Benseny et al. EPJ Quantum Technology (2017) 4:3
DOI 10.1140/epjqt/s40507-017-0056-x

RESEARCH

Open Access

Spatial non-adiabatic passage using
geometric phases
Albert Benseny1* , Anthony Kiely2 , Yongping Zhang1,3 , Thomas Busch1

and Andreas Ruschhaupt2*

Correspondence:
albert.benseny-cases@oist.jp
1
Quantum Systems Unit, Okinawa
Institute of Science and Technology
Graduate University, Okinawa,904-0495, Japan
Full list of author information isavailable at the end of the article

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 experimentallyrealistic, non-adiabatic protocols for spatial state preparation, which yield the same
ﬁdelity as their adiabatic counterparts, but on fast timescales. In particular, weconsider 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 thetunnelling 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 threetraps 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 [–].
STIRAP-like techniques have been successfully applied to a wide range of problems, andin 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 []. It has been examined for cold atoms in optical traps [–] and for electrons trapped in quantum dots[, ]. 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[, , ] and quantum information processing []. SAP has also been suggested for avariety of tasks such as interferometry [], creating angular momentum [], and velocity
ﬁltering []. It is also applicable to the classical optics of coupled waveguides [, ].
However, the high ﬁdelity and robustness of adiabatic techniques comes at the expenseof requiring long operation times. This is problematic as the system will therefore also havea long time to interact with an environment leading to losses or decoherence. To avoid thisproblem, we will show how one can speed-up processes that control the centre-of-massstate of quantum particles and introduce a new class of techniques which we refer to as© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.

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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 [, ]. Moreover, shortcuts are known to provide
the freedom to optimise against undesirable eﬀects such as noise, systematic errors ortransitions to unwanted levels [–].
Implementing the STA techniques for spatial control requires complex tunnelling amplitudes. However, tunnelling frequencies are typically real. To solve this, we show thatthe application of a magnetic ﬁeld to a triple well system containing a single charged particle (which could correspond to a quantum dot system [–]) can achieve complex
tunnelling frequencies through the addition of a geometric phase. This then allows one to
implement a counter-diabatic driving term [, , –] or, more generally, to design
dynamics using Lewis-Riesenfeld invariants [].
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 thecentre. In Section , we introduce the spatial adiabatic passage technique in a three-levelsystem and show that making one of the couplings imaginary allows the implementationof transitionless quantum driving. We then show, in Section ., how to create inverseengineering 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 . Section  presents a more realistic one-dimensional continuum model forthe system, where the same schemes are implemented. Finally, in Section , we review
and summarise the results.

2 System model
We consider a charged particle trapped in a system of three localised potentials, between
which the tunnel coupling can be changed in a time-dependent manner. In order to havecoupling between all traps, they are assumed to be arranged along a ring and a magnetic
ﬁeld exists perpendicular to the plane containing the traps, see Figure . The particle will
initially be located in one of the traps and we will show how to design spatial non-adiabatic
passage protocols where a speciﬁc ﬁnal state can be reached within a ﬁnite time and withhigh ﬁdelity. Such a model could, for example, correspond to an electron trapped in an
arrangement of quantum dots, where gate electrodes can be used to change the tunnelling
between diﬀerent traps []. Another option would be to use ion trapping systems [],where ring conﬁgurations have been recently demonstrated [–]. In these systems,
tunnelling of an ion has already been observed (and controlled by manipulating the radial
Figure 1 Diagram of the system consisting of three
coupled quantum wells and a localised magnetic ﬁeld in
the centre. The basis states and the couplings strengths
used in the three-level approximation are indicated. The
coordinate system for the continuous model in Section 5 isalso shown. The distance between two traps along the ring isdeﬁned as l, so that the total circumference of the ring is 3l.

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conﬁnement), as well as the Aharonov-Bohm phase [] acquired due to the presence ofan external magnetic ﬁeld [].
Let us start by considering the single-particle Schrödinger equationi∂ψ  ψ + V ψ,=(–i∇ – qA)
∂t
m

()

where 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 isoriginating from an idealised point-like and inﬁnitely long solenoid at the origin (creating = B êϕ (for r = ). Here r, ϕ, z are
a magnetic ﬂux B ) and it is therefore given by A
π rcylindrical coordinates and êϕ is a unit vector in the ϕ direction.
At low energies such a system can be approximated by a three-level (L) model, where
each basis state, |j, corresponds to the localised ground state in one of the trapping potentials (see Figure ). These states are isolated when a high barrier between them exists,
but when the barrier is lowered the tunnelling amplitude jk between states |j and |k
becomes signiﬁcant.
The presence of the magnetic ﬁeld leads to the particle acquiring an Aharonov-Bohm
phase [] whenever it moves (tunnels) between two diﬀerent positions (traps). This phase r r) · dr , where rj is the position of the jth trap, and for consistency,
is given by φj,k = q rj k A(we always chose the direction of the path of the integration to be anti-clockwise around thepole of the vector potential (at r = ). The eﬀects of this phase on the tunnelling amplitudes
is given through the Peierls phase factors [–], exp(iφj,k ), and the Hamiltonian for theL system can be written as⎛ eiφ, e–iφ,⎜
H = – ⎝ e–iφ, eiφ,⎞ e–iφ,⎟ eiφ, ⎠ .
()Here the jk are the coupling coeﬃcients in the absence of any vector potential. The total
phase around a closed path containing the three traps is then given by ≡ φ, + φ, + φ, =q r) · dl = q B ,
A(


()

 at the origin.
and is non-zero due to the pole of the vector potential A
To simplify the Hamiltonian () one can use the following unitary transformation, which
only employs local phases,⎛⎜U = ⎝
e–iφ,





⎞
⎟
⎠,

()

–i(φ, +φ, )eand transforms the Hamiltonian as
⎛


⎜
H → U † HU = – ⎝ 


i
 e


⎞
 e–i⎟ ⎠ ,
so that two of the tunnelling amplitudes become real-valued.

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A case of particular interest is when  = π/, i.e., when the magnetic ﬂux is B = π/q.
In this case the Hamiltonian becomes
H = – ( K +  K +  K ),()where each Kj is a spin  angular momentum operator deﬁned as⎛⎜K = ⎝ 






⎞

⎟
⎠ ,


⎛


⎜
K = ⎝






⎞

⎟
⎠ ,


⎛


⎜
K = ⎝ 
i





⎞
–i
⎟
⎠,


()

satisfying [Kj , Kk ] = i jkl Kl and jkl is the Levi-Civita symbol []. This means that the tunnel coupling between | and | becomes purely imaginary. We will show in the next
section that this allows for the implementation of spatial non-adiabatic passage processes
by either applying a transitionless quantum driving protocol or by using Lewis-Riesenfeld
invariants.

3 Processes in the three-level approximation
3.1 Adiabatic methods
A series of spatial adiabatic passage (SAP) techniques have been developed in recent years,
which allows one to manipulate and control the external degrees of freedom of quantum
particles in localised potentials with high ﬁdelity []. The standard SAP protocol for thetransport of a single particle in a triple well system [, ] is the spatial analogue of thequantum-optical STIRAP technique [–]. It involves three linearly arranged, degeneratetrapping states, |j with j = ,  and , that can be coupled through tunnelling by either
changing the distance between the traps or lowering the potential barrier between them.
The system in the L approximation is described by the HamiltonianH = – ( K +  K ),()which has a zero-energy eigenstate of the form|λ  = cos θ | – sin θ | with tan θ =  / .

()This state is often called the dark state and SAP consists of adiabatically following |λ from | (at t = ) to –| (at a ﬁnal time t = T), eﬀectively transporting the particle between the outer traps one and three. This corresponds to changing θ from  (   )to π/ (  ). Hence in the case of ideal adiabatic following, trap two (located in the
middle) is never populated.

3.2 Transitionless quantum driving
The main drawback of SAP is that it requires the process to be carried out adiabaticallyand therefore slowly compared to the energy gap []. If this requirement is not met, unwanted excitations will lead to imperfect transport. One way to speciﬁcally cancel possible
diabatic transitions in STIRAP was discussed in [] and a general approach for recovering adiabatic dynamics in a non-adiabatic regime is to use shortcuts to adiabaticity, such

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as transitionless quantum driving [–]. This technique consists of adding a counterdiabatic term to the original Hamiltonian, whose particular form is given as|∂t λn  λn | – λn |∂t λn |λn  λn | ,HCD = i()nwhere the |λn  are the eigenstates of H . For the reference Hamiltonian in Eq. () this gives[] (t)K ,HCD = –˙  –  
˙ 
 
.
with  (t) = θ̇ (t) =  + ()

We will see in Section . how this exact same scheme can also be obtained using LewisRiesenfeld invariants.
Shortcuts to adiabaticity have been studied in the context of STIRAP [, ], 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 (seethe deﬁnition of K in Eq. ()). However, the system we have presented here is ideal for
this, as the system Hamiltonian Eq. () is already equal to the total Hamiltonian H + HCD .
Other methods to implement the imaginary coupling could be, for example, the use ofartiﬁcial magnetic ﬁelds [] or angular momentum states [].
A heuristic but not rigorous explanation of why the coupling needs to be imaginary canbe obtained by examining the two ‘paths’ the particle can take to move from trap one totrap three. The ﬁrst is via SAP and leads to | → –| whereas the second is via the direct
coupling the shortcut introduces, which leads to | → iei |. One can then immediately
see that for constructive interference of these two terms the phase needs to have the value = π/, which corresponds to the required imaginary coupling between states | and |.
It is also interesting to note that the coupling between traps one and three in the shortcut
has the form of a π -pulse T T (t) dt = θ̇ (t) dt =  θ (T) – θ () = π.

()



3.3 Invariant-based inverse engineering
Another method of designing shortcuts to adiabaticity is by means of inverse-engineering
using Lewis-Riesenfeld (LR) invariants [, ]. In this section we will brieﬂy review thesemethods and then apply them to our particular system to both transport the particle andcreate a superposition between the three wells.
A LR invariant for a Hamiltonian H(t) is a Hermitian operator I(t) satisfying []∂I i
+ [H, I] = .
∂t 

()

Since I(t) is a constant of motion it can be shown that it has time-independent eigenvalues.
It can be further shown that a particular solution of the Schrödinger equation,
i∂t ψ(t) = H(t)ψ(t) ,()Benseny et al. EPJ Quantum Technology (2017) 4:3

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can be written asψk (t) = eiαk (t) φk (t) ,()where the |φk (t) are the instantaneous eigenstates of H(t) andαk (t) =


 t





φk (s) i∂s – H(s) φk (s) ds

()



are the LR phases. Hence a general solution to the Schrödinger equation can be written asψ(t) =ck ψk (t) ,

()

k

where the ck are independent of time.
The idea behind inverse engineering using LR invariants is not to follow an instantaneous eigenstate of the H(t) as one would in the adiabatic case, but rather follow an eigenstate of I(t) (up to the LR phase). To guarantee that the eigenstates coincide at the beginning and the end of the process, it is necessary that the invariant and the Hamiltonian
commute at these times, i.e., 
I(), H() = I(T), H(T) = .

()One is then free to choose how the state evolves in the intermediate time and once thisis ﬁxed, Eq. () determines how the Hamiltonian should vary with time to achieve thosedynamics.
A LR invariant for a three-level system described by Eq. () can be written as
I = – sin β sin αK – sin β cos αK + cos βK ,()where α and β are time dependent functions which must fulﬁl the following relations(imposed by Eq. ())α̇ = sin α +  cos α +, tan ββ̇ = ( sin α –  cos α).


()
()

The eigenstates of this invariant are⎛⎞– sin β cos α⎟φ (t) = ⎜⎝ –i cos β ⎠ ,sin β sin α⎛⎞
cos β cos α ± i sin α⎟φ± (t) = √ ⎜–i sin β⎝⎠,– cos β sin α ± i cos α

()

()

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with respective eigenvalues μ =  and μ± = ±. One solution of the time-dependent
Schrödinger equation is then given by |(t) = |φ (t) as the corresponding LR phase iszero in this case. Note that this invariant is a generalisation of the invariant considered in[] where a third coupling  was not taken into account.
After ﬁxing the boundary conditions using Eq. (), one is free to choose the functions
α(t) and β(t). Moreover, in this case, one is also free to directly choose the function  .
By inverting Eqs. () and (), the other coupling coeﬃcients are then given by = α̇ sin α tan β – β̇ cos α –  sin α tan β,() = α̇ cos α tan β + β̇ sin α –  cos α tan β.

()

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 L approximation. The ﬁrst one is thetransport between two diﬀerent traps, which is shown to be equivalent to the transitionlessquantum driving method from Section  in some cases. The second scheme will create anequal superposition of the particle in all three traps.
4.1 Transport
The ﬁrst example of control we examine is the population transfer determined by() = | → |target  = (T) = –|,()

which was considered in the optical regime in []. This can be achieved by choosing
auxiliary functions that fulﬁl the boundary conditionsπβ() = β(T) = – ,α() = ,and α(T) =

π
.


()

The experimentally required tunnelling frequencies are then explicitly given by Eqs. ()and ().
For the special choice of β(t) = –π/, one can show that |(t) =  for all times, i.e.
trap two is never occupied during the process. This choice then results in
tan α =




and

 = α̇.

()

By identifying α with θ (see Eq. ()) one can immediately see that this is the same pulse as
in the STA scheme derived in Section ..
The transport scheme can be implemented by the choosing the counterintuitive SAP
pulses  and  to have a Gaussian proﬁle [] (t) =  exp –(t/T – /) ,
 (t) =  exp –(t/T – /) ,

()
()

and then calculating  from Eq. (). The resulting pulses and associated dynamical
populations are shown in Figure . As expected the system follows exactly the dark state,transferring the population between states | and | without populating state |.

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Figure 2 Spatial non-adiabatic passage transport in the 3L approximation. T/τ = 100 for Ω0 τ = 0.25.
(a) Modulus of the tunnelling amplitudes. (b) Evolution of the populations Pi = | i|Ψ(t)|2 . The time unit τ isdeﬁned as τ = ml2 /h.

Figure 3 Transport process |1 → –|3 in the 3L approximation. (a) Fidelity as a function of the total time
and the total magnetic phase traversing the system. The green contour line is deﬁned by P3 = 99%.
(b) Probabilities of population in each of the traps for T/τ = 48 (indicated by a dashed white line in (a)) as afunction of the total magnetic phase traversing the system. The dashed black line indicates the optimal valueof the phase Φ = π/2.

The ﬁdelity of the transport process as a function of the total time and the phase 
generated by the magnetic ﬁeld is shown in Figure (a). Transport can be seen to occur
with perfect ﬁdelity for any value of the total time if the phase takes the appropriate value = π/. It can also be seen that the shortcut is successful for any value of the phase in
the limit of very short or very long times. The latter one is not surprising, as  can beneglected in the adiabatic limit, and hence its phase becomes irrelevant. A similar eﬀect
occurs for short total times, where the roles are reversed. In this limit  is the largestof all three couplings, and hence the phase relation between it and the other couplings
becomes inconsequential. As  is a 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  is inversely proportional to T. This sets a boundon how fast this scheme can be implemented, as any physical implementation will havea maximum tunnelling amplitude. Setting the maximum value of  to ./τ , the minimum process times T to achieve ﬁdelities above % are approximately τ for SAP
and τ for the shortcut scheme. These times are similar to the ones achievable in aspin-dependent transport scheme recently presented by Masuda et al. [], however the
setup in their work requires four traps and a constant and an AC magnetic ﬁeld.

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Figure 4 Spatial non-adiabatic superposition scheme |1 → √1 (|1 – i|2 – |3) in the 3L
3approximation. T/τ = 400. Sub-ﬁgures are the same as in Figure 2 and the ﬁdelity shown in (b) is deﬁned as
F = | Ψtarget |Ψ(t)|2 .

It is worth noting that this system also allows for the possibility of measuring the magnetic ﬂux B , as the amount of transferred population oscillates as a function of the total
phase , which is directly related to the magnetic ﬂux as  = q B . As an example weshow the occupation probabilities for T/τ =  in each trap at the end of the process as afunction of the phase in Figure (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
The second scheme we discuss highlights the generality of the LR invariant based method.
In this scheme we create an equal superposition state between the particles being in all
three traps, which means that the initial and target states are() = | → |target  = (T) = √ | – i| – | .
()This can be realised by imposing the boundary conditionsπβ() = – ,α() = ,√β(T) = – arctan ,α(T) =

π
,


()
()

on the auxiliary functions. A simple ansatz which fulﬁls these boundary conditions is afourth order polynomial for β(t) and third order polynomials for α(t) and  (t). The
pulses are then obtained from Eqs. () and () and their form is shown in Figure (a).
From Figure (b) it can be seen that this choice creates the target state at the ﬁnal timewith perfect ﬁdelity.

5 Spatial non-adiabatic passage in the continuum model
While the L 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

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Figure 5 Schematic of the potential used in the numerical simulations (black line) with the localised
states in each trap (coloured areas). The Gaussian shape of the traps is exaggerated here for clarity.

that are narrow enough to limit the system dynamics to an eﬀectively one-dimensional setting along the azimuthal coordinate, x = ϕR, i.e., around a circle of radius R, see Figure .
Moreover, we will assume that the magnetic ﬁeld is characterised by a vector potential in = Aêϕ .
the azimuthal direction, A
We are therefore dealing with a one-dimensional system of length πR with periodic
boundary conditions, whose dynamics are described by the following Schrödinger equation
i



∂
∂ψ
=
–i – qA ψ + V (x)ψ.
∂t
m
∂x

()

We assume a constant vector potential throughout the dynamical part of the protocols,
as any time-varying vector potential would produce an unwanted force due to the electricﬁeld E = –∂t A.
In order to be able to apply a well-deﬁned phase we model the trapping sites as highly
localised point-like potentials of depth j at the positions xj = jl – l/ (see Figure ). They
are separated by square barriers of heights Vjk (t) (and length l), giving a total potential

V (x, t) = –j (t)δ(x – xj ) +j=⎧⎪V (t)⎪⎪⎪⎪⎨V (t)
if  < x < x ,if x < x < x ,⎪⎪V (t) if x < x < x ,⎪⎪⎪⎩V (t) if x < x < l.

()Since point-like potentials are diﬃcult to implement numerically, in the simulations below
they are implemented as narrow Gaussians. It is important to note that this model is notdesigned to give realistic estimates for the ﬁdelities or exactly reproduce the dynamics ofthe L approximation. It is a toy model to validate the basic underlying processes and showthat our schemes also make sense in the continuum.
As mentioned above, the tunnelling amplitudes jk (t) in the L approximation are related to the barrier heights Vjk (t) of the continuum model, see the Appendix. However,changing the barrier heights in order to achieve tunnelling will also aﬀect the energies ofthe localised states in the neighbouring traps. Therefore, in order to reproduce the resonance of the L approximation (where the diagonal elements of the Hamiltonian are alwayszero) in the continuum model, the depths of the delta potentials j have to be adjusted as

Benseny et al. EPJ Quantum Technology (2017) 4:3

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Figure 6 Spatial non-adiabatic transport process in the continuum model. T/τ = 100. (a), (b) Barrierheights and trap depths obtained by mapping the couplings in Figure 2(a). (c) Evolution of the particle
density |ψ(x, t)|2 . (d) Corresponding populations Pi = | i|Ψ(t)|2 in each trap and of the target state. (e), (f) are
the same as (c), (d) but with the magnetic ﬂux ﬂowing in the opposite direction. The width of the Gaussian
traps is 10–4 l.

the barriers heights change, see Figure . Finally, to map the barrier heights Vjk and trap
depths j parameters of the continuum model to the tunnelling amplitudes jk of the Lapproximation, we numerically calculate the overlaps of neighbouring delta-trap eigenstates.
Results for transport of a particle using the shortcut scheme described in Section . areshown in Figure  and the barrier heights and trap depths used to match the pulses given
in Figure  are shown in Figure (a), (b). The probability density during the process canbe seen in Figure (c) and the populations in each trap are given in Figure (d). While theprocess is not perfect, one can see that the particle is transported to the ﬁnal trap with aﬁdelity of %. The eﬀect of the magnetic ﬁeld can be seen in Figure (e), (f ), where weshow results for the same process but with an inverted magnetic ﬁeld (using a total phaseof  = –π/). In this case the interference between the adiabatic and shortcut paths is
destructive, and almost no population ends up in the ﬁnal trap.
The results for the creation of the superposition state discussed in Section . are shownin Figure . The observed dynamics are very similar to the one in the L approximationand the process reaches a ﬁnal ﬁdelity of the target state of %.

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Figure 7 Same as Figure 6(a)-(d) but for the spatial non-adiabatic superposition scheme given in Eq.
(31) in the continuum model. T/τ = 400. F = | ψtarget |ψ(t)|2 is the ﬁdelity of the process.

Since the continuum model has many more degrees of freedom than the L model, it
is not surprising that the ﬁdelities obtained are lower. Nevertheless, the basic functioningof 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, goesbeyond the scope of the current work.

6 Conclusions and outlook
We have shown how complex tunnel frequencies in single-particle systems allow one todevelop 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 fromthe addition of a constant magnetic ﬁeld, and have allowed us to generalise adiabatic
state preparation protocols beyond the usual spatial adiabatic passage techniques []. 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 singleparticle system by using Lewis-Riesenfeld invariants. As an example, we have shown thatan equal spatial superposition state between the three wells can be created on a short timescale. 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 []). Often, if the transitionless quantum driv-

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ing technique is directly applied to complex quantum systems, the additional counteradiabatic terms become very complicated, hard to implement or even unphysical. Nevertheless, the steps outlined in our work (using a few-level approximation, applying theshortcut technique, and then mapping everything back to a continuous model) can inprinciple 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 bestudied on an individual basis.
It would also be very interesting to see the eﬀect of interactions in this system. For verystrong 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 []. In
this case, spatial non-adiabatic ideas can be straightforwardly transferred. For intermediate interaction strengths (but stronger than the tunnelling couplings), repulsively-boundpair processes have been shown to dominate the dynamics and single-particle-like dynamics can be recovered for the pair [, , ]. 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 canbe used to implement techniques based on composite pulses [].

Appendix: Mapping between three-level approximation and continuum model
In this appendix we give more details on how to connect the parameters of the threelevel (L) approximation and the continuum model. For clarity, we set  = m =  in thefollowing. Let us ﬁrst recall the eigenfunctions of a single asymmetric delta potential givenby

V (x) = – δ(x) +⎧⎨V ,x < ,
L⎩VR , x ≥ .

()This potential has only one bound state (as long as   > |VL – VR |) which is of the form⎧
⎨φ (x), x < ,Lψ(x) =⎩φR (x), x ≥ ,

()

where
 
 
 – (VL – VR )
( + VL – VR )x
φL (x) = exp
,

 /
 

(  + VR – VL )x – (VL – VR )
φR (x) = exp –
,

 /

()
()

with an energyE=–  + (VR – VL ) –   (VL + VR )
.
 

()

In our work we use these eigenstate as the localised basis states in each of the threedelta trap potentials. For example, the basis state ψ (x) for the ﬁrst trap can be constructedBenseny et al. EPJ Quantum Technology (2017) 4:3

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from the substitutions →  , VL → V , VR → V and x → x – l/ (see Figure ) and thestates ψ (x) and ψ (x) for the other two wells can be obtained in a similar manner. Whilechoosing a basis for the system this way does not necessarily lead to an orthogonal basis
set, we have checked numerically that the states are approximately orthogonal at all times
during our simulations. This allows us to approximate the Hamiltonian associated with
Eq. () as Eq. ().
The couplings constants between each pair of neighbouring two traps can be determined by calculating the overlap between the two respective trap states in the barrier
region between them, i.e.,  ≈ –ψ (x) – ∂x ψ (x) + V ψ (x) dx,

l/


 l/ 
ψ (x) – ∂x ψ (x) + V ψ (x) dx,
 ≈ –l/ l/ 
ψ (x – l) – ∂x ψ (x) + V ψ (x) dx.
 ≈ –l/ l/

()
()
()

Similarly, the on-site energies (or diagonal elements) are approximated by considering
only the regions for which the basis states are signiﬁcant, e.g.,


 
E ≈
ψ (x) – ∂x ψ (x) + V ψ (x) dx

l/


 l/ 
ψ (x) – ∂x ψ (x) + V ψ (x) dx+l/ l/

()and correspondingly for the energies E and E . We can then tabulate sets of values
for the barrier heights, {V , V , V }, trap depths, {  ,  ,  }, and tunnelling amplitudes,{ ,  ,  }, such that the energies all match a ﬁxed reference value, i.e., E = E = E =E where E is ﬁxed to some constant value. Since for a given protocol the required tunnelling amplitudes are known, we can ﬁnally numerically invert the table in order to determine how the barrier heights and trap depths have to vary in time.

Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
AB and AK performed the calculations. All authors participated in the design and discussion of the project, andcontributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.
Author details
1
Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, 904-0495, Japan.
2
Department of Physics, University College Cork, Cork, Ireland. 3 Department of Physics, Shanghai University, Shanghai,
200444, China.
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.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.
Received: 8 November 2016 Accepted: 15 March 2017

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