Quasi-integrability and nonlinear resonances in cold atoms under modulation

Quantum dynamics of a collection of atoms subjected to phase modulation has been carefully revisited. We present an exact analysis of the evolution of a two-level system (represented by a spinor) under the action of a time-dependent matrix Hamiltonian. The dynamics is shown to evolve on two coupled potential energy surfaces (PESs): one of them is binding, while the other one is scattering type. The dynamics is shown to be quasi-integrable with nonlinear resonances. The bounded dynamics with intermittent scattering at random moments presents a scenario reminiscent of Anderson and dynamical localization. We believe that a careful analytical investigation of a multi-component system that is classically non-integrable is relevant to many other fields, including quantum computation with multi-qubit systems.

Many years ago, an experiment carried out by the group led by Raizen [1] demonstrated the dynamical analog of Anderson Localization in a system of cold atoms.In this experiment, about one hundred thousand 23 Na atoms were trapped in a spherical volume of 300 µm at a temperature of 17 µK.At the end of the preparation step, the temperature was turned off and a modulated standing light field was switched on for 10 µs.The Hamiltonian describing the interaction of the sodium atom with the light field is given by [62] Here, H el contains the interaction of valence electrons with an atom.The last term denotes the electric dipole interaction of the electromagnetic field with an electron.Laser frequency and wavenumber are respectively denoted by ω L and k L , and ω is the modulation frequency.
Standing waves are generated by directing two counter-propagating laser beams into the trap, and, the modulation is achieved by passing one beam through an electro-optical phase modulator.The beam is made to strike a mirror in a cavity of length ∆L which is moving with the modulation frequency, ω.The laser frequency was chosen close to the D 2 line of sodium.The electronic Hamiltonian can be reduced to a two-level system containing ground state ψ − |g⟩ and an excited state, ψ + |e⟩.
Taking the energy average of the two states as zero energy, the matrix elements of H el and eF together give where the transition frequency is denoted by ω 0 , Ω denotes Rabi frequency, and σ ′ s are the Pauli matrices.Thus, H 0 may be written as where I denotes an identity matrix.
After we present the general Hamiltonian below, in §2, we present the Hamiltonian under Rotating Wave Approximation.Within this approximation, the case of adiabatic perturbation for the two cases of small and large detuning is considered.In §3, the exact solution for this matrix Hamiltonian is given.The method transforms the dynamics under the matrix Hamiltonian to dynamics on potential energy surfaces.Classical dynamics reveals the presence of nonlinear resonances in §4.The classical system obeys the Kolmogorov-Arnold-Moser (KAM) theorem [57], and hence is quasi-integrable [58].In a related context of quantum Rabi model, a discussion on integrability [60] and symmetries [61] has been presented relatively recently.Special solutions are discussed as they have been used to analyze experiments carried out by different groups.For each case discussed at the quantum mechanical level, we also present classical phase space pictures and show that this atomic system presents a very interesting and deep instance of the association of quasi-integrability and dynamical localization.The phase space pictures exhibit certain misleading features in the approximated Hamiltonian, compared to the exact Hamiltonian obtained by systematic expansion in powers of ℏ.

General Hamiltonian
We now transform to a frame which is rotating with ω L about the z-axis in spin space: ( Substituting ψ in the Schrödinger equation, iℏ∂ψ/∂t = H 0 ψ, we have the equation for the rotated wavefunction: Using the standard identity, e iω L σzt/2 σ x e −iω L σzt/2 = σ x cos ω L t − σ y sin ω L t, we have the transformed Hamiltonian: This is the general Hamiltonian for the physical situation described above where there are terms oscillating with twice the ω L .

Rotating Wave Approximation
The Schrödinger equation for H rot is usually solved under the Rotating Wave Approximation (RWA) [62,65].Here the terms oscillating with frequency 2ω L are neglected.This leads to a simplified Hamiltonian, where Let us rotate the state of this Hamiltonian further in the spin space by an angle (−α/2) about the y-axis, to obtain a new state, in which the second term is diagonal.Consequently, the equation satisfied by But this will transform the kinetic term as [66]: where I is an identity matrix.Now we can employ the well-known identity: While the "potential" part of the Hamiltonian becomes diagonal with these unitary transformations, the kinetic term modifies to (pI − ℏA) 2 .This has terms of order 1, ℏ, and ℏ 2 -thus, a semiclassical expansion (and not a perturbative expansion) appears naturally.Moreover, since A has non-zero diagonal matrix elements, there is a possibility of a geometric phase appearing in the state of the atoms as the system evolves.This is indeed due to the cavity modulation.Dimensionally, ℏA/e is a magnetic vector potential.H RWA eff can be written as: Except for terms of order O(ℏ 2 ), each of the terms can make a significant contribution.At this point, one of the possible simplifications occurs if α is slowly varying with time.This leads us to consider applying the adiabatic approximation, which we discuss now.

Adiabatic variation
We may neglect the term ℏσ y dα/dt.But note that in this case: The adiabatic Hamiltonian is: It matters a lot if the detuning is small or large.This is because So either for small or large detuning,

Small detuning
Here, ω 0 ∼ ω L , thus tan α → ∞ or α ∼ π/2.Considering (20) and keeping the terms upto O(ℏ), the adiabatic Hamiltonian further simplifies to Exploiting the smallness of detuning, we may expand binomially to obtain These provide the two potential energy surfaces on which the two-level system evolves, connected by tunneling.This can be seen by the fact that the intersection of the two curves occurs when Ω eff is zero, leading to for small detuning.The binding part of the potential in (22) supports eigenvalues.However, since the Hamiltonian is periodic in time, the eigenvalues are quasienergies.Owing to the imaginary part, these are more precisely "quasienergy resonances".

Large detuning
We consider the case where we have RWA and adiabatic approximation but δ L ≫ Ω.Then we have the Hamiltonian, This can be decomposed into two Hamiltonians: The potential energy curves intersect when Here the intersection points are real where the real part is the same as for small detuning.
The potential energy curves support sharp quasienergies.

Exact solution
We now return to the (7) and lift all the approximations considered in the last Section.The Hamiltonian is written as where The matrix, denoted by M in ( 27) can be diagonalized by a matrix S to get the diagonal matrix, J .The matrices are and Define ψ 1 = S −1 ψ rot with iℏ∂ψ rot /∂t = Hψ rot .The equation for the time evolution of Now, Here we again have a vector potential which is an artificial gauge field.
The Hamiltonian is thus written as an expansion [66,67], with H 0 has a simple form: Writing T with the superscript, T denoting the transpose, we have written the state with two components.The classical Hamiltonians corresponding to the states, ψ Usually, ψ is subjected to a binding potential and ψ is evolving on a scattering potential.There are two potential energy surfaces, ± a 2 + b 2 1 + b 2 2 on which the full two-component wavefunction, ψ 1 evolves.The potential energy surfaces meet at the solution of The solution is For small detuning (δ L ≪ Ω), the potential curves intersect at The complex value of crossing of the potential energy surfaces implies the tunneling of atoms.The tunneling across these surfaces where the underlying dynamics is nonlinear has some very interesting related phenomena like resonance assisted tunneling [63], which have been recently experimentally realized [64].
The Fig. 1 (a) and (b) show these crossings along the complex position plane.We note that the crossing gap at the null imaginary position plane vanishes as one reaches closer to resonance (at small detuning) and remains wide open at large detuning.
In (34), for large detuning, Ω 2 /(ω 0 − ω L ) 2 ≪ 1, a Taylor expansion immediately yields Among the two Hamiltonians, H (−) 0,l is binding; it can be seen that the second term in the Taylor expansion of cos[k L (x − ∆L sin ωt)] along with an overall negative sign will make this roughly parabolic for small arguments, at least.For the same reason, H (+) 0 is a scattering potential.The differences in Poincare sections for various cases can be seen in the following figure.We found that the 3 island ring which is present in both un-approximated case and RWA+Adiabatic case vanishes if we make a binomial approximation implying origin of this resonance is purely because of higher order terms of ( 38) and (22).We also note that the chaos is more apparent in the binomial case but less severe in all other cases.
We now study the classical mechanics of these Hamiltonians.

Quasi-integrability
In this Section, we study the classical dynamics of the Hamiltonians obtained above under different approximations.
We begin with the exact Hamiltonian, namely (32), and consider only in (34).We make the following transformations to convert it to a dimensionless form almost similar to [65].
where η is strength of Rabi resonance and δ L = ω 0 − ω L is the detuning of laser.The simplified Hamiltonian yields: Now, using the same transformations (39), we write the Hamiltonians for large detuning, neglecting the constant terms: This clearly implies a drastic change in the equation since if γ ≫ 1, thus even if we use ⟨cos 2 γt⟩ = 1/2, the second term contributes double compared to the contribution coming from the usual case with adiabatic and RWA approximation.
In order to understand the underlying phase space structure, we initialize 1000 ultracold atoms (blue dots) in one of the island in the Poincarè section taken in steps of modulation time period T as shown in Fig. 3 (top) and look at its stroboscopic evolution in multiples of the modulation time period.We find that after each modulation period, atoms move from one island to another lying around the same larger elliptic-like orbit (Fig. 3 (middle)).Similarly, we find that the number of islands is equal to (or twice if n is even) the number of modulation periods n for the marked islands in Fig. 3 (bottom).In other words, these islands satisfies T orbit = nT or Ω orbit /ω = 1/n.
To study the origin of these patterns in resonance structures, we write the dimensionless Hamiltonian (42) in action-angle variables.Let us write one of the RWA Hamiltonians as a perturbed harmonic oscillator: where ϵ is introduced for book-keeping (eventually, we shall put ϵ = 1).Employing the oscillator action-angle variables, (J, θ), with x = J πΩ sin(θ) and p = JΩ π cos(θ) with K = Ω 2 , the Hamiltonians are: Figure 3: Poincaré Sections taken in steps of modulation period using the same parameter as in [1].(a) 1000 ultracold atoms (purple dots) are loaded in one of the islands of stability in the Poincare section taken in steps of the driving period T. (b) stroboscopic evolution of the ultracold atoms reveals that they evolve with a period 4T.(c) Similarly, loading on different islands of stability shows the existence of 3T, 11T/3, 4T and 5T periods predominantly.
We use the classical time-dependent perturbation theory [57] to calculate the associated action of this Hamiltonian up to first order in perturbation.For this, we transform the action variables in a way that the new Hamiltonian H is only a function of the new action variable J alone.We obtain where J 0 (.) is the cylindrical Bessel function of order zero.The new frequency is where prime on the Bessel function denotes a derivative with respect to its argument.We subtract this ϵ⟨∆H⟩ from ϵ∆H to obtain the oscillating part ϵ{∆H}.For calculating the integral, we expand the potential term using Jacobi-Anger expansion [59] where both n, m are non-zero.The change in action ϵ∆S can be calculated as where Consequent to the above, The new action-angle variables can be calculated up to first order as Thus we have obtained the action with resonant denominators which leads to resonant condition n Ω( J) = mω (58) where ω is the modulation frequency and Ω( J) is the frequency of the orbit, ω is obtained when we substitute actual time, t in place of dimensionless time from (39).This explains the observed pattern in Fig. 3 : the orbital periods are integral multiples of the modulation period at the resonance.The strength of (n, m) th resonance is determined by the product of two Bessel functions J n ( J/Ωπ) and J m (λ).Using the first-order correction in the frequency Ω(J), we plot it as a function of J in Fig. 4. We see that only the 1:3 resonance is allowed under first-order correction.This means that all other resonances in Fig. 3 must originate from the higher-order perturbation terms in correction for Ω and J.That explains the dominance of primary islands in (n,m)=(3,1) resonance and the presence of secondary islands in other resonances.
For the expression without binomial approximation (42) where in Fig. 2 we saw (3,1) resonance to be dominantly present, but without binomial approximation (25), this resonance is suppressed and doesn't appear.This can lead to significant corrections for both quantum and classical equations despite being in large detuning limit.Similarly, very highordered resonances are enhanced by binomial approximation as the chaotic regime can be seen enhanced around the edges for this case.

Dynamical localization
Let us imagine that we prepare the initial state of the atoms as a localized wavepacket.As the system evolves, the wavepacket spreads.The wavefunction of the two-state system is shown to evolve, in all versions of description, on a pair of potential energy surfaces.The form of these potentials readily support bounded dynamics on one of the potentials.The complex intersection points provide paths for tunneling.The succession of these two dynamical features leads to localization of the wavepacket.The physics of this is nothing but the well-known argument by Mott [68] and Anderson [53], adapted in recent times in quantum chaos [54,55].

Conclusions
The matrix Hamiltonian driving a two-level atom has been unitarily transformed to a series of Hamiltonians arranged in the powers of Planck constant -which is the precise meaning of semiclassical expansion.A successive application of these transformations brings out an effective Hamiltonian to any desired level of accuracy.In principle, one could perform computations to all orders of ℏ.The system is shown to tunnel between two potential energy surfaces, the underlying dynamics is quasi-integrable in the KAM sense.
The analysis has been carried out in the past by employing physically appealing and rather standard approximations.We recapitulated these and then have provided exact solution where by "exact", we mean in the sense described in the preceding paragraph.We have seen that a matrix Hamiltonian for a spinor eigenstate.At different orders of Planck constant, there are different potential energy surfaces on which the system is shown to evolve.If one makes a binomial approximation in the Hamiltonian to treat the system, the detailed features in the Poincaré surfaces of section differ.The approximated analysis has certain appeal insofar as tunneling between islands is seen clearly.However, to establish that the existence of islands and tunneling, We show that the onset of islands of stability can be seen from the first-order perturbation theory.
The analysis reveals a vector potential that is related to an artificial gauge field.We believe that knowing the form for this could be useful for experiments with cold atoms and in developing fields of Hamiltonian engineering, quantum sensing and quantum interference.We have not developed these aspects here.
As referred to in the Introduction, our results add to the discussion of integrability in matrix models for atomic systems, in particular to the work on quantum Rabi model [60].In the future, by adding nonlinear terms to incorporate interactions that allow control of atomic states, these works could be useful for critical quantum metrology [69].Control of states of multi-qubit systems [70] and their protection [71] belongs to the present theme in a rather compelling manner.

Figure 1 :
Figure 1: Potential Energy Surface (PES) at (a) large detuning (δ L ≫ Ω) and (b) small detuning (δ L ≪ Ω).At large detuning the gap shrinks allowing a larger region for space for crossing of PES

Figure 2 :
Figure 2: Comparison of Poincare sections for Hamiltonians under different approximations for the case of large detuning for the same set of parameters used in Fig. 3. (a) Shows the un-approximated case corresponding to the exact solution.(b) Shows the application of binomial approximation to the exact solution.(c) Corresponds to the RWA+Adiabatic approximation and (d) corresponds to the RWA+adiabatic+Binomial approximation.Initial conditions and number of evolution steps are kept the same for all cases here.