A Scale-invariant eigenfunctions

Classical and quantum shortcuts to adiabaticity for scale-invariant driving


A shortcut to adiabaticity is a driving protocol that reproduces in a short time the same final state that would result from an adiabatic, infinitely slow process. A powerful technique to engineer such shortcuts relies on the use of auxiliary counterdiabatic fields. Determining the explicit form of the required fields has generally proven to be complicated. We present explicit counterdiabatic driving protocols for scale-invariant dynamical processes, which describe for instance expansion and transport. To this end, we use the formalism of generating functions, and unify previous approaches independently developed in classical and quantum studies. The resulting framework is applied to the design of shortcuts to adiabaticity for a large class of classical and quantum, single-particle, non-linear, and many-body systems.


I Introduction

Modern research in nanoengineering develops increasingly small devices, which operate in a regime described by effective classical dynamics Collins et al. (1997); Ming et al. (2003) or quantum mechanics Kane (1998); Kinoshita et al. (2006). Achieving a fast coherent control with high-fidelity Avron et al. (1988); Král et al. (2007) is a ubiquitous goal shared by a variety of fields and technologies, including quantum sensing and metrology Giovannetti et al. (2006), finite-time thermodynamics Andresen et al. (1984), quantum simulation Trabesinger (2012), and adiabatic quantum computation Nielsen and Chuang (2000). The quantum adiabatic theorem Messiah (1966), however, appears as a no-go theorem for excitation-free ultrafast processes. As a result, an increasing amount of theoretical and experimental research is targeting the design of shortcuts to adiabaticity (STA), i.e., non-adiabatic processes that reproduce in a finite-time the same final state that would result from an adiabatic, infinitely slow protocol Torrontegui et al. (2013).

A variety of techniques have been developed to engineer STA: the use of dynamical invariants Chen et al. (2010, 2011), the inversion of scaling laws del Campo (2011); del Campo and Boshier (2012), the fast-forward technique Masuda and Nakamura (2010, 2011); Torrontegui et al. (2012), and counterdiabatic driving, also known as transitionless quantum driving Demirplak and Rice (2003, 2005); Berry (2009). Among these techniques, counterdiabatic driving (CD) is unique in that it drives the dynamics precisely through the adiabatic manifold of the system Hamiltonian. In addition, it enjoys a wide applicability. In its original formulation Demirplak and Rice (2003, 2005); Berry (2009), one considers a time-dependent Hamiltonian with instantaneous eigenvalues and eigenstates . In the limit of infinitely slow variation of a solution of the dynamics is given by


In this adiabatic limit no transitions between eigenstates occur Messiah (1966), and each eigenstate acquires a time-dependent phase that can be separated into a dynamical and a geometric contribution Berry (1984), represented by the two terms inside the exponential in the above expression.

Now consider a non-adiabatic Hamiltonian . In the CD paradigm, a corresponding Hamiltonian is constructed, such that the adiabatic approximation associated with (1) is an exact solution of the dynamics generated by under the time-dependent Schrödinger equation. Writing the time-evolution operator as , one arrives at an explicit expression for Demirplak and Rice (2003, 2005); Berry (2009):


Here, the auxiliary CD Hamiltonian enforces evolution along the adiabatic manifold of : if a system is prepared in an eigenstate of and subsequently evolves under , then the term effectively suppresses the non-adiabatic transitions out of that would arise in the absence of this term. Note that the evolution is non-adiabatic with respect to the full CD Hamiltonian .

The CD Hamiltonian has been the object of intense study. It was found that the higher the speed of evolution, the larger is the intensity of the required auxiliary CD field Demirplak and Rice (2008); del Campo et al. (2012). Experimental demonstration of driving protocols inspired by the CD technique have recently been reported in single two-level systems Bason et al. (2011); Zhang et al. (2013). In the many-body case, generally includes non-local and multi-body interactions del Campo et al. (2012); del Campo (2013). Local driving protocols can be derived for unitarily equivalent Hamiltonians del Campo (2013), an approach which has proven useful as well in single-particle systems Bason et al. (2011); Zhang et al. (2013); Mostafazadeh (2001); Torrontegui et al. (2013).

However, the computation of the auxiliary term requires knowledge of the spectral properties of the instantaneous system Hamiltonian at all times. This constraint has limited the range of applicability of the method to the control of few-level systems Demirplak and Rice (2003, 2005); Berry (2009) and non-interacting matter-waves in time-dependent harmonic traps Muga et al. (2010); Torrontegui et al. (2011); Deng et al. (2013).

Recently a classical analogue of CD was proposed, namely dissipationless classical driving Jarzynski (2013). Here, for a time-dependent classical Hamiltonian , one seeks an auxiliary term such that under the Hamiltonian dynamics generated by , the classical adiabatic invariant of is conserved exactly. For systems with one degree of freedom an explicit solution of this problem, analogous to (1) above, was obtained (see Eq. (32) of Ref. Jarzynski (2013)). Moreover it was argued that this classical solution can be useful in constructing the quantal CD Hamiltonian , bypassing the spectral decomposition of . This was illustrated for arbitrary power-law traps (including the particle-in-a-box as a limiting case), for which simple expressions for in terms of position and momentum operators were obtained and quantized. Further progress was achieved using scaling laws in expansions and compressions for a wide-variety of single-particle, nonlinear, and many-body quantum systems del Campo (2013).

Our aim in this paper is to find an experimentally realizable CD Hamiltonian (2) for scale invariant processes, without using the explicit spectral decomposition of . Scale-invariant driving is generated by transformations of for which the density profile (and all local correlations in real space) is preserved up to scaling and translation. Using this property, we start with a single quantum particle in a one-dimensional potential, from which we will develop a general framework to find local CD protocols for multi-particle quantum systems, obeying both linear and non-linear dynamics. We will use methods from classical Hamiltonian dynamics, namely the formalism of generating functions, to treat dissipationless classical driving by the same means. Our approach also allows one to treat arbitrary external potentials, beyond the validity of perturbation theory.

The paper is organized as follows: we will begin in Sec. II by deriving an expression for the CD Hamiltonian (2) for the scale-invariant driving of a quantum system with one degree of freedom. Section III is dedicated to classical Hamiltonian dynamics, in which the classical version of can be rewritten in a local form using linear canonical transformations and the formalism of generating functions. These findings will be generalized and applied in Sec. IV to a broad family of many-body quantum systems. Specific protocols for arbitrary trapping potentials will be discussed in Sec. V. Section VI is dedicated to nonlinear systems, with emphasis in mean-field theories. In Sec. VII, we will discuss the relation of CD to more general scaling laws, before we explicitly engineer STA in Sec. VIII. We close with a summary and discussion in Sec. IX.

Ii Counterdiabatic Hamiltonian for scale-invariant driving

Generally, it appears to be hardly feasible to find closed form expressions, i.e., expressions that do not depend on the full spectral decomposition of , for the auxiliary term in the CD Hamiltonian (2). Recently, it has been shown that scale-invariance greatly facilitates this task for processes that describe self-similar expansions and compressions in a time-dependent trap del Campo (2013), including the family of power-law potentials as a special case Jarzynski (2013). More generally, scale-invariant driving refers to transformations of the system Hamiltonian associated with a set of external control parameters which can be absorbed by scaling of coordinates, time, energy, and possibly other variables to rewrite the transformed Hamiltonian in its original form up to a multiplicative factor. If only the potential term is modulated, its overall shape does not change under . For the time being, we focus on a quantum system with a single degree of freedom,


where and . Note that generally and are both allowed to be time-dependent, but we assume that they are independent of each other. This time-dependence encompasses transport processes (), dilations (such as an expansion or compression, with ) and combined dynamics, which are the focus of our attention and elements of the dynamical group of the system Hamiltonian, the universal covering group of , Gambardella (1975).

Our goal is to rewrite the auxiliary term (2) into a form that does not rely on the spectral decomposition of . Let be an eigenfunction of the Hamiltonian , then is an eigenfunction of , where is a normalization constant. The proof of this statement can be found in appendix A.

Now, we want to use this symmetry to simplify in Eq. (2). We have,


which reads in space representation


To simplify this expression, we note that


For the sake of clarity, let us treat both terms of in (5) separately. We obtain for the first term


while the second term reduces to


Note that the second component of does not contribute, since the wavefunction vanishes at infinity due to normalizability. In conclusion, we obtain the explicit expression of the auxiliary CD Hamiltonian,


where we used . Notice that in Eq. (9) is of the general form , which was found for a time-dependent harmonic-trap Muga et al. (2010) and more generally in Refs. Jarzynski (2013); del Campo (2013) for the class of potentials


where , and . See as well Jarzynski (2013); Di Martino et al. (2013) for a discussion of the limiting case , that of a box-like confinement. Obviously this class (10) belongs to the more general scale-invariant potentials introduced above in Eq. (3).

Equation (9) is our first main result. For all driving protocols under which the original Hamiltonian is scale-invariant, i.e., where the time-dependent potential is of the form (3), the auxiliary term takes the closed form (9). In particular, is independent of the explicit energy eigenfunctions, and only depends on the anticommutator, , the generator of dilations. As a result, CD applies not only to single eigenstates, but also to non-stationary quantum superpositions and mixed states. However, the expression (9) is still not particularly practical as non-local Hamiltonians 1 are hard to realize in the laboratory. We continue our analysis by explicitly constructing coordinate transformations, which allow us to write in local form, i.e., where depends only on position. In order to do so we will use the classical version of CD as a guide.

Iii Scale-invariant driving – a case for generating functions

We now turn to dissipationless classical driving Jarzynski (2013); Deng et al. (2013), the classical analogue of quantum counterdiabatic driving. For scale-invariant Hamiltonians the connection between the quantum and classical cases is particularly close, and the corresponding auxiliary CD terms and are essentially identical, up to quantization.

In complete analogy with the quantum case we consider a classical Hamiltonian with one degree of freedom,


where is a point in phase space. The classical adiabatic invariant is given by




is the volume of phase space enclosed by the energy shell of . In the adiabatic limit, remains constant along a Hamiltonian trajectory evolving under , just as the quantum number remains constant in the quantum case. We now consider non-adiabatic driving of the parameters , and we seek an auxiliary CD term


resembling Eq. (4), such that remains constant at arbitrary driving speed, for any trajectory evolving under the Hamiltonian .

It is useful to picture dissipationless driving in terms of an ensemble of trajectories evolving under , with initial conditions sampled from an energy shell of . Since the value of is preserved for every trajectory in this ensemble, at any later time these trajectories populate a single energy shell of , determined by the condition , which defines the adiabatic energy shell.

As discussed in Jarzynski (2013), it is useful to view as a generator of infinitesimal transformations , with


where is the Poisson bracket. Equation (15) provides a rule for converting a small change of parameters, , into a small displacement in phase space, . In order to achieve dissipationless classical driving, the energy shells of must be mapped, under Eq. (15), onto those of with


When this condition is satisfied, the term provides precisely the counterdiabatic driving required to preserve the value of . Thus, to construct the CD Hamiltonian, we must find the function that generates infinitesimal deformations of the adiabatic energy shell, as per Eqs. (15) and (16).

Our scale-invariant Hamiltonian




for any real and positive . Using these properties we can verify by direct substitution that the canonical mapping


satisfies Eq. (16). The change produces a coordinate translation, while under the change , the adiabatic energy shell is stretched along the coordinate and compressed along the momentum . The infinitesimal transformation (19) is generated by


as verified by substitution into Eq. (15), with and . Combining Eqs. (14) and (20) we arrive at


the classical counterpart of (9). With this auxiliary term, the value of remains constant along a trajectory evolving under the Hamiltonian , for any protocol . To illustrate this general result, we derive for an analytically solvable example, namely the parametric Morse oscillator in appendix B.

Equation (21) gives us a nonlocal CD Hamiltonian that accomplishes dissipationless classical driving. Our goal now is to find a coordinate transformation mapping to a set of new variables , and a corresponding Hamiltonian whose dynamics (in -space) is identical to that under , and for which is local, i.e. it is the sum of a kinetic energy term and a function . In classical mechanics such problems can be elegantly solved by the formalism of generating functions Goldstein (1959).

We briefly recall the main idea. Let be a time-dependent Hamiltonian, written in terms of coordinates in a two-dimensional phase space. Now consider new coordinates that are related to by a time-dependent canonical transformation:


Since canonical transformations are invertible, we can alternatively express the “old” coordinates as functions of the “new” ones, . If a function can be constructed such that the relationship between the two coordinate sets is given by


then is called a generating function of a type-2 canonical transformation Goldstein (1959). We can then define a Hamiltonian


that generates trajectories equivalent to those of . By this we mean that solutions to the equations


when rewritten in the new coordinates become solutions to


The function thus encodes the transformation of both the variables (23) and the Hamiltonian (24).

In what follows we shall apply this approach to three different sets of coordinates related by canonical transformations. The CD Hamiltonian for scale-invariant dynamics with the non-local term (21) reads


Now we define a type-2 generating function


and we use it with (23) and (24) to construct a canonical transformation to coordinates , obtaining




The Hamiltonians and generate equivalent trajectories, in the sense of Eqs. (25) and (26). Moreover, since these trajectories are identical in configuration space. This can be verified independently by considering the second-order differential equation for the coordinates and . In either case we have,


Comparing (27) and (30), we see that the non-local terms in the former are replaced by local terms in the latter, which we conceptually identify as a local formulation of (21). The first of these new terms in is an inverted harmonic oscillator whose stiffness is proportional to the acceleration of the scaling factor, cf. also Refs. del Campo and Boshier (2012); del Campo (2013). The second term is the classical analog of a Duru transformation in transport processes Duru (1989).

Now consider a protocol in which the parameters and are fixed outside some interval (as in the inset of Fig. 1), and imagine trajectories and that evolve under and , respectively, from identical initial conditions at . These equivalent trajectories are related by (29) at every instant in time. This immediately implies that and are identical in phase space for , then their momenta diverge during the interval , and finally they meet again at and remain identical thereafter. Since the adiabatic invariant is preserved exactly along the trajectory , it follows that along the trajectory the initial value of (at ) is identical to the final values of (at ), even if it varies at intermediate times. Thus the local Hamiltonian (30) provides a shortcut to adiabaticity, provided the parameters and are fixed initially and finally.

To gain further insight, let us construct a new canonical transformation, to variables , using


Applying (23) and (24) we get




The transformation (33) is a linear dilation of the coordinate and the reciprocal contraction in momentum space.

The fact that is time-independent, apart from the factor , has two interesting consequences. First, the quantity is a dynamical invariant, as follows from direct inspection of Hamilton’s equations. If we picture a level surface of as a closed loop in -space, then under a trajectory simply evolves round and round this loop, at a speed proportional to . The function can be expressed in any of the three sets of phase space coordinates considered above. The resulting functions


are all dynamical invariants, along Hamiltonian trajectories generated by , and , respectively. This follows from the equivalence of the trajectories , , and , but it can also be verified by inspection of Hamilton’s equations.

The invariance of allows us to visualize the evolution of these trajectories, as each one clings to a level surface of expressed in the given phase space coordinates. If is not varied with time, then a level surface of gets stretched along and contracted along as increases with time (or the other way around if decreases); and a level surface of additionally acquires a shear along the momentum direction, proportional to , as illustrated by the pairs of diagonal lines in Fig. 1. If is varied with time, then a level surface of undergoes translation along the coordinate , and level surface of additionally undergoes a displacement along by an amount .

Second, if we introduce the new time-like variable Berry and Klein (1984)


we obtain


which describe motion under a time-independent Hamiltonian, whose energy shells are the level surfaces of . Let denote a particular solution to these equations of motion. Inverting the canonical transformations in Eqs. (33) and (40), we can immediately use this solution to construct trajectories generated by the Hamiltonians and , namely:




Hence, trajectories generated by the time-dependent Hamiltonians and can be constructed directly from trajectories evolving under a time-independent Hamiltonian . This further emphasizes the equivalence between these trajectories. We will exploit these observations in the following discussion of shortcuts for time-dependent multi-particle quantum systems.

Energy-like dynamical invariants such as were intensely studied in the mathematical literature for classical and quantum dynamics. In particular, it can be shown that if (and only if) an energy-like invariant exists, then one can find a coordinate transformation as discussed in the present analysis Lewis (1968); Lewis and Riesenfeld (1969); Günther and Leach (1977); Leach (1977, 1977, 1977); Mostafazadeh (2001); Lohe (2009).

For completeness, we note that the transformation from to is generated by the function


for which we have


An illustrative example – particle in time-dependent box

For a particle in a time-dependent box the form of the new Hamiltonian (30) can be understood intuitively. Consider a particle of mass inside a one-dimensional box with hard walls at and , as described by the Hamiltonian


where is zero inside the box, and “infinite” outside. We further assume that changes with constant rate for times and is constant otherwise with and , cf. Fig. 1.

Figure 1: Shortcut to adiabaticity based on dissipationless classical driving. Energy shells for a particle in a one-dimensional box (42), in a time-dependent piston of width which changes at constant rate for (inset). Energy shells corresponding to are shown as a pair of parallel, dotted line segments of length , at momenta . The solid lines represent a level surface of the adiabatic invariant corresponding to the full, counter-adiabatic Hamiltonian (43). At the force (44) induces a “jump” of trajectories from dashed to solid line, for the adiabatic energy shell is deformed invariantly, and finally at force (44) induces jumps back to .

Now imagine the aforementioned adiabatic energy shell as a closed loop that is deformed as is varied with time. Then generates motion around this loop, and the auxiliary CD term adjusts each trajectory so that it remains on-shell Jarzynski (2013), see Fig. 1. The dashed lines represent the adiabatic energy shells corresponding to for a particular energy . Notice that particles at hit the hard wall, and are “boosted” from one branch to the other. In other words, particles hitting the hard wall with momentum are reflected at and travel back with , and so close the loop. The solid lines, and represent a level surface of the adiabatic invariant corresponding to the full counterdiabatic Hamiltonian


In the previous discussion we were asking for a set of coordinates and the corresponding Hamiltonian , for which here the solid lines represent the exact solution. For times and the energy shells for old and new coordinates are identical. This means that at the trajectories have to “jump” from to , where , and at back to the unperturbed shell. These jumps are induced by a force


which applies “impulses” at and . The latter force is the derivative of an auxiliary potential, ,


which we recognize as the additional potential term in the transformed Hamiltonian (30), with and .

Therefore, we conclude that the additional harmonic term in the Hamiltonian (30) with possible negative spring constant is nothing else but the term necessary to facilitate the transfer of the classical trajectories from the energy shells of to those invariant under (27) and eventually (30). Interestingly enough, this result agrees with the CD derived in the quantum case for a time-dependent box-like confinement using Lewis-Riesenfeld invariants and reverse engineering of scaling laws del Campo and Boshier (2012).

Iv Multi-particle quantum systems

In the previous section we showed how the auxiliary, classical term in the counterdiabatic Hamiltonian can be brought into a local form. We will next apply this finding to general multi-particle quantum systems. Let us consider the broad family of many-body systems described by the Hamiltonian


with unless stated otherwise ( denoting the effective dimension of the system), and where is the Laplace operator, and represents an external trap whose time-dependence is of the form (3), . As before (3), the trap can be shifted by the time-dependent displacement and simultaneously modulated by the scaling factor . We further assume that the two-body interaction potential obeys


which includes relevant examples in ultracold gases such as the pseudo-potential for contact interactions Yurovsky et al. (2008), e.g., the Fermi-Huang potential for s-wave scattering for which Huang (1996).

We define the dimensionless coupling constant at and consider a stationary state , with chemical potential , i.e., . The scale-invariant solution for this multi-particle quantum system, that generalizes the wavefunction for a single degree of freedom discussed earlier, reads


where is the time-like variable introduced above (36). By substituting the latter ansatz into the many-body Schrödinger equation we find that is actually the exact time-dependent solution for the dynamics generated by CD Hamiltonian


where the scaled spatial coordinate reads , as before. The scale-invariant solution to a related classical and restricted problem with was derived by Perelomov Perelomov (1978). We observe that in an interacting system (with ), there is an additional consistency condition for the dynamics to be scale-invariant


given the definition . Generally, inducing a scale-invaraint dynamics in an interacting system requires to tune interaction along the process. In ultracold atom experiments, this is a routine task in the laboratory assisted by means of a Feshbach resonance Timmermans et al. (1999) or a modulation of the transverse confinement in low-dimensional systems Staliunas et al. (2004); del Campo (2011). No interaction tuning is required in processes involving transport exclusively, this is, in protocols for which and . For processes with , there are relevant scenarios for which and no interaction tuning is required del Campo (2011); del Campo and Boshier (2012). In addition, for processes with and , whenever the scaling factor remains of order unity along the process, , a high-fidelity quantum driving is achieved even in the absence of interaction tuning, i.e. while keeping del Campo and Boshier (2012).

Provided that the consistency equation Eq. (50) is fulfilled (or approximately satisfied) so that , becomes a first integrand or constant of motion, and can be identified as an invariant operator satisfying


and that is the quantum equivalent of the classical, energy-like dynamical invariant (35).

The third line in Eq. (49) corresponds to the auxiliary CD Hamiltonian which in the original variables, , reads


Here, the curly brackets denote the anticommutator of two operators and , . In complete analogy to the classical case, the first term is the auxiliary CD term associated with transport along the trajectory , while the second-one is associated with the expansion. Equation (52) agrees with the single particle expression in Eq. (9) and previous results derived for power-law traps Chen et al. (2011); Jarzynski (2013); del Campo (2013).

In the previous section we found coordinate transformations, that allowed us to write the CD Hamiltonian for a system with one degree of freedom in local form. The crucial steps involved finding a generating function for the coordinate transformation, and a corresponding dynamical invariant. In the following, we will apply the same ideas to the multi-particle Hamiltonian (49). The representation in quantum mechanics of the group of linear canonical transformations has been discussed at length by Moshinsky, see e.g. Moshinsky and Smirnov (1996). We denote the quantum, multi-particle unitary transformation that plays the role of the classical generating function (28) by . It reads


The latter functions transform the “old” set of coordinates to a new set according to


Here, the new representation of the CD Hamiltonian becomes


which is the multi-particle quantum equivalent of the classical Hamiltonian (30). Under this canonical transformation the time-evolution of the initial state is mapped to . Finally, it follows that the dynamical invariant can be written in new coordinates ,