Spatial adiabatic passage: a review of recent progress
Adiabatic techniques are known to allow for engineering quantum states with high fidelity. This requirement is currently of large interest, as applications in quantum information require the preparation and manipulation of quantum states with minimal errors. Here we review recent progress on developing techniques for the preparation of spatial states through adiabatic passage, particularly focusing on three state systems. These techniques can be applied to matter waves in external potentials, such as cold atoms or electrons, and to classical waves in waveguides, such as light or sound.
I Introduction and motivation
The ability to coherently control the spatial degrees of freedom of matter waves is an important ingredient for the emerging field of quantum engineering micheli_single_2004 ; ruschhaupt_atom_2004 ; stickney_transistorlike_2007 ; thorn_experimental_2008 ; price_single-photon_2008 ; pepino_atomtronic_2009 ; benseny_atomtronics_2010 ; zozulya_principles_2013 ; ryu_experimental_2013 ; wright_driving_2013 ; jendrzejewski_resistive_2014 , with applications in matter wave interferometry, quantum metrology, and quantum computation. For many purposes, quantum transport of trapped matter waves is performed via direct tunneling through the manipulation of the potential barriers that separate the different traps. This direct transport is strongly dependent on the parameter values giving rise, in general, to very sensitive Rabi-type oscillations of the populations of the localized states of the traps. Alternatively, an efficient transfer of population between distant traps can be achieved using spatial adiabatic passage (SAP) processes, which consist of the adiabatic following of an energy eigenstate of the system that is spatially modified either in time or space. This transfer occurs with high fidelity regardless of the selected specific parameter values used to drive the system and their fluctuations. SAP processes are a particular case of adiabatic following, a concept arising from the adiabatic theorem born_beweis_1928 ; messiah_quantum_1976 : in the absence of level crossings, a system will remain in one of its eigenstates if the system is perturbed slowly enough.
The term adiabatic passage started to be commonly used around three decades ago in the field of quantum optics as two main techniques were introduced to adiabatically transfer population between internal atomic/molecular levels. These were rapid adiabatic passage (RAP) allen_optical_1987 ; vitanov_laser-induced_2001 and stimulated Raman adiabatic passage (STIRAP) bergmann_coherent_1998 ; fewell_coherent_1997 ; vitanov_laser-induced_2001 . The RAP technique is implemented in two-level atomic systems interacting with a chirped laser pulse to transfer the population between the two states by adiabatically following one of the two eigenstates of the system. The STIRAP technique is performed in a -type three-level atomic system interacting with two temporally delayed laser pulses in a counterintuitive sequence in order to completely transfer the population between the two atomic ground states by adiabatically following the so-called dark state alzetta_experimental_1976 ; arimondo_nonabsorbing_1976 ; gray_coherent_1978 ; alzetta_nonabsorption_1979 . These techniques have led to many relevant experimentally implemented applications shore_pre-history_2013 ; bergmann_perspective_2015 .
Following the success of techniques such as RAP and STIRAP, adiabatic passage processes were proposed to coherently transport quantum particles between localized states of spatially separated potential wells, leading to the term spatial adiabatic passage. While some proposals were reported in the early 2000s (see for example Ref. renzoni_charge_2001 ), the field gathered more interest after the publication of two seminal papers. The first one, by Eckert et al. eckert_three-level_2004 , analyzed the spatial adiabatic passage of a single cold atom in a system of three optical traps and the second one, by Greentree et al. greentree_coherent_2004 , described the transfer of an electron in a system of three quantum dots. Both proposals resembled the quantum-optical STIRAP technique, and used tunneling as a way to couple the different localized states of spatially separated wells. However, it was soon realized that SAP can be extended beyond the applications of STIRAP, as it allows for multi-dimensional configurations and many-particle systems. Therefore, due to the high efficiency and robustness inherited from STIRAP, many applications, such as vibrational state and velocity filtering busch_quantum_2007 ; loiko_filtering_2011 ; loiko_coherent_2014 , quantum tomography loiko_filtering_2011 , interferometry jong_interferometry_2010 ; menchon-enrich_single-atom_2014 , atomtronics lu_coherent_2011 ; benseny_atomtronics_2010 , and the generation of angular momentum states menchon-enrich_tunneling-induced_2014 have been proposed. Note that SAP in a triple-well system has also been referred as matter-wave STIRAP or coherent tunneling by adiabatic passage (CTAP).
In spite of the significant theoretical interest that they have attracted, SAP processes for matter waves have not been experimentally realized yet. It is worth noting, however, that due to the wave-like nature of the SAP processes, they can be extended to classical wave systems, and have been experimentally demonstrated with light beams in coupled waveguides. Since the initial works of Longhi et al. longhi_adiabatic_2006 ; longhi_coherent_2007 , SAP for light beams has also been studied in systems of more than three coupled waveguides longhi_optical_2006 ; valle_adiabatic_2008 ; rangelov_achromatic_2012 ; ciret_broadband_2013 , through the continuum longhi_transfer_2008 ; dreisow_adiabatic_2009 , and in the presence of nonlinearities and absorption barak_autoresonant_2009 ; graefe_breakdown_2013 . It has also lead to applications such as beam splitting dreisow_polychromatic_2009 ; rangelov_achromatic_2012 ; chung_broadband_2012 ; hristova_adiabatic_2013 , spectral filtering menchon-enrich_light_2013 , interaction free-measurements hill_parallel_2011 , quantum gates via long-range coupling hope_long-range_2013 , polarization rotation/conversion xiong_integrated_2013 , and photon pair generation wu_photon_2014 . Finally, SAP processes have also been recently proposed for sound propagation in sonic crystals menchon-enrich_spatial_2014 , investigating transfer and splitting of sound beams, as well as the use of the system as a phase analyzer.
This review is organized as follows. In Section II, we describe the concept of adiabatic following of an eigenvector and connect it to the different physical systems that we will study. We then, in Section III, review the spatial adiabatic passage of matter waves, which includes systems of single atoms, electrons, and Bose–Einstein condensates. We also discuss SAP in two-dimensional systems and address some practical considerations. This is followed by Section IV, where we discuss dark state adiabatic passage, and Section V, where recent developments of SAP for light and sound waves are summarized. Finally, in Section VI we conclude.
Ii Adiabatic following of an energy eigenstate
The concept of a quantum state evolution that adiabatically follows an energy eigenstate was introduced to quantum mechanics by Max Born and Vladimir Fock in 1928 born_beweis_1928 . They showed that a physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian’s spectrum. This insight is known by today as the adiabatic theorem.
To perform an adiabatic evolution it is therefore necessary that the dynamics do not allow transitions between eigenstates. While in most cases this can be achieved by changing the system parameters slowly enough, it also requires that all eigenstates evolve smoothly and that their first and second derivatives with respect to the temporal or spatial evolution parameter, , are well defined messiah_quantum_1976 . To quantify the adiabaticity of a process one can calculate the probability to excite a state while being in as messiah_quantum_1976
where with being the energy of the eigenstate . Since the value of has to be as small as possible for any , one can immediately see that the adiabatic following of an eigenstate does not succeed, in principle, when the corresponding energy eigenvalue becomes degenerate with any other eigenvalue of the system. Note however that adiabatic passage in the presence of quasi degenerate eigenstates is still possible if they are not coupled during the dynamics, i.e., when due to, for instance, some particular symmetries of the system.
SAP is a particular example of adiabatic following of a spatial eigenvector which is modified in either time or space. As an example for the first case, one can consider particles trapped in a local minimum of an external potential which is changed as a function of time. The second case (spatial evolution) usually refers to waveguides, where a localized wavepacket is travelling with finite velocity and encounters changing couplings between different waveguides along the propagation direction. It is worth noting that the concept of adiabatically following an energy eigenstate in quantum mechanics has a close analog in classical light and sound propagation, see Sect. V for a detailed discussion. There, a propagating light/sound wave can adiabatically follow a global spatially varying mode of the waveguide system, if the change of the mode profile is smooth along the propagation direction.
For SAP to work, three basic conditions have to be fulfilled. The first one is the existence of different and well-defined trapping regions, each of them supporting their own asymptotic eigenvectors, i.e, the trapped states/modes of each region isolated from the rest. The second requirement is the existence of a coupling mechanism between the asymptotic eigenvectors, and the third one is the ability to control the strength of the couplings and/or the eigenvalues of the asymptotic eigenvectors during the process. In a physical system where these three conditions are fulfilled, it is possible to perform a SAP process by following an eigenvector of the full system. The advantage of SAP with respect to other transfer techniques relies on its robustness: in the adiabatic regime the transfer between the asymptotic eigenvectors will be almost efficient no matter the total duration of the process or the particular parameter values chosen for the variation of the couplings and the eigenvalues.
Iii Spatial adiabatic passage of matter waves
Spatial adiabatic passage of matter waves can be achieved in a variety of trapping geometries by simply manipulating the height of the potential barriers and/or the trap distances in an adiabatic fashion. As two paradigmatic examples of SAP, we first review the formalism for adiabatic transport of matter waves between the outermost traps of a triple-well potential, and then we discuss the double-well case. The triple well requires the ability to control the tunneling coupling between the traps, while the double-well case in addition requires the manipulation of the energy bias between the localized eigenstates.
iii.1.1 SAP in a triple-well system
Consider a single quantum particle of mass trapped in a triple-well potential, see Fig. 1, which we want to move between the outermost potential minima, i.e., from state to . Since tunneling will take place along a fixed direction , we can describe the particle’s dynamics using the 1D Schrödinger equation
where is the trapping potential. As we are considering the low energy limit, i.e., only the ground states of each well are involved in the dynamics, we can write the atomic wavefunction as the superposition
Here are the localized ground state wavefunctions of each isolated trap centered at with accounting for the left, middle and right well, respectively. The probability amplitudes fulfill and to guarantee that the ground state wavefunctions satisfy it is necessary to orthonormalize them by means of, for instance, the Gram–Schmidt eckert_quantum_2002 ; loiko_filtering_2011 or the Holstein–Herring methods holstein_mobilities_1952 ; herring_critique_1962 . This lowest-band approximation is good as long as adiabatic evolution is maintained, which in many works we discuss below is supported by numerical integration of the full time-dependent Schrödinger equation.
with the three-mode Hamiltonian
where the energy of the asymptotic vibrational eigenstate and the tunnelling rate between states and are given, respectively, by
Without loss of generality we choose the eigenfunctions to be real and, therefore, .
Let us consider now the particular case for which (i) , i.e., the two outermost wells are in resonance; and (ii) , i.e., direct tunneling between the outermost traps is negligible. Then
where, for simplicity, the origin of energies has been taken as . Diagonalizing Eq. (8) one obtains three eigenstates
The mixing angles and are
To adiabatically transfer a particle in the considered triple-well system from the left to the right trap using SAP, one can follow state in Eq. (11), the so-called spatial dark state, by smoothly varying the tunneling rates such that the mixing angle evolves from to . This means to favor first the tunneling between the middle and right traps and then the tunneling between the left and middle traps. This temporal sequence of the tunneling couplings is named the counterintuitive coupling scheme. Since the spatial dark state only involves the localized ground states of the outermost wells, the signature of SAP in a triple well is that the middle well is negligibly populated during the whole transport process.
iii.1.2 Comparison to STIRAP
In quantum-optical STIRAP, two-laser pulses are applied in a counterintuitive temporal sequence to the two adjacent transitions of an atomic or molecular -type three-level system to transfer the population between the two lower energy internal states bergmann_coherent_1998 . Therefore, SAP in a triple-well potential is the matter wave analogue of STIRAP. From a physical point of view, SAP deals with the external (localized) degrees of freedom of trapped particles while STIRAP deals with the internal degrees of freedom. This implies some differences between both techniques:
(i) In STIRAP, to obtain the three-mode Hamiltonian with the two Rabi frequencies playing the role of the tunneling rates, the electric dipole and rotating wave approximations are needed. In the SAP case, the fact that the tunneling amplitudes must be smaller than the involved trapping frequencies is equivalent to the rotating wave approximation, and it ensures that, in the adiabatic limit, no transitions to excited vibrational states occur. Detrimental decoherence mechanisms in STIRAP such as Doppler broadening or photon recoil are not present for SAP although others appear such as trap shaking or finite trapping lifetimes. The role of some decoherence mechanisms in SAP is discussed in Sect. III.6.
(ii) The signature of both SAP in a triple-well system and STIRAP techniques is that the intermediate state is not being populated during the whole process. For SAP this seems to indicate that the local continuity equation associated to the Schrödinger equation fails. In Sect. III.2.2 we review this paradoxical issue.
(iii) Being a spatial transfer, SAP can be considered in systems for which it is possible to tunnel-couple all three traps by taking, for instance, three non-aligned wells in two and three-dimensional configurations. This introduces an additional tunneling rate which allows for potential applications in matter-wave interferometry and for the generation of angular momentum, see Sect. III.5. It is worth noting that this extra coupling has been proposed for cyclic STIRAP unanyan_laser_1997 ; fleischhauer_coherent_1999 and recently for superadiabatic STIRAP giannelli_three_2014 . In chiral molecules, whose states have ill-definite parity due to the lack of inversion symmetry, the possibility of coupling of all three levels appears naturally kral_cyclic_2001 ; kral_two-step_2003 .
(iv) At variance with STIRAP, there is no need to keep the resonance to perform SAP in a triple-well system. If the resonance condition does not hold during the dynamics, the spatial dark state becomes dressed by the localized state of the middle well. While in STIRAP the intermediate state is, typically, a fast decaying excited state that one does not want to populate at any time during the dynamics, SAP is free of this problem since the middle localized state is as stable as the localized states of the outermost wells. However, to achieve significant tunneling rates, it is convenient to keep the resonance condition between the traps. See Sect. III.6 for a detailed discussion on how to keep the resonance condition in practical implementations.
(v) In STIRAP, one can control the relative phase between the pump and Stokes laser pulses. However, it is not obvious how this can be achieved in SAP because the tunneling rate between two real-valued localized energy eigenfunctions is also real-valued. Potential control on the relative phase between the couplings is presently discussed to implement geometrical phases for quantum gates and holonomic quantum computation, see Sect. III.3.2 for a more detailed analysis.
(vi) To ensure the adiabatic transport it is necessary that , and . In STIRAP, because of the necessity to turn on a laser pulse, it is natural to consider pulses which are symmetric about their midpoint, such as Gaussian pulses GRB+88 , even though various pulse schemes have been described CH90 ; LS96 . In SAP, on the other hand, there is no requirement for the tunneling rates to be initially zero, and many proposals exploring error function pulses DGH07 , square sinusoidal pulses rab_spatial_2008 , sinusoidal pulses, linear variation vaitkus_digital_2013 and so on exist. The decision of choosing a pulse shape may come down to the availability and convenience of the control, provided that the counterintuitive pulse condition is maintained. Interestingly, although smooth variations of the control parameters would seem necessary for adiabatic passage, there are two pulse schemes that show good performances without the need for smoothly varying controls: piecewise adiabatic passage SMM07 ; SMS09 and digital adiabatic passage vaitkus_digital_2013 .
iii.1.3 SAP with three identical harmonic wells
To illustrate the SAP technique let us consider a triple-well potential modeled as three identical truncated harmonic wells eckert_three-level_2004 . This academic example allows for keeping the resonance between the three ground states of the triple-well potential and to obtain accurate analytical expressions for the tunneling rates. Other potentials, such as square wells opatrny_conditions_2009 , Gaussian eckert_three_2006 or Pöschl–Teller loiko_filtering_2011 are also considered in the literature, and lead to qualitatively similar results.
To control the tunneling rates in time, one can either adjust the well separation or the height of the barriers. Here we follow the former approach and assume that each well is centered at with such that the joint trapping potential reads
with being the trapping frequency. The distances between the well centers are given by and . We assume that at the initial time the three wells are negligibly coupled by tunneling and that the single quantum particle is prepared into the localized ground state of the left well, i.e.,
with being the width of the ground state of the harmonic potential.
The tunneling rate between the ground vibrational states of two truncated harmonic wells as a function of the separation between the two wells can be straightforwardly calculated as the energy splitting between the lowest symmetric and antisymmetric energy eigenstates of the double-well potential. Thus, for two piece-wise truncated harmonic wells one obtains razavy_quantum_2003 ; menchon-enrich_single-atom_2014
Here is the complementary error function and the distance between the two well centers. In the adiabatic limit and in order to avoid excitations to unwanted vibrational states, the distances between the traps must ensure that .
The counterintutive coupling sequence can then be implemented by the following temporal evolution of the trap centers
The total time is then , with being the time it takes to approach and separate the traps and the delay between the two approaching sequences, see Fig. 2(a). The corresponding tunneling rates calculated from Eq. (18) are shown in Fig. 2(b), and the resulting mixing angle can be seen to smoothly evolve from to in Fig. 2(c). This efficiently transports the quantum particle from the left to the right well, see Fig. 2(d).
For the three-mode case, it is possible to obtain a simple ‘global’ condition which guarantees the adiabaticity of the process. For tunneling rates with Gaussian-like temporal profiles of width and peak values , separated by an optimal temporal delay , this condition is given by bergmann_coherent_1998
The final population in the right well as a function of is shown in Fig. 3, demonstrating the high fidelity and the robustness of the SAP process for adiabatic time scales. In the non-adiabatic limit, i.e., for short durations of the process, two factors are inhibiting full transfer: Rabi oscillations between localized states and excitations to higher vibrational bands.
Finally, it is important to highlight that the validity of the three mode approximation to account for the adiabatic transport of a single quantum particle in a triple-well potential has been verified by the direct integration of the Schrödinger equation in one eckert_three-level_2004 ; cole_spatial_2008 , two menchon-enrich_single-atom_2014 ; menchon-enrich_tunneling-induced_2014 and three dimensions rab_spatial_2008 . Accurate quantitative agreement was not only obtained when SAP was applied to the adiabatic transport between the localized ground states, but also between excited states loiko_filtering_2011 . In addition, as will be discussed in Sect. III.4, SAP can be applied over a significant parameter range to the non-linear Gross–Pitaevskii equation to deal, in the mean field approximation, with the adiabatic transport of a Bose–Einstein condensate.
iii.1.4 SAP in a double-well system
Adiabatic passage techniques can also be implemented in double-well potentials to transfer a single particle from the left to the right trap. The two-mode Hamiltonian in terms of the left, , and right, , localized states is
where is the energy bias and the tunneling rate. The diagonalization of Eq. (21) gives the two eigenstates
with eigenvalues , and a mixing angle given by . Analogously to the quantum-optical RAP technique vitanov_laser-induced_2001 , it is therefore possible to transfer a single particle initially located in the left trap to the right one by adiabatically following either or . Alternatively it is straightforward to show that the time evolution of the probability amplitudes and for the quantum particle to be in the left and right traps, respectively, can be mapped to a three-variable model governed by the following equations of motion vitanov_stimulated_2006 ; ottaviani_adiabatic_2010
Here , and are, respectively, three real variables that correspond to two times the real and imaginary part of the spatial coherence and the population imbalance. The conservation of the norm implies that . The matrix on the r.h.s. of Eq. (24) is an odd real-valued skew symmetric matrix with one zero eigenvalue, , and two other ones . The eigenvector associated to the zero eigenvalue, named dark eigenvector, is given by
where . The dark eigenvector is stationary and decoupled from the rest of the eigenvectors. Thus, if the quantum particle is initially located in the left well (, ) and is smoothly varied from 0 to , the wavefunction splits equally between the two wells (, ). On the other hand, the particle can be adiabatically transferred from the left () to the right () well by smoothly varying from to . Finally, it is possible to inhibit the transport of a particle initially located in the left trap if the mixing angle evolves from to any arbitrary value to eventually reach again.
iii.2 Single atoms
iii.2.1 Three-level atom optics
Controlling the state of single quantum particles is a challenging task and a topic of significant present activity in the fields of quantum computation, quantum metrology, and quantum simulation. While there is a panoply of techniques to control the internal states of atoms, there is a need for the development of novel techniques to manipulate the external degrees of freedom of matter waves in optical and magnetic traps. To this aim, the transport of single atoms via SAP was proposed by Eckert et al. eckert_three-level_2004 in a set of techniques named three-level atom optics (TLAO). The name suggests an analogy between quantum-optical processes in electronic three-level systems and tunneling-based processes for cold atoms in triple-well potentials. The analogue of STIRAP in TLAO has already been presented in Sect. III.1. Two more analogies were presented in this initial work eckert_three-level_2004 : coherent population trapping (CPT), which allows to create a delocalized dark state between the outermost traps by finishing the evolution of the mixing angle at , and electromagnetically-induced transparency (EIT), which inhibits the transfer of the atom from the left to the right trap by strongly coupling the middle and right traps. The positioning sequences for the individual traps and the temporal evolution of the atomic density for these three TLAO techniques are shown in Fig. 4.
A follow-up work eckert_three_2006 focused on the robustness of the transfer, studied the transfer with two atoms in the triple-well system, introduced Gaussian potentials and also showed the possibility of applying SAP techniques to cold atoms propagating in optical waveguides. Examples of these processes are discussed later in this section.
iii.2.2 Population in the central well
As discussed above, for the transport of an atom between the two outermost traps in a triple-well following a dark state of the form of Eq. (11), the middle trap remains unpopulated. In real space this corresponds to the spatial dark state possessing a node in the middle trap. However, for finite times the dynamical state corresponds to the dark state being weakly dressed by other eigenstates, leading to a finite population in the central trap. The more adiabatic the process is performed, the better the following of the dark state, and thus, the smaller this population will be. This therefore allows for the paradoxical possibility that the transport of the atom between the outer traps can be achieved with a negligible population in the middle trap, see Fig. 4(a), which appears to contradict the quantum continuity equation.
This behaviour is not particular to systems of ultracold atoms in harmonic traps, but appears in any system where SAP can be implemented for three spatially separated wells following a spatial dark state of the form of Eq. (11). For instance, in Ref. rab_spatial_2008 this effect was discussed for an atomic Bose–Einstein condensate in a triple well created by adding two Gaussian barriers to a harmonic trap. SAP was then performed by lowering and raising these barriers in a counterintuitive fashion. An analytical study of the conditions for a vanishing central-well population was carried out in Ref. opatrny_conditions_2009 using three square wells separated by delta-function or square potential barriers. The authors found that in order to maintain a negligible occupation of the central trap, the depths of the outer wells need to be varied while the barrier heights are changed time-dependently.
However, how can the atom move between the outer wells without populating the middle trap? To clarify this apparently paradoxical behaviour, this process was studied in Refs. benseny_need_2012 and benseny_atomtronics:_2012 by means of Bohmian mechanics bohm_suggested_1952-1 ; bohm_suggested_1952-2 ; benseny_applied_2014 . Bohmian mechanics is a formalism equivalent to standard quantum mechanics in terms of predictions, and provides a good visualization of continuity because of its use of quantum trajectories. A particle’s velocity along a trajectory is given by
where is the probability density and
is its associated probability density current. For the SAP dynamics following the set up suggested in Ref. rab_spatial_2008 and described above, the trajectories are shown in Fig. 5 and can be seen to follow the wavefunction from the left region to the right one, transiting through the middle trap. However, as the dark state has a node in the middle region, indicated by the green dashed line in Fig. 5, the atomic density has a very low value at that position. Therefore, as can be seen in the figure, the velocity increases dramatically (cf. Eq. (26)) around this quasinode, reaching values which are orders of magnitude larger than the mean velocity of the wave packet. This means that, in the Bohmian picture, the atom transits through the central trap at a high velocity in order to keep the population low.
Slowing down the SAP process will therefore increase the Bohmian velocities benseny_need_2012 , because a better following of the dark state leads to a smaller population in the quasinode. There is thus a regime for a (finite) total time where the trajectories will approach and eventually surpass the speed of light. Since with a correct relativistic treatment, Bohmian trajectories for massive particles cannot surpass the speed of light leavens_are_1998 ; struyve_uniqueness_2004 , superluminal trajectories are an irrefutable indication of the application of the Schrödinger equation in a regime where it is not valid. It is remarkable that relativistic corrections are needed to properly address the transport and avoid superluminal propagation in a system where the associated average velocities are orders of magnitude smaller than the speed of light benseny_need_2012 . Using a relativistic formalism to study SAP, however, is still outstanding.
The vanishing of the central trap population also allows an atom to be transported between the outer wells even if a second atom is present in the central well. In Ref. gajdacz_transparent_2011 the transport was shown to be unaffected by the presence of a second atom of a different species that was considered to be deeply trapped, and thus remained essentially static in the central well. This proposal is also of interest because it is based on an optical superlattice with a unit cell consisting of three wells. Optical superlattices are the result of the combination of multiple laser beams with different frequencies/orientation and, at variance with regular lattices, consist of unit cells which can contain multiple traps. Superlattices give the opportunity to perform the same experiment multiple times simultaneously, once in each cell, which allows to easily scale up the size of more complex cold atomic systems.
iii.2.3 Quantum state preparation
Besides transport, SAP techniques have also been studied for the preparation of more complex quantum states from simpler ones. Such techniques can be very helpful to control and manipulate single neutral atoms for potential applications in quantum information processing. For instance, a configuration proposed in Ref. busch_quantum_2007 can be used to create a symmetric or antisymmetric superposition between the ground states of a double well. In this system of three traps, the rightmost trap is a double well consisting of two harmonic potentials. The state localized in the leftmost trap can be resonant with either the symmetric or antisymmetric combination of the ground states in the right trap wells, and SAP can be used to completely transfer the atom to that state. Which of these states is resonant can be chosen by detuning the trapping frequencies of the right-most wells either up to match the symmetric state or down to match the antisymmetric one.
The influence of the SAP dynamics on the phase of a quantum state was studied in Ref. mcendoo_phase_2010 , and was shown to allow to control the angular momentum of an atom by transporting it through three 2D harmonic traps. The initial state is chosen to carry a single unit of angular momentum, and therefore has a phase distribution that increases by for a closed loop around the centre of the state. While applying the counterintuitive SAP sequence of couplings leads to a complete transfer of the atom from the initial trap to the most distant one, the angular momentum of the final state oscillates continuously between clockwise and counterclockwise depending on the overall duration of the process, see Fig. 6. This shows that SAP is not robust with respect to the conservation of the phase mcendoo_phase_2010 . However, the dependence of the final angular momentum on the overall time of the process is deterministic, therefore it can be used to obtain the full spectrum of angular momentum superposition states, which can have applications for quantum information processing. Other SAP schemes for manipulating angular momentum when the traps are in a triangular configuration are discussed in Sect. III.5.
iii.2.4 State filtering
The control over the shape of the trapping potentials allows to create filtering mechanisms for vibrational states by engineering the energy spectrum of the system. A simple SAP mechanism in a triple-well can be devised by making the first vibrational state of the leftmost trap resonant with the ground states of the middle and right traps, see Fig. 7. For harmonic traps in 1D, this means making the trapping frequency of the middle and right traps three times larger than that of the left trap. In this situation, SAP transfers the population of the first excited state of the left trap to the ground state of the right trap, while the population of the ground state of the left trap remains unaffected.
Another filtering mechanism can be created by making use of the fact that for two identical potential traps at a fixed distance, the tunneling rate increases with the vibrational state of the traps loiko_filtering_2011 . As a consequence, the adiabaticity condition for SAP transport in a triple well is state-dependent.
Assuming that the energy separation between the different vibrational states of each trap is large enough to avoid cross tunneling among different vibrational states and that there is no significant coupling between the two outermost traps, the Hamiltonian of the system can be separated as
Here, is the Hamiltonian of the subsystem of the states in the -th vibrational level, which is analogous to the one in Eq. (8) and each has a spatial dark state which allows to transport an atom between the two outermost traps. However, each subsystem has a different tunneling rate with for , leading to a different adiabaticity condition, see Eq. (20). Therefore, for any given it is possible to find a parameter regime in which the process is adiabatic for levels and higher (and SAP transfers the atom to the right trap), but where the tunneling is too weak for those vibrational states lower than (and SAP leaves them in the left trap), see Fig. 8. This filtering process can be used, for instance, for the preparation of vibrational states on demand and to perform quantum tomography of the initial population of vibrational-states loiko_filtering_2011 . However, it cannot measure the relative phase between the different levels. In Ref. loiko_filtering_2011 this system was studied using Pöschl–Teller potentials, which approximate experimental Gaussian traps much better than the usually used truncated harmonic potentials and for which analytical expressions for the energy eigenvalues and eigenstates exist.
iii.2.5 Hole transport
The SAP dynamics can also be used for the transport of empty sites, i.e., holes benseny_atomtronics_2010 . For the hole description to be valid, each trap must contain, at most, one atom in its vibrational ground state at all times, which can be achieved by either considering Pauli’s exclusion principle (for spin-polarized identical fermions) or by introducing a large enough interaction between atoms. In the case of two atoms in three identical traps, one can then construct a three-level model for the hole state, see Fig. 9(a), which supports a spatial dark state for the hole and therefore allows for SAP dynamics.
The efficiency of the process was numerically determined for both spin-polarized fermions and interacting bosons by solving the two-particle Schrödinger equation with the trapping potential given in Eq. (16) and an interaction potential of the form
Here is the -wave scattering length of the interaction and is the trapping frequency in the transverse direction.
The dynamics of the SAP transport of a fermionic hole in a triple well are shown in Fig. 9(b). The initial state corresponds to the hole being in the left trap (atoms in the middle and right traps, see Fig. 9(b-i)) and the final state corresponds to the hole being in the right trap (atoms in the left and middle traps, see Fig. 9(b-vi)). During the entire evolution, the diagonal of the configuration space has a node due to the antisymmetrization of the wavefunction (or the strong contact interaction for bosons). The fact that the counterdiagonal also has a vanishing population is a signature of the SAP process because it means that state , i.e., hole in the middle trap (Fig. 9(a-ii)) is not being populated.
The control over the dynamics offered by the interaction between the atoms and their spin state makes this two-atom system more versatile than the single atom counterpart. Taking advantage of these two control parameters, the hole SAP transport was proposed to implement two atomtronic devices: a diode, where the hole transport succeeds in only one direction, and a transistor, where the transport efficiency depends on the spin state of the atoms benseny_atomtronics_2010 . Furthermore, the system’s Hamiltonian can be generalized for both fermions and hardcore bosons by means of a Hubbard model considering the hole as an effective particle. This allows, in a similar manner to multilevel STIRAP/SAP shore_multilevel_1991 ; jong_coherent_2009 , to implement hole transport in arrays of (odd) traps containing atoms.
iii.2.6 Multiple particle transport
Another system that shows particle-induced nonlinearities in SAP is the Bose–Hubbard model bradly_coherent_2012 . For the case of particles across 3 wells, using the canonical SAP geometry, it is possible to express the allowed states in a triangular diagram, with the initial state , final state , and the state at the apex. Particles hopping from the left to centre wells are represented by diagonal lines going from bottom left to top right, while particles hopping from the centre to the right well are represented by lines from the top left to the bottom right. By using such a representation, Bradly et al. were able to show that the lowest two levels of the non-interacting Bose–Hubbard SAP process were equivalent to the alternating SAP process jong_coherent_2009 , and hence could be treated following the results given in Eq. (33) in Sect. III.3.1 (below). The results showed the usual hallmarks of SAP, namely particles moving from left to right wells without transient occupation of the central well.
Including particle-particle interactions into the model introduced energy gaps to the intermediate states, which in turn led to an increase in the time required for high-fidelity transport, and central well detuning could be used to mitigate the effects of particle-particle interaction to some extent. Since the interaction raises the overall energy of the system, additional care needs to be taken in order not to excite transitions to higher-lying bands.
iii.2.7 Coupled matter waveguides
Up to now we have seen that the external wavefunction of trapped ultracold atoms can be manipulated by a temporal variation of the coupling between potential traps. Analogously, one can engineer the couplings in space between matter waveguides by appropriately designing a fixed guiding structure, which allows to manipulate the propagation of an atomic wave packet. In particular, one can implement SAP techniques in a three-waveguide system eckert_three_2006 to transport an atomic wave packet between the two outer waveguides by engineering the couplings with position-dependent distances as shown in Fig. 10(a).
Considering that the separation between waveguides in the transverse direction, , varies slowly along the longitudinal direction, , the velocity of the atom along any of the waveguides can be approximated as its projection onto the -axis, . This ensures that the atomic longitudinal motion can not excite the transversal modes and it can therefore be decoupled from the transversal dynamics. Thus, the system can be effectively reduced in the -direction to a 1D triple well potential, analogous to the system presented in Sect. III.1.1. The validity of this assumption has been checked by the direct numerical integration of the time-dependent 2D Schrödinger equation with the waveguides modeled as truncated harmonic potentials in the transverse direction loiko_coherent_2014 . Figure 10(b) shows three consecutive snapshots of the 2D atomic probability distribution where, with the SAP counterintuitive approaching sequence, the atom is completely transferred from the transverse vibrational ground state of the waveguide to the same state of the waveguide. Realistic models close to present experiments for the implementation of this technique have been proposed for cold atoms in atom-chip waveguides osullivan_using_2010 ; morgan_coherent_2013 , or radiofrequency traps (for both cold atoms and BECs) which allow for a better control of the asymptotic energy states of each waveguide morgan_coherent_2011 , see Sect. III.6.
The SAP process depicted in Fig. 10(a,b) constitutes an injection protocol of a single cold neutral atom into a ring trap if we assume that the waveguide is part of a ring, see Fig. 10(c-left). It is easy to realize that extraction from the ring can be achieved by exchanging the role of the curved and ring waveguides, as depicted in Fig. 10(c-center) loiko_coherent_2014 . The two waveguides coupled to the ring can be switched on or off at will by simply turning on or off the laser field that generates them. As a consequence, these processes can be applied selectively when needed and with higher robustness and efficiency than in the case of simply spatially overlapping the ring and the input/output waveguides kreutzmann_coherence_2004 .
It is also possible to implement a velocity filter if one takes into account that the adiabaticity of the process is controlled by the atom’s injection velocity. Then, the adiabaticity condition, Eq. (20), depends on through the length of the interaction region, , which allows for the definition of a threshold longitudinal velocity loiko_coherent_2014
where is the maximum tunneling rate for the -th vibrational state. For injection velocities below , the process is adiabatic and the transfer succeeds, but atoms with higher velocities end up spread over the three waveguides. While the switch between these two behaviours is smooth, gives an estimate of the region of parameters for which SAP is efficiently performed. The velocity filter can then be implemented with the structure shown in Fig. 10(c-right), designed to perform a double SAP process: from the ring to the external waveguide and back to the ring. In this situation, slow atoms are able to adiabatically follow the spatial dark state and return to the ring trap with high fidelity, while faster atoms that do not fulfill the adiabaticity condition spread among the three waveguides.
The challenge of scalable quantum computing using spins LD98 ; Kan98 or charges HDW+04 in semiconductors introduced the need for long-range quantum transport. This need was either to satisfy the DiVincenzo criteria around flying qubits DiV00 , or to address gate density issues with donor in silicon approaches to quantum computing COI+03 . Electronic SAP provides one potential avenue to long-range transport in a quantum circuit and examples are discussed below. However the possibility of engineering the placement of sites, and hence tailoring the Hilbert space accessible to the electrons, provides other opportunities. Although it is not discussed in detail here, there is also related work on SAP in superconducting systems SB04 ; SBF06
The idea of adiabatically transferring electrons in an electrical circuit appears to have originated around 2000/2001 TRM2000 ; RTA2000 ; BRB01 . These early works considered a STIRAP-like process to transfer a charge through a double quantum dot system, and as such differ qualitatively from the SAP schemes that we are concentrating on systems that do not utilize electromagnetic driving. Nevertheless, this scheme forms an important bridge between STIRAP and SAP, and it is therefore instructive to review it in some detail.
Spatial STIRAP considers an engineered double quantum dot, as shown in Fig. 11, in which the electron is moved between the two molecular states. These two molecular states are labelled and , and each can be coupled to an excited state . Only the excited state is coupled to the source-drain leads, so that and are long-lived. Resonant radio-frequency fields are applied to induce STIRAP dynamics, which leads to adiabatic population control. However, this scheme does not provide net population transfer for complete transport, as the molecular states have equal populations in each of the dots, although it is straightforward to consider schemes where full population transfer would be realized. Because the excited state is coupled to source and drain leads, the proposal is analogous to STIRAP via ionizing states PH07 .
In contrast to STIRAP, SAP schemes involve spatially distinct start and end points, and the electronic proposals involve localized sites (e.g. dopants or quantum dots) with control of inter-site coupling via modulation of the tunnelling barrier, usually via electrostatic surface gates. The other important distinction between electronic SAP and SAP of neutral particles is the role of decoherence, and this will be discussed below. The first proposal that explored such transport was that of Ref. greentree_coherent_2004 , which considered long-range transport of an electron through either a chain of dopant ions (in particular, phosphorus in silicon) or quantum dots.
While the three-site transport mirrors other SAP proposals, the importance of Ref. greentree_coherent_2004 was the extension to chains of more than three sites via the straddling scheme MT97 , which has applications in phosphorus-in-silicon quantum computing. This scheme allows for long-range transport to be achieved without the necessity of applying gate control to the dopant sites within the chain, but only to the end-of-chain dopants. In this way, the overall gate density can be reduced, leading to a proposal for a scalable architecture utilizing adiabatic electronic transport for spin-based quantum computation HGF+06 . Adiabaticity arguments show that the time for transport across a chain of sites scales with to leading order GCH+05 . Extending electronic transport via the straddling scheme has also been discussed petrosyan_coherent_2006 ; jong_coherent_2009 .
Another major reason for exploring SAP for charge-based transport in phosphorus-in-silicon quantum architectures is related to the timescales for control and tunnelling. For a proposed inter-dopant spacing of about 20 nm, the hopping time is expected to be of order of 100 ps HGF+06 . For non-adiabatic control of population on such timescales, a bandwidth at least an order of magnitude faster than this is required, a task highly non-trivial for classical, cryogenic control electronics. Conversely, using adiabatic passage requires control pulses with a bandwidth at least an order of magnitude or two smaller than the hopping time. In this way, the full bandwidth implied by strong coupling between the dopants can be utilized without control at this bandwidth, provided the decoherence rates permit the longer adiabatic timescales (see below).
Hydrogenic arguments have been used for determining the relevant energy scales and disorder effects greentree_coherent_2004 ; HGF+06 ; VGA+10 . More sophisticated treatments have explored the microscopic properties of phosphorus in silicon for adiabatic passage more rigorously using the NEMO3D tight binding code rahman_atomistic_2009 , including disorder rahman_coherent_2010 . The main result from the NEMO3D simulations is the existence of extra molecular states for realistic phosphorus atoms, which are in general not detrimental to SAP.
Quantum dot systems provide considerable opportunities for transport protocols. In addition to the idealized proposal of greentree_coherent_2004 , more realistic transport in a triple square well system was analyzed in Refs. cole_spatial_2008 ; huneke_steady-state_2013 . SAP in exchange-only coupled quantum dots has also been proposed FMM+15 . Whilst electronic SAP is still yet to be demonstrated in any system, recent developments in coupled quantum dot systems suggest that this milestone will soon be achieved. The first designed triple dots in the one electron domain were shown by Schröer et al. SGG+07 , with further developments in Refs. BBR+2013 ; PTS+2015 .
As mentioned above, the key challenge to electronic SAP is to maintain the coherence of the system over a timescale long compared with the transport time. A naive estimate of the error rate yields the standard result that the transport error is simply the product of the total time with the decoherence rate, provided that the adiabatic limit is satisfied. A slightly more sophisticated calculation gives the error just less than this product, as the population is not in an equal superposition throughout the protocol HGF+06 . However, such calculations ignore some more realistic aspects of decoherence in the solid state, which highlight the fact that the role of practical decoherence is interesting and non-trivial.
To go beyond phenomenological models for decoherence requires a microscopic treatment. The first such treatment was given by Kamleitner et al. who considered both spatially-registered and non-Markovian noises kamleitner_adiabatic_2008 . Subsequent work by Rech and Kehrein rech_effect_2011 considered measurement backaction and Vogt et al. considered the related problem of STIRAP in the presence of two-level fluctuators vogt_influence_2012 . In all of these cases, the analyses show that although decoherence is important when applied to the ends of the chain, SAP is relatively immune to the effect of decoherence, whatever the cause (measuring devices or a fluctuating environment), that is applied solely to the central ‘bus’ states.
The relative robustness from decoherence arises from the suppression of population in the bus states, and is strongest in the case of the straddling scheme. For a multi-state chain with sites, labelled in increasing order from left () to right (), the general solution for the dark state is petrosyan_coherent_2006 ; jong_coherent_2009
where we have introduced as the (possibly time-varying) tunnel matrix element between site and , and the normalization
Note that this result guarantees that all of the even numbered sites will have precisely zero population in the adiabatic limit. To understand the robustness to decoherence, one can consider the case of the straddling scheme, where with . In this limit, and assuming , Eq. (31) reduces to the simple result MT97 ; greentree_coherent_2004
for . As can be seen, the form of this straddling dark state is the same as that for conventional three-state SAP, except for the correction term, which is suppressed by the factor. Now if we consider just the case of isolated ‘decohering centres’ (for example localized measurement devices or two-level fluctuators coupled to the moving charge), which are considered one of the main sources of decoherence in the solid state, then the only parts of that can be effectively measured are the ends of the chain, because these are the only parts of the system with appreciable population. This has significant importance for the design of high-fidelity quantum wires in the solid state, since the fragility of a SAP chain does not increase linearly with the length of the chain. Note, however, that the increased chain length implies a increase in transport time, and hence the decoherence environment of the ends of the chain needs to be proportionally longer GCH+05 .
iii.3.2 Gates, branches and electronic interferometers
Engineered quantum systems provide the opportunity to design interesting topologies, and in the following we review a few of the suggestions for multi-qubit gate operations, branched and interferometric configurations. The branched topology approach greentree_quantum-information_2006 ; WC15 , which is also referred to as mutliple-recipient adiabatic passage, MRAP, is a simple extension of conventional SAP, with the consideration of multiple end destinations for the excitation. These end points connect to a shared ‘bus’, and due to symmetry, an excitation can be distributed to an equally weighted superposition state of the end points. In its simplest form, where the ‘bus’ comprises a single site, and there are two final states, this scheme is very similar to the Unanyan, Shore and Bergmann (USB) approach to geometric gates unanyan_laser-driven_1999 (see also KR02 ; FSF03 ), which we discuss below.
The USB approach was introduced for realizing robust unitary gates for laser-driven tripod atoms, shown in Fig. 12. The motivation for this scheme is that in laser driven systems, although absolute intensities, and hence Rabi frequencies, are not always easy to control, relative intensities can be maintained with high accuracy, as can predefined phase shifts. changeThis overall motivation does not always translate across to SAP systems, for example in evanescently coupled waveguides where errors in position are expected to be uncorrelated. Nevertheless, this scheme is still explored for adiabatic gates.
The four-state USB Hamiltonian can be written in matrix form with basis ordering , , , , where is the shared excited state, as
and where it is assumed that the Rabi frequencies are real. The USB scheme assumes that the system is initially in state , and it seeks an adiabatic pulsing scheme for the that will perform a geometric gate operation. When and are non-zero, can be expressed in the canonical dark/bright basis,
where is the dark and is the bright state. Note that these states only depend on the ratio , which is kept constant during the USB scheme. The standard approach for understanding problems involving bright and dark states is to rewrite the Hamiltonian in this basis RLA99 . The USB scheme now works by employing a pulse sequence with applied first, and and together later, i.e. , and . This will adiabatically move the population in to , while the population in will not evolve. To achieve a non-trivial gate operation on the qubit subspace and , the evolution is then reversed with the addition of an extra phase shift. This is achieved in the optical implementation of the USB scheme by a non adiabatic phase shift in the Rabi frequency of the field effecting . In SAP, implementing this phase shift is more difficult, and seemingly restricted to purely real variation. Hope et al. HNM+15 proposed varying coupling to a ‘slab’ mode (discussed further in Sect. V.4). Another option would be to follow a return pathway via an extra two intermediate states, to pick up an additional minus sign, as can be seen from Eq. (31) below, although this does increase the complexity of the scheme.
The net result of the forward and backwards SAP with the symmetry breaking phase shift is a rotation gate acting on the qubit space defined by , , that is HNM+15
Because the are real, this gate can only perform rotations in the X-Z plane of the Bloch sphere, and hence arbitrary gate operations cannot be achieved using this method. The USB scheme can be extended to consider more general Morris–Shore type operations MS83 ; RVS06 .
Intriguingly, MRAP allows for a symmetry-breaking approach DGH07 , qualitatively different from what has been explored in the USB scheme. Starting from the simple example where a superposition state can be created by effectively performing SAP with two endpoints, rather than one, one immediately realizes than the reverse process trivially returns the system to its initial state. The symmetry between the forward and backward paths, however, is broken in the USB approach by introducing an additional -phase shift, which allows for the nontrivial gate rotation. However a mechanism for breaking symmetry between the pathways can also be provided by two-qubit interactions, see Fig. 13.
The system comprises a particle, for example an electron, which can exist in one of four possible states: initial state (for Alice), final states and (for Bob 1 and Bob 2), and central state . The Bobs also have qubits, and , and the idea is to use the transported electron as a means of mediating a two-qubit operator measurement between the qubits.
At the start of the protocol, the system is initialised with the electron in state , and the qubits and are in the states and respectively. Adiabatic passage through is then used create a superposition state at sites and , via
where we have introduced as the tunnel matrix element between and , and is the tunnel matrix element between and , and and . This brings the system into state
which is fully separable.
Because the electron is in a superposition of states, the application of two-qubit gates between the electron and the qubits will in general entangle the qubits. If we assume that the two-qubit gate takes the form of a controlled operation (e.g. CNOT) where the electron is the control and the qubit the target (), the system transfers to
While this state appears similar to , it is no longer separable, and therefore the reversal of the SAP process does not lead to the electron trivially returning to . Instead, it leaves the system in the state
The protocol then proceeds by performing a measurement to determine if the electron is at . If the electron is found at , then the qubits are projected into the entangled state . If the electron is not at , then a local rotation of the phase of the electron at (for example) is performed, and then the electron can be returned to via a reversal of the initial SAP process. In this case, the qubits are projected to the orthogonal (but known) entangled state . This approach can be used as a primitive for realizing more general operator-measurements, and hence may be useful for quantum error correction protocols.
A more direct approach to realising two-qubit gates has been proposed by Kestner and das Sarma kestner_proposed_2011 . In this protocol they consider a triangular triple dot configuration with two spins acting as qubits. Their aim is to effect a CNOT interaction between the qubits. This is achieved by a combination of SAP pulses, with spin-dependent tunnelling and local spin flips. The key advantage of the Kestner and das Sarma proposal is that due to the use of SAP, they predict significant robustness against fluctuations in the control parameters, and low frequency environmental noise.
A concept for interferometer-like behavior in a quantum dot network using a a Mach–Zehnder style configuration was studied in Ref. jong_interferometry_2010 . The authors considered a central ‘ring’ of four connected quantum dots, with initial and final dots connected to the outside of the ring. The system shows an interesting interplay between adiabatic and non-adiabatic features as the degeneracy (introduced by an additional pair of control gates) between the two arms of the interferometer is increased. For anti-symmetric detunings of the arms, this effects a kind of adiabatic electrostatic Aharanov–Bohm loop Boy73 , superimposed on robust adiabatic transport for all other detuning situations.
iii.4 Bose–Einstein condensates
Spatial adiabatic passage is not limited to the transport of single quantum particles and it has been considered, for instance, for Bose–Einstein condensates (BECs). A BEC is obtained when a dilute gas of identical bosons is cooled down to quantum degeneracy, and it exhibits a non-linear behavior due to interparticle interactions. In the limit of zero temperature and within the mean-field approximation, the BEC dynamics in a one-dimensional geometry can be described by a wavefunction that obeys the 1D Gross–Pitaevskii equation (GPE) pitaevskii_Bose_2003
where is the trapping potential and is the 1D non-linear interaction constant. The total number of atoms in the BEC is given by , () is the trapping frequency for the () harmonic confinement, and is the -wave scattering length. The normalization of the wavefunction is chosen to be .
The three-mode Hamiltonian for the BEC in a triple well potential can be derived in a similar manner as in Sect. III.1.1 by writing down the BEC wavefunction as a superposition of the localized eigenstates for the isolated wells with the probability amplitudes , where . Here, and are the number of atoms and the phase of the BEC in each well. Making use of the definitions (6) and (7) one then obtains the BEC three-mode Hamiltonian which is analogous to the three mode Hamiltonian (5), with the diagonal terms replaced by , where is the atomic self-interaction energy ottaviani_adiabatic_2010 .
Note that, as a consequence of the non-linear contributions in the diagonal of BEC three-mode Hamiltonian, the localized states in each trap are in general not resonant during the SAP sequence. Moreover, the nonlinearity leads to additional non-linear energy eigenstates, level crossing scenarios, and bifurcations that can break up the adiabatic following of the spatial dark state. Graefe et al. graefe_mean-field_2006 showed that by imposing , a complete SAP transfer of the BEC between the outermost wells of a triple-well potential can be achieved for and , where (see Fig. 14). Additionally, as discussed in morgan_coherent_2011 , time-dependent trapping frequencies can be considered to compensate for the time-dependent energy shifts that the non-linear interaction produces in each well. In the context of the three-mode Hamiltonian, the SAP sequence has also been discussed in a cyclic triple well potential nesterenko_stirap_2009 . The numerical integration of the GPE was used to study the fidelities for the SAP process in different parameter regimes rab_spatial_2008 and to show that nonlinearities can improve the sensitivity of matter-wave interferometers based on SAP rab_interferometry_2012 .
For adiabatic passage of a BEC within the two mode approximation nesterenko_adiabatic_2009 ; ottaviani_adiabatic_2010 ; nesterenko_adiabatic_2010 , one obtains the same Hamiltonian as for the single particle case, see Eq. (24), but now replacing by . Thus, there exists a dark variable with a mixing angle that now reads . In Ref. ottaviani_adiabatic_2010 , this model is used to investigate the robust splitting (varying from to ), transport (varying from to ), or inhibiting transport (varying from to through arbitrary intermediate values of ) of a BEC initially located in the left site of a double-well potential. Temporal control of the mixing angle can now be achieved via a temporal variation of either the energy bias or the nonlinear interaction with respect to the tunneling rate. The dynamics were investigated from a nonlinear systems point of view, by deriving the stationary solutions, performing a linear stability analysis, and discussing possible bifurcation scenarios. In nesterenko_adiabatic_2009 ; nesterenko_adiabatic_2010 protocols generalizing the Landau–Zener and Rosen–Zener schemes were presented, stressing the role of the nonlinearity for transport in double-well potentials.
iii.5 SAP in two dimensions
In the following, we will review recent results related to SAP in two-dimensional systems. Considering more spatial dimensions can lead to new scenarios where additional couplings can be obtained.
In fact, in the triple-well case, the simultaneous coupling of all traps during the SAP procedure can be achieved by designing triangular trapping geometries. In quantum optics, to couple the two ground states of a system requires advanced electric or magnetic couplings giannelli_three_2014 or the use of chiral molecules kral_cyclic_2001 ; kral_two-step_2003 .
iii.5.1 Triangular trap configuration
The simplest two-dimensional system that can be considered is formed by three harmonic potentials (labeled , and ) with equal trapping frequencies forming a triangle menchon-enrich_single-atom_2014 , as schematically shown in Fig. 15. The tunneling rates between traps and are denoted by , and their explicit dependence on the distance is given by Eq. (18) in Sect. III.1.1. Assuming that the dynamics of the system are restricted to the space spanned by the localized ground states of the three traps, (see Sect. III.1.1), the Hamiltonian that governs the particle’s evolution can be written as
Diagonalizing this Hamiltonian gives the energy eigenvalues
where and , and . The corresponding eigenstates read
For , which means , the system recovers the same expressions as for the 1D SAP case, see Sect. III.1.1.
Starting with a single atom located in the vibrational ground state of trap , and keeping trap fixed, the SAP sequence consists in initially approaching and separating traps and . Later on, and with a certain temporal delay , traps and are approached and separated, keeping the angle fixed. This variation of the distances between traps leads to the familiar counterintuitive sequence of couplings and also introduces a coupling between traps and , see Fig. 16(a).
From the analytical expression of the eigenstates (44) one can see that at the particle is in state . Following this eigenstate, see Fig. 16(b), at the end of the adiabatic process, the atom will be in state , and full transfer is achieved. As one can see from Fig. 16(d), this process is successful in the interval . For , the three tunneling rates become identical at a certain time during the dynamics, which corresponds to the appearance of a level crossing in the spectrum, see Fig. 16(c). In this case, it is no longer possible to adiabatically follow , and the system is instead transferred to . At the end of the process, the particle state will therefore be in an equal superposition of the ground states of traps and . For the level crossing is again avoided but for large the transfer of the atom between traps and fails again, because the coupling strength is significant during the first stage of the process and leads to unwanted Rabi-like oscillations. This behaviour has been confirmed by numerically integrating both the Hamiltonian (42) and the 2D Schrödinger equation, obtaining a very good agreement as shown in Fig. 16(d).
The coherent splitting of the atomic wavefunction between the traps and occurring for in the triangular configuration of traps can be used to design an interferometer to measure spatial field inhomogeneities. After the splitting, a relative phase is picked up by the parts of the wavefunction in traps and , leading to a state of the system of the form , which can be decomposed in a superposition of and . Reversing the temporal evolution of the SAP couplings, the contribution of will be transferred to at the level crossing and end up in trap , while the contribution of will evolve backwards and at the end of the process will be in a superposition of traps and . By then measuring the population of the three traps, one can infer the relative phase before the recombination. Simulating this process, very good agreement between a numerically imprinted phase and the one obtained from the populations has been obtained, demonstrating the excellent performance of this system as a matter-wave interferometer menchon-enrich_single-atom_2014 .
iii.5.2 Generation of angular momentum
In addition to the already discussed possibilities for quantum state preparation through SAP processes in 1D (see Sect. III.2.3), SAP in 2D can also be used to create and control angular momentum menchon-enrich_tunneling-induced_2014 . For this, one can consider a geometrical arrangement of three 2D harmonic traps, analogous to the one depicted in Fig. 15, but with the trapping frequency of trap chosen as half that of traps and . In this configuration, the ground energy levels in traps and are resonant with the first excited level in trap , allowing for a large tunnel coupling between them. Since the first excited energy level of trap is doubly degenerate, it supports an angular momentum carrying state through a superposition of the two energy eigenstates and in the chosen – reference frame. In particular, maximum angular momentum, , occurs when these two degenerate states are equally populated and have a phase difference of .
In the considered triangular configuration, tunneling into the first excited states of trap is described by the rates , , and menchon-enrich_tunneling-induced_2014 . Therefore, the dynamics of the system in the basis of the asymptotic states of the traps, , can be described by the Hamiltonian
The generation of angular momentum occurs during the SAP transfer of the particle from trap to trap , with the same trap movement as described in Sect. III.5.1. The initial state of the system (particle in trap ) can be written as a superposition of two of the eigenstates of the system. Thus, if the process is adiabatic and level crossings are absent, this superposition of eigenstates is followed all through the process, leading to a final state that only involves the asymptotic states of trap , and . In fact, the two asymptotic states of trap are equally populated at the end of the process, with a phase difference, , proportional to the total time of the process . The expectation value of the angular momentum can be found to be , which allows to control the angular momentum of the system by choosing an appropriate value for , see Fig. 17.
iii.5.3 Two-dimensional lattices
SAP can also be applied between distant sites of 2D rectangular and triangular lattices by dynamically controlling the tunneling rates. For this, one maps the motion of a particle in the 2D lattice to the Fock space dynamics of a second-quantized Hamiltonian for appropriate bosonic fields longhi_coherent_2014 . To illustrate the procedure, one can consider a rectangular lattice in the tight binding and nearest-neighbor approximations. In this case, the temporal evolution equations for the probability amplitudes for finding the particle in each of the sites can be derived from the time-dependent Schrödinger equation with a quadratic Hamiltonian. This requires the introduction of six coupled bosonic oscillators consisting of products of annihilation and creation operators of independent bosonic modes and of a state vector which is a linear superposition of nine states resulting from the application of two different creation bosonic operators onto the vacuum. The Heisenberg equations of motion for the six bosonic field operators are formally analogous to two sets of three-level equations for SAP, provided that the operators are replaced by c-numbers longhi_coherent_2014 . Thus, the adiabatic evolution of the operators leads to adiabatic passage in Fock space. This analysis can also be extended to rectangular lattices (with an odd number of sites in each direction) and to triangular lattices longhi_coherent_2014 .
iii.6 Practical considerations
Implementing SAP techniques requires identification of experimental systems that have localized asymptotic states between which a controllable time-dependent coupling exists. To achieve this, one typically considers trapping potentials in which the low energy part of the spectrum is experimentally accessible. Changing either the distance between adjacent traps or the height of the separating barrier between them allows to create a time-dependent tunneling coupling. The first possibility is usually considered in microtrap systems for atoms or ions eckert_three-level_2004 , whereas the latter one is more suitable for electrons in quantum dots greentree_coherent_2004 .
A second requirement is that the system must posses a state, such as the spatial dark state, that allows transfer. From the three-state model it is clear that this is fulfilled if the asymptotic states for each potential at any point during the process are in resonance. However the exact diagonalisation of the Schrödinger equation shows that high fidelity transport can still be achieved in the presence of a small detunings between the traps. If these detunings are too large, however, effective tunneling between the potentials is prevented and the transport will break down.
From an experimental point of view the second requirement means that the trapping potentials should not change significantly over the whole process. Changing the distance between microtraps, however, leads to certain overlap between neighboring traps which can alter the shape of the individual potentials. This can lead to level crossings in the spectrum that can make following of the dark state harder, or detunings that become too large for the process to be efficient. Ensuring that this does not happen usually involves significant experimental resources or restrictions on the parameter space eckert_three-level_2004 ; eckert_three_2006 ; rab_spatial_2008 .
Finally, the whole SAP process has to be carried out adiabatically, so that the system does not leave the lowest band or the dark state at any point during the evolution. This requires precise dynamical control as the time scales for the process need to be chosen such that they fulfill an adiabaticity condition, like the one given in Eq. (20), and are shorter than the life- and coherence times of the system.
iii.6.1 Optical traps
One of the first experimental systems suggested for observing SAP were optical potentials generated through a microlens array eckert_three-level_2004 . By illuminating the lenses with red-detuned laser light, ultracold atoms can be trapped in the focus above the lens, with typical trapping frequencies for Rb atoms on the order of s in the transverse directions and s along the laser beam direction birkl_atom_2001 ; dumke_micro-optical_2002 . This means that the traps can be adiabatically approached in the millisecond range or even faster by using optimization techniques.
The distances between the individual traps can be adjusted by using separate laser beams illuminating each lens. However, this also leads to an increased overlap of the optical intensities and therefore to a significant distortion of the resulting potential. The resonances between the individual traps are then no longer guaranteed and one needs to consider the use of additional compensation techniques to restore them. For example, a simple and robust manner to do this in a system of three Gaussian-shaped dipole traps is to dynamically adjust the depth of the center trap eckert_three_2006 , see Fig. 18.
iii.6.2 Atom chips
As discussed in Sect. III.2.7, SAP processes can also be implemented for particles moving in a waveguides system by changing the distances between three different waveguides in the direction of propagation. Particle waveguides can be created using optical or magnetical potentials and in morgan_coherent_2013 a system of three waveguides on an atom chip was investigated. Atom chips are versatile experimental tools that are by today used extensively in experiments with ultracold atoms. A small current flowing through nanofabricated wires on the substrate produces a magnetic field gradient in such a way that cold atoms can be trapped very close to the surface. Because the layout of the nanowires can be chosen during the chip production process and the currents are small, they allow for highly stable potential generation AtomChipReview .
An example of a waveguide potential for Li atoms and for experimentally realistic parameters is shown in Fig. 19. If an atom is injected in the left waveguide, and moves in the positive direction, these waveguides provide the desired counterintuitive tunnel coupling needed to transfer it to the right waveguide. However, this system also suffers from the effect that the overlap between neighboring waveguides leads to a deformation of the individual potentials (see Fig. 19(a)), and consequently to a breakdown of resonance. Similarly to the optical traps discussed above, a reduction of the current in the middle wire allows to compensate for this (see Fig. 19(b)).
If one considers a wavepacket traveling in the waveguides, a couple of additional points need to be considered: the wavepacket must be launched with an initial velocity, and its dispersion in the longitudinal direction needs to be compensated. Both of these tasks can be dealt with by adding a harmonic potential along the direction centered at the middle of the chip. This will allow to launch the wavepacket and also lead to a refocusing at the classical turning point on the other side of the chip. In morgan_coherent_2013 it was shown that the process is adiabatic if its total time is taken to be much larger than the inverse of the transverse trapping frequencies of the individual waveguides. Choosing realistic parameters for the atom chip (waveguide length in the direction 1 mm, initial separation between waveguides 7 m, transverse trapping frequencies kHz, frequency of the longitudinal harmonic potential Hz), this time was shown to be 0.1 s. Even though the bend in the waveguide couples the longitudinal and transversal directions, the large difference in trapping frequencies ensures that this coupling is small.
Finally, the bend in the wires will also lead to a potential from the currents in the -direction (note that this is absent in the optical waveguide realization, discussed below), which requires the atom to have enough kinetic energy to overcome it. This has direct consequences for being able to fulfill the adiabaticity condition. However, this local potential maximum can be reduced by increasing the length of the atom chip (-direction) and therefore reducing the curvature of the wires.
Full 3D simulations with experimentally realistic parameters were carried out in morgan_coherent_2013 and confirmed that such systems allow to achieve SAP with very high fidelities for appropriate tunings of the currents in the individual waveguides.
iii.6.3 Radio-frequency traps
A system that is more robust against distortion from the overlap of two neighbouring potentials than optical traps can be constructed by radio-frequency (rf) traps, which rely on coupling magnetic sublevels in the presence of an inhomogeneous magnetic field Zobay_RF_Traps_2001 ; Schumm:05 ; Lesanovsky:07 ; Zimmermann:06 . This coupling leads to avoided crossings in the energy spectrum and therefore to minima and maxima in the potential landscape the atoms see. As the couplings are spatial resonances, changes in the eigenspectrum only alter the potentials locally and do not affect the overlap with neighbouring traps.
To produce an rf potential with three minima along the -direction, it is necessary to employ six different radio frequencies. Moving the traps can be achieved by changing the individual rf frequencies that are associated with each trap. An exact sequence that allows to realize the SAP transfer was given in morgan_coherent_2011 , where it was also shown that high transfer fidelities can be achieved without the need for any additional compensation potentials.
The use of rf traps also allows to design a realistic setup to extend SAP to non-linear systems, for example Bose–Einstein condensates (BECs) (see Sect. III.4). Here the Hamiltonian in the three-level approximation has a non-linear term in the diagonal which is as a function of the particle numbers in each trap. As these numbers are a function of time, the non-linear terms will change during the SAP process and therefore modify the resonances between the traps. A straightforward way to compensate for this is to allow for the trapping frequencies to be functions of time as well morgan_coherent_2011 . Starting with the BEC in the left trap, will decrease during the process, while will increase. Adjusting the trapping frequencies and can restore the resonance between the uncoupled traps by ensuring that is approximately constant at all times. However, in order to be able to make the three-level approximation, one must ensure that for all values of and . This means in practice that the process is limited to cold atomic clouds with small nonlinearities. A detailed description of this process is given in morgan_coherent_2011 . Note that a different setup for SAP transfer of BECs using optical traps was described in graefe_mean-field_2006 .
iii.6.4 Speeding up adiabatic techniques
As discussed above, the robustness of adiabatic processes such as SAP make them very useful tools to prepare and manipulate quantum states. However, the price for this are the long time scales these processes require, i.e., the fact that the inverse of the total time has to be much larger than all characteristic frequencies of the system. At first look, this makes adiabatic processes uninteresting for quantum information processing because they not only require highly efficient protocols, but also fast transport processes between gate operations. Slow processes can be problematic because decoherence and noise can affect the system, leading to final states with reduced fidelity. Therefore, it is desirable to develop techniques which are fast and lead to high fidelities.
One such group of techniques are based on optimal control theory (OCT) algorithms. These work by determining the time dependence of a number of control parameters which minimizes a cost function, e.g. the infidelity of a process, subject to some constraints such as the initial state or the maximum coupling strengths. OCT has been applied to speed up many adiabatic processes salamon_maximum_2009 ; hohenester_optimal_2007 ; murphy_high-fidelity_2009 ; rahmani_optimal_2011 , and in particular SAP transport of a BEC in a realistic model of Gaussian-shaped optical microtraps negretti_speeding_2013 . In this work, the authors divided the entire SAP process into three steps and optimized them individually via the CRAB algorithm caneva_chopped_2011 ; doria_optimal_2011 . These three steps consisted of (i) the initial approach between all traps before tunneling set in, (ii) the actual SAP process, and (iii) the final separation of the traps once the SAP dynamics are finished. Furthermore, the optimization algorithms allowed to take the atomic interactions into account and showed that a significant speed up could be achieved without compromising high fidelities.
Another group of techniques are shortcuts to adiabaticity (STA), which were first introduced in Ref. chen_fast_2010 to describe different protocols that allow to speed up adiabatic processes torrontegui_shortcuts_2013 . These rely on a number of different approaches, such as counterdiabatic or transitionless tracking algorithms demirplak_adiabatic_2003 ; demirplak_assisted_2005 ; demirplak_consistency_2008 ; berry_transitionless_2009 , the use of the Lewis–Riesenfeld invariants Lewis_1969 ; chen_fast_2010