Abstract
A method is proposed to drive an ultrafast nonadiabatic dynamics of an ultracold gas trapped in a timedependent box potential. The resulting state is free from spurious excitations associated with the breakdown of adiabaticity and preserves the quantum correlations of the initial state up to a scaling factor. The process relies on the existence of an adiabatic invariant and the inversion of the dynamical selfsimilar scaling law dictated by it. Its physical implementation generally requires the use of an auxiliary expulsive potential. The method is extended to a broad family of interacting manybody systems. As illustrative examples we consider the ultrafast expansion of a TonksGirardeau gas and of BoseEinstein condensates in different dimensions, where the method exhibits an excellent robustness against different regimes of interactions and the features of an experimentally realizable box potential.
Introduction
“Fast good” is a new culinary concept envisioned by the chef F. Adrià aimed at creating a diet enjoying of two apparently mutuallyexclusive features: fastservice and highquality. When similar ideas are invoked in the quantum realm, one faces the adiabatic theorem which imposes a price. The preparation of a given target state with highfidelity, free from spurious excitations, generally demands a long time of evolution and the implementation of an adiabatic dynamics. As a result, it comes as no surprise that the recent development of shortcuts to adiabaticity (STA)^{1}, has led to a surge of theoretical^{2,3,4,5,6,7,8,9,10} and experimental activity^{11,12,13}.
It is the purpose of this work to show that STA can be implemented in a timedependent box, the paradigmatic model of a quantum piston^{14}. The relevance of this confinement is enhanced by the development of experimental techniques to create optical billiards for ultracold gases^{15,16}, paint arbitrary potential shapes for BoseEinstein condensates^{17} and realize alloptical boxes^{18} and analogous traps in atom chips^{19}. It has the potential to greatly advance the field of quantum simulation, facilitating the connection between ultracold atom experiments and condensed matter systems.
The prospects of finding STA for box confinement seem challenging at the very least, based on the following considerations. Since the early insight by M. Moshinsky in 1952, it is known that the rapid expansion of matterwaves initially localized in a region of space exhibits quantum transients and excitations in the form of density ripples, a phenomenon referred to as diffraction in time^{20,21,22}. Moreover, reflections from the walls of the trap generally lead to Talbot oscillations and the formation of a quantum carpet in the time evolution of the density profile^{23,24}. On top of that, a series of work during the last decades have shown that the suppression of excitations in an expanding box is inevitably constrained by the adiabatic theorem both in the noninteracting^{25,26,27,28} and meanfield regime^{29}.
In spite of these results preventing ultrafast excitationfree dynamics in box traps, we show that fast nonadiabatic expansion or compression is in fact possible, allowing preparation in a predetermined finite time of the same target state than the adiabatic evolution. The key to this shortcut to adiabaticity is the use of an auxiliary potential term superimposed on the box trap.
We shall start by introducing a dynamical invariant in a timedependent box trap at the singleparticle level and use it to derive a scaling law that governs the time evolution. The inversion of this scaling law will be the key to engineer the STA. This result will be extended to manybody systems with a broad family of interactions. Finally, we shall illustrate the method with some examples relevant to experiments with ultracold gases, showing that the STA is robust in the presence of interactions and experimental imperfections.
Results
Dynamical invariants and selfsimilar dynamics
For a timedependent box of width ξ(t), the scaling laws governing the dynamics of the expanding eigenstates reported to date are associated with trajectories of the form (with a, b, c real constants)^{25,26,27,28}, which turn out to be unsuited for engineering a STA (see below). Nonetheless, given a timedependent Hamiltonian , it is possible to build a dynamical invariant^{30}, such that
with spectral decomposition in terms of the set of eigenmodes φ_{n}(t)〉 with eigenvalues λ_{n}. This is a particularly useful basis to describe the time evolution of an initial state Ψ, by the superposition , where the LewisRiesenfeld phase is given by and can be understood as the sum of the dynamical phase and the AharanovAnandan phase^{28}. For a timedependent box of width ξ(t) and initial width ξ(0) = ξ_{0}, a dynamical invariant exists^{31},
with eigenvectors and eigenvalues , with k_{n} = nπ/ξ_{0}. The LewisRiesenfeld phase can be computed to be with . The condition in Eq. (1) for to be an invariant requires the box potential to be supplemented with an auxiliary harmonic term
whose frequency is dictated by the ratio of the acceleration of the trajectory ξ(t) and the trajectory itself. Note that ξ(t) > 0, so that for , Ω(t) is purely imaginary and the auxiliary term U^{aux} is a repulsive harmonic potential. If , so and have common eigenstates. Further, if holds as well, then U^{aux}(x, 0) = 0 and an eigenstate Ψ_{n}(x, 0) of the box at t = 0 evolves into Ψ_{n}(x, t) = exp[iα_{n}(t)]φ_{n}(t), a key observation to engineer a STA as we shall see. We note that the experimental implementation of U^{aux}(x, t) can be assisted by the same techniques used to create the box potential: the use of a bluedetuned laser^{32} or direct painting of the required trap^{15,16,17}.
Shortcuts to adiabaticity: inverting the dynamical scaling law
We next discuss how to implement a nonadiabatic expansion of the box by a factor γ(τ) = ξ(τ)/ξ_{0} in a given finitetime τ suppressing excitations in the final state. We shall impose the condition U^{aux}(x, 0) = U^{aux}(x,τ) = 0. As in the adiabatic case, in a STA the time evolution of an eigenmode of the initial box should reproduce an eigenmode of the final trap. As at t = 0, this can be enforced by imposing the condition . The set of boundary conditions at t = 0, τ excludes the possibility of a linear ramp, as well as the family of trajectories, , considered so far in the literature^{25,26,27,28}. However, it suffices to determine a polynomial ansatz for the trajectory , i.e. a scaling factor of the form , with s = t/τ. This further determines the required timedependent frequency of the auxiliary harmonic potential U^{aux}(x, t) according to Eq. (3),
The trajectory, displayed in Fig. 1 shows that during an expansion U^{aux}(x,t) becomes an expulsive potential in an early stage (t < τ/2), providing the speedup required to achieve the STA in an arbitrary finite time τ (bounds in the presence of perturbations will be discussed below). In a subsequent stage, t > τ/2, Ω^{2}(t) changes sign and U^{aux}(x, t) becomes a trapping potential, slowing down the expanding mode and reducing it to an eigenstate of the final Hamiltonian at t = τ. Precisely the opposite behavior is exhibited during a fast nonadiabatic compression. Provided that an arbitrary Ω^{2}(t) dependence can be implemented, a STA has no lower bound for τ (notice however that Ω(t) ~ τ^{–1}). By contrast, the adiabatic condition
leads to the requirement .
Note that the energy of the expanding mode
has two contributions, the first one being the adiabatic energy E_{n}(t) = E_{n}(0)ξ(0)^{2}/ξ(t)^{2} and the second one depending explicitly on , so that given that a STA demands . This relation illustrates the fact that a STA is associated with a nonadiabatic evolution, which reproduces the adiabatic result at the end of the process. Moreover, STA work as well for excited states: the time evolution of the nth eigenstate of the initial trap leads to the nth eigenstate of the final trap at t = τ. As a result, STA in boxes pave the way for fast populationpreserving cooling in the following sense. Given a system described by the canonical ensemble with a density matrix , where β = 1/k_{B}T, k_{B} is the Boltzmann constant and T is the temperature, the final temperature reads
These results strictly hold for a box with infinite walls at x ∈ [0, ξ(t)]. However, in the alloptical trap reported in^{18}, the endcap lasers providing the box walls have a Gaussian profile which smooths the potential in a length scale σ, i.e. a box trap of the form
A similar smoothing occurs in other physical realizations^{17}. Since this smoothing is expected to be the most significant deviation of laboratory potentials from the ideal infinite box, it is important to consider its effect on the selfsimilar dynamics required for a STA. This can be quantified by the overlap between the states resulting from the expansion in an idealized and realistic box trap, Ψ_{n}(t)〉 and Ψ_{nσ}(t)〉 respectively. Clearly the role of σ decreases (increases) during a expansion (contraction) of the box. The numerical solution of the timedependent Schrödinger equation for this box potential is shown in Fig. 2, where the STA is compared with both the polynomial and linear expansion of the box in the absence of U^{aux}(x,t). It is seen that the STA is the only successful strategy and that the process is robust even for a substantial smoothing of the potential barriers, where the target states deviate from those of an idealized box.
We have further explored numerically the dynamics under “concatenated STA”, in which the overall expansion is splitted into a sequence of k STA with either constant expansion factor γ or constant box size increment between consecutive steps. The efficiency of the process exhibits a nonmonotonic improvement with increasing k, suggesting a natural scenario where STA techniques could be combined with optimal quantum control^{8,9}.
Beyond adiabatic invariants: Shortcuts to adiabaticity in interacting manybody systems
Knowledge of the adiabatic invariants for a single particle in a timedependent box has provided us with the insight of assisting the dynamics with an auxiliary potential to design STA for expansions and compressions in a finite time, without inducing quantum transients associated with diffraction in time, which are ubiquitous in this type of scenario^{22}. The technique can be directly applied to noninteracting gases and other manybody quantum fluids which can be mapped to noninteracting systems. It further suppresses the Talbot dynamics associated with quantum carpets woven by the density profile typically observed in boxes^{23} and the question naturally arises as to its applicability to interacting systems^{24}. The presence of interactions, e.g. a twobody potential, hinders the exploitation of the superposition principle in terms of the eigenmodes of the LewisRiesenfeld invariant. However, we note that to design a STA it suffices to enforce a selfsimilar dynamics and ultimately no knowledge of adiabatic invariants is required. As a result, we next consider a broader family of manybody systems, confined in a box, defined by the Hamiltonian
where , Δq_{i} is the Laplace operator in dimension D, the auxiliary term is now given by
and the twobody interaction potential obeys V(λq) = λ^{–α}V(q), e.g. for the FermiHuang pseudopotential describing swave scattering in ultracold gases, α = D. For a hardwall box, r_{i} = q_{i} ∈ [0,ξ(t)]; we shall relax below this approximation and consider realistic potential boxes as those created in alloptical setups. The case D = 1 corresponds to a box with onewall moving (the symmetric case in which both walls move in opposite directions can be obtained by a Duru transformation^{33}). For D = 2,3 cylindrical and spherical symmetry is assumed respectively. Without loss of generality, we choose the dimensionless timedependent coupling constant to satisfy . A stationary state Ψ(t) = Ψ(q_{1}, …, q_{N};t) of N particles and chemical potential μ follows for t > 0 the evolution
with the boundary conditions Ψ(t) = 0 for q_{i} = ξ(t) (i = 1,…,N, in addition to Ψ(t) = 0 for q_{i} = 0 in D = 1), as long as
which can be implemented exploiting a Feshbach resonance or modulating the transverse confinement in anisotropic systems^{4,34}.
The selfsimilar dynamics in a STA leads to a scaling of all local correlation functions. In particular the density of a given manybody state follows the law . By contrast, nonlocal correlation functions exhibit a nontrivial dynamics. The onebody reduced density matrix of a state obeying Eq. (11), follows the scaling law , analogous to that observed under harmonic confinement^{43}. The additional phase factor induces a major distortion of the momentum distribution, . However, a STA ensures that at t = τ the LewisRiesenfeld phase factor vanishes, so that the final state exhibits the same correlations of the initial state scaled by a factor γ(τ),
Examples
In the following we shall illustrate different aspects of shortcuts to adiabaticity in some paradigmatic models.
We shall first consider the evolution of correlations in a onedimensional cloud of ultracold bosons in the limit of hardcore contact interactions, this is, in the TonksGirardeau (TG) regime^{35}. This system, as well as its latticeversion, has become a favorite testbed to study the breakdown of thermalization and adiabaticity^{36}. Its manybody ground state is given by the BoseFermi mapping^{35}, , where if x > 0 (< 0) and . In a STA, the selfsimilar dynamics is inherited from the singleparticle orbitals Ψ_{j}(q_{k}, t) whence it follows that no tuning of interactions is required. Its density exhibits a scaling law for all t. The same holds true for the entanglement entropy with respect to a bipartition [0, aξ(t)] (a < 1)^{37}. The selfsimilar dynamics breaks down for the momentum distribution, which can be computed efficiently^{38} and we shall focus on its evolution along a STA. Different snapshots are depicted in Fig. 3A and confirm that during an expansion the cloud is accelerated during the interval [0, τ/2] and slowed down during [τ/2, τ]. The reverse sequence, is observed in a fast frictionless compression. The axis are scaled up by the expansion factor γ(t) in such a way that for an adiabatic dynamics, curves at different times would collapse into a single curve. Along a STA, the width and mean of the momentum distribution do not remain constant and change along the process.
A similar distortion of correlations, known as dynamical fermionization, occurs in the dynamics of a cloud suddenly released from an arbitrary trap^{39}. Under ballistic dynamics the asymptotic momentum distribution in a 1D expansion evolves to that of the dual system, a spin polarized Fermi gas. In particular for a cloud released from a box the exact time evolution is not selfsimilar^{40} but dynamical fermionization is observed^{41,42}. However, under a selfsimilar scaling law, the asymptotic n(k) maps to the density profile of the initial state^{43} and no dynamical fermionization occurs. This is the case of relevance to STA, where the dynamical scaling law in Eq. (11) holds. (We note that the case of the initial harmonic confinement is singular in that the free expansion is selfsimilar and that the the singleparticle eigenstates can be written in terms of Hermite polynomials, which are eigenfunctions of the continuous Fourier transform. As a result the asymptotic momentum distribution can be related to both the initial density profile and the momentum distrubtion of noninteracting fermions. See [5] for a discussion of STA in harmonic traps.) Moreover, this distortion of correlations is not restricted to expansion processes. Along a STA, this is shown in Fig. 3 for both expansions (A) and compressions (B). This is a spurious effect for the purpose of STA, which is to reproduce the adiabatic result in a finite short time. Indeed, the distortion induced during the first half of the STA associated with the accelerated expansion or compression, is compensated in the second half of the dynamics, in such a way that the correlations of the initial state are reconstructed at t = τ and scaled by a factor γ(τ).
We next turn our attention to the design of STA for a BEC in timedependent box trap, where different strategies can be adopted depending on the dimensionality and the regime of interactions. The timedependent GrossPitaevski equation (TDGPE) governs the evolution of the normalized condensate wavefunction Φ(q, t),
for which adiabaticity conditions have been reported^{29}. The ansatz
satisfies the TDGPE provided that
These relations constitute the box analogue of the wellknown CastinDumKaganSurkovShlyapnikov relations in harmonic traps^{44,45}. The twodimensional case is special since the scaling law holds when g_{2D}(t) is kept constant.
Figure 4 is a set of numerical solutions of the timedependent GrossPitaeveskii equation that illustrate the robustness of the STA for realistic BEC experiments. We consider a box trap with Gaussian barriers and in all numerical simulations interactions are kept constant, i.e. g_{D}(t) = g_{D}(0), deviating from the ideal prescription in Eq. (16). The top row illustrates the dynamics for a quasi1D BEC. The (onebody) fidelity between the resulting state Φ(τ) and the ground state of the final box is . For smaller values of γ, the fidelity is even higher as expected, given that implementation of the exact STA requires a smaller tuning of g_{1}. The bottom row shows the dynamics of a quasi2D cloud, which requires no interaction tuning in a STA, but is more sensitive to the smoothness of the box boundaries, .
It is noteworthy that in the ThomasFermi regime, the kinetic term contribution can be neglected and it is possible to induce an exact selfsimilar dynamics (and a STA) exclusively with the help of an external field. Then, the scaling ansatz is a solution of the TDGPE provided that
This regime is particularly robust against the smooth boundaries of physically realizable box potentials. The simulations correspond to the most delicate regime with moderate meanfield interactions, both far from the noninteracting and ThomasFermi limits.
Discussion
In conclusion, we have presented a method to drive an ultrafast dynamics in a timedependent box trap which reproduces the adiabatic result at the end of the evolution. The method is assisted by an auxiliary external harmonic potential which provides the speedup and is applicable to a large family of both noninteracting and interacting manybody systems supporting dynamical scaling laws, where it not only leads to a robust expansion of the density but also preserves the nonlocal correlation functions of the initial state, up to an expansion factor. The proposal is applicable to realistic box potentials and can be implemented in the laboratory with wellestablished technology. Its applications range over all scenarios requiring a shortcut to adiabaticity, i.e., probing strongly correlated phases, preventing decoherence, the effect of perturbations and atomic losses. The method can be directly applied as well to ultrafast populationpreserving cooling methods, quantum heat engines and refrigerators^{46} providing an alternative to the paradigmatic model of a quantum piston^{14}.
References
Chen, X., Ruschhaupt, A., Schmidt, S., del Campo, A., GuéryOdelin, D. & Muga, J. G. Fast optimal frictionless atom cooling in harmonic traps. Phys. Rev. Lett. 104, 063002 (2010).
Chen, X., Lizuain, I., Ruschhaupt, A., GuéryOdelin, D. & Muga, J. G. Shortcut to adiabatic passage in two and three level atoms. Phys. Rev. Lett. 105, 123003 (2010).
Muga, J. G., Chen, X., Ruschhaupt, A. & GuéryOdelin, D. Frictionless dynamics of BoseEinstein condensates under fast trap variations. J. Phys. B: At. Mol. Opt. Phys. 42, 241001 (2009).
del Campo, A. Fast frictionless dynamics as a toolbox for lowdimensional BoseEinstein condensates. EPL 96, 60005 (2011).
del Campo, A. Frictionless quantum quenches in ultracold gases: a quantum dynamical microscope. Phys. Rev. A 84, 031606(R) (2011).
Torrontegui, E., Ibáñez, S., Chen, X., Ruschhaupt, A., GuéryOdelin, D. & Muga, J. G. Fast atomic transport without vibrational heating. Phys. Rev. A 83, 013415 (2011).
Torrontegui, E., Chen, X., Modugno, M., Schmidt, S., Ruschhaupt, A. & Muga, J. G. Fast transport of BoseEinstein condensates. New J. Phys. 14, 013031 (2012).
Stefanatos, D., Ruths, J. & Li, J.S. Frictionless atom cooling in harmonic traps: a timeoptimal approach. Phys. Rev. A 82, 063422 (2010).
Stefanatos, D., Schaettler, H. & Li, J.S. MinimumTime Frictionless Atom Cooling in Harmonic Traps. SIAM J. Control Optim. 49, 2440–2462 (2011).
Choi, S., Onofrio, R. & Sundaram, B. Optimized sympathetic cooling of atomic mixtures via fast adiabatic strategies. Phys. Rev. A 84, 051601(R) (2011).
Schaff, J.F., Song, X.L., Vignolo, P. & Labeyrie, G. Fast optimal transition between two equilibrium states. Phys. Rev. A 82, 033430 (2010).
Schaff, J.F., Song, X.L., Capuzzi, P., Vignolo, P. & Labeyrie, G. Shortcut to adiabaticity for an interacting BoseEinstein condensate. EPL 93, 23001 (2011).
Bason, M. G. et al. Highfidelity quantum driving. Nature Phys. 8, 147–152 (2012).
Quan, H. T. & Jarzynski, C. Validity of nonequilibrium work relations for the rapidly expanding quantum piston. Phys. Rev. E 85, 031102 (2012).
Milner, V., Hanssen, J. L., Campbell, W. C. & Raizen, M. G. Optical billiards for atoms. Phys. Rev. Lett. 86, 1514 (2001).
Friedman, N., Kaplan, A., Carasso, D. & Davidson, N. Observation of chaotic and regular dynamics in atomoptics billiards. Phys. Rev. Lett. 86, 1518 (2001).
Henderson, K., Ryu, C., MacCormick, C. & Boshier, M. G. Experimental demonstration of painting arbitrary and dynamic potentials for BoseEinstein condensates. New J. Phys. 11, 043030 (2009).
Meyrath, T. P., Schreck, F., Hanssen, J. L., Chuu, C.S. & Raizen, M. G. BoseEinstein Condensate in a Box. Phys. Rev. A 71, 041604(R) (2005).
van Es, J. J. P., Wicke, P., van Amerongen, A. H., Rétif, C., Whitlock, S. & van Druten, N. J. Box traps on an atom chip for onedimensional quantum gases. J. Phys. B: At. Mol. Opt. Phys. 43, 155002 (2010).
Moshinsky, M. Diffraction in time. Phys. Rev. 88, 625631 (1952).
Gerasimov, A. S. & Kazarnovskii, M. V. Possibility of observing nonstationary quantummechanical effects by means of ultracold neutrons. Sov. Phys. JETP 44, 892–897 (1976).
del Campo, A., GarcíaCalderón, G. & Muga, J. G. Quantum transients. Phys. Rep. 476, 150 (2009).
Friesch, O. M., Marzoli, I. & Schleich, W. P. Quantum carpets woven by Wigner functions. New. J. Phys. 2, 4 (2000).
Ruostekoski, J., Kneer, B., Schleich, W. P. & Rempe, G. Interference of a BoseEinstein condensate in a hardwall trap: From the nonlinear Talbot effect to the formation of vorticity. Phys. Rev. A 63, 043613 (2001).
Berry, M. V. & Klein, G. Newtonian trajectories and quantum waves in expanding force fields. J. Phys A 17, 1805–1815 (1984).
Dodonov, V. V., Klimov, A. B. & Nikonov, D. E. Quantum particle in a box with moving walls. J. Math. Phys. 34, 3391–3404 (1993).
Chen, X., Muga, J. G., del Campo, A. & Ruschhaupt, A. Atom cooling by nonadiabatic expansion. Phys. Rev. A 80, 063421 (2009).
Mostafazadeh, A. Dynamical invariants, adiabatic approximation and the geometric phase, (New York: Nova, 2001).
Band, Y. B., Malomed, B. & Trippenbach, M. Adiabaticity in nonlinear quantum dynamics: BoseEinstein Condensate in a timevarying box. Phys. Rev. A 65, 033607 (2002).
Lewis, H. R. & Riesenfeld, W. B. An exact quantum theory of the timedependent harmonic oscillator and of a charged particle in a timedependent electromagnetic field. J. Math. Phys. 10, 1458–1473 (1969).
Cerveró, J. M. & Lejarreta, J. D. The timedependent canonical formalism: Generalized harmonic oscillator and the infinitesquare well with a moving boundary. Europhys. Lett. 45, 6 (1999).
Khaykovich, L. et al. Formation of a MatterWave Bright Soliton. Science 296, 1290–1293 (2002).
Duru, I. H. Quantum treatment of a class of timedependent potentials. J. Phys. A: Math. Gen. 22, 4827–4833 (1989).
Staliunas, K., Longhi, S. & de Valcárcel, J. Faraday patterns in lowdimensional BoseEinstein condensates. Phys. Rev. A 70, 011601(R) (2004).
Girardeau, M. Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension. J. Math. Phys. 1, 516–523 (1960).
Polkovnikov, A., Sengupta, K., Silva, A. & Vengalattore, M. Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys. 83, 863883 (2011).
Calabrese, P., Mintchev, M. & Vicari, E. The entanglement entropy of onedimensional gases. Phys. Rev. Lett. 107, 020601 (2011).
Pezer, R. & Buljan, H. Momentum distribution dynamics of a TonksGirardeau gas: Bragg reflections of a quantum manybody wavepacket. Phys. Rev. Lett. 98, 240403 (2007).
Cazalilla, M. A., Citro, R., Giamarchi, T., Orignac, E. & Rigol, M. One dimensional Bosons: From Condensed Matter Systems to Ultracold Gases. Rev. Mod. Phys. 83, 1405–1466 (2011).
del Campo, A. & Muga, J. G. Dynamics of a TonksGirardeau gas released from a hardwall trap. Europhys. Lett. 74, 965–971 (2006).
del Campo, A. Fermionization and bosonization of expanding onedimensional anyonic fluids. Phys. Rev. A 78, 045602 (2008).
Jukić, D., Klajn, B. & Buljan, H. Momentum distribution of a freely expanding LiebLiniger gas. Phys. Rev. A 79, 033612 (2009).
Gritsev, V., Barmettler, P. & Demler, E. Scaling approach to quantum nonequilibrium dynamics of manybody systems. New J. Phys. 12, 113005 (2010).
Castin, Y. & Dum, R. BoseEinstein Condensates in Time Dependent Traps. Phys. Rev. Lett. 77, 5315 (1996).
Kagan, Y., Surkov, E. L. & Shlyapnikov, G. V. Evolution of a Bosecondensed gas under variations of the confining potential. Phys. Rev. A 54, R1753 (1996).
Feldmann, T. & Kosloff, R. Quantum lubrication: Suppression of friction in a firstprinciples fourstroke heat engine. Phys. Rev. E 73, 025107(R) (2006).
Acknowledgements
It is a pleasure to thank H. Buljan, B. Damski, M. G. Raizen, E. Timmermans and W. H. Zurek for discussions and useful comments on the manuscript. This work was supported by the U.S. Department of Energy through the LANL/LDRD Program and a LANL J. Robert Oppenheimer fellowship.
Author information
Authors and Affiliations
Contributions
A.d.C. initiated the project, developed the theoretical analysis and prepared the manuscript. Both authors carried out the numerical simulations and interpretation of the numerical data.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
This work is licensed under a Creative Commons AttributionNonCommercialShareALike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/byncsa/3.0/
About this article
Cite this article
Campo, A., Boshier, M. Shortcuts to adiabaticity in a timedependent box. Sci Rep 2, 648 (2012). https://doi.org/10.1038/srep00648
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep00648
This article is cited by

Speeding up quantum dynamics by adding tunable timedependent Hamiltonians
Quantum Information Processing (2022)

Enhancing quantum annealing performance by a degenerate twolevel system
Scientific Reports (2020)

Ordered spacetime structures: Quantum carpets from Gaussian sum theory
Science China Physics, Mechanics & Astronomy (2019)

Shortcuts to adiabatic for implementing controllednot gate with superconducting quantum interference device qubits
Quantum Information Processing (2018)

A quantum particle in a moving finite harmonic potential
Journal of Mathematical Chemistry (2017)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.