Realistic Shortcuts to Adiabaticity in Optical Transfer

Shortcuts to adiabaticity (STA) are techniques allowing rapid variation of the system Hamiltonian without inducing excess heating. Fast optical transfer of atoms between different locations is a prime example of an STA application. We show that the boundary conditions on the atomic position, which are imposed to find the STA trajectory, lead to highly non-practical boundary conditions for the optical trap. Our experimental results demonstrate that, as a result, previously suggested STA trajectories generally do not perform well. We develop and demonstrate two complementary methods that solve the boundary conditions problem and allow the construction of realistic and flexible STA movements. Our technique can also account for non-harmonic terms in the confining potential.

the interaction of atoms with a far-off-resonance laser light introduces negligible dissipation and can be effectively described as a conservative potential for the atoms [28][29][30], with potential depth that is proportional to the laser intensity. Ultracold atoms can be trapped in the vicinity of potential minimum, which for the lowest energy optical mode, namely a Gaussian beam, is at the focal point ("waist"). Rapid changes may be desired in the trap shape [11,20,21,23] or position [14,15,17,18,31]. Optical transfer of ultracold atoms can be used to move atoms between different sites or implement quantum gates [32][33][34]. Shorter transfer duration is advantageous considering the finite coherence time of any experimental system, but at the same time, it might have an adverse effect on the fidelity of single operations [34]. STA techniques applied to optical transfer can alleviate this conflict.
In a practical implementation of invariant-based STA, a problem arises: the boundary conditions are given for the atomic trajectory while the experimental control is over the position of the trap, which in turn is derived from the atoms position through the equation of motion. This leads to boundary conditions for the trap which may be very difficult to satisfy in reality. For example, it may require the trap and atoms to start at the same position, with the trap having an initial velocity while the atoms are at rest. Here we study experimentally non-adiabatic optical transfer with ultracold atoms and find that indeed this issue almost always harms the performance of known STA trajectories. We show that in order to lift conflicting boundary conditions, it is necessary to increase the number of degrees of freedom in the trajectory. We describe and demonstrate two complementary approaches to achieve this: One, by an addition of a "correction" spectral component at the trapping frequency with a specific phase and amplitude. This method can be applied to any existing trajectory. Two, by using a polynomial trajectory with high enough order and incorporating the boundary conditions by a correct choice of the coefficients.
The structure of this paper is as follows: In section II we briefly introduce the STA formalism, the required boundary conditions and the use of the final sloshing mode as a measure of undesired excitations added by the movement. In section III, we introduce the apparatus, and describe the experimental sequence and probing technique. We then present in section IV our experimental study of several known STA trajectories for optical transfer. Our results show that almost always the movement results in considerable heating. This leads us in section V to develop and demonstrate two new methods to construct proper STA trajectories. We conclude and give our outlook in section VI.

II. SHORTCUT TO ADIABATICITY IN TRANSPORT PROBLEMS
The optical dipole potential induced by a Gaussian beam can be written as [28]: where z denotes the atoms coordinate in the laboratory frame of reference and z ∪ the position of the potential minimum, both of which will later become time-dependent, r is the radial distance from the beam path, U 0 is the maximal potential depth, and z R = πσ 2 /λ is the Rayleigh range, with σ the waist radius and λ the wavelength. We expand the potential to fourth order in powers of z − z ∪ which leads to the following Hamiltonian (up to a constant): where ω 0 = 2U 0 /mz 2 R 1/2 is the harmonic axial trapping frequency. In our experiment we induce motion along the axial direction and in addition the radial trapping frequency is much higher than the axial one (ω r /ω z ≈ 90). Hence, we average out the radial motion and retain only the axial dependence in the Hamiltonian. This averaging is done over a period of ω −1 r for which the axial movement is negligible, and the terms 1 − 2 r 2 σ 2 e −2r 2 /σ 2 and 1 − 4 r 2 σ 2 + 2 r 4 σ 4 e −2r 2 /σ 2 in Eq. (2) are replaced by their ensemble averaged values. At low temperatures the atoms lie very close to the potential minimum and r 2 /σ 2 1, where · denotes the ensemble average. Thus, we expand the exponent, retaining terms up to first order, and write the Hamiltonian as: where we also averaged out the fourth order of the axial displacement z−z ∪ , as (z − z ∪ ) 2 /z 2 R 1 for this temperature regime.
Since the temperature in our experiments is very low, in most of what follows we will assume that the two last terms in the rightmost parentheses can be omitted, and we are left with a harmonic Hamiltonian in the axial direction: However, as we show in section V A 2, at higher temperatures the non-Harmonic terms become important, and we account for them when constructing the STA trajectories.
We impose the following boundary conditions on the atoms motion: That is, the atomic movement of d meters during t f seconds begins and ends with zero velocity and acceleration. As was shown by Lewis and Riesenfeld [35], with these boundary conditions, and for a general class of Hamiltonians that includes the one in Eq. (4), there exists a dynamical invariant I (t) that satisfies: which can be written explicitly in the harmonic case as: The third equalities in Eq. (5a) and in Eq. (5b) guarantee that [ The state which we would like to conserve can therefore be written at t = 0 as a superposition of I eigenstates, and since I is invariant under the motion, their state at t f is the same as at t = 0.
However, the boundary conditions Eq. (5) are given for the atomic position, while experimentally the control is over the trap position. The latter can be derived from the former using the equation of motion: The boundary conditions Eq. (5a) imply that z ∪ (0) = 0 andż ∪ (0) = ...
z (t f ) are generally not zero, they require initial and final non-zero velocities for the trap. The first condition is very hard to implement in most physical realizations, as it requires that the atoms are initially at rest and that their position coincide with the trap minimum but there is a finite initial velocity to the trap. In most cases, both the trap and the atoms are at rest before the transport commences. The second condition means that even if the atoms are brought to the adiabatically connected state at the end of the motion, since the trap is still moving it will soon induce atoms excitation. Evidently, we need to require that the trap will also be at rest at the beginning and at end of the transport.
This implies: Most STA trajectories do not generally satisfy this extra set of boundary conditions for ... z (t) and, as we later demonstrate, their experimental implementation might leave excess energy in the cloud The only motion excited mode in a harmonic potential is the center-of-mass oscillation (sloshing mode). Non-harmonic terms in the Hamiltonian and interactions between the atoms couple between the sloshing mode and higher order modes which eventually lead to increase of temperature.
Hence, the sloshing mode and temperature are the observables one is required to measure when characterizing realistic implementations of STA in optical transport. As an example, we present in Fig. 1 a series of absorption images of atomic clouds taken during and following a non-adiabatic transport. The upper panel presents the result of a trajectory originally suggested in [15]. Considerable sloshing is apparent after the end of the trap motion. This is because the trajectory does not respect the third boundary condition given in Eq. (9). In contrast, the lower panel shows the same trajectory with our spectral correction method applied. The final state shows no detectable sloshing and minimal increase in temperature.
In the harmonic case, the sloshing mode amplitude can be calculated by integrating over the response function [41]: This is merely the Fourier component at the trap frequency of the trap velocity trajectory. For realistic trap trajectories that maintain Eq. (9) and Eq. (5a), the conditions in Eq. (5b) translate into zero ultimate sloshing amplitude. This can be used as a guideline for constructing STA trajectories: they should result in zero final sloshing.

III. THE EXPERIMENTAL APPARATUS
The system is composed of three interconnected vacuum chambers. In the first chamber, a two-dimensional magneto-optical trap (MOT) [42] generates a stream of cold fermionic potassium atoms that fly through a narrow nozzle to the second chamber. There, the atoms are captured and cooled in a three-dimensional dark SPOT MOT [43] on the D 2 line and get farther cooled using gray molasses on the D 1 line [44]. Then, we optically pump the atoms and load them into a QUIC magnetic trap [45], where we perform forced microwave evaporation. Next, around 25 · 10 6 atoms at T /T F ≈ 4.5, with T F the Fermi temperature, are loaded into a far-off-resonance optical dipole trap of λ = 1064 nm. This trap is made of a single Gaussian beam, with a waist of 39 µm and power of about 2.5 W. The atoms are then transported [46] in approximately a second to the third chamber. This is done by moving a single lens which is a part of an optical relay system that creates the optical trap. The actual movement is performed with an air-bearing translation stage, to reduce vibrations and heating of the ensemble, as depicted in Fig. 2. Upon arrival at After the cloud reaches equilibrium conditions with a negligible sloshing of a sub-micron amplitude, we execute the STA trajectory by moving again the lens mounted on the air-bearing stage.
The stage specifications guarantee that the position of the trap minimum is accurate to within 1 µm, and the trapping frequencies are constant to within 1.5% during the movement. At each point along the translation we can stop and record the sloshing mode. This is done by waiting for some duration, then abruptly shutting off the trap, letting the atoms expand ballistically and then recording the atomic density distribution using absorption imaging. From these images we can extract the number of atoms, the center-of-mass position and T /T F . The sloshing mode can be reconstructed by fitting a series of such images taken at different waiting times (see Fig. 1) with a decaying sine. A typical dataset together with the fit is shown in Fig. 3, that depicts the resemblance of the center-of-mass motion to a decaying oscillation. The typical decay time is of the blue curve designates the extracted sloshing amplitude 68% certainty (1σ) that will be presented as the amplitude error in following graphs, and the width of its green background represents the error in the other fitting parameters.
obtained from eleven different durations, for each we averaged over three repetitions. Note that the time-of-flight expansion before imaging acts effectively as a magnification of the sloshing mode amplitude relative to in situ. For the 12 ms expansion employed here, the magnification is ×1.09.

IV. EXPERIMENTAL TEST OF THREE UNCORRECTED STA TRAJECTORIES
We have implemented three non-adiabatic trajectories that maintain Eq. (5a) and Eq. (9) but not necessarily Eq. (5b). Their velocity profiles are depicted in the insets of Fig. 4. The first trajectory is a Sine with a velocity profile given byż ∪ (t) = πd 2t f sin πt t f . It satisfies Eq. (5b) only for t f · f 0 = 1 2 + n where f 0 = ω 0 /2π and n is an integer n ≥ 1. The second trajectory has a Triangular velocity profile with a constant acceleration and deceleration given by 4d/t 2 f [14]. It satisfies Eq. (5b) only for t f · f 0 = 2 · n with integer n ≥ 1. The third trajectory is a Polynomial, which generally can be written for the atomic position as z (t) = d ∞ n=1 a n t t f n . The lowest order polynomial to respect the boundary conditions in Eq. (5) is given by [15]: Each of these trajectories was executed for a total movement of d = 1.29 mm. The resulting sloshing amplitudes A (t f ) for three non-adiabatic durations t f are depicted in Fig. 4. For reference, we also plot the calculated sloshing amplitude as given by Eq. (10). There is a satisfactory agreement between the experiment and theoretical calculations. As expected, zero sloshing is obtained only at specific t f · f 0 values. This places a strict constraint on potential applications of STA. Moreover, it puts a lower limit on the duration of the trajectory which is on the order of In what follows we develop two methods to construct STA trajectories that satisfy all eight boundary conditions. In principle, for an ideal harmonic potential these trajectories can be constructed for any desired duration above the fundamental limits [47]. A more practical limit on the shortest possible trajectory stems from the finite trap depth: a faster movement coherently drives the population during the motion via higher energy levels.

V. REALISTIC AND FLEXIBLE NON-ADIABATIC TRAJECTORIES
For a trajectory to be realistic we require that both the trap and atoms will start and end at rest. As we explained earlier, this implies boundary conditions Eq. (5) and Eq. (9). For a trajectory to be flexible, we require that it could be constructed for a wide range of distances d and durations Let us denote the original trajectory byz ∪ , then the "corrected" trajectory is given by: with A (t) and φ 0 being the amplitude and phase of the correction term, respectively. Our method thus added two new degrees of freedom: the amplitude and phase of the new spectral component. If we plug Eq. (12) into the boundary conditions, we get that both A (t) and its first derivative needs to vanish at the starting and ending points. We have chosen to ramp up the correction amplitude as A t < 0.5f −1 0 = A 0 sin 2 ω 0 2 t , and its time-reversed version when decelerating towards the end. Other choices are also possible. According to Eq. (8) and Eq. (9), the atomic trajectory then satisfies: The boundary conditions Eq. (13a) are automatically fulfilled when there is no initial sloshing, as Eq. (5a). There are two more conditions that the atomic trajectory needs to fulfill given by Eq. We have tested experimentally our method with polynomial and sinusoidal trajectories. To obtain the minimal sloshing amplitude, we scanned the correction parameters around the calculated optimal values. The measured sloshing amplitudes (blue squares) and phases (gray circles) are plotted in Fig. 6 for the polynomial (upper panel) and sine (lower panel) trajectories. A clear minimum in the sloshing amplitude can be observed in both cases. At the optimal correction amplitude, the measured excess energy in the sloshing mode due to the non-adiabatic trajectory is consistent with zero. In contrast, the uncorrected trajectories (A 0 = 0) display a substantial sloshing. The atomic cloud dynamics in the non-corrected and corrected (A 0 = 105 µm) polynomial trajectories are presented in Fig. 1 (a) and (b) respectively. When measuring the temperature after the center-of-mass motion has ceased, we do find an increase of about 200 nK for all non-adiabatic transports (corrected and uncorrected). This is probably due to high frequency errors in the actual executed motion that couple through the anharmonic terms to higher vibrational modes of the cloud. We also measure an additional increase in temperature due to the excess energy in uncorrected trajectories compared to corrected ones.
Theoretical calculations shown as ribbons with matching colors in Fig. 6 agree reasonably well with the experiments at low sloshing amplitudes, but at higher values they deviate, probably due to contributions from non-harmonic terms in the potential. For both types of trajectories, the sloshing mode phase jumps sharply by π when crossing the optimal correction amplitude, as expected from an over-compensated driven harmonic oscillator. We find that the measured optimal correction phase deviates from the theoretical calculation by 23 • and 41 • for the polynomial and sinusoidal trajectories, respectively. This is most likely due to experimental imperfections in the execution of the trajectory, to which the phase is most sensitive. This exemplifies another advantage of our method: it can correct for experimental imperfections easily by parameters tuning.

Anharmonic potential
Even in the anharmonic case, it is still true that the sloshing mode is the first to be excited from a rapid shift of the trap. Hence, nullifying the sloshing amplitude will provide us with a trajectory very close to optimum. According to Eq. (3), the anharmonicity can be incorporated into an effective harmonic frequency ω 0 = ω 0 1 − 4 In the case of a Gaussian beam, the frequency decreases with increasing temperature, an effect referred to as "softening". To study how anharmonicity affects the STA, we repeated the experiments of Fig. 6(a) with temperature increased by ×1.6 using parametric excitation [28]. The axial in situ variance of the atomic cloud density (z − z ∪ ) 2 before the transport is about 229 µm and 272 µm for the cold and hot clouds, respectively. A good estimate for the radial variance r 2 can be obtained using the known aspect ratio of the trapping frequencies in the axial and radial directions. Using this data together with Eq. (3), we calculate the ratio between the effective harmonic frequencies in the two experimental conditions and obtain ω cold ω hot = 1.027. For this, we can numerically find a new optimum value for correction amplitude A 0 . The correction phase φ 0 , however, is unaffected by this variation of the effective frequency.
In Fig. 7 we present the measured sloshing amplitudes for both hot (red triangles) and cold (blue squares) clouds as function of A 0 together with theoretical calculations based on ω cold ω hot and Eq. (10) (ribbons in matching colors). We find that using the same correction frequency and phase we get a amplitudes at t f versus the harmonic correction amplitude, A 0 , for cooler (blue squares) and warmer (red triangles) atomic ensembles. As the temperature increases, the atoms experience more the non-harmonic terms and the effective harmonic trapping frequency decreases. The blue data is the same as in Fig. 6(a). The temperature and number of atoms are T = 320 (30)  clear minimum with respect to the correction amplitude also for the warmer case, correlative to the calculated amplitudes for ω 0 = 2π·7.16 (15) Hz and ω 0 = 2π·6.98 (15) Hz for the cooler and warmer cases, respectively, as calculated from the extents of the clouds before the experiment. In addition, the sloshing frequencies as obtained directly from the measured data following the transport agree with this shifted value of ω 0 . This demonstrates that the spectral correction technique can account for non-harmonic terms by using an effective trapping frequency.

B. Method II: Septic polynomial trajectory
In the second method, in order to comply with all of the eight boundary conditions, we use a polynomial trajectory of the seventh order, written in terms of the normalized time as: This path for the atoms respects the invariant necessitated boundary conditions and its associated trap trajectory results with zero velocities at motion ends, so it is feasible to be implemented experimentally. The trap trajectory is then given by Eq. (8), z ∪ (t) = z 7 (t) +z 7 (t) /ω 2 0 . Due to this dependency of the desirable path on the trapping frequency ω 0 , one is required to provide the later with great accuracy in order to respect the boundary conditions and accomplish the transport with zero sloshing amplitude. In Fig. 8 we present the sloshing amplitudes following such a trajectory where we scan the value of the frequency parameter. Indeed, for the correct value of the frequency parameter we observe a sloshing amplitude consistent with zero. The right-hand side data points are in fact the case where we used the trajectory z 7 (t) directly for the trap itself, which results in a considerable excess energy.

VI. DISCUSSION AND OUTLOOK
While trying to implement previously suggested non-adiabatic trajectories in optical transfer, we have found that in most cases they leave excess energy in the cloud, which we measure as sloshing of the center-of-mass after the trap has stopped. We have identified the source for this behavior in a gap between the required boundary conditions for the trap and realistic constraints.
In reality, the trap holding the atoms is at rest before and after the movement. Formerly proposed STA trajectories, however, require the trap to have some initial and final velocity while the atoms are at rest. In principle, it is physically possible, for example by working with two precisely synchronized traps, one holding the atoms at rest and the other moving, then switching between them abruptly when they exactly spatially overlap as the motion commences. Clearly, it is not a practical solution. Hence, we introduced two new boundary conditions requiring the trap to be at rest before and after the motion. We also presented two methods to construct STA trajectories regime where the transfer duration is on the order of the inverse trapping frequency. We have also shown that our technique can account for non-harmonic terms in the trapping potential. Thus, our approach is both realistic and flexible, and we anticipate it will be useful in the wide range of applications that can benefit from STA.