# Spatial non-adiabatic passage using geometric phases

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## Abstract

Quantum technologies based on adiabatic techniques can be highly effective, but often at the cost of being very slow. Here we introduce a set of experimentally realistic, non-adiabatic protocols for spatial state preparation, which yield the same fidelity as their adiabatic counterparts, but on fast timescales. In particular, we consider a charged particle in a system of three tunnel-coupled quantum wells, where the presence of a magnetic field can induce a geometric phase during the tunnelling processes. We show that this leads to the appearance of complex tunnelling amplitudes and allows for the implementation of spatial non-adiabatic passage. We demonstrate the ability of such a system to transport a particle between two different wells and to generate a delocalised superposition between the three traps with high fidelity in short times.

## Keywords

shortcuts to adiabaticity geometric phases complex tunnelling## 1 Introduction

Adiabatic techniques are widely used for the manipulation of quantum states. They typically yield high fidelities and possess a high degree of robustness. One paradigmatic example is stimulated Raman adiabatic passage (STIRAP) in three-level atomic systems [1, 2, 3]. STIRAP-like techniques have been successfully applied to a wide range of problems, and in particular, to the control of the centre-of-mass states of atoms in microtraps. This spatial analogue of STIRAP is called spatial adiabatic passage (SAP) and it relies on coupling different spatial eigenstates via a controllable tunnelling interaction [4]. It has been examined for cold atoms in optical traps [5, 6, 7, 8, 9, 10, 11, 12] and for electrons trapped in quantum dots [13, 14]. The ability to control the spatial degrees of freedom of trapped particles is an important goal for using these systems in future quantum technologies such as atomtronics [9, 15, 16] and quantum information processing [17]. SAP has also been suggested for a variety of tasks such as interferometry [11], creating angular momentum [12], and velocity filtering [18]. It is also applicable to the classical optics of coupled waveguides [19, 20].

However, the high fidelity and robustness of adiabatic techniques comes at the expense of requiring long operation times. This is problematic as the system will therefore also have a long time to interact with an environment leading to losses or decoherence. To avoid this problem, we will show how one can speed-up processes that control the centre-of-mass state of quantum particles and introduce a new class of techniques which we refer to as spatial non-adiabatic passage. The underlying foundation for these are shortcuts to adiabaticity (STA) techniques, which have been developed to achieve high fidelities in much shorter total times, for a review see [21, 22]. Moreover, shortcuts are known to provide the freedom to optimise against undesirable effects such as noise, systematic errors or transitions to unwanted levels [22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

Implementing the STA techniques for spatial control requires complex tunnelling amplitudes. However, tunnelling frequencies are typically real. To solve this, we show that the application of a magnetic field to a triple well system containing a single charged particle (which could correspond to a quantum dot system [32, 33, 34, 35, 36, 37]) can achieve complex tunnelling frequencies through the addition of a geometric phase. This then allows one to implement a counter-diabatic driving term [21, 22, 38, 39, 40] or, more generally, to design dynamics using Lewis-Riesenfeld invariants [41].

The paper is structured as follows. In the next section, we present the model we examine, namely a charged particle in a triple well ring system with a magnetic field in the centre. In Section 3, we introduce the spatial adiabatic passage technique in a three-level system and show that making one of the couplings imaginary allows the implementation of transitionless quantum driving. We then show, in Section 3.3, how to create inverse-engineering protocols in this system using Lewis-Riesenfeld invariants. Results for two such protocols, namely transport and generation of a three-trap superposition, are given in Section 4. Section 5 presents a more realistic one-dimensional continuum model for the system, where the same schemes are implemented. Finally, in Section 6, we review and summarise the results.

## 2 System model

*m*and

*q*are the mass and charge of the particle, respectively, and

*V*corresponds to the potential describing the trapping geometry. We assume that the vector potential is originating from an idealised point-like and infinitely long solenoid at the origin (creating a magnetic flux \(\Phi_{B}\)) and it is therefore given by \(\vec {A} = \frac{\Phi_{B}}{2 \pi r} \hat{e}_{\varphi}\) (for \(\vec{r} \neq0\)). Here

*r*,

*φ*,

*z*are cylindrical coordinates and \(\hat{e}_{\varphi}\) is a unit vector in the

*φ*direction.

At low energies such a system can be approximated by a three-level (3L) model, where each basis state, \(|j\rangle\), corresponds to the localised ground state in one of the trapping potentials (see Figure 1). These states are isolated when a high barrier between them exists, but when the barrier is lowered the tunnelling amplitude \(\Omega_{jk}\) between states \(|j\rangle\) and \(|k\rangle\) becomes significant.

*j*th trap, and for consistency, we always chose the direction of the path of the integration to be anti-clockwise around the pole of the vector potential (at \(\vec{r} = 0\)). The effects of this phase on the tunnelling amplitudes is given through the Peierls phase factors [48, 49, 50], \(\exp (i \phi_{j,k} )\), and the Hamiltonian for the 3L system can be written as

*A⃗*at the origin.

## 3 Processes in the three-level approximation

### 3.1 Adiabatic methods

*dark state*and SAP consists of adiabatically following \(|\lambda _{0}\rangle\) from \(|1\rangle\) (at \(t=0\)) to \(-|3\rangle\) (at a final time \(t=T\)), effectively transporting the particle between the outer traps one and three. This corresponds to changing

*θ*from 0 (\(\Omega_{23} \gg \Omega_{12}\)) to \(\pi/2\) (\(\Omega_{23} \ll\Omega_{12}\)). Hence in the case of ideal adiabatic following, trap two (located in the middle) is never populated.

### 3.2 Transitionless quantum driving

Shortcuts to adiabaticity have been studied in the context of STIRAP [40, 53], i.e., population transfer between internal levels. Its spatial analogue is more challenging as it requires that the additional tunnelling coupling between sites one and three is imaginary (see the definition of \(K_{3}\) in Eq. (7)). However, the system we have presented here is ideal for this, as the system Hamiltonian Eq. (6) is already equal to the total Hamiltonian \(H_{0} + H_{\mathrm{CD}}\). Other methods to implement the imaginary coupling could be, for example, the use of artificial magnetic fields [54] or angular momentum states [55].

*π*-pulse

### 3.3 Invariant-based inverse engineering

Another method of designing shortcuts to adiabaticity is by means of inverse-engineering using Lewis-Riesenfeld (LR) invariants [41, 56]. In this section we will briefly review these methods and then apply them to our particular system to both transport the particle and create a superposition between the three wells.

*α*and

*β*are time dependent functions which must fulfil the following relations (imposed by Eq. (13))

## 4 Examples of spatial non-adiabatic passage schemes

In the following we will discuss two examples of spatial non-adiabatic passage derived from LR invariant based inverse engineering in the 3L approximation. The first one is the transport between two different traps, which is shown to be equivalent to the transitionless quantum driving method from Section 3 in some cases. The second scheme will create an equal superposition of the particle in all three traps.

### 4.1 Transport

*α*with

*θ*(see Eq. (9)) one can immediately see that this is the same pulse as in the STA scheme derived in Section 3.2.

*π*pulse, perfect population transfer in this regime can be achieved regardless of the phase.

However, in order to maintain this pulse area, a strong coupling is required for very short processes, as the strength of \(\Omega_{31}\) is inversely proportional to *T*. This sets a bound on how fast this scheme can be implemented, as any physical implementation will have a maximum tunnelling amplitude. Setting the maximum value of \(\Omega_{31}\) to \(0.25/\tau\), the minimum process times *T* to achieve fidelities above 99% are approximately 880*τ* for SAP and 100*τ* for the shortcut scheme. These times are similar to the ones achievable in a spin-dependent transport scheme recently presented by Masuda et al. [58], however the setup in their work requires four traps and a constant and an AC magnetic field.

It is worth noting that this system also allows for the possibility of measuring the magnetic flux \(\Phi_{B}\), as the amount of transferred population oscillates as a function of the total phase Φ, which is directly related to the magnetic flux as \(\Phi=\frac{q}{\hbar}\Phi_{B}\). As an example we show the occupation probabilities for \(T/\tau= 48\) in each trap at the end of the process as a function of the phase in Figure 3(b). One can see that the populations strongly depend on the phase and over a large range of values one can therefore determine the magnetic flux. The exact relationship between the probabilities and the magnetic flux differs for different total times *T*.

### 4.2 Creation of a three-trap superposition

## 5 Spatial non-adiabatic passage in the continuum model

While the 3L approximation discussed above gives a clear picture of the physics of the system, it does not include effects such as excitations to higher energy states that can occur during the process. We will therefore in the following test the approximation by numerically integrating the full Schrödinger equation in real space. For this, we will consider traps that are narrow enough to limit the system dynamics to an effectively one-dimensional setting along the azimuthal coordinate, \(x = \varphi R\), i.e., around a circle of radius *R*, see Figure 1. Moreover, we will assume that the magnetic field is characterised by a vector potential in the azimuthal direction, \(\vec{A} = A \hat {e}_{\varphi}\).

*l*), giving a total potential

As mentioned above, the tunnelling amplitudes \(\Omega_{jk}(t)\) in the 3L approximation are related to the barrier heights \(V_{jk}(t)\) of the continuum model, see the Appendix. However, changing the barrier heights in order to achieve tunnelling will also affect the energies of the localised states in the neighbouring traps. Therefore, in order to reproduce the resonance of the 3L approximation (where the diagonal elements of the Hamiltonian are always zero) in the continuum model, the depths of the delta potentials \(\epsilon_{j}\) have to be adjusted as the barriers heights change, see Figure 5. Finally, to map the barrier heights \(V_{jk}\) and trap depths \(\epsilon _{j}\) parameters of the continuum model to the tunnelling amplitudes \(\Omega_{jk}\) of the 3L approximation, we numerically calculate the overlaps of neighbouring delta-trap eigenstates.

Since the continuum model has many more degrees of freedom than the 3L model, it is not surprising that the fidelities obtained are lower. Nevertheless, the basic functioning of our spatial non-adiabatic techniques is clearly established from the calculations shown above. Optimising the fidelity in the continuum is an interesting task which, however, goes beyond the scope of the current work.

## 6 Conclusions and outlook

We have shown how complex tunnel frequencies in single-particle systems allow one to develop spatial non-adiabatic passage techniques that can lead to fast and robust processes for quantum technologies. In particular, we have discussed the case of a single, charged particle in a microtrap environment. The complex tunnelling couplings are obtained from the addition of a constant magnetic field, and have allowed us to generalise adiabatic state preparation protocols beyond the usual spatial adiabatic passage techniques [4]. This demonstrates that non-adiabatic techniques can be as efficient as their adiabatic counterparts, without requiring the long operation times.

In particular, we have discussed the implementation of the counter-diabatic term for spatial adiabatic passage transport via a direct coupling of all the traps. This was, in a second step, generalised to a flexible and robust method for preparing any state of the single-particle system by using Lewis-Riesenfeld invariants. As an example, we have shown that an equal spatial superposition state between the three wells can be created on a short time scale. Finally, we have presented numerical evidence that spatial non-adiabatic processes work also in a one-dimensional toy model by introducing a mapping between the discrete three-level approximation and a continuum model.

While in this work we have focused on a three-trap system, an interesting extension would be to investigate similar schemes in larger systems, or in different physical settings (for example, superconducting qubits [59]). Often, if the transitionless quantum driving technique is directly applied to complex quantum systems, the additional counter-adiabatic terms become very complicated, hard to implement or even unphysical. Nevertheless, the steps outlined in our work (using a few-level approximation, applying the shortcut technique, and then mapping everything back to a continuous model) can in principle be applied to any trap configuration. These steps might lead to schemes which are much easier to implement experimentally than the direct application of the transitionless quantum driving. However, each of these generalised configuration would need to be studied on an individual basis.

It would also be very interesting to see the effect of interactions in this system. For very strong interactions such that double occupancy of a site is suppressed and a single empty site is present, one might expect to observe similar dynamics but for the empty site [9]. In this case, spatial non-adiabatic ideas can be straightforwardly transferred. For intermediate interaction strengths (but stronger than the tunnelling couplings), repulsively-bound pair processes have been shown to dominate the dynamics and single-particle-like dynamics can be recovered for the pair [10, 60, 61]. In this case the presented techniques might be extended for a particle pair.

Finally, it is also worth noting that these complex tunnelling couplings we introduce can be used to implement techniques based on composite pulses [62].

## Notes

### Acknowledgements

This work has received financial support from Science Foundation Ireland under the International Strategic Cooperation Award Grant No. SFI/13/ISCA/2845 and the Okinawa Institute of Science and Technology Graduate University. We are grateful to David Rea for useful discussion and commenting on the manuscript.

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