Shortcuts to adiabaticity
Quantum adiabatic processes –that keep constant the populations in the instantaneous eigenbasis of a time-dependent Hamiltonian– are very useful to prepare and manipulate states, but take typically a long time. This is often problematic because decoherence and noise may spoil the desired final state, or because some applications require many repetitions. “Shortcuts to adiabaticity” are alternative fast processes which reproduce the same final populations, or even the same final state, as the adiabatic process in a finite, shorter time. Since adiabatic processes are ubiquitous, the shortcuts span a broad range of applications in atomic, molecular and optical physics, such as fast transport of ions or neutral atoms, internal population control and state preparation (for nuclear magnetic resonance or quantum information), cold atom expansions and other manipulations, cooling cycles, wavepacket splitting, and many-body state engineering or correlations microscopy. Shortcuts are also relevant to clarify fundamental questions such as a precise quantification of the third principle of thermodynamics and quantum speed limits. We review different theoretical techniques proposed to engineer the shortcuts, the experimental results, and the prospects.
keywords:adiabatic dynamics, quantum speed limits, superadiabaticity, quantum state engineering, transport engineering of cold atoms, ions, and Bose-Einstein condensates, wave packet splitting, third principle of thermodynamics, transitionless tracking algorithm, fast expansions
- 1 Introduction
- 2 General Formalisms
- 3 Expansions of trapped particles
- 4 Transport
- 5 Internal State engineering
- 6 Wavepacket Splitting
- 7 Discussion
The expression “shortcuts to adiabaticity” (STA) was recently introduced in Chen et al. (2010b), to describe protocols that speed up a quantum adiabatic process, usually, although not necessarily, through a non-adiabatic route.111 The word “adiabatic” may have two different meanings: the thermodynamical one (no heat transfer between system and environment) and the quantum one, 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”. Here we shall always understand “adiabatic” in the quantum mechanical sense. There the Lewis-Riesenfeld invariants were used to inverse engineer the time dependence of a harmonic oscillator frequency between predetermined initial and final values so as to avoid final excitations. That paper and its companion on Bose-Einstein condensates (Muga et al., 2009) have indeed triggered a surge of activity, not only for harmonic expansions (Chen and Muga, 2010; Muga et al., 2010; Stefanatos et al., 2010; Schaff et al., 2010, 2011a; del Campo, 2011a; Schaff et al., 2011b; Stefanatos et al., 2011; Torrontegui et al., 2012c; Fasihi et al., 2012; Torrontegui et al., 2012a; del Campo and Boshier, 2012; Stefanatos and Li, 2012), but for atom transport (Torrontegui et al., 2011, 2012d; Chen et al., 2011b; Bowler et al., 2012), quantum computing (Sarandy et al., 2011), quantum simulations (Lau and James, 2012), optical lattice expansions (Yuce, 2012; Ozcakmakli and Yuce, 2012), wavepacket splitting (Torrontegui et al., 2012b), internal state control (Chen et al., 2011a; Ibáñez et al., 2011; Ruschhaupt et al., 2012; Ban et al., 2012; Ibáñez et al., 2012a; Güngördü et al., 2012), many-body state engineering (del Campo, 2011b; del Campo and Boshier, 2012; del Campo et al., 2012; Juliá-Díaz et al., 2012), and other applications such as sympathetic cooling of atomic mixtures (Choi, Onofrio and Sundaram, 2011; Choi et al., 2012), or cooling of nanomechanical resonators (Li et al., 2011; Zhang et al., 2012a). In fact several works had previously or simultaneously considered to speed up adiabatic processes making use of different techniques. For example Demirplak and Rice (2003, 2005, 2008) and Berry (2009) proposed the addition of counterdiabatic terms to a reference Hamiltonian to achieve adiabatic dynamics with respect to . This “transitionless tracking algorithm” (Berry, 2009) has been applied to manipulate the populations of two-level systems (Demirplak and Rice, 2005; Berry, 2009; Chen et al., 2010c; Bason et al., 2012; Zhang et al., 2012b). Another technique to design laser pulses for fast population transfer is parallel adiabatic passage (Gulérin et al., 2002; Vasilev et al., 2009; Dridi et al., 2009; Gulérin et al., 2011). Couvert et al. (2008a) designed trap motions in order to perform non-adiabatic fast transport of atomic cold clouds. Also, Masuda and Nakamura (2010) developed a “fast-forward technique” for several manipulations on wavepackets such as expansions, transport or splitting of Bose-Einstein condensates. Related work had also been carried out for wave packet splitting making use of optimal control (Hohenester et al., 2007; Grond et al., 2009a, b), and in the context of quantum refrigerators, to find fast “frictionless” expansions (Salamon et al., 2009; Rezek et al., 2009). For recent developments on this line stimulated by invariant-based engineering results see Kosloff and Feldmann (2010); Hoffmann et al. (2011); Levy and Kosloff (2012); Feldmann and Kosloff (2012).
The considerable number of publications on the subject, and a recent Conference on “Shortcuts to adiabaticity” held in Bilbao (16-20 July 2012) demonstrate much current interest, not only within the cold atoms and atomic physics communities but also from fields such as semiconductor physics and spintronics (Ban et al., 2012). Indeed adiabatic processes are ubiquitous, so we may expect a broad range of applications, even beyond the quantum domain, since some of the concepts are easy to translate into optics (Lin et al., 2012; Tseng and Chen, 2012) or mechanics (Ibáñez et al., 2011). Apart from the practical applications, the fundamental implications of shortcuts on quantum speed limits (Bender et al., 2007; Bason et al., 2012; Uzdin et al., 2012), time-energy uncertainty relations (Chen and Muga, 2010), multiple Schrödinger pictures (Ibáñez et al., 2012a), and the quantification of the third principle of thermodynamics and of maximal cooling rates (Salamon et al., 2009; Rezek et al., 2009; Chen and Muga, 2010; Kosloff and Feldmann, 2010; Hoffmann et al., 2011; Levy and Kosloff, 2012; Feldmann and Kosloff, 2012) are also intriguing and provide further motivation.
In this review we shall first describe different approaches to STA in Sec. 2. While the main goal there is to construct new protocols for a fast manipulation of quantum states avoiding final excitations, additional conditions may be imposed. For example, ideally these protocols should not be state specific but work for an arbitrary state222Contrast this to the quantum brachistochrone (Bender et al., 2007), in which the aim is to find a time-independent Hamiltonian that takes a given initial state to a given final state in minimal time. Studies of “quantum speed limits” adopt in general this state-to-state approach, as in Uzdin et al. (2012).. They should also be stable against perturbations, and keep the values of the transient energy and other variables manageable throughout the whole process. Several applications are discussed in Secs. 3 to 6. We have kept a notation consistency within each Section but not throughout the whole review, following when possible notations close to the original publications.
2 General Formalisms
2.1 Invariant Based Inverse Engineering
Lewis-Riesenfeld invariants.– The Lewis and Riesenfeld (1969) theory is applicable to a quantum system that evolves with a time-dependent Hermitian Hamiltonian , which supports a Hermitian dynamical invariant satisfying
Therefore its expectation values for an arbitrary solution of the time-dependent Schrödinger equation , do not depend on time. can be used to expand as a superposition of “dynamical modes” ,
where ; are time-independent amplitudes, and are orthonormal eigenvectors of the invariant ,
The are real constants, and the Lewis-Riesenfeld phases are defined as (Lewis and Riesenfeld, 1969)
We use for simplicity a notation for a discrete spectrum of but the generalization to a continuum or mixed spectrum is straightforward. We also assume a non-degenerate spectrum. Non-Hermitian invariants and Hamiltonians have been considered for example in Gao et al. (1991, 1992); Lohe (009); Ibáñez et al. (2011).
Inverse engineering.– Suppose that we want to drive the system from an initial Hamiltonian to a final one , in such a way that the populations in the initial and final instantaneous bases are the same, but admitting transitions at intermediate times. To inverse engineer a time-dependent Hamiltonian and achieve this goal, we may first define the invariant through its eigenvalues and eigenvectors. The Lewis-Riesenfeld phases may also be chosen as arbitrary functions to write down the time-dependent unitary evolution operator ,
obeys where the dot means time derivative. Solving formally this equation for , we get
According to Eq. (6), for a given invariant there are many possible Hamiltonians corresponding to different choices of phase functions . In general does not commute with , so the eigenstates of , , do not coincide with the eigenstates of . does not necessarily commute with either. If we impose and , the eigenstates will coincide, which guarantees a state transfer without final excitations. In typical applications the Hamiltonians and are given, and set the initial and final configurations of the external parameters. Then we define and its eigenvectors accordingly, so that the commutation relations are obeyed at the boundary times and, finally, is designed via Eq. (6). While the may be taken as fully free time-dependent phases in principle, they may also be constrained by a pre-imposed or assumed structure of . Secs. 3, 4 and 5 present examples of how this works for expansions, transport and internal state control.
A generalization of this inverse method for non-Hermitian Hamiltonians was considered in Ibáñez et al. (2011). Inverse engineering was applied to accelerate the slow expansion of a classical particle in a time-dependent harmonic oscillator without final excitation. This system may be treated formally as a quantum two-level system with non-Hermitian Hamiltonian (Gao et al., 1991, 1992).
Quadratic in momentum invariants.– Lewis and Riesenfeld (1969) paid special attention to the time-dependent harmonic oscillator and its invariants quadratic in position and momentum. Later on Lewis and Leach (1982) found, in the framework of classical mechanics, the general form of the Hamiltonian compatible with quadratic-in-momentum invariants, which includes non harmonic potentials. This work, and the corresponding quantum results of Dhara and Lawande (1984) constitutes the basis of this subsection.
A one-dimensional Hamiltonian with a quadratic-in-momentum invariant must have the form ,333 and may denote operators or numbers. The context should clarify their exact meaning. with the potential (Lewis and Leach, 1982; Dhara and Lawande, 1984)
, , , and are arbitrary functions of time that satisfy the auxiliary equations
where is a constant. Their physical interpretation will be explained below and depends on the operation. A quadratic-in- dynamical invariant is given, up to a constant factor, by
and the function can be written as (Dhara and Lawande, 1984)
in terms of the solution (normalized in -space) of the auxiliary Schrödinger equation
The strategy of invariant-based inverse engineering here is to design and first so that and commute at initial and final times, except for launching or stopping atoms as in Torrontegui et al. (2011). Then is deduced from Eq. (7). Applications will be discussed in Secs. 3 and 4.
2.2 Counterdiabatic or Transitionless Tracking Approach
For the transitionless driving or counterdiabatic approach as formulated by Berry (2009), and equivalently by Demirplak and Rice (2003, 2005, 2008)444Berry’s transitionless driving method is equivalent to the counterdiabatic approach of Demirplak and Rice (2003, 2005, 2008). In Section 2.4 we shall see how to further exploit this scheme together with “superadiabatic iterations”., the starting point is a time-dependent reference Hamiltonian,
The approximate time-dependent adiabatic solution of the dynamics with takes the form
where the adiabatic phase reads
The approximate adiabatic vectors in Eq. (15) are defined differently from the dynamical modes of the previous subsection, but they may potentially coincide, as we shall see. Defining now the unitary operator
a Hamiltonian can be constructed to drive the system exactly along the adiabatic paths of , as , where
is purely non-diagonal in the basis.
We may change the , and therefore itself, keeping the same . We could for example make all the zero, or set (Berry, 2009). Taking into account this freedom the Hamiltonian for transitionless driving can be generally written as
Subtracting , the generic is
It is usually required that vanish for and , either suddenly or continuously at the boundary times. In that case the become also at the extreme times (at least at and ) eigenstates of the full Hamiltonian.
Using Eq. (1) and the orthonormality of the we may write invariants of with the form For the simple choice , then .
In this part and also in Sec. 2.1 the invariant-based and transitionless-tracking-algorithm approaches have been presented in a common language to make their relations obvious. Reinterpreting the phases of Berry’s method as , and the states as , the Hamiltonians in Eqs. (6) and (19) may be equated. As well, the implicit in the invariant-based method is given by Eq. (20), so that the dynamical modes can be also understood as approximate adiabatic modes of (Chen et al., 2011a). An important caveat is that the two methods could coincide but they do not have to. Given and , there is much freedom to interpolate them using different invariants, phase functions, and reference Hamiltonians . In other words, these methods do not provide a unique shortcut but families of them. This flexibility enables us to optimize the path according to physical criteria and/or operational constraints.
Non-Hermitian Hamiltonians.– A generalization is possible for non-Hermitian Hamiltonians in a weak non-hermiticity regime (Ibáñez et al., 2011, 2012b). It was applied to engineer a shortcut laser interaction and accelerate the decay of a two-level atom with spontaneous decay. Note that the concept of “population” is problematic for non-Hermitian Hamiltonians (Leclerc et al., 2012). This affects in particular the definition of “adiabaticity” and of the shortcut concept. It is useful to rely instead on normalization-independent quantities, such as the norm of a wave-function component in a biorthogonal basis (Ibáñez et al., 2012b).
Many-body Systems.– Following del Campo et al. (2012), the transitionless quantum driving can be extended as well to many-body quantum critical systems, exploiting recent advances in the simulation of coherent -body interactions (Müller et al., 2011; Barreiro et al., 2011). In this context STA allow a finite-rate crossing of a second order quantum phase transition without creating excitations. Consider the family of quasi-free fermion Hamiltonians in dimension , , where the -mode Pauli matrices are and are fermionic operators, and the sum goes over independent -modes. Particular instances of quantum critical models within this family of Hamiltonians are the Ising and XY models in (Sachdev, 1999), and the Kitaev model in (Lee et al., 2007) and (Sengupta et al., 2008). The function is specific for each model (Dziarmaga et al., 2010). All these models can be written down as a sum of independent Landau-Zener crossings, where the instantaneous -mode eigenstates have eigenenergies It is possible to adiabatically cross the quantum critical point driving the dynamics along the instantaneous eigenmodes of provided that the dynamics is driven by the modified Hamiltonian , where (del Campo et al., 2012)
is typically highly non-local in real spaces and involves many-body interactions in the spin representation. However, it was shown in the 1D quantum Ising model that a truncation of with interactions restricted to range is efficient to suppress excitations on modes (del Campo et al., 2012).
2.3 Fast-forward Approach
Based on some earlier results (Masuda and Nakamura, 2008), the fast-forward (FF) formalism for adiabatic dynamics and application examples were worked out in Masuda and Nakamura (2010, 2011); Masuda (2012) for the Gross-Pitaevskii equation or the corresponding Schrödinger equation. The aim of the method is to accelerate a “standard” system subjected to a slow variation of external parameters by canceling a divergence due to an infinitely-large magnification factor with the infinitesimal slowness due to adiabaticity. A fast-forward potential is constructed which leads to the speeded-up evolution but, as a consequence of the different steps and functions introduced, the method is somewhat involved, which possibly hinders a broader application. The streamlined construction of fast-forward potentials presented in Torrontegui et al. (2012a) is followed here.
The starting point is the 3D time-dependent Gross-Pitaevskii (GP) equation (Dalfovo et al., 1999)
Using the ansatz () we formally solve for in (22) and get for the real and imaginary parts
Imposing , i.e.,
Eq. (23) gives a real potential. In the inversion protocol it is assumed that the full Hamiltonian and the corresponding eigenstates are known at the boundary times. Then we design , solve for in Eq. (25), and finally get the potential from Eq. (23). In Torrontegui et al. (2012a) it was shown how the work of Masuda and Nakamura (2008, 2010, 2011) relates to this streamlined construction.
Since the phase that solves Eq. (25) depends in general on the particular , Eq. (23) gives in principle a state-dependent potential. However, in some special circumstances, the fast-forward potential remains the same for all modes. This happens in particular for the Schrödinger equation, , and Lewis-Leach potentials associated with quadratic-in-momentum invariants. In other words, the invariant-based approach can be formulated as a special case of the simple inverse method (Torrontegui et al., 2012a).
2.4 Alternative Shortcuts Through Unitary Transformations
Shortcuts found via the methods described so far or by any other approach might be difficult to implement in practice. In the cd approach, for instance, the structure of the complementary Hamiltonian could be quite different from the structure of the reference Hamiltonian . Here are three examples, the first two for a particle of mass in 1D, the third one for a two-level system:
where is a polar angle and .
In all these examples the experimental implementation of is possible, but the realization of the counter-diabatic terms is problematic. A way out is provided by unitary transformations that generate alternative shortcut protocols without the undesired terms in the Hamiltonian (Ibáñez et al., 2012a). A standard tool is the use of different interaction pictures for describing one physical setting. Unitary operators connect the different pictures and the goal is frequently to work in a picture that facilitates the mathematical manipulations. In this standard scenario all pictures describe the same physics, the same physical experiments and manipulations.
The main idea in Ibáñez et al. (2012a) is to regard instead the unitary transformations as a way to generate different physical settings and different experiments, not just as mathematical transformations. The starting point is a shortcut described by the Schrödinger equation , our reference protocol. (In all the above examples .) The new dynamics is given by , where , and , where . If the final states will coincide, i.e., for a given initial state . If, in addition, , then , and . Let us now list the unitary transformations that provide for the three examples realizable Hamiltonians (Ibáñez et al., 2012a):
- Example 1: Harmonic oscillator expansions,
- Example 2: Harmonic transport,
- Example 3: Population inversion in a two-level system,
Why do the s in Eqs. (29-31) have these forms? The answer lies in the symmetry possesed by the Hamiltonian. Transformations of the form based on generators of the corresponding Lie algebra produce operators within the algebra and, by suitably manipulating the function undesired terms may be eliminated.
Superadiabatic iterations.– As discussed in Sec. 2.2, Demirplak, Rice, and Berry proposed to add a suitable counterdiabatic (cd) term555This is the term of Sec. 2.2. The superscript is added now to distinguish it from higher order cd terms introduced below. to the time dependent Hamiltonian so as to follow the adiabatic dynamics of . The same also appears naturally when studying the adiabatic approximation of the original system, i.e., the one evolving with . This system behaves adiabatically, following the eigenstates of , precisely when the counterdiabatic term is negligible.
This is evident in an interaction picture (IP) based on the unitary transformation such that . In this IP, the new Hamiltonian is and . If is zero or negligible, becomes diagonal in the basis , so that the IP equation is an uncoupled system with solutions
The same solution, which, for a non-zero , is only approximate, is found exactly by adding to the IP Hamiltonian the counterdiabatic term . This requires an external intervention and changes the physics of the original system. In the IP the modified Hamiltonian is and in the Schrödinger picture (SP) the modified Hamiltonian is , so we identify . In other words, a “small” coupling term that makes the adiabatic approximation a good one also implies a small counterdiabatic manipulation. However, irrespective of the size of , provides a shortcut to slow adiabatic following because it keeps the populations in the instantaneous basis of invariant, in particular at the final time .
Looking for generalized adiabatic approximations, Garrido (1964), Berry (1987) or Deschamps et al. (2008) have investigated further iterative interaction pictures and the corresponding approximations. The idea is best understood by working out explicitly the next iteration: one starts with and diagonalizes to produce its eigenbasis . A unitary operator plays now the same role as in the previous IP. It defines a new IP wave function that satisfies , where and . If is zero or “small” enough, i.e. if a (first order) superadiabatic approximation is valid, the dynamics would be uncoupled in the new interaction picture, namely,
We may get the same result by changing the physics and adding to (Demirplak and Rice, 2008; Ibáñez et al., 2012a). In the SP the added interaction becomes a first order counterdiabatic term . Transforming back to the SP and using the state (33) becomes
Quite generally the populations of the final state in the adiabatic basis will be different from the ones of the adiabatic process, unless and , up to phase factors. The first condition is satisfied if and the second one if . Then the superadiabatic process will actually lead to the same final populations as an adiabatic one, possibly with different phases for the individual components. Similarly, the first-order counterdiabatic term would provide a shortcut with in the SP, different from the one carried out by . Moreover, if , then , at . Further iterations define higher order superadiabatic frames. Is there any advantage in using one or another counterdiabatic scheme? There are two reasons that could make higher order schemes attractive in practice: one is that the structure of the may change with . For example, for a two-level population inversion , whereas , where is the polar angle corresponding to the the Cartesian components of (Ibáñez et al., 2012a). The second reason is that, for a fixed process time, the cd-terms are smaller in norm as increases, up to a value in which they begin to grow, see e.g. Deschamps et al. (2008). One should pay attention though not only to the size of the cd-terms but also to the feasibility of the boundary conditions at the time edges to really generate shortcuts in this manner.
2.5 Optimal Control Theory
Optimal control theory (OCT) is a vast field covering many techniques and applications. As for STA, fast expansions (Salamon et al., 2009), wavepacket splitting (Hohenester et al., 2007; Grond et al., 2009a, b), transport (Murphy et al., 2009) and many-body state preparation (Rahmani and Chamon, 2011) have been addressed with different OCT approaches. The combination of OCT techniques with invariant-based engineering STA is particularly fruitful since the later provides by construction families of protocols that achieve a perfect fidelity or vanishing final excitation, whereas OCT may help to select among the many possible protocols the ones that optimize some physically relevant variable (Stefanatos et al., 2010, 2011; Chen et al., 2011b; Stefanatos and Li, 2012). In this context the theory used so far is the maximum principle of Pontryagin (1962). For a dynamical system , where is the state vector and the scalar control, in order to minimize the cost function , the principle states that the coordinates of the extremal vector and of the corresponding adjoint state formed by Lagrange multipliers, fulfill Hamilton’s equations for a control Hamiltonian . For almost all times during the process attains its maximum at and , where is constant. We shall discuss specific applications in Secs. 3 and 4.
3 Expansions of trapped particles
Performing fast expansions of trapped cold atoms without losing or exciting them is important for many applications: for example to reduce velocity dispersion and collisional shifts in spectroscopy and atomic clocks, decrease the temperature, adjust the density to avoid three body losses, facilitate temperature and density measurements, or to change the size of the cloud for further manipulations. Of course trap compressions are also quite common.
For harmonic traps we may address expansion or compression processes with the quadratic-in- invariants theory by setting in Eq. (7). This means that Eq. (9) does not play any role and the important auxiliary equation is the “Ermakov equation” (8). The physical meaning of is determined by its proportionality to the standard deviation of the position of the “expanding (or contracting) modes” .
Here we shall discuss the expansion from to (Chen et al., 2010b). Choosing
and commute. They actually become equal, and have common eigenfunctions. Consistent with the Ermakov equation, holds as well for a continuous frequency. At we impose666If the final frequency would not be but . If discontinuities are allowed and the frequency is changed abruptly from to the excitations will also be avoided, at least in principle. A similar discontinuity is possible at if and the frequency jumps abruptly from to .
In this manner the expanding mode is an instantaneous eigenvector of at and , regardless of the exact form of . To fix , one chooses a functional form to interpolate between these two times, flexible enough to satisfy the boundary conditions. For a simple polynomial ansatz (Palao et al., 1998), where .
The next step is to solve for in Eq. (8). This procedure poses no fundamental lower limit to , which could be in principle arbitrarily small. There are nevertheless practical limitations and/or prices to pay. For short enough , may become purely imaginary at some (Chen et al., 2010b) and the potential becomes a parabolic repeller. Another difficulty is that the transient energy required may be too high, as discussed in Chen and Muga (2010) and in the following subsection. Since actual traps are only approximately harmonic, large transient energies will imply perturbing effects of anharmonicities and thus undesired excitations of the final state, or even atom losses.
3.1 Transient Energy Excitation
Knowing the transient excitation energy is also important to quantify the principle of unattainability of zero temperature, first enunciated by Nernst. This principle is usually formulated as the impossibility to reduce the temperature of any system to the absolute zero in a finite number of operations, and identified with the third law of thermodynamics. Kosloff and coworkers in (Salamon et al., 2009) have restated the unattainability principle in quantum refrigerators as the vanishing of the cooling rate when the temperature of the cold bath approaches zero, and quantify it by the scaling law that relates cooling rate and cold bath temperature. We shall examine here the consequences of the transient energy excitation on the unattainability principle in two ways: for a single, isolated expansion, and considering the expansion as one of the branches of a quantum refrigerator cycle (Chen and Muga, 2010).
A lower bound for the time-averaged energy of the n-th expanding mode (time averages from to will be denoted by a bar) is found by applying calculus of variations (Chen and Muga, 2010), so that . If the final frequency is small enough to satisfy , and , the lower bound has the asymptotic form A consequence is that When is limited, because of anharmonicities or a finite trap depth, the scaling is fundamentally the same as the one found for bang-bang methods with real frequencies (Salamon et al., 2009), and leads to a cooling rate in an inverse quantum Otto cycle (the proportionality factor may be improved by increasing the allowed ). This dependence had been previously conjectured to be a universal one characterizing the unattainability principle for any cooling cycle (Rezek et al., 2009). The results in Chen and Muga (2010) provide strong support for the validity of this conjecture within the set of processes defined by ordinary harmonic oscillators with time-dependent frequencies. In (Hoffmann et al., 2011) a faster rate is found with optimal control techniques for bounded trap frequencies, allowed to become imaginary. There is no contradiction with the previous scaling since bounding the trap frequencies does not bound the system energy. In other words, achieving such fast cooling is not possible if the energy cannot become arbitrarily large.
Independently of the participation of the harmonic trap expansion as a branch in a refrigerator cycle, we may apply the previous analysis also to a single expansion, assuming that the initial and final states are canonical density operators characterized by temperatures and . These are related by for a population-preserving process. In a harmonic potential expansion, the unattainability of a zero temperature can be thus reformulated as follows: The transient excitation energy becomes infinite for any population-preserving and finite-time process when the final temperature is zero (which requires ). The excitation energy has to be provided by an external device, so a fundamental obstruction to reach in a finite time, is the need for a source of infinite power (Chen and Muga, 2010).
The standard deviation of the energy was also studied numerically (Chen and Muga, 2010). There it was found that the dominant dependences of the time averages scale with and in the same way as the average energy. These dependences are different from the ones in the Anadan and Aharonov (1990) (AA) relation where
3.2 Three Dimensional Effects
The previous discussion is limited to one dimension (1D) but actual traps are three-dimensional and at most effectively 1D. Torrontegui et al. (2012c) worked out the theory and performed numerical simulations of fast expansions of cold atoms in a three-dimensional Gaussian-beam optical trap. Three different methods to avoid final motional excitation were compared: inverse engineering using Lewis-Riesenfeld invariants, which provides the best overall performance, a bang-bang approach with one intermediate frequency, and a “fast adiabatic approach”777 The adiabaticity condition for the harmonic oscillator is . An efficient, but still adiabatic, strategy by Chen et al. (2010b) is to distribute uniformly along the trajectory, i.e., , being constant. Solving this differential equation and imposing we get . This may be enough for some applications. This function was successfully applied in Bowler et al. (2012)..
The optical trap considered in Torrontegui et al. (2012c) is formed by a laser, red detuned with respect to an atomic transition, and is characterized in the harmonic approximation by longitudinal and radial frequencies. To fourth order in the coordinates the effective potential includes anharmonic terms and radial-longitudinal coupling terms. While magnetic traps allow for an independent control of longitudinal and radial frequencies (Schaff et al., 2010, 2011a, 2011b), this is not the case for a simple laser trap. In Torrontegui et al. (2012c) it was assumed that the time-dependence of the longitudinal frequency is engineered to avoid final excitations with a simple 1D harmonic theory. The main conclusion of the study is that the transitionless expansions in optical traps are feasible under realistic conditions. For the inverse engineering method, the main perturbation is due to the possible adiabaticity failure in the radial direction, which can be suppressed or mitigated by increasing the laser waist. This waist increase would also reduce smaller perturbing effects due to longitudinal anharmonicity or radial-longitudinal coupling. The simple bang-bang approach fails because the time for the radial expansion is badly mismatched with respect to the ideal time, and the fast adiabatic method fails for short expansion times as a result of longitudinal excitations. Complications such as perturbations due to different noise types, and consideration of condensates, gravity effects, or the transient realization of imaginary trap frequencies are still open questions. Other extensions of Torrontegui et al. (2012c) could involve the addition of a second laser for further control of the potential shape, or alternative trap shapes. Optical traps based on Bessel laser beams, for example, may be useful to decouple longitudinal and radial motions.
3.3 Bose-Einstein Condensates
In this section we shall discuss the possibility of realizing STA in a harmonically trapped Bose-Einstein condensate using a scaling ansatz. A mean-field description of this state of matter is based on the time-dependent Gross-Pitaevskii equation (GPE) (Dalfovo et al., 1999),
Here, is the -dimensional Laplacian operator and is the -dimensional coupling constant. For a three-dimensional cloud, using the normalization , , for a condensate of a number of atoms of mass , interacting with each other through a contact Fermi-Huang pseudopotential parameterized by a -wave scattering length . In the corresponding expression for can be obtained by a dimensional reduction of the 3D GPE (Salasnich et al., 2002). As a mean-field theory the GPE overestimates the phase coherence of real Bose-Einstein condensates. The presence of phase fluctuations generally induces a breakdown of the dynamical self-similar scaling law that governs the dynamics of the expanding cloud and the formation of density ripples. The conditions for quantum phase fluctuations to be negligible for STA were discussed in del Campo (2011a). In the following we shall ignore phase fluctuations. The results of this section will be generalized to strongly correlated gases in Sec. 3.4, including as a particular case, the microscopic model of ultracold bosons interacting through s-wave scattering.
STA in the mean-field regime were designed in Muga et al. (2009) based on the classic results by Castin and Dum (1996), and Kagan, Surkov and Shlyapnikov (1996), who found the exact dynamics of the condensate wavefunction under a time-modulation of the harmonic trap frequency. Consider a condensate wavefunction , a solution of the time-independent GPE with chemical potential in a harmonic trap of frequency , i.e., . Under a modulation of the trap frequency the scaling ansatz
is an exact solution of the time-dependent Gross-Pitaevskii equation provided that
It follows that the scaling factor must be a solution of the Ermakov equation, precisely as in the single-particle harmonic oscillator case. This paves the way to engineer a shortcut to an adiabatic expansion or compression from the initial state to a target state by designing the trajectory . The modulation of the coupling constant required in can be implemented with the aid of a Feshbach resonance (Muga et al., 2009), or, in , by a modulation of the transverse confinement (Staliunas et al., 2004; Engels et al., 2007; del Campo, 2011a). The requires no tuning in time of the coupling constant as a result of the Pitaevskii-Rosch symmetry (Pitaevskii and Rosch, 1997). It has recently been suggested that this symmetry is broken upon quantization, constituting an instance of a quantum anomaly in ultracold gases (Olshanii et al., 2010). To date no experiment has provided evidence in favor of this observation. We point out that observing a breakdown of shortcuts to expansions of 2D BEC clouds would help to verify this quantum-mechanical symmetry breaking.
An important simplification occurs in the Thomas-Fermi regime, where the mean-field energy dominates over the kinetic part. Assuming the validity of this regime along the dynamics, the scaling ansatz (38) becomes exact as long as the following consistency equations are satisfied,
Hence, in the Thomas-Fermi regime, it is possible to engineer a shortcut exactly, while keeping the coupling strength constant (Muga et al., 2009). Optimal control theory has been recently applied in this regime to find optimal protocols with a restriction on the allowed frequencies Stefanatos and Li (2012).
Dimensional reduction and modulation of the non-linear interactions.– For low dimensional BECs, tightly confined in one or two directions, an effective tuning of the coupling constant can be achieved by modulating the trapping potential along the tightly confined axis, see e.g. Staliunas et al. (2004), a proposal experimentally explored in Engels et al. (2007). In a nutshell, the tightly confined degrees of freedom decoupled from the weakly confined ones, are governed to a good approximation by a non-interacting Hamiltonian. It is then possible to perform a dimensional reduction of the 3D GPE, and derive a lower-dimensional version for the weakly confined degrees of freedom, where the effective coupling constant inherits a dependence of the width of the transverse modes which have been integrated out. Adiabatically tuning the transverse confinement leads to a controlled tuning of the effective coupling constant. A faster-than-adiabatic modulation can be engineered by implementing a shortcut in the transverse degree of freedom. Consider the 3D mean-field description
with . For tight transverse confinement ( and ), the transverse excitations are frozen. The transverse mode can be approximated by the single-particle harmonic oscillator ground state , so that the wavefunction factorizes . Integrating out the transverse modes, and up to a time-dependent constant which can be gauged away, one obtains the reduced GPE
with the effective coupling A general trajectory can be implemented by modifying the frequency of the transverse confinement according to