Shortcuts to adiabaticity in three-level systems using Lie transforms
Sped-up protocols (shortcuts to adiabaticity) that drive a system quickly to the same populations than a slow adiabatic process may involve Hamiltonian terms difficult to realize in practice. We use the dynamical symmetry of the Hamiltonian to find, by means of Lie transforms, alternative Hamiltonians that achieve the same goals without the problematic terms. We apply this technique to three-level systems (two interacting bosons in a double well, and beam splitters with two and three output channels) driven by Hamiltonians that belong to the four-dimensional algebra U3S3.
pacs:32.80.Qk, 37.10.Gh, 42.79.Fm
“Shortcuts to adiabaticity” are manipulation protocols that take the system quickly to the same populations, or even the same state, than a slow adiabatic process review (). Adiabaticity is ubiquitous to prepare the system state in atomic, molecular and optical physics, so many applications of this concept have been worked out, both in theory and experiment review (). Some of the engineered Hamiltonians that speed up the adiabatic process in principle, may involve terms difficult or impossible to realize in practice. In simple systems, such as single particles transported transport_2011 () or expanded by harmonic potentials expansions_2010 (), or two-level systems 2ls_2010 (); IP (); 2ls_2011 (), the dynamical symmetry of the Hamiltonian could be used to eliminate the problematic terms and provide instead feasible Hamiltonians. In this paper we extend this program to three-level systems whose Hamiltonians belong to a four-dimensional dynamical algebra. This research was motivated by a recent observation by Opatrný and Mølmer Molmer (). Among other systems they considered two (ultra cold) interacting bosons in a double well within a three-state approximation. Specifically the aim was to speed up a transition from a “Mott-insulator” state with one particle in each well, to a delocalized “superfluid” state. The reference adiabatic process consisted on slowly turning off the inter particle interaction while increasing the tunneling rate. To speed up this process they applied a method to generate shortcuts based on adding a “counterdiabatic” (cd) term to the original time-dependent Hamiltonian Rice (); Berry09 (); 2ls_2010 (), but the evolution with the cd-term turns out to be difficult to realize in practice Molmer (). In this paper we shall use the symmetry of the Hamiltonian (its dynamical algebra) to find an alternative shortcut by means of a Lie transform, namely, a unitary operator in the Lie group associated with the Lie algebra. Since other physical systems have the same Hamiltonian structure the results are applicable to them too. Specifically the analogy between the time-dependent Schrödinger equation and the stationary wave equation for a waveguide in the paraxial approximation Longhi_2008 (); Rev_Longhi (); Rev_Nolte (); Longhi_2011 (); Vitanov_2012 (); Tseng_2013 () is used to design short-length optical beam splitters with two and three output channels.
In Sec. II we describe the theoretical model for two indistinguishable particles in two wells. In Sec. III we summarize the counterdiabatic or transitionless tracking approach and apply it to the bosonic system. Sec. IV sets the approach based on unitary Lie transforms to produce alternative shortcuts. In Sec. V we introduce the insulator-superfluid transition and apply the shortcut designed in the previous section. In Sec. VI we apply the technique to generate beam splitters with two and three output channels. Section VII discusses the results and open questions. Finally, in the Appendix A some features of the Lie algebra of the system are discussed.
Ii The model
where () are the bosonic particle annihilation (creation) operators at the -th site and is the occupation number operator. The on-site interaction energy is quantified by the parameter and the hopping energy by . They are assumed to be controllable functions of time. For two particles the Hamiltonian in the occupation number basis , and , is given by Molmer ()
This Hamiltonian belongs to the vector space (Lie algebra) spanned by , , and two more generators,
with nonzero commutation relations
This 4-dimensional Lie algebra, U3S3 Mac (), is described in more detail in the Appendix A. To find the Hermitian basis we calculate , and then all commutators of the result with previous elements. This operation is repeated for all operator pairs until no new, linearly independent operator appears.
Iii Counterdiabatic or transitionless tracking approach
The approximate time-dependent adiabatic solutions are
where the adiabatic phase reads
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 and the dot represents time derivative.
For our system (), the counterdiabatic term takes the form
Implementing this interaction is quite challenging as discussed in detail in Molmer (). In particular, a rapid switching between and , to implement through their commutator, is not a practical option Molmer (). Our goal in the following is to design an alternative Hamiltonan to perform the shortcut without .
Iv Alternative driving protocols via Lie transforms
The main goal here is to define a new shortcut, different from the one described by , where . A wave function , that represents the alternative dynamics, is related to by a unitary operator ,
and obeys , where
These are formally the same expressions that define an interaction picture. However, in this application the “interaction picture” represents a different physical setting from the original one IP (). In other words, is not a mathematical aid to facilitate a calculation in some transformed space, but rather a physically realizable Hamiltonian different from . Similarly represents in general different dynamics from . The transformation provides indeed an alternative shortcut if , so that for a given initial state . Moreover, if also the Hamiltonians coincide at initial and final times, and . These boundary conditions may be relaxed in some cases as we shall see.
We carry out the transformation by exponentiating a member of the dynamical Lie algebra of the Hamiltonian,
where is a time dependent real function to be determined. This type of unitary operator constitutes a “Lie transform”. Lie transforms have been used for example to develop efficient perturbative approaches that try to set the perturbation term of a Hamiltonian in a convenient form both in classical and quantum systems classical (); Bambusi ().
Note that in Eq. (23) becomes and commutes with . Then, , given now by
depends only on , , and its repeated commutators with , so it stays in the algebra. If we can choose and so that the undesired generator components in cancel out and the boundary conditions for are satisfied, the method provides a feasible, alternative shortcut. In the existing applications of the method IP (); review (), and in this paper we proceed by trial an error, testing different generators. In the present application we want the Hamiltonian to keep the structure of the original one, with non-vanishing components proportional to and . We may quickly discard by inspection , , and as candidates for . Choosing in Eq. (24), and substituting into Eqs. (22) and (25), the series of repeated commutators may be summed up. becomes
To cancel the term, we choose
which has the same structure (generators) as the reference Hamiltonian but with different time-dependent coefficients.
V Insulator-Superfluid transition
Changing the ratio, the system may go from a “Mott insulator” (the two particles isolated in separate wells) to a “superfluid” state (in which each particle is distributed with equal probability in both wells). From Eq. (11), the Mott-insulator ground state is and in the superfluid regime the ground state becomes . To design a reference process (one that performs the transition when driven slowly enough) we consider polynomial functions for and . Since we want to drive the system from to , we impose in Eq. (11)
To have the wells isolated at but connected (allowing the particles to pass from one to the other) at we also set
so that and . Moreover, for a smooth connection with the asymptotic regimes (, ) we put
This implies that , see Eq. (20). The condition
is also needed to implement alternative shortcuts, in particular, to satisfy . At intermediate times, we interpolate the functions as and , where the coefficients are found by solving the equations for Eqs. (29), (30), (31) and (32). These functions are shown in Fig. 1. In this and other figures , where is the maximum value of .
The actual time evolution of the state
is given by solving Schrödinger’s equation with the different Hamiltonians. For this particular transition, and the ideal target state is (up to a global phase factor) .
The dynamics versus time is shown in Fig. 2 for . For this short time fails to drive the populations to and , whereas when is added the intended transition occurs successfully. As for the alternative Hamiltonian in Eq. (28), with , and in Eq. (27), we find
(Eq. (32) is necessary to have and consequently ), whereas
However so and provides the desired shortcut.
Solving numerically the dynamics for we obtain a perfect insulator-superfluid transition (see Fig. 2 (b)). Notice that, as is diagonal in the bare basis, the bare-populations are the same for the dynamics driven by and , see Fig. 2 (b).
In order to compare our approach with other protocols we reformulate as
where and . The inverse transformation is
Consider a simple protocol with and a linear from and Molmer (). Setting the value of so that , it is found that the simple protocol needs to perform the transition with a 0.9999 fidelity. In other words, the protocol based on is times faster according to this criterion.
Vi Beam splitters
The three-level Hamiltonian (2) describes other physical systems apart from two bosons in two wells. For example it represents in the paraxial approximation and substituting time by a longitudinal coordinate three coupled waveguides Longhi_2008 (); Rev_Longhi (); Rev_Nolte (); Longhi_2011 (); Vitanov_2012 (); Tseng_2013 (), where is controlled by waveguide separation and by the refractive index. In particular and may be manipulated to split an incoming wave in the central wave guide into two output channels (corresponding to the external waveguides) or three output chanels Vitanov_2012 (); Tseng_2013 (). The Hamiltonian also represents a single particle in a triple well Mompart_2004 (), with representing the bias of the outer wells with respect to the central one and the coupling coefficient between adjacent wells. The beam splitting may thus represent the evolution of the particle wave function from the central well either to the two outer wells or to three of them with equal probabilities.
For either of these physical systems111The Hamiltonian (2) also describes a three-level atom under appropriate laser interactions, see Longhi_2008 (). the minimal channel basis for left, center and right wave functions is , and .
vi.1 1:2 beam splitter
To implement a beam splitter, see Fig. 4, the goal is to drive the eigenstate from to . As in the previous section we use polynomial functions for and to set a reference process. We impose
in Eq. (11). The wells (waveguides) should be isolated at initial and final times. If morever all wells are at equal heights at those times we set
to satisfy . We also impose
to smooth the functions at the time boundaries and make . In addition
Figure 6 shows the dynamics for . This time (corresponding to the splitter length in the optical system) is too short for the reference Hamiltonian to drive the bare-basis populations to and . Adding the transition occurs as desired. As in Sec. IV, we construct an alternative shortcut without using the transformation . With in Eq. (27), , whereas
This is enough for our objective as , and .
To compare the new shortcut and the simple approach with and we set . The constant- protocol needs to achieve fidelity, so the protocol driven by is times faster.
vi.2 1:3 beam splitter
We also describe briefly a beam splitter, see Fig. 8. The aim is to drive the system from to equal populations in , , and . To design a reference protocol we use polynomial interpolation for and , see Fig. 9, with the same boundary conditions of the splitter but with and the additional condition (to satisfy so that ). The Lie transform may be applied as before on the protocol with the counterdiabatic correction, see Fig. 10 (b).
A simple protocol with and needs , if , for a 0.9999 fidelity, so the protocol based on is 11 times faster.
Starting from shortcuts to adiabaticity for three-level systems with U3S3 symmetry (a four-dimensional Lie algebra) that include Hamiltonian terms difficult to implement in the laboratory, we have found alternative shortcuts without them by means of Lie transforms. These transformations are formally equivalent to interaction picture (IP) transformations. However the resulting IP-Hamiltonian and state represent a different physical process from the original (Schrödinger) Hamiltonian and dynamics. We have found shortcuts for different physical systems. For two particles in two wells we have implemented a fast insulator-superfluid transition. For coupled waveguides or a particle in a triple well we have implemented fast beam splitting with one input channel and two or three output channels. In all cases the IP Hamiltonian involves only two realizable terms (generators).
In a companion paper we have worked out a related method Lie (). Both approaches rely on Lie algebraic methods and aim at constructing shortcuts to adiabaticity. However we do not use dynamical invariants explicitly in the current approach, whereas the bottom-up approach in Lie () engineers the Hamiltonian making explicit use of its relation to dynamical invariants. In contrast we start here from an existing, known shortcut –for example the one generated by a counter-diabatic method–; then, a Lie transform is applied to generate alternative, feasible or more convenient shortcuts, as in IP (). A connection between the transformation method and dynamical invariants is sketched briefly in the Appendix but it deserves an extensive separate study.
Finally, further applications of this work may involve systems with Lie algebras of higher dimension. Within the scope of the algebra U3S3, other physical systems that could be treated are in quantum optics (three level atoms) Chen_3ls (); rev_Shore (), nanostructures (triple wells or dots) Kiselev (), optics (mode converters) Tseng_2012 (); Chen_2012 (), or Bose-Einstein condensates in an accelerated optical lattice Dou ().
This work was supported by the National Natural Science Foundation of China (Grant No. 61176118), the Grants No. 12QH1400800, IT472-10, BFI-2010-255, 13PJ1403000, FIS2012-36673-C03-01, and the program UFI 11/55. S. M.-G. acknowledges a fellowship by UPV/EHU. E. T. is supported by the Basque Government postdoctoral program.
Appendix A Lie algebra
The algebra of this three-level system is a four-dimensional Lie Algebra U3S3 according to the classification of 4-dimensional Lie algebras in Mac (). (For comparison with that work it is useful to rewrite the generators in the skew-Hermitian base , .) is a direct sum of the one dimensional algebra spanned by the invariant , that commutes with all members of the algebra, and a three-dimensional SU(2) algebra spanned by . Notice that this realization of the 3D algebra is not spanned by the matrices
that correspond, in the subspace , to the operators
In particular we cannot get the matrices for or by any linear combination of our matrices (see Eqs. (3-4)). A second-quantized form for the consistent with the matrices includes quartic terms in annihilation/creation operators:
Notice that these second-quantized operators do not form a closed algebra under commutation but their matrix elements for two particles do.
An invariant (defined in a Lie-algebraic sense) commutes with any member of the algebra. There are generically two independent invariants for patera (). For the matrix representation in Eqs. (3) and (4) they are
, which is not in the algebra, has eigenvalues
and , a member of the algebra, has eigenvalues
The two invariants have the same eigenvectors,
with and spanning a degenerate subspace.
Lie-algebraic invariants constructed with time-independent coefficients satisfy as well the equation
so they are also dynamical invariants LR () (i.e., operators that satisfy Eq. (54) whose expectation values remain constant). The degenerate subspace of eigenvectors allows the existence of time-dependent eigenstates of time-independent invariants. In particular, in all the examples in the main text, the dynamics takes place within the degenerate subspace: the initial state is at and ends up in some combination of and at . The specific state as a function of time is known explicitly, , see Eq. (21). Note that and in Eqs. (11) and (13) are two orthogonal combinations of and . Also , see Eq. (12). In the non degenerate subspace spanned by “nothing evolves”, other than a phase factor, but the initial states in the examples do not overlap with it.
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