The generalized harmonic potential theorem in the presence of a time-varying magnetic field

We investigate the evolution of the many-body wave function of a quantum system with time-varying effective mass, confined by a harmonic potential with time-varying frequency in the presence of a uniform time-varying magnetic field, and perturbed by a time-dependent uniform electric field. It is found that the wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. In other words, we generalize the harmonic potential theorem to the case when the effective mass, harmonic potential, and the external uniform magnetic field with arbitrary orientation are all time-varying. The results reduce to various special cases obtained in the literature, particulary to that of the harmonic potential theorem wave function when the effective mass and frequency are both static and the external magnetic field is absent.

Scientific RepoRts | 6:35412 | DOI: 10.1038/srep35412 In light of above facts, in this work we try to investigate the HPT in the most general form, i.e. the case when the effective mass, confining frequencies, and the external uniform magnetic field are all TV. After stating the generalized HPT, we then give the proof via two different approaches, i.e. the operator method and the accelerated frame approach 9 . We show that the WF is still comprised of a phase factor times the solution to the unperturbed TD Schrödinger equation with the latter being translated by a TD value that satisfies the classical driven equation of motion.

Results
Hamiltonian and the generalized HPT. Consider a system of N identical particles with a TV effective mass m = m(t) under an external TV magnetic field with arbitrary orientation B(t) = (B 1 (t), B 2 (t), B 3 (t)), confined in an external TD harmonic potential = ⋅ ⋅  v t t r r K r ( , ) () 1 2 , with K(t) an symmetric positive 3 × 3 matrix. The TD harmonic potential can be used to describe many experimental situations. For instance in BEC experiments, it can describe rotating the quadratic trap 51 , modulated trapping 52 , or reflecting fact that the trap is perturbed to obtain the response spectrum of the condensate [53][54][55] . The two-body interaction between the particles u(r i − r j ) can be of arbitrary form. A uniform TD driving electric field E(t) = f(t)/q is turned on at time t = 0 with q = − e the charge of an electron. Thus, in the coordinate representation the Hamiltonian reads where the unperturbed component is with  r i the transpose of the position vector r i . π i is the physical momentum operator Choosing the symmetry gauge such that the vector potential A(r i , t) = (B(t) × r i )/2, and substituting eq. (3) into eq. (2) yields where L i is the angular momentum operator of the i-th particle, and is still a 3 × 3 real positive symmetric matrix with The model above considered is obvious an open system. Generally there exist two main approach for such systems. The first is so called system-plus-bath approach [56][57][58] , and the second one is the effective Hamiltonian approach 24,59,60 . The effective Hamiltonian usually has a TV mass that arises due to the interaction between the system and the bath 48 , and governed by the TD Schrödinger equation within the adiabatic approximation. Hence, the Hamiltonian of eq. (1) can be regarded as an effective Hamiltonian for some open system and obey the following TD Schrödinger equation, H t t r r r r r r r r r ( , , , ; ) ( , , , ; ) ( , , , ; ) (7) The core of the generalized HPT is the solution to the TD Schrödinger equation eq. (7). We refer to this solution as HPT WF. The generalized HPT states that the following WF Note that the phase factor in eq. (8) has the form similar to classical action. In eq. (8), M(t) = Nm(t), ξ(t) is the translation vector, P ξ (t) the corresponding momentum vector (see eq. (26) below), and = ∑ N R r/ i i the center of mass coordinate. The translation vector ξ(t) satisfies the classical equation of motion (10) is just the classical equation of motion for a harmonically trapped particle with a TV mass in the presence of TV external magnetic field B(t), perturbed by an external force F(t).
Proof of the theorem via derivation. Next we prove the generalized HPT by derivation. Using the center of mass (CM) and relative coordinates and momentums 61-63 and similarly for Y (2) , … Y (N) , Z (2) , … Z (N) , and P (2) , … P (N) , the Hamiltonian of eq. (1) can be decomposed into the CM and relative motion parts, x y z 1 is the perturbation term due to the external electric field, and L R the angular momentum operator for the CM coordinate R. The relative motion part Ĥ t ( ) rel contains only the relative coordinates, Consequently, the CM motion and the relative motion are separable. Therefore, the total WF of the Hamiltonian is the product of the WFs of CM motion and relative motions: ( ) and the CM motion WF Φ (R, t) satisfy their own Schrödinger equations, respectively, with certain initial conditions. In the following, we shall focus on the CM motion Hamiltonian of eq. (14), and try to find its WF Φ (R, t). Similar to the structure of the HPT WF, let us assume where Φ 0 (R, t) is the WF for the unperturbed CM motion Hamiltonian, i.e. which satisfies the following Schrödinger equation, cm  Inserting eq. (18) into eq. (20) and using eq. (14), we have since R now is the eigenvalue of the coordinate operator R , whose hat has been dropped since we work in the coordinate representation. With the ansatz that the phase factor can be cast into the following form,   (23) and (24) into eq. (21) yields By comparing the coefficients of ∇ R on both sided of eq. (25), we have and similarly for the coefficients of R, we obtain Inserting eq. (26) into eqs (27) and (28), we immediately find that the translation vector satisfies eq. (10). Note is the induced electric force by the TV magnetic field, and t 0 0 2 is just the classical action without the electric field term. Hence, from eqs (17), (18), (22), (10) and (29), we obtain the final WF of eq. (8). In other words, we have proved via derivation that the HPT WF eq. (8) is the solution of the TD Schrödinger equation eq. (7). The HPT WF is the key result of this paper. Note that if one requires that the initial state is the eigenstate of the unperturbed Hamiltonian, i.e. Ψ (t = 0) = Ψ 0 (t = 0), then usually one has the initial conditions: ξ(0) = 0, ξ =  (0) 0. We stress that the HPT WF can reduce to various special cases existed in literature 1,[9][10][11][12] . Thus, we have extended the HPT to the case when the quantum systems have a TV effective mass and TV confining frequencies, in the presence of a uniform TV magnetic field with arbitrary orientation.
The Hamiltonian and wave function in the accelerated frame. Inspired by the method of Vignale 9 , we next show that our results can also be obtained by transforming the system to an accelerated reference frame. Making the acceleration transformations i i i on the system with ξ(t) governed by eq. (10), hence the connection between the original WF Ψ (t) and the accel- The WFs Ψ (t) and Ψ ′ (t) in the above equation satisfy the following Schrödinger equations respectively: And the explicit form of the unitary operator ξ U t ( ) is 64 From eqs (8), (31) and (33), the connection of the accelerating WF and the WF in the absence of the external electric field can be written in a simpler form as after a long calculation, above Hamiltonian can be recast into the following form  (34) and (36), one can readily see that the uniform time dependent electric field is eliminated by performing the acceleration transformations. This proves again the generalized harmonic potential theorem.

Discussion
In summary, we have presented the detailed analytical form of the evolution of the WF for an quantum system with TV effective mass trapped in a harmonic potential with TV frequency, in the presence of a TV uniform magnetic field with arbitrary orientation, and driven by a TD uniform electric field. It is found that the WF is comprised of a phase factor times the solution to the unperturbed TD Schrödinger equation with the latter being translated by a TD value that satisfies the classical equation of motion for a driven harmonic oscillator with TV mass in the presence of an external TV magnetic field. The analytical form of the phase is also given. The results can reduce to various special cases existed in the literature. We also show that our results can be obtained by transforming the system to an accelerated frame. Moreover, we stress that our results are applicable to both the fermionic and bosonic systems with general effective masses and external magnetic fields that can be described by some smooth functions of time, since the derivations do not rely on the statistical properties of the WF or any specifically choice of the TD terms and parameters. However, the external TD electric field must be uniform. Finally, we briefly discuss some real physical systems that our results might shed lights on. Notice that if one identifies the angular velocity Ω = e m t c B t ( ) 2 ( ) and the gravity with the external driving force, then the model Hamiltonian of eq. (1) can be used to describe atoms trapped in a harmonic potential rotating instantaneous around the z axis 51 , the related experiment has been done at ENS 65 . In above case of a vertical axis of rotation, the only effect of gravity is a displacement of the equilibrium position 66 thus can be ignored. When the axis of rotation was titled away from the trap axis such those experiments done in refs 67,68, the effect of gravity must be taken into account. For instance, in a uniformly rotating trap, it can causes resonances hence the escape of the center of mass for a collection of interacting particles from the trap 66 . Our results implies that even the rotation is titled and TV, the effect of gravity is solely to transport rigidly the center of mass, or density distribution of the system. This is expected to be confirmed experimentally.