Quantum recurrences in the kicked top

The correspondence principle plays an important role in understanding the emergence of classical chaos from an underlying quantum mechanics. Here we present an infinite family of quantum dynamics that never resembles the analogous classical chaotic dynamics irrespective of dimension. These take the form of stroboscopic unitary evolutions in the quantum kicked top that act as the identity after a finite number of kicks. Because these state-independent temporal periodicities are present in all dimensions, their existence represents a universal violation of the correspondence principle. We further discuss the relationship of these periodicities with the quantum kicked rotor, in particular the phenomenon of quantum anti-resonance.


I. INTRODUCTION
The quantum-classical correspondence principle broadly states, in its commonly understood form, that the predictions of a dynamically evolving quantum system should reproduce the predictions of a classical system under appropriate circumstances [? ].This sometimes takes the form of a particular limit of some set of parameters that characterize the quantum system (i.e.large quantum numbers, vanishing Planck action, etc.).In such situations the transition may be called a classical limit of the quantum system [2].It is well know that classical systems can display chaotic behaviour -broadly defined as exponential sensitivity to initial conditions.Interestingly, in quantum systems that have such a chaotic classical limit, the correspondence principle is not well understood [3][4][5][6].Exploring such systems can thus provide insight to the structural differences between quantum and classical dynamics as well as the fundamental origin of chaotic phenomena.
A useful model studied in this context is the quantum kicked top [7].This model is a spin-j system subject to a Floquet evolution (i.e. a stroboscopic dynamics).It is of interest because it lives in a finite-dimensional Hilbert space, its dynamics have a well-defined classical limit (j → ∞) with an easily tunable degree of chaos via its Hamiltonian parameters, and it is experimentally feasible [8][9][10].Furthermore, the alternative representation of any spin-j system as a many-body system of indistinguishable qubits has lead to much work on understanding the surprisingly subtle relationship between dynamical entanglement, Hilbert space dimension, and emergent chaos in the kicked top model [6,[11][12][13][14][15][16].
In this paper, we probe quantum-classical correspondence in the kicked top, and present a startling result.For certain system parameters, classical chaotic behaviour is not recovered no matter how large the value of the spin quantum number, in contradiction to Bohr correspondence.We analytically and numerically show that in all dimensions (all values of the spin j), the kicked top displays several state-independent, temporal periodici-ties/recurrences: three for integer spin values and two for half-integer spin values.Whereas previous work has explored a specific temporal periodicity in the semiclassical limit [17], the general set of recurrences derived in our analysis have not been previously identified.Because these recurrences are state-independent and generally occur at large chaoticity values, they have no classical analog and so represent a violation of the correspondence principle.Furthermore, our results show that the transition to classical behaviour does not smoothly vary with the size of the system.Our analysis also resolves previous conflicting results on how the chaoticity parameter κ in the kicked top influences the presence of quantum temporal periodicity [14].In addition we establish a relationship between our kicked top periodicities and the quantum resonances identified in the kicked rotor.Our results highlight the complex nature of quantum chaos and challenge typical notions of quantum-classsical correspondence.

A. Kicked Top Model
The quantum kicked top (QKT) is a finite-dimensional dynamical model used to study quantum chaos, known for its compact phase space and parameterizable chaoticity structure [7].The time-dependent, periodically-driven system is governed by the Hamiltonian where {J x , J y , J z } are the generators of angular momentum: [J i , J j ] = iϵ ijk J k .It describes a spin of size j precessing about the y-axis together with impulsive statedependent twists about the z-axis with magnitude characterized by the chaoticity parameter κ.The period between kicks is τ , and p is the amount of y-precession within one period.The associated Floquet time evolution operator for one period is The classical kicked top can be obtained by computing the Heisenberg equations for the re-scaled angular momentum generators, J i /j, followed by the limit j → ∞ [7].In the commonly considered case of (τ = 1, p = π/2), the classical map is As the chaoticity parameter κ is varied the classical dynamics ranges from completely regular motion (κ ≤ 2.1) to a mixture of regular and chaotic motion (2.1 ≤ κ ≤ 4.4) to fully chaotic motion (κ > 4.4) [6].The classical stroboscopic map in polar coordinates for a set of initial conditions with κ = 2.5 and κ = 3.0 is given in Fig. 1a and Fig. 1b respectively.

B. Husimi function
To study the quantum-classical correspondence in the quantum kicked top, the Husimi function is often used as an aid to compare quantum vs. classical dynamics [6,18].It is a non-negative quasiprobability distribution defined as subject to the normalization condition 2j + 1 4π where |θ, ϕ⟩ are the standard spin coherent states associated with SU(2) dynamical symmetry [19].

III. PERIODICITY IN TWIST STRENGTH
As pointed out in [20], there is a recurrent relationship between unitaries separated by an amount κ = 2πj: The unitary e −iπJ 2 z characterizes the difference between the actions of U k and U k+2πj on Hilbert space.We will show that this operator acts as a symmetric local unitary in the qubit picture and so does not modify any correlations between the qubits. Denoting z , consider the operator e −iπJ 2 z in the qubit picture: where ⃗ k = (k 1 , ..., k n ) is a multi-index of positive integers that sums to 2, and 2 ⃗ k is a multinomial coefficient.Separate the multi-indices into those with a single k i = 2 and those that don't; the former will happen n times, and the associated Pauli operator squares to the identity: The remaining indices each have exactly two different slots equal to 1 and so the multinomial coefficient is always 2. The exponentials consequently reduce to It is already clear from Eq. ( 8) that e −iπJ 2 z is a local unitary and so does not affect any correlations between the qubits.
Hence, the entanglement generated between the qubits is periodic in the chaoticity parameter with period ∆κ = 2πj [20].That is to say, where LO refers to (symmetric) local operations over the global Hilbert space of the qubits.Eq. ( 8) can be written in more compact form as which is clearly symmetric.This breaks into three cases of spin IV. TEMPORAL PERIODICITY Here we derive the temporal periodicity of the kicked top evolution for three special values of twist strength: {2πj, πj, πj 2 }, each of which are split into cases of integer and half-integer spins.

A. Twist strength κ = 2πj
The Floquet operator in the case of κ = 2πj is where e −iπJ 2 z is a symmetric local unitary in the qubit picture (10).Like many of the results here, the consequences on temporal periodicity strongly depends on whether the spin is integer or half-integer.

Integer spin
In the case of integer spin the evolution squares to the identity regardless of the y-rotation angle: This can be seen using either integer form of Eq. ( 11) and writing the y-rotation in the qubit picture as And because in this case U 2 2πj is simply a composition of symmetric local unitaries, it suffices to consider a single tensor factor: with the last line coming from the anti-commutation relations of Pauli matrices.See Fig. 2 for a visual interpretation of Eq. ( 13) by tracking the collectively shared Bloch vector.After the first y-rotation the twist effectively acts as a π-rotation about the z axis.Consequently, the second y-rotation then undoes the first and the second twist rotates back to the starting point.Note that this demonstration depends on each step being a symmetric local unitary, meaning that a spin coherent state will remain so throughout and no correlations are ever generated.FIG.2: Evolution of the Bloch vector associated to any reduced qubit state for κ = 2πj.After two kicks the state returns to its initial point, showing a two-step temporal periodicity.The initial point is (θ, ϕ) = (2.25,2.0).

Half-integer spin
In the case of half-integer spin the twist becomes a scaled identity operator (11), leading to a p-dependent temporal periodicity (up to an irrelevant global phase) of N kicks under the condition Hence only when p is a rational fraction of π does there exist a temporal periodicity at this twist strength.It is interesting that in the half-integer case the rotation angle p is critical for determining the existence of a temporal periodicity while in the integer case p has no effect.

B. Twist strength κ = πj
The case of twist strength κ = πj yields a more interesting temporal periodicity that also depends on the spin being integer or half-integer.Details of calculations can be found in Supplementary I.

Integer spin
Following a similar argument from the previous section, the general expression for the twist unitary e which reduces to when κ = πj.This appears to be a difficult expression to evaluate but simplifies to as can be verified by comparing the two actions on the computational basis in (C 2 ) ⊗2j .This can also be found using the Gaussian sum decomposition result from [17].With this in mind, and writing the y-rotation as in Eq. ( 14), the Floquet operator can be shown to exhibit the finite-time periodicity This can be done by establishing through repeated use of the Pauli group commutation relations.As n is an even integer and Y 2 = I, this is enough to give Eq.( 20).The same calculation may be repeated for κ = πj + 2πj = 3πj which also shows the period 8 periodicity.While expected from the 2πj periodicity in correlations [20], this additional calculation is necessary to conclude the stronger notion of temporal periodicity of the state itself, possibly up to global phase.For example, any SU(2) rotation with an angle incommensurate to π will produce a sequence of spin coherent states -and therefore a period-1 recurrence in the quantum correlations -that never returns to the original state exactly.
In contrast to the previous twist strength of κ = 2πj, here entanglement is generated (and destroyed) throughout the period-8 orbit.This can be seen from Eq. ( 19) which is clearly not a symmetric local unitary.Fig. 3 shows the orbit in the Husimi representation (4) of a j = 50 spin coherent state initially centred at (θ = 2.25, ϕ = 2.0).After the initial rotation about the yaxis, we see the action of ( 19) "splitting" the state into a cat-like superposition.A second round of rotation-twist iteratively produces a balanced superposition of four spin coherent states distributed over phase space.Another two kicks recombines this state into the initial spin coherent state but reflected about the y-axis, matching (21).Finally another four kicks repeats this process, resulting in a recurrence of the initial state.This regular, periodic dynamical behaviour appears to have no analogue in the classical kicked top (not least of which at κ = πj) and so represents a departure from the classical-quantum correspondence.
It should also be noted that while the above is the generic temporal periodicity, certain states related to the Hamiltonian symmetries will experience a shorter orbit.In particular if we take the initial state as |+⟩ y , i.e. (θ, ϕ) = (π/2, π/2), then the rotation part of the unitary will be ineffective.The twist (19) will create the superposition of |+⟩ y and |−⟩ y ; it can be shown that the evolution reduces to a period-4 orbit for even integer spins and a period-2 orbit for odd integer spins.See also [15] for a related analysis.

Half-integer
In the case of half-integer spin and κ = jπ the general twist operator, Eq. ( 17), is equivalent to the following unitary in the qubit picture: Similar to the integer case, repeated and iterated use of the Pauli group commutation relations show that Eq. ( 23) raised to the 6th power yields a π-rotation up to phase: The full state recurrence comes after 12 kicks: which is a finite temporal periodicity up to global phase.To our knowledge this recurrence was first discovered for the spin-3 2 case in Ref. [16]; here we have shown that it exists in all dimensions.Also, as expected, a similar calculation shows another period-12 recurrence at κ = πj + 2πj = 3πj, similar to the integer-spin case.
Again starting with a generic spin coherent state (i.e. a symmetric product state in the qubit picture), entanglement is generated and destroyed throughout its 12-state orbit.Similar to Fig. 3, the generation occurs during the recursive splitting of the state into successive catlike superpositions, and the destruction occurs during the subsequent recombination into new, displaced spin coherent states.States associated with Hamiltonian symmetries again experience a reduced orbit length.For the initial state as |+⟩ y , i.e. (θ, ϕ) = (π/2, π/2), the evolution reduces to a period-3 orbit.
The case of κ = πj 2 has the most apparent difference between integer and half-integer spin.

Integer spin
Using the Gaussian sum decomposition [17], for integer spin the twist operator e −i π 4 J 2 z splits into the superposition of rotations In the qubit picture this becomes where we have used the fact that n is even to simplify.Numerical calculations suggest a temporal periodicity with period 48, This has been confirmed up to spin j = 500 where the Hilbert-Schmidt distance ∥U 48 πj 2 − I∥ remains zero within the working error tolerance of 10 −10 .And similar to the previous κ = πj case (both integer and half-integer) here the Floquet operator raised to half the periodicity (i.e.24) also acts as an effective π-rotation about the y-axis up to some global phase.Cat-like splitting and recombination cycles were furthermore observed in the Husimi function tracking of a generic spin coherent state.Part of the difficulty in showing this analytically comes from determining the twist operator in the qubit picture as in Eqs. ( 19) and (23).Numerics also confirm a period-48 recurrence at κ = πj 2 + 2πj = 5πj 2 .We also note what appears to be two higher-frequency temporal recurrences present in low dimensions at this chaoticity value: for j = 1 and j = 3 the evolution repeats after only 16 kicks rather than 48.This observation is distinct from the continued theme of the special states |±⟩ y experiencing a reduced orbit of 24 for even values of j and 4 for odd values of j, which we numerically verified.

Half-integer spin
In the half-integer case we surprisingly find no temporal periodicity for κ = πj 2 .This was numerically concluded by computing the entanglement entropy of any one of the reduced constituent qubits, via the collective spin observables {S z , S ± = S x ± iS y } where S i = J i /j [21].This approach was used instead of Hilbert-Schmidt distance to avoid optimizing over the angles φ that could have a priori appeared in a hypothetical periodicity of the form U n = e iφ I.
All that is needed to conclude the lack of a global recurrence is the identification of a spin coherent state that never returns to product form.We thus focus on our running example of |θ, ϕ⟩ = |2.25,2.0⟩.We found that up to spin j = 50 1  2 , the single qubit dynamical entropy never falls below 10 −5 within the first 5000 kicks.In fact, the entropy generally increased with dimension.Fig. 4 plots the smallest entanglement entropy obtained by any of qubits throughout the first 5000 kicks.As can be seen, higher spins experience a highly entangled orbit, remaining close to the upper bound of S max = ln 2.
Further evidence supporting the lack of a recurrence can be found in the specific case of spin j = 3  2 , the smallest possible kicked top applicable to this scenario.Recently, many aspects of this low-dimensional system were solved exactly, including the single-qubit linear entropy of various initial spin states as a function of twist strength and kick number N [16].In particular, the single-qubit linear entropy of the state U N κ |+⟩ y was found to be where are the Chebyshev polynomials with arguments related to the twist strength via Eq. ( 31) may be efficiently computed using symbolic programming and we found that the linear entropy does not exactly vanish within the first million kicks.
Given that a recurrence is almost certainly not present in the j = 3/2 system at this twist strength, it seems highly unlikely that a family of recurrences exist, one for each half-integer j > 3/2.This argument has an added strength by focusing on the special state |+⟩ y , which, due to the Hamiltonian symmetry of the system, has a pattern of experiencing a reduced recurrence time when a global periodicity exists.
The lack of periodicity at this κ value also shows that in general, not all twist strengths commensurate to π yield an exact recurrence.

D. Summary and other resonances
Table I summarizes our results.With these recurrences established, a natural question to ask is if there are others.To this end, we have performed a numerical search for such recurrences characterized by κ = πj r s for all coprime 1 ≤ r, s ≤ 10, for all integer and half-integer spin, upto 15.5 and have found none.This was done by computing the von-Neuman entropy for the initial state given by |θ, ϕ⟩ = |2.25,2.0⟩, upto 500 kicks and found that minimum value of von-Neuman entropy for the different sets of r and s upto j = 15.5 never falls below 10 −7 .Numerical simulations suggests that there are no other sets of r and s that shows the temporal periodicity, than what we have found.This therefore places constraints on any additional values of κ that yield a state-independent finite periodicity.Here × signifies the non-existence of periodicity and (*) represents results from numerical simulation.The numbers are specific to p = π 2 with the exception of the integer-spin period-2 orbit for κ = 2πj, which is independent of p.

E. Relation to kicked rotor
It is interesting to compare our results to the quantum resonance behaviour found in the quantum kicked rotor [22][23][24].This purely quantum dynamics occurs when one of the Hamiltonian parameters takes the form 4π r s and is characterized by quadratic growth of the wavefunction in momentum space.In contrast, the classical kicked rotor at the same parameter value only has linear scaling.An interesting exception to this quadratic growth behaviour is the case of r/s = 1/2, which yields a period-2 stateindependent orbit.This special case is known as quantum anti-resonance due to the complete lack of momentum growth [22].
Ref. [17] proposed a kicked top version of the quantum resonance condition as for coprime integers r and s, where it may be assumed r/s < 1 without loss of generality due to the global symmetry U κ = U κ+4πj .This proposal is motivated by the well-known contraction from the quantum kicked top to the quantum kicked rotor [25], effected via the simultaneous scaling Here the kicked top parameter κ becomes the relevant parameter in the kicked rotor that controls the existence of resonances.(Also note that despite j → ∞ the above is not to be considered a classical limit as the quantum kicked rotor is a fully quantum object -hence contraction.) The periodicities examined here do not satisfy the p ∼ 1/j scaling (35) and therefore are not to be seen as "pre-contracted" phenomena, at least not in a strict sense.It is thus interesting that despite only having a partial relationship to the resonance behaviour found in the kicked rotor we still observe non-standard dynamics in the kicked top at these special chaoticity values.
The lone case of κ = 2πj (i.e.r/s = 1/2) for integer spin discussed in sec.IV A actually can be seen as being "pre-contracted".This is because the period-2 orbit does not depend on the rotation angle p, and so without loss of generality we may set it to scale as p ∼ 1/j.Thus the peculiar behaviour of quantum anti-resonance found in the rotor may be seen as originating in the quantum kicked top.Previous works focusing on quantum correlations [20] or the pseudo-classical framework [17] do not fully capture this specialized anti-resonance effect: both approaches depend on the rotation angle p and both predict a higher than necessary orbit period [26].

V. CONCLUSION
Previous studies comparing classical and quantum dynamics in the kicked top largely validate the correspondence principle in the semiclassical regime [11,27].Other works have gone into characterizing if and when the correspondence principle may be applied in the deep quantum regime [9,10,14,20,28].Here we have demonstrated a general violation of the correspondence principle by finding various sets of state-independent, finite-time periodicities that have no classical analog, and which exist for all spins (i.e. both the deep quantum and semiclassical regimes).Some of these recurrences had been identified earlier for specific spins [14,16] or in a semiclassical context [17], but here we have generalized these results.We have analytically shown the existences of sets of recurrences and numerically introduced others.A preliminary search for additional "simple" periodicities indicate that if they exist the recurrence time must be relatively large.
Our analysis resolves a confusion over the general relationship between the rationality of the chaoticity parameter κ and the existence of a recurrence in the quantum kicked top.Ref. [14] argued that whenever this value is a rational multiple of π the evolution will be periodic in the sense that any initial state will only explore a finite subset of Hilbert space.Ref. [20] on the other hand maintained that this is only true for spin-1 systems; i.e. higher dimensional kicked tops do not experience such finite-orbit periodicity regardless of the chaoticity value.Here we have demonstrated the answer lies somewhere in-between.In particular, while recurrences do exist in all dimensions, and these recurrences do come from a rational κ value, not all rational κ values may yield a recurrence.
We further established a relationship to the quantum resonance phenomenon of the quantum kicked rotor [22], and showed that the peculiar anti-resonance effect (i.e.U 2 = I) in the kicked rotor may have its origins in the quantum kicked top.Given the link to the kicked rotor and the simple, general criterion for periodicity, it would seem reasonable to expect such non-classical resonances to occur in other periodic or kicked systems as well.Future work in this direction would help shed light on the complicated route to correspondence in chaotic systems.
To show the temporal periodicity of the full Floquet operator, we will use the above expression of the twist operator given in Eq. ( 23) along with the rotation part of the unitary given by Eq. (S12) and we took its sixth power.This gives us We use the similar trick as in integer-spin case of pulling the rotation part of U 6 towards the end using group commutator relations.This leads us to an expression given by where A is composed of the terms in side the square bracket.Now we will compute the A 2 and we will show that A 3 just an I with some coefficient.For the next few steps, we have repeatedly used the following relations,

FIG. 1 :
FIG. 1: Stroboscopic map showing the classical time evolution over 150 kicks for a. κ = 2.5 and b. κ = 3.0 for several hundred initial points.

FIG. 3 :
FIG. 3: Stroboscopic Husimi evolution at κ = πj of a spin coherent state starting at (θ, ϕ) = (2.25,2.0) over 8 kicks.The state splits and becomes entangled then recombines back to the original unentangled position.Q max corresponds to the maximum height of the Husimi distribution in each plot.Here j = 50.

TABLE I :
Recurrence periods for different κ values.