Shortcut to adiabatic control of soliton matter waves by tunable interaction

We propose a method for shortcut to adiabatic control of soliton matter waves in harmonic traps. The tunable interaction controlled by Feshbach resonance is inversely designed to achieve fast and high-fidelity compression of soliton matter waves as compared to the conventional adiabatic compression. These results pave the way to control the nonlinear dynamics for matter waves and optical solitons by using shortcuts to adiabaticity.

dependence of nonlinearity is specified even for BEC atoms. Here fast compression of the bright soliton for matter waves, as an extension, is proposed by tuning time-dependent nonlinearity, contributing from atom interaction, in terms of Feshbach resonance.

Results
Model and Hamilton. It is well known that at zero temperature the dynamics of weakly interacting Bose gas trapped in a time-dependent harmonic trap is well described by Gross-Pitaevski equation (GPE). Here we consider the cigar-shaped harmonic trapping potential with the elongated axis in the x-direction, where ψ(x, t) is the wavefunction of the condensate, g(t) is a time-varying function of nonlinearity. g > 0 (g < 0) corresponds to the case of negative (positive) scattering length a s (t). Considering physical potential, V(x, t) = ω 2 [x − x 0 (t)] 2 is a real function with trap frequency ω. For convenience, all variables have been dimensionless in Eq. (1).
(transverse harmonic-oscillator length), ω ⊥ is the transverse frequency, and m is the atomic mass. g(t) = 4πħ 2 a s (t)/m characterizes the strength of interatomic interaction, which can be effectively modulated by s-wave scattering length a s with Feshbach resonances in the experiment.
We begin with the scaling solution of Eq. (1), which is the well-known bright solitary wave solution 29 , represented as , and φ ≡ φ(t) represent the amplitude, width, chirp, velocity, center position and phase, respectively. They are all time dependent and real functions. The normalization is Here two main parameters a and ζ satisfy the following equations: Obviously, the width of wave packet a is connected with time-dependent interaction g(t) through Eq. (3). Equation (4) also provides the relation between the center of the mass of wave packet ζ(t) and trap center x 0 (t).
Here we are concerned about the compression, instead of transport, so the harmonic trap is assumed to be static, x 0 (t) = 0, and the solution of Eq. (4) gives ζ(t) ≡ 0. In what follows we will only play with Eq. (3) for inverse engineering, thus achieving fast and perfect soliton compression by designing the atom interaction appropriately, from initial state 2 to the final one Adiabatic Reference. First of all, we discuss the adiabatic reference of soliton compression before using shortcuts to adiabaticity in terms of inverse engineering. To this end, we consider the analogy exists between Eq. (3) and the dynamical equation of a fictitious (classical) particle with position x, as a perturbed Kepler problem 27,29 . The particle moves in the potential of the form, Newton's equation takes indeed the form of Eq. (3) with certain kinetic energy  a /2 2 . This analogy helps us to achieve the adiabatic reference for further STA design. When ω = 0, ∂ U(t)/∂ a = 0 gives the minimum point of potential On the other hand, when ω ≠ 0, the above equation gives the numerical solution a c (t) from the following expression: Scientific RepoRts | 6:38258 | DOI: 10.1038/srep38258 Assuming ω ≪ 1, we can obtain the analytical solution, by using perturbation, as 27 which provides the minimum of perturbed Kepler potential of an effective particle. Thus, a c (t) in Eq. (10) provides the adiabatic reference, when the interaction g(t) is given, since trajectory a c (t) always gives the solution of non-adiabatic energy minimization 17 . Without loss of generality, we choose the switching function of g(t) as follows,  In adiabatic case, final width a(t f ) approaches asymptotically the adiabatic reference a c (t f ), see also Fig. 1(a). Furthermore, numerical calculations are performed in Fig. 1(b) by solving the GP equation (1) with time-dependent nonlinear term g(t), using the Crank-Nicholson scheme 31 . By using adiabatic protocol, we can compress the soliton from the initial width a(0) = 0.494 to the final one a(t f ) = 0.0834. The results presented here almost coincide with the evolution of a(t) in Fig. 1(a). The slight difference results from the property of switching function (11), see the discussion below. However, for non-adiabatic process, the wave function of soliton at final time t f is completely different from what we expected from the adiabatic reference a c (t). Actually, multi-soliton could happen when the process is short.
Inverse Engineering. Now we shall apply the auxiliary differential equation (3) to construct STA protocol in terms of concept on inverse engineering. To speed up the adiabatic process but achieving the same target as the adiabatic reference, we have to set the following boundary conditions at the time edges t = 0 and t = t f , c f c f In addition, more boundary conditions, c f c f are required to make  a t ( ) and ä t ( ) continuous at t = 0 and t = t f through Eq. (3). From Eqs (5) and (6), the boundary conditions guarantee that during fast adiabatic-like compression of soliton the initial and final states should be the same as the adiabatic references with specified width a c (0) and a c (t f ). Otherwise, the initial and final states are not stationary solution of GPE at initial and final times, so the STA design will be broken down. Once the boundary conditions are settled down, we can use simple polynomial ansatz to interpolate the function of a(t), where the coefficients b j are solved by using the boundary conditions Eqs (14)(15)(16). Finally the function of nonlinearity, g(t), can be constructed from Eq. (3). Since the boundary conditions are fixed in advance, the final result should be consistent with the adiabatic reference for an arbitrarily short time. This guarantees how the STA works for fast adiabatc-like compression of soliton. Noting that we consider the nonlinearity g(t) as a free parameter to control. This is different from the previous shortcut design for BEC by using inverse engineering 13 and counter-diabatic driving 24 , where the non-interacting limit is assumed or the specific driving of the interaction, g(t) = 1/a, has to be imposed. Figure 2(a) demonstrates that the evolution of a(t) designed by STA does not follow the adiabatic reference a c (t), but the initial and final values do coincide with the adiabtic reference. This means the soliton can be compressed via STA. In Fig. 2(b) the designed the function of nonlinearity, g(t), are also depicted, in which the values of g(0) and g(t f ) are consistent with the values predicted by adiabatic reference, hyperbolic tangent function (11). Mathematically, there is no limit for shortening the time by using STA, when the boundary conditions are satisfied. However, physical constraint comes from the experimental implementation of time-dependent nonlinearity g(t). Since the time-dependent interaction is tuned by using Feshbach resonances in presence of magnetic or electric fields 32 , one can not modulate atom interaction as rapidly as we wish. In addition, when decreasing t f the maximum value of g(t) becomes larger, see Fig. 3. Supposing operation time is quite short, i.e. t f ≪ 0.2, we will have negative value of g(t) at certain time, and crossing resonance could induce atom loss. Anyway, here we shorten the time by ten times as compared to adiabatic compression. The smooth function of nonlinearity, g(t), is reasonable for experimental implementation. Figure 4(a) shows initial and final width of matter wave soliton, to demonstrate the shortcut to adiabatic compression. Numerical calculation has a good agreement with the theoretical prediction. STA can exactly compress the soliton to specific final width, as compared to adiabatic process. In order to visualize it, we also show the dynamical evolution of soliton matter waves during the shortcut process, in Fig. 4(b).

Discussion
Finally, we shall turn to the stability of STA with respect to different values of A s . As mentioned above, the amplitude of compression can be actually determined by the parameter A s . The fidelity, defined as F = |〈 ψ ad (x, t f ) |ψ(x, t f )〉 | 2 , is numerically calculated in Fig. 5, where |ψ(x, t f )〉 is the numerical result and |ψ ad (x, t f )〉 is the ideal target state predicted from adiabatic reference. Remarkably, by using STA protocol, we can compress the soliton to exact final width that we expected, when t f = 10 and s = 10, and fidelity is always perfect, equal to 1, showing the advantage of STA on robustness. On the contrary, when t f = 100 and s = 1, the process is adiabatic, and in general the fidelity is above 90%, but there exists the oscillation due to the switching function g(t). This oscillation is relevant to hyperbolic tangent function for nonlinearity g(t) (11), similar to Allen-Eberly scheme in two-level system 33 , and thus can be somehow suppressed by choosing other smooth switching function. Additionally, we do not show the results for the parameters, t f = 10 and s = 1 by using the switching function, g(t) (11), since the fidelity is far away from 1, due to non-adiabatic transition. In fact, when decreasing t f further, the fidelity becomes imperfect, less than 1. This is due to the fact the values of a c (t) and its derivative at t = t f calculated from Eq. (9) become inaccurate. One has to modify the boundary conditions (14)(15)(16) by replacing the numerical values calculated from Eq. (9), instead of Eq. (10).

Conclusion
In summary, we have proposed the shortcut to adiabatic control of soliton matter waves in harmonic trap by using inverse engineering. The variational method used here provides new auxiliary differential differential equation (3) for engineering inversely the time-dependent nonlinearity g(t), implemented by tunable Feshbach resonance. As an example, we first fix boundary conditions from adiabatic reference, design the function of nonlinearity g(t)  and finally achieve fast adiabatic-like compression. This is completely different from other STA protocols for BEC manipulation, in which the time-dependent nonlinearity has to be imposed in both methods of inverse engineering and counter-diabatic driving. The physical constraint on shortening time is discussed, and the time-optimal problem and other optimization will be interesting for further investigation by combing STA and optimal control theory, see example in ref. 34. In addition, several extensions could be done in the near future. For instance, one can transport the soliton matter wave 4,5 simultaneously along with compression/expansion. Optical soliton can be compressed in optical fibre by modulating gain and loss 35,36 . These results presented here pave the way to study the fast and robust control of nonlinear dynamics in classical and quantum systems 26,30 .

Methods
The Lagrangian density is 37,38  , where p presents one of the parameters a, b, c and ζ, give the following coupled differential equations: Note that δ δφ = / 0  is tantamount to the conservation of the norm of the wave function, which is the single dynamical invariant of Eq. (2), thus the absolute phase variable φ(t) plays no role in the variational dynamics. Finally, the above coupled differential equations (18)(19)(20)(21) become Eqs (3) and (4). Here we choose hyperbolic secant ansatz, see Eq. (2), the solution of nonlinear GPE, which is different from the Gaussian one as the ground state in the linear limit (no interactions) 38 .