf. Fig. 5, left panel) with a single Wannier wave-packet is wi 2 = 0.90, 0.95, 0.99, respectively, and the corresponding maximal interaction energy per particle is Uii/(2J)  0.8, 1.1, 2.5. Such g1DN values correspond to scattering lengths of as = 0.54a0, 0.78a0, 1.78a0 for N = 5000 and /(2t0) = 98 Hz. Experimental signatures for the formation of a discrete time crystal for the case s = 40 were presented in Figs. 3 and 4 of [6].

Discussion and Conclusions
We have investigated the range of sizes s of discrete time crystals (where s is the ratio of response period to driving period) that can be created for a BEC of attractively interacting atoms bouncing resonantly on an oscillating mirror. We considered the effects of having a realistic soft Gaussian potential mirror, such as that produced by a repulsive light-sheet, and suitable atomic systems for performing time crystal experiments.
We find that for reflection from a soft Gaussian potential mirror the optimal amplitude of the mirror oscillations ( 75 nm for 39 K atoms and s = 30, /(2t0) = 2.8 kHz) is about an order of magnitude larger, and hence more readily accessible in an experiment, than the optimal oscillation amplitude for a hard-wall mirror. For reflection from a Gaussian potential mirror the trajectories of the atoms are smooth at the reflection position, so that smaller amplitude harmonics are created and, as a result, larger mirror oscillation amplitudes are needed to create a sufficiently large band gap (E/J  10) and stable resonance islands, compared with a hard-wall mirror. Our estimates show that perturbations due to mechanical vibrations transmitted by a typical optical table are negligible when we want to control the motion of the atom mirror with amplitudes of a few nanometers at frequencies of a few kilohertz.
We find that using a 39 K BEC and realistic experimental parameters, it should be possible to create discrete time crystals with sizes in the range s  20 -100. For s  20, the drop height starts to become small ( 80 m for the parameters considered), which may make it difficult to probe the atom density at different fixed positions with high spatial resolution. For s  100, the optimal mirror oscillation amplitude becomes small ( 30 nm for a soft Gaussian mirror for the parameters considered) and the tunnelling times in the absence of interactions start to become long ( 1.3 s), which increases the time needed to perform an experiment compared with the lifetime of a bouncing BEC.
The robustness and stability of these discrete time crystals against small perturbations are summarised in the form of a phase diagram of the detuning parameter h0 against interaction strength g1DN in an accompanying paper in this issue [7].
Time crystals involving resonantly bouncing ultracold atoms provide a platform for investigating a broad range of non-trivial condensed matter phenomena in the time domain. These include Mott insulator-like phases in the time-domain [23]; Anderson localization [6,23] and many-body localization [24] due to temporal disorder; dynamical quantum phase transitions in time crystals [6,25]; many-body systems with exotic long-range interactions [26]; time quasi-crystals  which are ordered but not periodic in time [26,27]; and topological time crystals [28].