Shortcuts to adiabaticity in the double well
Sofía Martínez Garaot
Prof. Juan Gonzalo Muga Francisco
Departamento de Química-Física
Facultad de Ciencia y Tecnología
Universidad del País Vasco/Euskal Herriko Unibertsitatea
Leioa, Febrero 2016
“El aprendizaje es experiencia, todo lo demás es información.”
Acknowledgements.Al final de este largo camino me gustaría agradecer a todos aquellos que de alguna forma u otra han estado ayudándome y apoyándome. Mi primera idea era poner una larga lista de nombres en la que quedasen reflejadas todas y cada una de las personas que me han acompañado durante esta etapa pero finalmente he pensado que todo aquel que me conoce encontrará un hueco donde ubicar su nombre en estas líneas. En primer lugar tengo que agradecer a mi director por haber confiado en mí y haberme guiado durante estos años. A mis colaboradores por haberme enseñado tanto. A mis compañeros de despacho porque todos ellos son parte de este trabajo. Al resto de compañeros de la universidad por todos los momentos vividos. A todas mis amigas y amigos por estar siempre a mi lado. A mi familia, a mis padres y a Pau. Y por último, a ti, por animarme cada día. Gracias a todos. Esta Tesis ha sido financiada a través de una beca predoctoral concedida por la Universidad del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU).
- Published Articles
- \thechapter Engineering fast and stable splitting of matter waves
- \thechapter Shortcuts to adiabaticity in three-level systems using Lie transforms
- \thechapter Fast quasi-adiabatic dynamics
- \thechapter Vibrational mode multiplexing of ultracold atoms
- \thechapter Compact and high conversion efficiency mode-sorting asymmetric Y junction using shortcuts to adiabaticity
- \thechapter Fast bias inversion of a double well without residual particle excitation
- \thechapter Interaction versus asymmetry for adiabatic following
- \thechapter Lie algebra
List of Figures
- 1 Contour plot of in units from Eq. (11) for (a) a three-well interpolation and (b) a -shaped form. Parameters: rad/s, and ms.
- 2 Different fidelities versus the perturbation parameter for the FF approach (lines) and the two-mode model (symbols). : (blue) long-dashed line and circles; : (red) solid line and squares; : (black) short-dashed line and triangles; : (green) dotted line and diamonds. The vertical (orange) line is at . (a) ms. (b) ms. (c) ms. rad/s.
- 3 Fidelities vs dimensionless coupling constant for , ms, and m. Lines are the same as in Fig. 2. Symbols are for a two-level model like Eq. (13) with the nonlinear diagonal terms added, where and are populations for left and right states . The vertical line is at ; see the Appendix \thechapter.
- 4 Fidelities for a Bose-Einstein condensate; lines are the same as in Fig. 2. Equation (23) is used to design the potential . Parameters: m, rad/s, =0.138, and ms.
- 5 Functions in : (a) and (b) . Parameters: where is the maximum value of and .
- 6 Bare-state populations for (a) ; (b) and . (red circles), (short-dashed blue line) and (solid black line). Parameters: with the maximum value of , .
- 7 (a) Interaction energy for the reference Hamiltonian (solid green line) and for (short-dashed green line). (b) Hopping energy for (solid magenta line) and (short-dashed magenta line). The same parameters as in Fig. 5.
- 8 Schematic representation of a beam splitter.
- 9 (a) and (b) . where is the maximum value of . .
- 10 Bare-state populations for (a) , and (b) and . (red circles), (short-dashed blue line) and (solid black line). Parameters: with the maximum value of , and .
- 11 (a) Interaction energy for (solid green line) and (short-dashed green line). (b) Hopping energy for (solid magenta line) and (short-dashed magenta line). The same parameters as in Fig. 9.
- 12 Schematic representation of the beam splitter.
- 13 (a) and (b) . , where is the maximum value of . .
- 14 Bare-state populations for (a) , and (b) and . (red circles), (short-dashed blue line) and (solid black line). Parameters: with the maximum value of , and .
- 15 (a) Interaction energy for (solid green line) and (short-dashed green line). (b) Hopping energy for (solid magenta line) and (short-dashed magenta line). The same parameters as in Fig. 13
- 16 (a) Bias vs for linear-in-time bias (green triangles), pulse (short-dashed red line), and FAQUAD (solid black line). (b) Final ground-state population vs for linear-in-time bias (green triangles), pulse (short-dashed red line), and FAQUAD (solid black line). (c) Bias vs for FAQUAD (solid black line), LA approach (blue dots), and UA approach (long-dashed magenta line). The inset amplifies the kink of the UA approach. (d) vs for FAQUAD (solid black line), LA approach (blue dots), and UA approach (long-dashed magenta line). The stars in (b) and (d) correspond to integer multiples of the characteristic FAQUAD time scale . , .
- 17 (a) Schematic representation of splitting from to . (b) Cotunneling from to .
- 18 (a) Energy levels vs . For : (solid magenta line), (long-dashed green line), and (short-dashed orange line). . (b) vs for linear-in-time bias (green triangles) and FAQUAD (solid green line). , , and .
- 19 (a) Time dependence of the bias with FAQUAD. (b) vs for linear-in-time bias (green triangles) and FAQUAD (solid green line). , , and .
- 20 (a) Single-particle energy levels for (dashed lines) and (solid lines) in units of . The ordering is . (b) for , from the bottom up to the top.
- 21 (a) Fidelity for [FAQUAD (solid black line) and linear (short-dashed red line)] and [FAQUAD (blue circles) and linear (green triangles)]. is the time-evolved TG state starting from the ground state for , and is the ground state of the TG gas at . (b) Fidelity vs if FAQUAD is applied following a wrong for (solid black line) and (short-dashed red line). Here .
- 22 Population inversion using trap deformations in three steps: demultiplexing, bias inversion, and multiplexing.
- 23 (a) and (b) . Hz, s, s, and ms.
- 24 Lattice height , and trap frequency using invariant-based engineering and mapping. nm.
- 25 (a): Ground state at (long-dashed, blue line); final state with the shortcut (solid, blue line, indistinguishable from the ground state of the final trap); final state with linear ramp for and Hz (short-dashed, magenta line). (b): Same as (a) for the first excited state. Parameters like in Fig. 24 at ms. The linear ramp for ends in the same value used for the shortcut.
- 26 Fidelities with respect to the final ground state starting at the ground state (a) and with respect to the final first excited state starting at the excited state (b) versus final time , via shortcuts ( and , blue circles), or linear ramping of ( and , red triangles). The fidelity is computed at 2 ms less than the nominal . Other parameters as in Figs. 23, 24, and 25.
- 27 Populations of the states for the shortcuts (a) and the linear ramp for (b). Ground state (, solid blue line); first excited state (, long-dashed red line); second excited state (, short-dashed black line). Parameters as in Fig. 25 (a).
- 28 Schematic of the asymmetric Y junction.
- 29 Conversion efficiencies of a linearly separating Y junction using the second mode as the input for different device lengths
- 30 Parameters for the invariant-based Y junction.
- 31 Mode-sorting operation of the invariant-based Y junction. Input (a) fundamental mode (b) second mode.
- 32 Mode-sorting operation of the linearly separating Y junction. Input (a) fundamental mode (b) second mode.
- 33 Output field profile of the Y junctions. Solid: invariant-based. Dashed: linearly separating. Dash-dotted: waveguide walls. Input (a) fundamental mode (b) second mode.
- 34 Conversion efficiencies as a function of width variation using the second mode as the input.
- 35 Schematic representation of demultiplexing (left arrow), bias inversion (framed in dashed line, central arrow), and multiplexing (right arrow). The densities of two one-atom eigenstates are represented in all potentials. In the harmonic potentials (unframed potentials on the left and right charts) the states are the ground state and first excited state. In the two central charts with tilted double wells the states are the lowest for each well. The color (white or gray) indicates how they would evolve sequentially following the fast protocol described in the text. For example, the gray state is initially the ground state of the harmonic oscillator, then it becomes the lowest state of the left well, and remains being the lowest state of that well after the bias inversion. In the last step it becomes the first excited state of the final harmonic oscillator.
- 36 versus for the polynomial in Eq. (180) (solid black line) and FAQUAD (short-dashed red line). zN, , pN/m, and mN/m. Also shown are the different effective slopes adding a compensation to the polynomial, , for the mass of Be and times s (long-dashed blue line); s (green dots); and s (magenta squares).
- 37 Left: Ground state of the left well at (long-dashed magenta line) and at (magenta triangles), and final state with the compensating force applied on the double well (solid blue line). Right: Ground state of the right well: at (short-dashed red line) and at (red dots) and final state with the compensating force applied (solid black line). ns and other parameters as in Fig. 36 for Be.
- 38 (a) Fidelity , where is the lowest state located in the left well in the final time configuration, and is the evolved state following the shortcut at final time. (b) Final excitation energy for the process on the left well using compensating-force (blue dots), fifth degree polynomial in Eq. (180) (solid black line), and FAQUAD (short-dashed red line). The parameters are for Be as in Fig. 36.
- 39 Evolution of the wave function densities following the shortcut in Eq. (191) for states in left and right wells. The parameters are for Rb: m, Hz, kHz, nm, and s.
- 40 (a) Fidelity , where is the lowest state predominantly of the left well at final time (the first excited state of the double well) and is the evolved state following the shortcut at final time. (b) Final excitation energy. Compensating-force approach (blue dots), fith degree polynomial in Eq. (180) with the change without compensation (solid black line), and FAQUAD approach (short-dashed red line). The parameters are chosen for Rb: m, Hz, kHz, and nm.
- 41 Structural fidelities for the Bose-Einstein condensate. From left to right, , and . In all curves m and rad/s. Equation (23) was used to design the potential .
Chapter \thechapter List of publications
I) The results of this Thesis are based on the following articles
E. Torrontegui, S. Martínez-Garaot, A. Ruschhaupt, and J. G. Muga
Shortcuts to adiabaticity: Fast-forward approach
Phys. Rev. A 86, 013601 (2012).
E. Torrontegui, S. Martínez-Garaot, M. Modugno, X. Chen, and J. G. Muga
Engineering fast and stable splitting of matter waves
Phys. Rev. A 87, 033630 (2013).
S. Martínez-Garaot, E. Torrontegui, X. Chen, M. Modugno, D. Guéry-Odelin, S.-Y. Tseng, and J. G. Muga
Vibrational Mode Multiplexing of Ultracold Atoms
Phys. Rev. Lett. 111, 213001 (2013).
S. Martínez-Garaot, S.-Y. Tseng, and J. G. Muga
Compact and high conversion efficiency mode-sorting asymmetric Y junction using shortcuts to adiabaticity
Opt. Lett. 39, 2306 (2014).
S. Martínez-Garaot, E. Torrontegui, and J. G. Muga
Shortcuts to adiabaticity in three-level systems using Lie transforms
Phys. Rev. A 89, 053408 (2014).
S. Martínez-Garaot, A. Ruschhaupt, J. Gillet, Th. Busch, and J. G. Muga
Fast quasiadiabatic dynamics
Phys. Rev. A 92, 043406 (2015).
S. Martínez-Garaot, M. Palmero, D. Guéry-Odelin, and J. G. Muga
Fast bias inversion of a double well without residual particle excitation
Phys. Rev. A 92, 053406 (2015).
II) Other articles produced during the Thesis period
Published Articles not included in this Thesis
S. Ibáñez, S. Martínez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga
Shortcuts to adiabaticity for non-Hermitian systems
Phys. Rev. A 84, 023415 (2011).
S. Ibáñez, S. Martínez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga
Erratum: Shortcuts to adiabaticity for non-Hermitian systems
Phys. Rev. A 86, 019901 (2012).
E. Torrontegui, S. Martínez-Garaot, and J. G. Muga
Hamiltonian engineering via invariants and dynamical algebra
Phys. Rev. A 89, 043408 (2014).
M. Palmero, S. Martínez-Garaot, J. Alonso, J. P. Home, and J. G. Muga
Fast expansions and compressions of trapped-ion chains
Phys. Rev. A 91, 053411 (2015).
M. Palmero, S. Martínez-Garaot, U. G. Poschinger, A. Ruschhaupt, and J. G. Muga
Fast separation of two trapped ions
New J. Phys. 17, 093031 (2015).
III) Review Articles
E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga
Shortcuts to adiabaticity
Adv. At. Mol. Opt. Phys. 62, 117 (2013).
During the last three decades, the research in quantum optics has experienced a phenomenal boost, largely driven by the rapid progress in microfabrication technologies, precision measurements, coherent radiation sources, and theoretical work. Many quantum optical systems are employed to test and illustrate the fundamental notions of quantum theory. They have also practical applications for communications, quantum information processing, metrology and the development of new quantum-based technologies, whose physical aspects have by now become an integral part of quantum optics. Frequently, the manipulated systems are quite simple, such as one or a few ions or neutral atoms in harmonic or double wells. Bose-Einstein condensates involve of course many more atoms, but may still be described by mean-field theories. Controlling these systems accurately has become a major goal in contemporary Physics. Serge Haroche and David J. Wineland won the Nobel Prize in 2012 after developing methods for manipulating individual ions in Paul traps or photons in cavities while preserving their quantum-mechanical nature.
This Thesis contributes to this goal by proposing fast operations for one to few ultra cold atoms, or Bose-Einstein condensates, in a double well potential, extending the results as well to optical waveguide systems. “Fast” is to be understood with respect to adiabatic processes. The “adiabatic” concept may have two different meanings: the thermodynamical one and the quantum one. In thermodynamics, an adiabatic process is the one in which there is no heat transfer between system and environment. In quantum mechanics, as stated by Born and Fock (1928) in the adiabatic theorem: “a physical system remains in its instantaneous eigenstate when 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”. In terms of the instantaneous eigenvalues and their corresponding instantaneous eigenvectors , the adiabaticity condition, i.e., the condition that has to be satisfied to follow the adiabatic dynamics, can be written as
In this Thesis, we shall always understand “adiabatic” in the quantum-mechanical sense. Quantum adiabatic processes are in principle useful to drive or prepare states in a robust and controllable manner, and have also been proposed to solve complicated computational problems. However, they are prone to suffer noise and decoherence or loss problems due to the long times involved. This is often problematic because some applications require many repetitions or too long times.
Shortcuts to adiabaticity (STA) are alternative fast processes that reproduce the same final populations, or even the same final state, as the adiabatic process in a finite, shorter time. The expression “shortcut to adiabaticity” was introduced in 2010 by Chen et al.  to describe protocols that speed up a quantum adiabatic process, usually, although not necessarily, through a non-adiabatic route. There are many different approaches to design the shortcuts. For example, the counterdiabatic or transitionless tracking approach formulated by Demirplak and Rice (2003, 2005, 2008) [2, 3, 4] or independently by Berry (2009) , based on adding counterdiabatic terms to a reference Hamiltonian to achieve adiabatic dynamics with respect to . Moreover, Lewis-Riesenfeld invariants (1969)  were used to inverse engineer a time-dependent Hamiltonian from the invariant . Masuda and Nakamura (2010) developed a “fast-forward technique” for several manipulations . There are also alternative methods that use the dynamical symmetry of the Hamiltonian or based on distributing the adiabaticity parameter homogeneously in time, or Optimal Control Theory (OCT) . In this Thesis I will not only apply these existing methods but also develop new ones.
Since adiabatic processes are ubiquitous, the shortcuts span a broad range of applications in atomic, molecular, and optical physics, such as fast transport, splitting and expansion of ions or neutral atoms; internal population control, and state preparation (for nuclear magnetic resonance or quantum information), vibrational mode multiplexing or demultiplexing, cooling cycles, many-body state engineering or correlations microscopy . The Thesis focuses on the double well potential, which is an interesting model to study some of the most fundamental quantum effects, like interference or tunneling. Using utracold atoms it has become possible to study the double well at an unprecedented level of precision and control. This has allowed the observation of Josephson oscillations, nonlinear self-trapping and recently, second-order tunneling effects. Few-body systems are lately of much interest as they enable us to study finite-size effects for a deeper understanding of the microscopic mechanics in utracold atoms, and for the possibility of realizing operations involving a few qubits. Also, double wells for single atoms and Bose-Einstein condensates have been used for precise measurement in interferometry experiments. For trapped ions, the double well is used to implement basic operations for quantum information processing, for example, separation or recombination of ions, Fock states creation, or tunable spin-spin interactions and entanglement.
The Thesis is divided into six chapters: The first chapter is devoted to fast splitting of matter waves. The fast-forward approach is applied to speed up the process and a two dynamical-mode model is introduced. This two-mode model will be an important test-bed model during the whole Thesis. Linear and non-linear matter waves (interacting Bose-Einstein condensates) are studied. Chapter 2 deals with an interacting few-body boson gas in a two-site potential. In particular, we investigate how to accelerate an insulator-superfluid transition and the implementation of a and beam splitter. To achieve these goals, a new STA method based on Lie transforms is worked out. In chapter 3 I present one more new STA method that uses the time dependence of a control parameter to delocalize in time the transition probability among adiabatic levels. Some general properties are described and the approach is used to speed up basic operations in three different systems: a two-mode model, interacting bosons in a double well and a few-particle system on a ring. In chapter 4 the invariant-based inverse engineering approach is used to accelerate multiplexing or demultiplexing processes. The shortcut is designed in the two-mode model and then it is mapped into a realizable coordinate potential. Chapter 5 extends the results of the previous chapter to optical wave guides systems. Finally, chapter 6 provides a strategy based on the compensating force-approach to implement a fast bias inversion both in neutral atoms and in trapped ions. Combining this fast bias inversion with fast multiplexing and demultiplexing processes, population inversions using only trap deformations can be achieved.
Due to the length of the manuscript and the different topics discussed, the notation is consistent within each chapter, but not necessarily throughout the Thesis.
Chapter \thechapter Engineering fast and stable splitting of matter waves
When attempting to split a coherent noninteracting atomic cloud by bifurcating the initial trap into two well separated wells, slow adiabatic following is unstable with respect to any slight trap asymmetry, and the matter wave “collapses” to the deepest well. A generic fast chopping splits the wave but it also excites it. Shortcuts to adiabaticity engineered to speed up the unperturbed adiabatic process through nonadiabatic transients provide, instead, quiet and robust balanced splitting. For a Bose-Einstein condensate in the mean-field limit, the interatomic interaction makes the splitting, adiabatic or via shortcuts, more stable with respect to trap asymmetry. Simple formulas are provided to distinguish different regimes.
The splitting of a wave packet is an important operation in matter wave interferometry [9, 10, 11, 12]. A strategy to improve the interferometer performance is to suppress the interaction [13, 14], so let us first consider a non-interacting Bose-Einstein condensate. For this system, a complete wave splitting into two separated branches is a peculiar operation because adiabatic following, rather than robust, is intrinsically unstable with respect to a small external potential asymmetry . The potential is assumed here to evolve from a single well to a final double-well where tunnelling is negligible . The ground-state wave function “collapses” into the final lower well (or more generally into the one that holds the lowest ground state as in ) and a very slow trap potential bifurcation fails to split the wave except for perfectly symmetrical potentials. A fast bifurcation remedies this but the price is typically a strong excitation which is also undesired, as it produces loss of contrast in the interference patterns when recombining the two waves . We propose here a way around these problems by using shortcuts to adiabaticity that speed up the adiabatic process along a nonadiabatic route . Wave splitting via shortcuts avoids the final excitation and is significantly more stable with respect to asymmetry than the adiabatic following. Specifically we shall use a streamlined version  of the fast-forward (FF) technique of Masuda and Nakamura  applied to the Gross-Pitaevskii (GP) or Schrödinger equations. There have previously been found some obstacles to apply the invariants-based method (quadratic-in-momentum invariants do not satisfy the required boundary conditions ) and the transitionless-driving algorithm  (because of difficulties in implementing counter-diabatic terms in practice).
In Sec. 2 we summarize the FF approach for condensates (interacting or not) in one dimension and its application to splitting. In Sec. 3 the effect of a small asymmetric perturbation is studied for noninteracting matter waves, and Sec. 4 analyzes and interprets the results with the aid of a moving two-mode model. Sec. 5 studies the remarkable stability with respect to the asymmetry achieved due to interatomic interactions in the mean-field limit, and different regimes are distinguished. Finally, Sec. 6 discusses the results and open questions.
2 Fast-forward approach
The FF method [18, 7, 19] may be used to generate external potentials and drive the matter wave from an initial single well to a final symmetric double-well. The starting point is the three-dimensional (3D) time-dependent GP equation,
where includes kinetic energy , external potential , and mean field potential . We are assuming an external local potential, where “local” means here . The kinetic and mean field terms in the coordinate representation have the usual forms,
The GP equation (1) is used to describe a Bose-Einstein condensate within the mean field approximation and it takes into account the atom-atom interaction through , the atom-atom coupling constant. In the case of vanishing coupling constant the GP equation simplifies to the Schrödinger equation.
By solving Eq. (1) in coordinate space, may be written as
with . By introducing into Eq. (4) the ansatz
where the dot means time derivative. The real and imaginary parts are
Our purpose is to design a local and real potential such that an initial eigenstate of the initial Hamiltonian, , typically the ground state, but it could be otherwise, evolves in a time into the corresponding eigenstate of the final Hamiltonian, . We assume that the full Hamiltonian and the corresponding eigenstates are known at the boundary times.
By construction the potential of Eq. (6) is local. If we impose , i.e.,
then we get from Eq. (7) a local and real potential.
In the inversion protocol is designed first, and Eq. (9) is solved for to get from Eq. (7). To ensure that the initial and final states are eigenstates of the stationary GP equation we impose at and . Then Eq. (9) has solutions independent of at the boundary times . Using this in Eq. (7) at , and multiplying by , we get
The initial state is an eigenstate of the stationary GP equation with chemical potential . Note that the above solution of (with at boundary times) admits the addition of an arbitrary function that depends only on time and modifies the zero of energy. A similar result is found at .
In the remainder of this chapter we will restrict to the one dimensional case so the potential in Eq. (4) is reduced to
with and . The primes denote derivatives with respect to , is the effective 1D-coupling constant of the Bose-Einstein condensate, and is the number of atoms. For the numerical examples we consider Rb atoms. Using in Eq. (10) the ansatz , the real and imaginary parts will be
where the dot means time derivative.
In the following two sections we consider first and split an initial single Gaussian state into a final double Gaussian . In previous works [18, 7] use has been made of the interpolation , where is a smooth, monotonously increasing function from 0 to 1, and is a normalization function. This produces a triple-well potential at intermediate times. Here we use instead the two-bump form which generates simpler -shaped potentials (see Fig. 1). We impose , so at the boundary times. In the numerical examples , where , and m (see e.g. ); Equation (12) is solved with the initial conditions .
3 Effect of the perturbation
Assume now a perturbed Hamiltonian with , where is the step function and the potential imbalance. Except in the final discussion, we assume that is some uncontrollable and hard-to-avoid small perturbation, typically unknown, due to imperfections of the experimental setting. The adiabatic splitting becomes unstable, as we shall see, but the instability does not depend strongly on this particular form, chosen for simplicity. It would also be found, for example, for a linear-in- perturbation, a smoothed step, slightly different frequencies for the final right and left traps, or a shifted central barrier . In the final potential configuration, with negligible tunneling, the two wells are independent, and the global ground state is localized in one of them.
To analyze the effects of the perturbation on the wavefunction structure and on the shortcut dynamics, we compute several wavefunction overlaps:
, the (black) short-dashed line in Fig. 2, is the “structural fidelity” between the (perfectly split) ground state of the unperturbed potential and the final ground state of the perturbed potential . This would be the fidelity found with the desired split state if the process were adiabatic. decays extremely rapidly from 1 at to , which corresponds to the collapse of the ground state of the perturbed potential into the deeper well.
, the (blue) long-dashed line in Fig. 2, is the fidelity between the state dynamically evolved with , , and . is the initial ground state with . If is used instead, the results are indistinguishable; see the overlap , [green dotted line] in Fig. 2. The flat at small , in sharp contrast to the rapid decay of , demonstrates the robustness of the balanced splitting produced by the shortcut. Shorter process times make the splitting more and more stable [compare Figs. 2(a)-2(c)]. (We assume condensate lifetimes of the order of seconds; see e.g., .) In principle, may be reduced arbitrarily. In practice, this reduction implies an increase in transient energy excitation that requires accurate potential engineering for higher energies . Considering that the time-averaged standard deviation of the energy should be limited at some value a general bound is . For the trap frequency in the examples (780 rad/s) and setting the bound saturates for a time ms, times shorter than our shortest time in Fig. 2.
[solid (red) line in Fig. 2] is the fidelity between the dynamically evolved state and the final ground state for the perturbed potential. If the process is adiabatic, then . For very small perturbations . In this regime the dynamical wave function is not affected by the perturbation and becomes , up to a phase factor; note that there. We understand and quantify below this important regime as a sudden process in a moving-frame interaction picture. As increases, the energies of the ground and excited states of separate and the process becomes less sudden and more adiabatic. In Fig. 2(c) for ms and for large values of , approaches 1 again, the final evolved state collapses to one side and becomes the ground state of . For the shorter final times in Figs. 2(a) and 2(b), larger values are needed so that approaches 1 adiabatically.
4 Moving two-mode model
Static two-mode models have been previously used to analyze splitting processes or double-well dynamics [11, 24, 25]. Here we add the separation motion of left and right basis functions to provide analytical estimates and insight. In terms of a (dynamical) orthogonal bare basis , our two-mode Hamiltonian model is
where is the tunneling rate [11, 24, 25]. We may consider constant through a given splitting process, for the time being, and equal to the perturbative parameter that defines the asymmetry. A more detailed approach discussed later does not produce any significant difference. The instantaneous eigenvalues are
and the normalized eigenstates take the form
where is the mixing angle given by .
The bare basis states are symmetrical and orthogonal-moving left and right states. Initially they are close to each other and . The instantaneous eigenstates of are the symmetric ground state and the antisymmetric excited state of the single well. At we distinguish two extremes:
i) For the final eigenstates of tend to symmetric and antisymmetric splitting states ;
ii) For the final eigenfunctions of collapse and become right-and left-localized states: and .
Since is set as a small number to avoid tunneling in the final configuration, the transition from one to the other regime explains the sharp drop of at small .
4.1 Moving-frame interaction picture
We define now a moving-frame interaction-picture (IP) wave function , where and is the Schrödinger-picture wave function. obeys
but for real and , the symmetry makes .
Inverting Eq. (15) the bare states may be written in terms of the ground and first excited states and energies. The two-level model approximates the actual dynamics by first identifying and with the instantaneous ground and excited states and energies of the unperturbed FF Hamiltonian.111Contrast this with the variational approach in . We combine them to compute the bare basis in coordinate representation and then the matrix elements , for . From Eq. (13), .222 For , we may consistently calculate , where is a shift to match the zero-energy point between the FF and the two-mode models. differs slightly from the constant at short times, but the results of substituting by are hardly distinguishable in the calculations, so the treatment with is preferred for simplicity. Once all matrix elements are set we solve the dynamics in the moving frame for the two-mode Hamiltonian. The initial state may be the ground state of the perturbed or unperturbed initial potential. The agreement with the exact results is excellent (see the symbols of Fig. 2), which denotes the absence of higher excited states. This two-level model thus provides a powerful interpretative and control tool. To gain more insight we now perform further approximations.
4.2 Sudden and adiabatic approximations
The fidelities at low may be understood with the sudden approximation in the IP. Its validity requires 
where . We take and , where the matrix elements of in the basis coincide with the matrix elements of in Eq. (13), when the latter are expressed in the basis . The condition for the sudden approximation to hold becomes
Vertical lines mark in Fig. 2 and demonstrate that indeed this condition sets the range in which so that the fast protocol provides balanced splitting in spite of the asymmetry.
5 Interacting Bose-Einstein condensates
We now generalize the results of the two previous sections for a condensate with interatomic interaction in the mean-field framework. We calculate the ground states and of a harmonic trap that holds a Bose-Einstein condensate with and atoms and define , where . is constructed as
where is a normalization factor and . We then get from Eq. (11) and evolve the initial ground state with the GP equation using the perturbed potential .
The fidelities are shown in Fig. 3 versus the dimensionless coupling constant . Note the stabilization of towards upon increasing the interaction (this implies more stable shortcuts). increases too, as the dynamics tends to be more adiabatic. The structural fidelity jumps to around from the linear case value , i.e., balanced splitting by adiabatic following is robust versus trap asymmetry for (see the Appendix \thechapter). The extra filling of the lower well increases the nonlinear interaction there opposing the external potential imbalance.
The two-level model may also be extended to interacting condensates with minor modifications, also providing an accurate description (see Fig. 3). Adiabaticity fails eventually when decreasing and/or , but the shortcut provides then balanced splitting (see the example of Fig. 4): for small , adiabatic following would be stable (see and compare to the sharp drop in Fig. 2 for linear dynamics), but the process is not quite adiabatic () for the chosen time, ms -more time would be needed. The shortcut is nevertheless more stable than the hypothetical adiabatic process ().
We have designed simple -shaped (position and time dependent) potential traps to fully split noninteracting matter waves rapidly without final excitation, avoiding the instability of the adiabatic approach with respect to slight trap asymmetries. We also avoid or mitigate in this manner the decoherence and noise that affect slow adiabatic following [16, 17]. The bifurcation may be experimentally implemented with optical traps created with the aid of spatial light modulators . A simpler approximate approach would involve two Gaussian beams. Further manipulations, such as the application of differential ac Stark phase shifts could be combined with the proposed technique . Also, a differential phase among the two final wave parts will develop due to the imbalance, allowing for precision metrology [20, 21], without the time limitations of methods based on adiabatic splitting . In addition, optimal control methods [9, 10, 11] complement the present approach to further improve stability and/or optimize other variables such as the transient excitation.
A unique feature of the above application of shortcuts to adiabaticity, compared to previous ones [1, 31, 32, 33], is that the shortcut does not attempt to reproduce the result of an adiabatic following of the perturbed asymmetrical system in a shorter time. (The assumption has been made so far that the perturbation is uncontrolled and, possibly, unknown.) Instead, the shortcut reproduces the balanced splitting of the adiabatic following corresponding to the unperturbed, perfectly symmetrical system. In other words, shortening the time here is not really the goal, but it means to achieve stability.
Other operations may actually make positive use of the instability due to potential asymmetries. In particular, the ground- and first-excited-state components of the initial trap could be spatially separated by a controlled, slightly asymmetrical adiabatic bifurcation. Moreover, both states would become ground states of the right and left final traps, so the process may as well be used as a population inversion protocol from the excited to the ground state.
We have also analyzed and exemplified the effect of interatomic interactions for a condensate in the mean-field regime. The interaction changes the behavior of the system with respect to asymmetry, stabilizing dramatically balanced splitting. The total adiabatic collapse of the wave onto one of the two final separated wells requires, in this case, a significant perturbation, proportional to the coupling constant. Compared to the noninteracting case, this offers different manipulation opportunities, in particular, the possibility of considering the asymmetric perturbation as a known, controllable parameter, so that the imbalance between the two wells may be prepared at will. Examples of this type of manipulation may be found in [34, 35, 36]. Shortcuts to adiabaticity and, in particular, the FF approach may be readapted to that scenario by designing the fast protocol taking into account the known, controlled asymmetry. The emphasis would be again, as in most applications of the shortcuts, on accelerating and reproducing the result of a slow process.
Shortcuts to adiabaticity could play other roles in systems described by a double-well with varying parameters. They have been applied, in particular, to speed up the generation of spin-squeezed many-body states in bosonic Josephson junctions . Here we suggest other applications: for example, Stickney and Zozulya  have described a wave-function recombination instability due to the weak nonlinearity of the condensate. Specifically, they consider an initially weak ground symmetric mode of the double-well which is exponentially amplified at the expense of an initially strong excited asymmetric mode when the wells are recombined. Similarly to the instability due to asymmetry described in this chapter for noninteracting waves, the nonlinear instability is in fact enhanced by adiabatic following. A shortcut-to-adiabaticity strategy as the one followed in this chapter would stabilize the recombination. Our present results may as well be applied to design Y-junctions in planar optical waveguides [39, 40, 41], since the equation that describes the field in the paraxial approximation is formally identical to the linear Schrödinger equation, with the longitudinal coordinate playing the role of time. Finally, partial splitting, in which the final two wells are not completely separated and tunnelling is still allowed, may as well be considered.
Chapter \thechapter Shortcuts to adiabaticity in three-level systems using Lie transforms
Sped-up protocols that drive a system quickly to the same populations that can be reached by a slow adiabatic process may involve Hamiltonian terms which are difficult to realize. We use the dynamical symmetry of the Hamiltonian to find, by means of Lie transforms, alternative Hamiltonians that achieve the same goals without the problematic terms. We apply this technique to three-level systems (two interacting bosons in a double well, and beam splitters with two and three output channels) driven by Hamiltonians that belong to the four-dimensional algebra U3S3.
“Shortcuts to adiabaticity” are manipulation protocols that take the system quickly to the same populations, or even the same state, that can be reached by a slow adiabatic process . Adiabaticity is ubiquitous in preparing a system state in atomic, molecular, and optical physics, so many applications of this concept have been worked out, in both theory and experiment . Some of the engineered Hamiltonians that speed up the adiabatic process in principle may involve terms which are difficult or impossible to realize in practice. In simple systems the dynamical symmetry of the Hamiltonian can be used to eliminate the problematic terms and provide instead feasible Hamiltonians. Examples are single particles transported or expanded by harmonic potentials [42, 43], or two-level systems [31, 44, 33]. In this chapter we extend this program to three-level systems whose Hamiltonians belong to a four-dimensional dynamical algebra. This research was motivated by a recent observation by Opatrný and Mølmer . Among other systems they considered two (ultracold) interacting bosons in a double well within a three-state approximation. Specifically, the aim was to speed up a transition from a “Mott-insulator” state with one particle in each well, to a delocalized “superfluid” state. The reference adiabatic process consisted in slowly turning off the interparticle interaction while increasing the tunneling rate. To speed up this process they applied a method of generating shortcuts based on adding a “counterdiabatic” (cd) term to the original time-dependent Hamiltonian [31, 2, 3, 4, 5], but the evolution with the cd term turns out to be difficult to realize in practice . In this chapter we shall use the symmetry of the Hamiltonian (its dynamical algebra) to find an alternative shortcut by means of a Lie transform, namely, a unitary operator in the Lie group associated with the Lie algebra. Since other physical systems have the same Hamiltonian structure the results are applicable to them too. Specifically, the analogy between the time-dependent Schrödinger equation and the stationary-wave equation for a waveguide in the paraxial approximation [46, 47, 48, 49, 50, 51] is used to design short-length optical beam splitters with two and three output channels.
In Sec. 8 we describe the theoretical model for two bosons in two wells. In Sec. 9 we summarize the counterdiabatic or transitionless tracking approach and apply it to the bosonic system. Section 10 sets the approach based on unitary Lie transforms to produce alternative shortcuts. In Sec. 11 we introduce the insulator-superfluid transition and apply the shortcut designed in the previous section. In Sec. 12 we apply the technique to generate beam splitters with two and three output channels. Section 13 discusses the results and open questions. Finally, in the Appendix \thechapter some features of the Lie algebra of the system are discussed.
8 The model
where () are the bosonic particle annihilation (creation) operators at the th site and is the occupation number operator. The on-site interaction energy is quantified by the parameter and the hopping energy by . They are assumed to be controllable functions of time. For two particles the Hamiltonian in the occupation number basis , , and is given by 
This Hamiltonian belongs to the vector space (Lie algebra) spanned by , , and two more generators,
with nonzero commutation relations
This four-dimensional Lie algebra, U3S3 , is described in more detail in the Appendix \thechapter. To find the Hermitian basis we calculate , and then all commutators of the result with previous elements. This operation is repeated for all operator pairs until no new linearly independent operator appears.