Shortcuts to adiabaticity for an ion in a rotating radially-tight trap

We engineer the fast rotation of a quantum particle confined in an effectively one-dimensional, harmonic trap, for a predetermined rotation angle and time, avoiding final excitation. Different schemes are proposed with different speed limits that depend on the control capabilities. We also make use of trap rotations to create squeezed states without manipulating the trap frequencies.

In this paper we study rotations of a single ion as depicted in figure 1. Our aim is to inverse engineer the timedependence of the control parameter(s) to implement a fast process, free from final excitations. We assume for simplicity that the ion is trapped in a linear, harmonic trap, tightly confined in a radial direction so that it moves effectively along a one-dimensional (1D) axial direction, hereafter 'the line'. The trapping line is set horizontally and is rotated in a time t f up to an established final angle (q p = 2 f in all examples) with respect to a vertical axis that crosses the center of the trap. Such an operation would be useful to drive atoms through corners and junctions in a scalable quantum processor [20,21]. It is also a first step towards the more complicated problem of rotating an ion chain [17,20,22], which would facilitate scalability in linear segmented traps, and be useful to rearrange the ions, e.g. to locate a cooling ion at the right position in the chain [17].
We shall first find the classical Hamiltonian. Let s denote a point on the line. s may take positive and negative values. A time dependent trajectory s(t) has Cartesian, laboratory frame components = x x s t , ( ), = y y s t , Note that the angular velocity of the rotation q must be real but could be negative, whereas w 2 may be positive or negative, making ω purely imaginary in the later case. Unless stated otherwise, the following physically motivated boundary conditions are also assumed: the initial and final trap should be at rest, and we also impose continuity of the angular velocity where the last line follows from the second one using equation (

( )
We will from now on work with this quantum Hamiltonian (possibly with a more general potential) and corresponding quantum states. It represents formally a harmonic oscillator with time-dependent frequency, but there are significant differences with an actual harmonic oscillator when the inverse engineering of w t ( ) is considered. For an actual harmonic oscillator a fast and safe expansion or compression in a time t f should take the system from an initial value to a final value of ω without final excitation, in principle without further conditions. By contrast, in the rotation process, according to equation (6), the initial and final effective frequencies are the same, but the conditions in equations (4) and (5) must be satisfied. This implies an integral constraint on ω ò ò where the square root branch should be chosen to satisfy continuity. One further difference is that in a physical expansion/compression w t ( ) is controlled directly whereas in the rotation there are several options. If w 0 is constant, only q ṫ ( ) is controlled, so that w t ( ) is an 'effective' frequency. In general both w 0 and q could be controlled as time-dependent functions, see the next section. As for the final excitation, the expression for the energy of a state that begins in the nth eigenstate of the trap at rest can be found making use of the Lewis-Riesenfeld invariants [12,23] see the corresponding time-dependent wave function in the appendix Here b is a scaling factor, proportional to the width of the invariant eigenstates, that satisfies the Ermakov equation Figure 1. Schematic representation of the rotation process. The ion is confined along a line (where it is subjected to an effective onedimensional-longitudinal-potential), which is rotated by an angle θ up to q f in a time t f , so that the final state is not excited. 5 This is easily generalized for a potential U(s), not necessarily harmonic, as To avoid any final excitation, it is required that for the initial conditions . The boundary conditions for b and equations (4)-(6) imply that = H H t 0 f ( ) ( ) commutes with the corresponding Lewis-Riesenfeld invariant [23], so that the nth initial eigenstate is dynamically mapped onto itself (but rotated) at time t f . In equations (10) and (11) both the excitation energy and the wave packet width are mass independent, so that inverse-engineered rotation protocols will be independent of the species. In the following sections we shall analyze different methods to perform the rotation without final excitation.

Control of trap frequency and angular velocity
If both the trap angular frequency w 0 and the angular velocity q are controllable functions of time, a simple family of solutions to the inverse problem is found by setting a q ṫ ( ) that satisfies equations (4) and (5), and compensating the time dependence of q 2 with a corresponding change in w t ( ) ( ) remains constant during the whole process. From the point of view of the effective harmonic oscillator 'nothing happens' throughout the rotation, so that the effective state remains unexcited at all times.
We may apply the Lewis-Leach theory of quadratic in momentum invariants [24,25] to extend the above results to arbitrary potentials 6 . The family of Hamiltonians where U is an arbitrary function, and Ω depends on time, has the invariant where p = bp mbṡ , and W 0 is a constant, provided the Ermakov equation as a constant for all times, it follows that W = t 0 ( ) . However, as we saw in the previous section, the rotation of a line with the potential U(s) produces in the line frame a centrifugal term q s m 2 2 2 . To cancel the total harmonic term, we have to add to the trap potential a compensating harmonic term, w s m 2 . The resulting Hamiltonian and invariant (in this case they are equal) are simply i.e., time independent. No excitation occurs at any time in spite of the fact that a rotation is taking place. For some applications it may be interesting to consider in equation (13) the more general case in which b depends on time (for example to achieve a squeezed state), and w w q = -2 c 2 2 , corresponding to an auxiliary harmonic term and the centrifugal term. The inverse engineering in this case proceeds by designing q t ( ), so that , and then b(t) obeying the boundary conditions ) that guarantee the commutation between invariant and Hamiltonian at boundary times. Once θ and b are set we design the auxiliary harmonic term considering, as before, W = 0 0 in equation (15) 6 The theory was first formulated for classical systems in [24] but is applicable to quantum systems as well [25]. Incidentally this means that the rotation protocols designed in this paper-in this and the following sections-are valid for classical particles as well. The difference appears only when considering which states are valid or not for classical and quantum particles, e.g., when using phase-space formulations of quantum states and classical ensembles.
The auxiliary harmonic term vanishes at both boundary times according to the boundary conditions imposed on b and q . In fact W 2 vanishes as well at the boundary times so that before and after the rotation the atom is confined only in the potential U(s). This type of protocols, where both the rotation speed and the potential have to be controlled (the latter in space and time) may be quite demanding experimentally. In the rest of the paper we shall assume the simpler scenario in which only the rotation speed q is controlled, and the trap potential is purely harmonic with constant angular frequency w 0 .

Bang-bang
It is possible to perform rotations without final excitation satisfying equations (4) and (5) keeping q constant or piecewise constant. Here we consider the simplest one-step case Note that equations (5) and (6) are only satisfied now as one-sided limits. A bang-bang approach may admitedly be difficult to implement because of the sharp changes involved, but it sets a useful, simple reference for orders of magnitude estimates of rotation speeds which may be compared to smoother approaches that will be presented later. Integrating q we find q = ct .

Optimal control by Pontryagin's maximum principle
While the previous bang-bang method with just one time segment provides a simple guidance, we are also interested in knowing the absolute time minimum that could in principle be achieved (even if the 'optimal' protocol ends up being hardly realizable). Unlike ordinary expansions/compressions, the shortest time protocol for bounded control is not of a bang-bang form. To find it we first rescale the time with w 0 by setting s w Then, we can write a control system describing the Ermakov equation (15) and the constraints in (4)- (6), and formulate the time-optimal control (OC) problem for rotation of a quantum particle on a line as Note that we assume that the boundary conditions for u at t=0 and = t t f can be fulfilled by the use of a sudden switch.

Unbounded control
We apply the Pontrygin's maximum principle [26] to solve the time-OC problem (30), where the Hamiltonian is given by ( ( ) ( ) ( )) ( ), which might be an indication that the system is not controllable. Figure 2 (a) shows the time evolution of u for this case following equation (33) but forcing it to be 0 in the boundary times.

Bounded control
Because the Hamiltonian (32) is quadratic in u, the optimal control that minimizes H is of the form ( ) The bounded time-optimal control and the resulting optimal trajectory are illustrated in figure 2 (b). The minimum (dimensionless) time that completes the desired rotation is T=11.9984 and the calculated final state following the optimal control is = x T x T x T , , 1.0083, 0.0382, 1.5708 ], from (30) we see that q > 0 , and hence the rotation is always forward. In this case, x 3 reaches the desired q p = 2 f at s = 11.9028, and the control is turned off. Then, the states x 1 and x 2 are oscillating to reach the desired terminal state 1, 0 ( ). Figure 2 (b) shows the time evolution of u for this solution.

Smooth inverse engineering
An alternative inversion route that provides smooth solutions is depicted in the following scheme First, q t ( ) is designed to satisfy equations (4) and (5) with some free parameters. The corresponding q and final energy are calculated, and the parameters are changed until the minimum energy (and excitation) is found.
A convenient choice for θ is a fifth order polynomial ansatz q = å = a t t n n n n 0 5 f . In order to satisfy the boundary conditions in equations (4) and (5)  ) . The other two parameters, a a , 4 5 , are left free in order to satisfy the remaining two boundary conditions in equation (12) and suppress the final excitation energy. In practice we solve numerically equation (11) to find the final energy (10) for each pair a a , 4 5 , and use MATLAB's 'fminsearch' function to find the values of the free parameters that minimize the final excitation energy.
In figure 3 the values of the free parameters that result from this process are given, and in figure 4 we depict the corresponding excess energy with respect to the ideal target state (as in previous examples, w p = 2 2 0 ( ) MHz). Vanishing residual excitations are found for times shorter than half an oscillation period up to a timẽ t 0.23 f μs, not much larger than the unbounded-optimal-control minimum of 0.18 μs. Figure 5 depicts the difference between the ideal value of b t f ( ) and the actual value, and makes evident the sharp change that marks the shortest time for which a solution exists. Since we have limited the possible solutions by imposing a functional form of the function q t ( ), this time is larger than the one found via OC. Note also that the shortest final time is slightly better than the one provided by the simple bang-bang protocol. Table 1 summarizes the results.      is the displacement operator. Note that the phase at t f , a arg(˜), is controllable by means of the gfunction that depends on the process history, whereas the squeezing parameter g 1 is controlled by the imposed boundary condition. If necessary, a controlled tilt of the squeezed state in phase space is easy to achieve by letting it evolve, after its formation at t f , in the fixed, non-rotating trap.

Wave packet squeezing
As a simple example let us consider the generation of squeezed vacuum states starting from the ground state of the initial trap, so that a = 0. To design the squeezing process we may follow a similar procedure as in the previous section, but minimizing the cost function Since, due to the centrifugal force during the rotation, the wave packet tends to spread first, the squeezed states with g > 1 may be achieved in shorter times than the ones needed without squeezing in the previous section. Figure 6 depicts the free parameters that optimize a rotation with a final squeezed state for the same parameters in the previous subsection, but g = 3 2 , and figure 4 the excess energy with respect to the target state. The excitation in a process with a final moderate squeezing is smaller than for the simple rotation without squeezing. Figure 5 depicts the difference between the target value of the function b (proportional to the width of the wavepacket) and its actual value at final time for rotations without and with squeezing. Again, the minimizations change suddenly to a different solution that cannot satisfy the conditions at a critical time, see also figures 3 and 6.

Discussion
We have worked out different schemes to perform fast rotations of a 1D trap without any final excitation of the confined particle, which we have considered to be an ion throughout but could be a neutral particle as well by setting the proper trapping interaction. Apart from excitation-free rotations it is also possible to generate squeezed states in a controllable way. For an arbitrary trap, the fast processes could in principle be performed in an arbitrarily short time if an auxiliary harmonic potential with time dependent frequency could be implemented. In a simpler setting, where only the rotation speed may be controlled, the rotation time cannot be arbitrarily short, as demonstrated by inverse engineering or bang-bang approaches, and confirmed by OC theory. Bang-bang and OC protocols provide useful information and time bounds but are difficult to implement experimentally due to the sudden kicks in the angular velocity of the trap. Smooth protocols designed by invariant-based inverse engineering have also been worked out. They achieve negligible excitations for times close to the minimum times given by OC theory.
The analysis may be generalized for a two-dimensional (2D) trap but it becomes considerably more involved [27] and will be considered separately. The 1D approximation used here will be valid for total energies well below the transversal confinement energy w =Ê . For the shortest final times considered in our simulations, excitation energies are never larger than w 2 0 so that w ŵ  0 would be enough for their validity. Rotations are elementary manipulations which together with transport, splitting, and expansions, may help to build a scalable quantum information architecture. In particular, they provide a mechanism for connecting sites by changing transport directions in 2D networks. Rotations have been demonstrated experimentally for trapped ions [17] and improving the capability to control the parameters involved is feasible with state-of-theart trapped-ion technology. To extend the present analysis to ion chains [17], an approach similar to that in [9,13,15] could be applied, working out the dynamical modes of the system and taking into account the dipoledipole interaction due to the rotation of the charged particles. The present results set a first step towards accurately controlling rotating ion chains which would allow for fast reordering.