Fast atom transport and launching in a nonrigid trap

We study the shuttling of an atom in a trap with controllable position and frequency. Using invariant-based inverse engineering, protocols in which the trap is simultaneously displaced and expanded are proposed to speed up transport between stationary trap locations as well as launching processes with narrow final-velocity distributions. Depending on the physical constraints imposed, either simultaneous or sequential approaches may be faster. We consider first a perfectly harmonic trap, and then extend the treatment to generic traps. Finally, we apply this general framework to a double-well potential to separate different motional states with different launching velocities.

species) or electric fields (for ions), is their broad range of applicability, beyond the very restricted class of atoms with a cycling transition that can be treated by standard laser cooling approaches. The opposite process, launching, is also of much current interest: launching ions with a specific speed is used in particular for their implantation or deposition 28 . Accurately controlled launching can contribute to different quantum technologies such as ion microscopy, those using a controlled "soft landing" of slow ions on a surface, and those controlling the location of defects (NV centers) that have been proposed for sensors and also as the basis of a possible architectures for quantum information processing. Deterministic sources of single cold ions have been proposed and demonstrated 28,29 that limit the position-momentum uncertainty only due to the Heisenberg principle. Our goal here is to control of the velocity, and its dispersion. This is facilitated by the possibility to change the trap frequency along the shuttling. Differential launching of different motional states is also possible as we shall demonstrate with a double well.
While the mathematical framework of this work is equally applicable to neutral atoms or trapped ions, the numerical examples make use of parameters adapted to trapped ions 1, 2 . Invariant-based inverse engineering. Lewis and Riesenfeld 30 noted that the solutions of the Schrödinger equation for a time-dependent Hamiltonian can be written as superpositions of eigenstates of its dynamical invariants. Dhara and Lawande 31 and Lewis and Leach 32 worked out the details for a particle of mass m that evolves according to Hamiltonians of the form ω ρ where F(t) is a homogeneous force, ω π t ( )/(2 ) the frequency of a harmonic term, U an arbitrary function, and α(t) and ρ(t) are auxiliary functions. x and p represent conjugate position and momentum operators of the particle.
The Hamiltonian in Eq. (1) has the quadratic-in-momentum invariant where the dot means time derivative. I satisfies indeed the invariance equation provided the scaling factor ρ and α satisfy the Ermakov and Newton equations, 2 where ω 0 is a constant. For simplicity we choose ω 0 = ω(0). Any wavefunction ψ(t) driven by the Hamiltonian (1) may be written in terms of eigenvectors ψ n of the invariant (2), x space) are the solutions of the auxiliary, stationary Schrödinger equation The physical meaning of α is made evident in Eq. (7) as a centroid for the dynamical wavefunctions that satisfies the Newton equation (5). α is also the center of the potential term ρ α ρ To inverse engineer the interaction between the initial time, t = 0, and a final time t f , we first set the initial and final Hamiltonians. For transport between stationary traps, commutativity is imposed between the Hamiltonian and the invariant at boundary times so that they share eigenstates. Thus the dynamics maps eigenstates of H(0) onto eigenstates of H(t f ) via the corresponding invariant eigenstates, even though at intermediate times diabatic transitions may occur. The commutation of H and I at boundary times implies boundary conditions for α, ρ, and their derivatives. We design these functions to satisfy the necessary boundary conditions, and then, from the auxiliary Eqs (4) and (5) the control parameters ω t ( ) and F t ( ) are found. For launching/stopping processes the invariant and Hamiltonian do not commute at final time in the laboratory frame, but the states may be chosen as eigenstates of the Hamiltonian in the comoving and coexpanding frame.

Results
Dual-task transport in a nonrigid harmonic trap. Let us assume first that the external trap is purely harmonic, i.e., we take U = 0 and is the position of the trap center. Then, the Hamiltonian in Eq. (1) becomes, adding a purely time-dependent term that does not affect the physics to complete the square, The average energy for this system in the nth state (7) is given by For rigid transport 6 , ω is constant and Eq. (4) is trivially satisfied for ρ = t ( ) 1. Here, the goal is to transport a particle a distance d, and additionally change the angular frequency of the trap from the initial value ω 0 to the final value ω ω ω γ The control parameters are the frequency ω(t) and the position of the center of the trap x t ( ) 0 . Figure 1 shows schematically this process. The auxiliary functions α(t) and Now, we may propose ansatzes that satisfy all boundary conditions in Eqs (11), (12) and (13) Fixing the coefficients ρ i and α i to satisfy the boundary conditions, the auxiliary functions become Substituting ρ in Eq. (4), the time dependent frequency in (9) takes the form whereas, from Eq. (5), the transport function (position of the trap center) is that can be now calculated with Eqs (14) and (15). The form of the polynomial for ρ in Eq. (14) is not affected by the transport, so the function for the frequency in Eq. (15) is the same as the one used for pure expansions 4 . Similarly, the form of α t ( ) is not affected by the expansion, but the trap position x t ( ) 0 is different from the one in rigid transport 6 due to the time dependence of the frequency. The dual task protocol is thus not just a simultaneous superposition of recipes for pure expansions and rigid transport but a genuinely different process.
We performed a number of tests to compare the times required by the sequential or dual protocols. In principle, both the sequential and the dual drivings can be done arbitrarily fast, if no limitations are imposed. However, subjected to technical limitations the minimal times may be different. One of the bounds will be to keep the frequency always real, ω > t ( ) 0 2 , since a repulsive parabola may be difficult to implement in some trapping methods. Other natural constraint is to limit the trap position bounded within the "box" [0, d].
We carry out the comparisons for a 9 Be + ion, shuttled over a distance = d 370 μm in a trap with initial frequency ω π = /(2 ) 2 0 MHz expanded by a factor of 10, γ 2 = 10. For these parameters and polynomial ansatzes, the simple expansion has a minimal final time μs, below which imaginary frequencies appear. Note that this will also be the limit time before getting imaginary frequencies in the dual process, as Eq. (15) gives exactly the same evolution for ω in a simple expansion or a dual process. For rigid transport, carried out before the expansion at the highest trap frequency, the limit time is μs before exceeding the box. Thus, the total minimal time for the sequential protocol is = .  Table 1.
If the only restriction is to keep real frequencies, dropping the limitation on the domain of the trap position, the minimal final time is in principle μs for both the sequential and dual protocols, but in the sequential protocol this is a really challenging limit since the transport should be done in zero time. In both protocols the transport function exceeds the box [0, d]. In Fig. 2 we compare the ratio between the exceeded distance beyond [0, d] and d for the sequential and the dual drivings, with respect to the total process time. The exceeded distance is defined in terms of the maximum (x 0 max ) and the minimum (x 0 min ) values of the trajectory as . The figure shows that the dual protocol is much more robust. As the minimal possible time is approached, the ratio in the sequential protocol increases dramatically. In contrast, the ratio in the dual protocol is very stable, making potentially easier to perform the dual protocol for short times.

Dual-task launching in a harmonic trap.
We study now launching processes where the frequency of the trap is time dependent (stopping processes may be designed by inverting the launching protocols). If the ion is to be launched adiabatically with a very precise velocity, the trap should have a small final frequency to minimize the uncertainty. STA protocols will achieve the same goal in a shorter time.
The order of the sequence plays a relevant role to compare sequential or dual launching protocols. In the previous subsection, when the final state is at rest, the sequential protocol may be faster than the dual one when transport is done first, then the expansion. For the launching process, the only meaningful sequential process implies to expand first, and then to transport, but a small trap frequency does not enable us to implement a fast MHz.

Figure 2.
Ratio of the exceeded distance x e and the transport distance d for the dual (blue circles) and sequential (red diamonds) non-rigid harmonic tranport protocols, for final times that do not require imaginary frequencies. Parameters used are = d 370 μm, γ = 10 , and ω π = /(2 ) 2 0 MHz.
launching. It is therefore useful to combine the time dependences of frequency and displacement of the trap in a dual protocol.
The boundary conditions to be imposed for this launching protocol are the same as in Eqs (11), (12) and (13), except that the first derivative of α at final time, is now the final launching velocity v f , Additionally, boundary conditions are imposed on the third derivative of α, where (n) means nth derivative, so that, according to Eq. (16), the velocity of the trap  x 0 and the velocity of the wave packet α  are the same at the boundary times. In order to satisfy the additional boundary conditions, we consider a higher-order polynomial ansatz for α, α α = ∑ = s i i i 0 7 , which upon fixing parameters to satisfy all boundary conditions gives Boundary conditions for ρ are the same as in the previous subsection, so the same ansatz used in Eq. (14) is valid here. Thus, the evolution of the frequency is given in Eq. (15), while the evolution of the trap position is found substituting Eqs (15) and (19) Table 2. Here the dual protocol clearly outperforms the sequential one. A control possibility we have for the dual process, which does not exist for the sequential one, is to design the launching with a given constant expanding velocity, i.e., we impose α =  t v ( ) f f as before and also ρ ε = .  t ( ) (20) f Additionally, boundary conditions may be imposed on the third derivative,  With the evolutions considered in this section, either for the expanding or the nonexpanding launching, a state which is initially an eigenstate of H(0) will not become an eigenstate of the Hamiltonian H t ( ) f . Instead, the state of the system at the end of the process is, see Eq. , can be shown to correspond to the Hamiltonian eigenstate in the moving and expanding reference system of the trap (see Methods).
The expectation value of the velocity for ψ x t ( , ) n f is v f and its dispersion is Much smaller spreads can be achieved by the dual protocol, but γ cannot be made arbitrarily small in a fixed process time. In particular, the requirement of keeping the frequency real implies the bound 9,14 γ ω A constant electric field has its own, different limitations, in particular, with constant acceleration the time is fixed as = t d v 2 / f f to reach a given final velocity v f in a distance d.
Dual-task shortcuts in an arbitrary trap. Now, we extend the analysis to move and expand or compress an arbitrary confining potential from U x To stay within the family of processes described by Eq. (1), so that invariants are known, we must impose that the harmonic and linear terms depending on ω 2 and F vanish at the boundary times. We thus set ω =0 0 hereafter. If initial and final potentials are at rest, by imposing commutativity between the Hamiltonian (1) and the invariant (2) and continuity at the boundary times, we get the same boundary conditions as in Eqs (12) and (13). We must also impose the boundary conditions in Eq. (11) for the system to be displaced and expanded or compressed, noting that now the constant γ is not related to ω 0 . With these boundary conditions, using the auxiliary Eqs (4) and (5), That is, the only non vanishing term of the potential at the boundary times = We design the functions α t ( ) and ρ t ( ) polynomially as before, so that they satisfy all boundary conditions, and introduce them in the auxiliary equations to inversely obtain the control parameters. The auxiliary functions can be the same as in Eq. (14). Substituting ρ in Eq. (4), 2 and substituting this result and α in Eq. (5) we get In other words, the protocol requires auxiliary time-dependent linear and quadratic potential terms apart from the scaled potential . This protocol is of course technically more demanding than the one designed for the simple harmonic trap, because of the need to implement and control all terms (linear, quadratic, and U-term) of the Hamiltonian (1).
The results can be extended to a launching scenario. To be specific, we shall consider the double well, a paradigmatic quantum model that has been used, for example, to study and control some of the most fundamental quantum effects, like interference or tunneling. With the advent of ultracold-atom-based technology, it also finds applications in metrology, sensors, and the implementation of basic operations for quantum information processing, like separation or recombination of ions 24 , as well as Fock state creation 33 , and multiplexing/demultiplexing vibrational modes 34,35 . Here, we explore the possibility of using it for differential launching of vibrational modes. We in σ-space, and effective angular frequency Limiting the linear coefficient as µ β < m (2 / ) 1/2  , the first excited and ground states lie in different wells 34 . We want to implement a protocol with a nonzero final expansion velocity, such that the effective launching velocities for ground and first excited states are different so that they separate further. We choose the boundary conditions for the auxiliary functions in Eqs (11) and (13) and for the first derivatives Here the boundary conditions for the third derivatives [Eqs (18) and (21)] are not necessary. With these conditions, using fifth-order polynomial ansatzes, the auxiliary functions are finally given by These parameters directly give us the evolution of the potential term ρ α ρ The auxiliary harmonic and linear terms in the total Hamiltonian (1) are found by substituting α and ρ in Eqs (24) and (25), respectively. The resulting potential (the sum of the three potential terms in Eq. (1)) is depicted in Fig. 3 as a function of α − x d ( )/ , with α depicted in Fig. 4. For this evolution, we can calculate the average final velocity of the ground states in each well, and the final dispersion,   Fig. 3. which is the same in both wells, as the effective frequency is also equal. Details of these calculations are displayed in Methods. Choosing the parameters so that − > ∆ + − v v v 2 , guarantees that the wave packets of each well will never overlap.

Discussion
In this paper, we have used the invariant-based inverse-engineering method to design shortcuts to adiabaticity for nonrigid driven transport and launching. Shortcuts for a harmonic trap are designed first, and then the analysis is extended to an arbitrary trapping potential. Compared to rigid transport 6 , nonrigid transport requieres a more demanding manipulation, but it also provides a wider range of control opportunities, for example to achieve narrow final velocity distributions in a launching process, suitable for accurate ion implantation or low-energy scattering experiments. A further example is the possibility to launch the ground states of each well in a double well with different velocities. In a previous work 34 processes to separate the ground and the first-excited states of a harmonic trap into different wells of a biased double well using STA were described. The processes discussed here can be applied to different systems such as neutral atoms in optical traps, or classical mechanical oscillators, for which, mutatis mutandis, most of the results apply.  (40) and (42) the expressions for σ ± and Ω, Eqs (27) and (28), and the final values of the auxilary functions and their derivatives in Eqs (11) and (29).