Quantum Chaos Measures for Floquet Dynamics

Home , Standard map

arXiv:2007.07283v2 [quant-ph] 31 Jul 2020 h ikdrtr egv ersnaintertcderiva theoretic si representation a a For give rotor. we kicked rotor, quantum kicked the the for operator alternative Floquet equivalent the of determine unitar also the We of eigensystem systems. the quantum of terms in measures and these function to pute spectral transition function, the Wigner (or echo, Loschmidt of the presence the of measures qua non-integrable diagnostic For chaos. of model simple illustrative ∗ eidclykce lqe ytm uha h ikdroto kicked the as such systems Floquet kicked Periodically mi:[email protected] email: unu ho esrsfrFloquet for measures chaos Quantum eateto hsc,Ahk nvriy India University, Ashoka Physics, of Department mnA Nizami A. Amin dynamics Abstract 1 ∗ tmdnmc hr r several are there dynamics ntum ino t dynamics. its of tion hoi eaiu including behaviour chaotic ) TC eaayial com- analytically We OTOC. lqe prtro driven of operator Floquet y om fteeigen-equation the of forms pe nerbevratof variant integrable mpler r aaimtcand paradigmatic a are r 1 Introduction

Classical chaos is diagnosed through two characteristic features: local instability - an exponential divergence of nearby trajectories in phase space and global mix- ing - intuitively a strong delocalisation or spreading of concentrated initial conditions throughout phase space, a property which implies ergodicity. The Poincare- Bendixson theorem rules out chaotic behaviour in the phase plane, that is, for autonomous systems with one degree of freedom (d.o.f) - or equivalently a 2 dimensional phase space. Seeking chaos, one may turn to the non-linear dynamics of two coupled d.o.f (4 dimensional phase space, for example that of 2 coupled non-linear oscillators) or a driven system - this has a 3 dimensional phase space (1.5 d.o.f). Al- ternatively one may study discrete dynamical systems (classical or quantum maps) where quite simple models can show extremely rich dynamics. Delta function kicked driven systems, the subject of our study, are examples of this kind. To ﬁnd chaotic behaviour one has to look at many-body systems (at any rate with more than one d.o.f) which are strongly interacting. As a result, perturbative methods are usually of not much help since the KAM theorem implies that integrable dynamics is robust to small perturbations (weak nonlinearities). It is only when appropriate coupling constants in the hamiltonian are tuned to suitably large values, thereby exiting the perturbative regime, that the KAM tori start to break apart and the phase space typically starts exhibiting expanding islands of chaotic behaviour as inferred by studying, for example, the Poincare sections. For this reason, most studies of chaotic behaviour are numerical. In the quantum case, one quantizes the dynamical system which in its classical limit can show both integrable and chaotic behaviour, and looks at suitable diagnostic measures with qualitatively diﬀerent features in the two cases [1]. We may expect that for simpler systems with only a few d.o.f at least some of these measures of chaos may be calculated exactly, thereby giving an analytical handle to study the onset of quantum chaos by tuning parameters in the hamiltonian and looking for characteristic behaviour. In this paper we revisit the quantum kicked rotor (QKR), a much studied simple model of classical and quantum chaotic behaviour [2, 3]. The dynamics of this system is encapsulated by the time dependent Hamiltonian:

∞ p2 H(t)= H + H (t)= + V (θ) δ(t pT ) (1) 0 int 2M − p=1 X Here V (θ) is periodic in θ. The choice V (θ) = K cos θ generates the so-called standard or Chirikov map. This represents a particle moving on a circle, kicked periodically in time with a Dirac delta function potential, and with a non-linear spatial part which is also periodic and has a strength parameter (K). Such a set-up can be realised experimentally [7] - for example, by suitably conﬁning cold atoms/molecules kicked periodically by a pulsed laser. The (non-relativistic) motion

2 of a particle in a circular accelerator boosted repeatedly through periodic energy kicks can be modelled similarly. Classically, the integrated Hamilton’s equations of motion of this system (with V (θ)= K cos θ ) give the well-known Chirikov map:

θj = θj−1 +(T/M)Pj−1 , Pj = Pj−1 + K sin θj (2) where θj, Pj denote the value of the observable right after the jth kick at t = jT . As the kicking strength, determined by the parameter K, is increased the system transitions from regular to chaotic behaviour. See [4] for an overview of the QKR. Quantum mechanically, this model exhibits dynamical localisation - a one- dimensional variant of Anderson localisation - where the classical energy diffusion, as measured by P 2 , is suppressed after a certain time due to quantum interfer- h i ence effects [2] and the dynamics is quasi-periodic and bounded. This is seen to be a characteristic feature of quantized systems whose classical limit is chaotic - a quantum mechanical “softening” of classical chaotic effects. There are multifarious quantitative measures used for detecting chaotic behaviour when studying the quantized dynamics of classically non-integrable systems. As is well known, the state overlap given by the inner product ψ(0) ψ(t) can not h | i be used since the linear unitary evolution of quantum states is norm-preserving and therefore shows no sensitive dependence on initial conditions1. An alternative is to detect sensitivity of the hamiltonian to small perturbations [9]. This leads to the fidelity/Loschmidt echo studied in section 3. Also of much utility is the thermal out of time order correlator (OTOC) or chaos correlator first considered by [19] and more recently studied, from the perspective of holography and quantum information scrambling in black holes, as a chaos diagnostic [20]. We will derive analytical expressions for several quantum chaos measures in terms of the eigensystem of the Floquet operator. These are known to show char- acteristically different behaviour in the regular and chaotic regime - for example, the Loschmidt echo and the thermal OTOC typically have an exponential form up until the Ehrenfest time in the presence of chaos. One reason the chaos measures are useful is that one can study, numerically or analytically, the transition from regular to chaotic behaviour in the quantum domain. In section 2 , after a brief review of the salient features of the classical and quantum kicked rotor, we discuss how Floquet theory simplifies its study and formulate the eigen-equation for the Floquet operator in various forms. In section 3 we analytically compute the OTOC, Loschmidt echo, Wigner function and spectral form factor in terms of the eigensystem of the unitary Floquet operator of this system. In section 4 we look at an integrable simpler variant of the kicked rotor first studied in [22] and use group representation theory to solve for its dynamics. We conclude with a brief discussion in section 5. 1Other measures of state distinguishability in Hilbert space could still be used - see [12] for a definition using relative state complexity.

3 2 The kicked rotor: basics and Floquet theory

The particle on a ring is among the simplest quantum systems and yet quite rich in its physics. The Hamiltonian for a free particle (mass M) on a ring (radius R) is

p2 ~2 ∂2 H = = − (3) 0 2M 2MR2 ∂θ2 The energies and normalised eigenfunctions are:

n2~2 einθ En = , ψn(θ)= (4) 2MR2 √2π and a general state can be expanded in a basis of these momentum/energy eigen- n~ states (P n = R n ) | i | i i(nθ−ωnt) Ψ(θ, t)= αne (5) n X The Hamiltonian for a kicked rotor is

∞ ∞ p2 ~2 ∂2 H(t)= H + H (t)= + V (θ) δ(t pT )= − + V (θ) δ(t pT ) 0 int 2M − 2MR2 ∂θ2 − p=1 p=1 X X (6) We will set M = R = 1 and often use V (θ)= K cos θ which generates the quantum standard map. The analysis of periodically driven quantum systems is greatly simpliﬁed by using Floquet theory. Given a hamiltonian

H(t)= H0 + V¯ (t) (7) where the interaction part is periodic in time- V¯ (t + T ) = V¯ (t), one decomposes the state at time t in terms of the Floquet eigenstates ψ (t) | n i ψ(t) = c ψ (t) (8) | i n | n i n X One can show using Floquet theory that

iǫ t ψ (t) = exp n φ (t) (9) | n i − ~ | n i where ǫn - deﬁned modulo 2π/~ are the quasi-energies and φn(t) - the Floquet modes - are periodic and need only be deﬁned over one time period - [0, T ].2 The Floquet operator is the single step time evolution operator (generating time evolution over a time period T ) and will be denoted by UF (T ). To solve for the state at time t: Diagonalise U (T ) to obtain eigenvalues: exp iǫnT , and eigenvectors: φ (0) . • F − ~ | n i 2This is the analogue of the well known Bloch theorem for spatially periodic potentials.

4 The initial state is decomposed in the Floquet eigenbasis : •

ψ(0) = cn φn(0) (10) | i n | i X it is easy to show that the state at any time t is then given by •

iǫnt ψ(t) = cn exp ~ φn(t) (11) | i n − | i X where iǫ t c = φ (0) ψ(0) φ (t) = exp n U(t) φ (0) (12) n h n | i | n i ~ | n i where in the above equation t is measured mod T using Floquet periodicity. The dynamics of a kicked system comprises of free propagation alternating with localised kicks. One can thus look at the system stroboscopically, right after a kick, at a time t = jT . Time evolution is generated by iterations of the Floquet Operator j U(t = jT )= UF (T )

i i U (T ; K)= U U (T ), U = exp − V (θ) , U (T ) = exp − P 2T (13) F V 0 V ~ 0 2~ where periodic kicks alternate with free propagation. To show this, write the evo- −i lution operator U(t) = UV (t)U0(t) where U0(t) = exp ~ H0t . Since U(t) satisﬁes i~ dU(t) = H(t)U(t), it follows that U (t) satisﬁes i~ dUV (t) = V¯ (t)U (t) and there- dt V dt V −i t ′ ¯ ′ fore, UV (t) = T exp ~ 0 dt V (t ) . The integral in the Dyson series is easily −i evaluated for a delta functionR kick and we get UV = exp ~ V (θ) . For the kicked rotor with V (θ) = K cos θ the matrix form of the Floquet oper- 3 ator, in the H0 eigenbasis, is known to be

im2T U = n U(T ; K) m = exp ( i)m−nJ (K/~) (14) nm h | | i 2~ − m−n Before proceeding, a few side remarks about the interesting form of this matrix.

Due to the Bessel function, away from the diagonal the magnitude of the • elements decreases quite fast - it is eﬀectively a banded matrix. Except for a diagonal part (the exponential factor), this is a Toeplitz matrix 4 • - the matrix elements Amn are of the form Am−n, that is, all elements on any ﬁxed super/sub - diagonal are the same.

3This follows easily from the generating function of the Bessel function

∞ exp(ik sin θ)= Jn(k) exp(inθ) n=−∞ X 4I thank Rajendra Bhatia for pointing this out.

5 There are two natural time scales in this problem - the time period for motion • around the circle and that of periodic forcing. When the ratio of the associated angular frequencies is a rational number ω ~T M 0 = = (15) ω 4πMR2 N we get the quantum resonance condition and it was shown by [6] that in this case the problem for the Floquet operator can be effectively reduced to one for a finite dimensional N N matrix. × Consider the Euclidean group in 2 dimensions - E(2), generated by rota- • tions Rθ (parametrised by angle θ) about a fixed point and translations T~a (parametrised by a and φ - giving the magnitude and direction of the trans- lation vector ~a) on a plane. A general element of this group can be written

as g = T~aRθ. Its inﬁnite dimensional unitary irreducible representations are known and given by

= e−imθei(m−n)φJ (λa) (16) Unm m−n λ is a continuous positive parameter labelling the representation and m labels the basis states in a particular representation space. This is quite close struc-

turally to our Unm, again except for the diagonal part - we will use the above form to solve for a simpler variant of the kicked rotor in section 4.

For the Floquet operator UF (T ), we denote its eigenvalues by µn, corresponding eigenvectors by µ and the eigenfunctions by µ (θ). | ni n The general time-evolved state (after j kicks) is given by

j j ψ(t) = UF ψ(0) = cnµn µn (17) | i | i n | i X with c = µ ψ(0) . The wave-function at time t = jT is then given by: n h n| i

j Ψ(θ, t)= cn(µn) µn(θ) (18) n X where imθ (m) µn(θ)= e µñ (19) m X (m) andµ ñ = m µ is the mth Fourier coefficient of the Floquet eigenfunction µ (θ). h | ni n The momentum space wave function after j kicks is (with p = m~)

j j (m) Ψ(m, t = jT )= µn m µn µn ψ(0) = cn µn µ˜n (20) n h | ih | i n X X

6 Forms of the Floquet operator eigen-equation

The eigen-equation for the Floquet operator is:

U µ = µ µ (21) | mi m | mi (n) In the H eigenbasis, this reads withµ ˜m = n µ 0 h | mi ′2 im T m−m′ (n) (n) exp ( i) J ′ (K/~)˜µ ′ = µ µ˜ (22) 2~ − m−m m m m m′ X an eigen-equation for an inﬁnite dimensional unitary matrix. We will see in the next subsection that this eigen-equation can be transformed to one for a Toeplitz hermitian matrix. The Floquet operator is usually presented as a matrix in the momentum eigenbasis as above. It is possible, however, to write it in diﬀerent equivalent forms. For example, in the position basis the eigen-equation reads:

i i~T exp − K cos θ exp ∂2 µ µ(θ)=0 (23) ~ 2 θ − ! an inﬁnite order linear ODE. Quantisation of µ would be imposed by requiring the wave function µ(θ) to be single-valued. A Discrete Fourier transform of µ(θ) leads to the matrix eigen-equation eq. (22) above. A usual integral Fourier transform of the above equation to frequency space leads to (after using the convolution theorem for Fourier transforms) the relation

µ−1 exp i~Tω2/2 φ˜ (ω)=(f˜ φ˜ )(ω) (24) − µ ∗ µ where denotes integral convolution, f(θ) = exp(iK cos θ/~) - whose Fourier tans- ∗ form gives a Bessel function. We therefore get the following (type 2 Fredholm) linear integral equation:

ω−ν 2 −1 √2π dν i Jω−ν(K/~) exp i~Tω /2 φ˜µ(ν)= µ φ˜µ(ω) (25) Z These equations should be of use in obtaining more information about the spectrum of the QKR Floquet operator.

Unitary to Hermitian Mapping of the Floquet Operator eigen- problem

It was shown by [5] that the eigen-problem for the Floquet operator can be mapped to an equivalent problem of a particle hopping on a 1D lattice. A unitary operator U = exp(ih) can be written in terms of a hermitian operator H in the alternative way (Cayley transform): 1+ iH U = , H = tan(h/2) (26) 1 iH − 7 One can use this to map the eigen-problem for the unitary Floquet operator to one for an associated hermitian operator. Let exp( iλ), φ be the eigenvalue and − λ corresponding eigenvector of UF (T ). Deﬁne:

uλ = (exp(iλ)Uf + 1)φλ (27)

The Floquet operator is UF (T )= UV U0(T ), if we absorb the U0 part in the eigenfunction and use the Cayley transform of UV , the eigen-equation for UF (T ) can be cast in the form TH λ V (θ) tan 0 u = tan u (28) 2~ − 2 λ 2~ λ In terms of the Fourier modes of uλ, one can write the eigenvalue equation in the H0 eigenbasis m2T λ f un = tan um (29) m−n λ 4~ 2 λ n − X V (θ) where fk is the k-th Fourier coefficient of f(θ) = tan 2~ : 1 2π V (θ) f = dθ tan e−ikθ (30) k 2π 2~ Z0 Note that we have transformed the eigen-equation of a unitary matrix to one of a hermitian matrix 5. For the family of periodic potentials V (θ)= 2~ tan−1(W (θ)), − the Fourier coefficients are simple (f = W ). The eigenvalues, in such cases, may k − k often be calculated analytically - for example [5] for W (θ) = k cos θ E, the so − called Lloyd’s model - it gives a nearest neighbour hopping tight binding model - we have λ m2 (mod 2π). m ∝ Note also that the hermitian matrix is now Toeplitz. Although we do not pursue this further here, Toeplitz matrices have special properties [8] which are of use in the study of this problem. For example, to calculate sums over functions of eigenvalues (such as a spectral form factor) of an infinite dimensional hermitian Toeplitz matrix we may use Szego’s theorem

n 1 1 2π lim F (λk,n)= dθ F (f(θ)) (31) n→∞ n 2π k 0 X=1 Z Here F is a function of the eigenvalues λk,n, and f(θ) is the symbol of the Toeplitz matrix fmn - it is the (periodic) function whose Fourier coeﬃcients are the elements of the Toeplitz matrix.

3 Measures of quantum chaos for driven systems

In this section we will derive expressions for a number of chaos diagnostic measures in Floquet systems in terms of the eigensystem of the associated Floquet Operator. 5We ﬁnd it convenient to write the above equation in this form - it is usually cast in a form describing the hopping amplitude of a particle in a 1D tight binding lattice model [5]

8 These expressions hold for general Floquet systems which are probed stroboscopically and not just delta-function kicked systems like the quantum standard map.

Loschmidt echo

One way of diagnosing chaos in the quantum regime is to study the sensitivity of the hamiltonian (rather than the initial state) to perturbations, which may be due to small changes in its parameters. This was done first by A. Peres [9] and leads naturally to studying the time variation of the quantum fidelity or Loschmidt Echo [10]. This quantity is important also in the study of quantum decoherence [11]. Consider a fixed initial state ψ(0) at t = 0 and evolution through two “neigh- | i bouring” hamiltonians H and H′ = H + δH to respective states ψ(t) and ψ′(t) . | i | i In particular the perturbation may be due to a variation of a parameter K in the hamiltonian which controls the interaction strength (with K′ = K + δK). The fidelity of the evolution is defined by

2 L(t)= ψ ′ (t) ψ (t) (32) |h K | K i| and naturally measures the deviation, after a time t, induced by the small initial perturbation. This can also be interpreted as forward evolution for time t of the initial state ψ(0) by the hamiltonian H followed by backward evolution for time | i K t by HK′ (imperfect time reversal) and then taking the overlap with the initial state. Thus viewed, the same quantity is called the Loschmidt echo. This function typically has an exponential/power-law decay in the chaotic/regular regime [9, 11]. To calculate the Loschmidt echo/fidelity for a periodically driven system we will take the initial state to be ψ(0) - which could be the the free eigenstate m - | i | i and then consider forward/backward evolution through j kicks. We have the echo † ′ j j operator Ej = UF (T,K ) UF (T,K) and the Loschmidt echo is defined in terms of the expectation value of the echo operator by L(t)= E(t) 2. |h i| For example for the QKR with V (θ)= K cos θ, for a single kick the expectation value of the echo operator in the state m is | i ′ E = J ′ (K /~)J ′ (K/~) (33) h i m−m m−m m′ X with K′ = K + δK. For evolution in a general Floquet system for time t which includes j full time periods, U(t)= U (t jT )U U (T )...... U U (T )U U (T ) (34) 0 − K 0 K 0 K 0 with j factors of UK and, E = ψ(0) U j†(T,K′)U j (T,K) ψ(0) (35) h ij h | F F | i Using Floquet theory, one can get a simpler expression. By introducing the completeness relation 1 = µ µ between each Floquet operator insertion in k | kih k| the above expression, it follows that P 9 ′ ∗ ∗ ′ j E(t) = µ ′ (K ) ψ(0) µ (K) ψ(0) µ (K)µ ′ (K ) (36) h ij h m | i h m | i m m m,m′ X (n) ′ (n)∗ ∗ ′ j = µ˜m′ (K )˜µm (K) µm(K)µm′ (K ) m,m′ X with the second equality above for the initial state ψ(0) = n . By definition, | i | i L(t)= E(t) 2. Here µ (K) and µ (K) are the eigenvalues and eigenvectors of |h i| n | n i the Floquet operator UF (T ; K).

Wigner function

The Wigner function [13] is a quasi-probabilty distribution in phase space, and is often used in semiclassical analysis. It makes the connection to the classical limit particularly transparent. The “volume” of its negative part can be used to quantify how non-classical a state is [15] and it is used as a chaos diagnostic as well [16]. See [18] for a numerical study of the QKR Wigner function. It also provides a route to an alternative phase space formulation of quantum mechanics [14]. For a review of its main properties and uses see [16, 17]. It is deﬁned (in terms of position space wave functions) by:

1 W (x, p; t)= dy Ψ(x + y, t)Ψ∗(x y, t) exp( 2ipy/~) (37) π~ − − Z For the QKR, using the explicit form of the wave function after j kicks - eq.(18), the integral can be performed giving the following form in terms of the Floquet eigendata:

1 j (2pl−m) 2i(m−pl)θ (m)∗ ∗j ∗ W (θ,p ; t)= c µ µ˜ e µ˜ ′ µ ′ c ′ (38) l π~ n n n n n n m,n,n′ X where pl is an integer labelling the momentum eigenvalue p = ~pl. Alternatively one may start with the Wigner function deﬁned in terms of the momentum space wave function. Adapted to the case of a discrete momentum spectra [16] this deﬁnition is 1 W (θ,p ; t)= Ψ(p + m, t)Ψ∗(p m, t) exp(2imθ) (39) l π l l − m X where the momentum space wave function Ψ(n, t) is given by (20). This gives the same expression for the Wigner function as above.

Spectral functions

To study operator growth and complexity in a kicked chaotic system, one natural object to study is A(t) = Tr P (t)P (0) where P (t) is the Heisenberg picture 10 momentum operator at time t. Probing the Floquet dynamics stroboscopically at †j j t = jT , we have Pj = UF P0UF . Calculating the trace in the Floquet eigenbasis gives A = µ U †j P U j P µ (40) j h n| F 0 F 0| ni n X Inserting the completeness relation 1 = µ µ in between each operator k | kih k| yields, after simplifying, the sought relation P A = µ P µ 2(µ∗ µ )j (41) j |h m| 0| ni| n m m,n X Likewise, the (inﬁnite temperature) spectral form factor after time t = jT is given in terms of the quasi-energies by

i(ǫm ǫn)jT S(t)= exp −~ (42) m,n X OTOC

Along the same lines as above, one may compute the OTOC

C(t)= ψ(0) [P (t),P (0)]2 ψ(0) (43) −h | | i starting with

[P ,P ] ψ(0) = (µ∗ µ )j µ P µ µ P ψ(0) P µ ψ(0) µ (44) j 0 | i n m h n| 0| mi h m| 0| i − 0 h m| i | ni m,n X The norm of the above state gives the expression for the OTOC. For the special case of an initial state ψ(0) = n =0 or ψ (θ)=1/√2π which is uniform on the | i | i 0 circle, this takes the form

∗ ∗ j (0) (0)∗ 2 C = (µ µ µ ′ µ ′ ) µ˜ ′ µ˜ µ P µ µ ′ P µ ′ µ ′ P µ (45) j n m n m m m h n| 0| mi h n | 0| m i h n | 0 | ni m,n,m′,n′ X An alternative expression for the OTOC, with the initial state ψ(0) = n , is | i | i C = ~2 (n m)2 m P n 2 (46) j − |h | j| i| m X with

m P n = (µ∗µ∗)jµ˜(m)µ˜(n)∗ µ P µ (47) h | j| i k l k l h k| 0| li k,l X The thermal OTOC of the QKR was studied numerically in [21].

11 4 A simpler kicked rotor

In this section we look at a variant of the kicked rotor which is linear rather than quadratic in the momentum, the hamiltonian being:

∞ H(t)= αp + K cos θ δ(t pT ) (48) − p=1 X This system was studied in [22]. It is an integrable system and its quasi energies and eigenfunctions have been analytically determined, see [22] for details and also how the classical and quantum dynamics is distinguished for rational vs. irrational values of α. Here we give a representation theoretic derivation of its quantum dynamics by solving for the unitary evolution operator for j kicks.

The free part of the hamiltonian H0 = αP has eigenvalues En = nα~ whereas the eigenfunctions (which we use as a basis) are the same as in eq. (4) above. The Floquet operator corresponding to the hamiltonian above is:

iK cos θ iαTP U (T ) = exp exp (49) F − ~ − ~ which, in the momentum eigenbasis, has the matrix form:

iαT m (U ) = exp im−nJ (K/~) (50) F nm − ~ m−n Compare this with the form of the irreducible inﬁnite dimensional unitary matrix representations of the Euclidean group in 2 dimensions - E(2) - given in eq. (16). With θ = αT/~, φ = π/2, a = K/~, and λ = 1, we can identify the two expressions. The realisation that the Floquet operator generating quantum dynamics forms a representation of the symmetry group E(2) is helpful in solving this problem. This is because the geometric group action of E(2) in the plane:

(R2,~a2)(R1,~a1)=(R2R1, R2~a1 + ~a2) (51) can be leveraged together with the group representation property that enables to write products of several U’s as a single U(R, A~). We get for the time evolution operator after j kicks

j j ~ Uj = UF (Rθ,~a) = UF Rθ, Aj (52) where ~ j j−1 j−2 Aj =(Rθ + Rθ + Rθ + ... + Rθ)~a (53)

Here Rθ is the orthogonal rotation matrix (counterclockwise rotation by angle θ) T and ~a = (0, a) . The vector A~j has magnitude Aj = a sin(jθ/2)/ sin(θ/2) and angle Φj = π +(j +1)θ /2 with the x-axis. This geometrically corresponds to an alternating sequence of j rotations (each of angle θ) and translations (parallel to

12 the y axis, each of magnitude a) in the plane. Thus we can write the full ﬁnal form of the time evolution operator after j kicks as:

i K sin(jαT/2~) (U ) = exp (j + 1)m αT im−nJ (54) j nm −2~ m−n ~ sin(αT/2~) This completely determines the dynamics. Note how the use of the group representation condition gives the above compact form. Measures like the Loschmidt echo can be now computed. For example the echo operator can be evaluated using a similar method: after j translations and rotations, we subsequently perform j in- verse transformations which partially undo the ﬁrst sequence. The Loschmidt echo for j kicks, with the initial state being any H0 eigenstate, can be calculated to be given by the simple expression

2 δK sin(jθ0) ~ Lj = J0 | | , θ0 = αT/2 (55) ~ sin θ0 Of course, in this case no chaotic behaviour is expected - this function oscillates periodically. The Wigner function for this system - in the initial state n - can also | i be computed (starting from eq. (39)) in the form of a Fourier mode expansion

n−pl ( 1) 2irθ K sin(jθ0) K sin(jθ0) W (θ,p ; t)= − e J l J l (56) n l π n−p −r ~ sin θ n−p +r ~ sin θ r 0 0 X Note that the method used in this section has a wider scope. It can be applied whenever the unitary Floquet operator of the system forms a group representation, with the group having a natural geometric action on a space.

5 Discussion

We have studied aspects of Floquet quantum dynamics for chaotic systems. The main new results include explicit expressions for various quantum chaos measures in terms of the Floquet operator eigensystem, alternative forms of the QKR Floquet operator eigen-equation and a representation-theoretic derivation of the dynamics of an integrable kicked rotor. Although we did not solve the Floquet eigen-problem analytically for the QKR, the results obtained should be helpful also in numerical studies of the various diagnostic measures of quantum chaos. It is possible that a deeper understanding of QKR dynamics can be obtained also from the alternative forms of the eigen-equation in section 2. The representation theoretic treatment for the integrable variant of the kicked rotor (section 4) was based on the insight that its Floquet operator forms a representation of E(2). For the QKR (standard map), as noted in section 2, the form of the Floquet matrix is very similar - see eq.(14). It would be interesting if representation theoretic methods can be wielded also in this more complex chaotic case to gain further insights into its dynamics [24]. There is a beautiful geometrical picture, discovered by M. Berry, of semiclassical quantum dynamics in terms of the Wigner function. Berry [16] investigated the

13 classical and semiclassical limit of the Wigner function for the case of hamiltonian systems with a 2n dimensional phase space (n d.o.f) . For the integrable case, n constants of motion are in involution and the Wigner function localises in the classical limit (~ = 0) as a delta function on the n - dimensional KAM torus on which the classical motion is confined. In the semiclassical limit (~ small but not zero) there is a non-singular localisation of the Wigner function near the surface of the KAM torus and an exponential/oscillatory behaviour on either sides of the surface. In other words, the delta function softens to have finite height and non-zero width in the vicinity of the surface of the KAM torus - there can however be other singularities away from the torus surface in the integrable case. For non-integrable hamiltonian dynamics, in general H is the only constant of motion and the classical motion fills a 2n 1 dimensional submanifold in phase − space. This case is not as well studied but the Wigner function is expected to also delocalise over this hypersurface. It would be interesting to explicitly evaluate the Wigner function for specific quantum maps, for which the results of this paper may be helpful, and study these aspects of the dynamics in detail. This will help in better understanding the quantum analogue of the classical KAM theory for the integrable to non-integrable transition. One measure of chaos that we did not study here is quantum complexity [23]. In the chaotic regime this is expected to increase linearly with time before saturating and it would be insightful to calculate it for this simple model. It should also be interesting to study other simple Floquet systems where some analytical computa- tions are feasible, and at least some measures of chaos can be computed exactly and used to investigate the classical to quantal and regular to chaotic transition [24].

Acknowledgements

I would like to thank Rajendra Bhatia for pointing out the relevance of Toeplitz matrices in this work. I also thank Arpan Bhattacharya for discussions and suggesting some useful references.

References

[1] For a pedagogical treatment of quantum chaos, see for example: F. Haake, Quantum signatures of chaos (Springer, 2010); H.J. Stockmann, Quantum Chaos: An Introduction (C.U.P. Cambridge, 2007); G.Casati and B. Chirikov, Quantum Chaos: Between Order and Disorder (C.U.P. Cam- bridge, UK, 1995); M. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer NY, 1990) [2] G. Casati, B. V. Chirikov, F. M. Izraelev and J. Ford, in “Stochastic Behavior in Classical and Quantum Hamiltonian Systems”, Vol. 93 of Lecture Notes in Physics, edited by C. Casati and J. Ford (Springer, New York, 1979).

14 B. V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52, 263 (1979); J. Greene, Method for determining a stochastic transition, J. Math. Phys. 20, 1183 (1979) [3] V. Chirikov, F. Izrailev and D. Shepelyansky, Sov. Sci. Rev. Sec. C 2, 209 (1981); T. Hogg and B.A. Huberman, Phys. Rev. Lett. 48 (1982) 711-714 ; S. Fishman, D.R. Grempel and R.E. Prange, Phys. Rev. Lett. 49 (1982) 509-512. [4] F.M.Izrailev, Simple models of quantum chaos: spectrum and eigenfunctions, Phys. Rep. 196: 299 (1990) [5] D. R. Grempel, R. E. Prange, and S. Fishman, Quantum dynamics of a non-integrable system, Phys. Rev. A 29, 1639 (1984). [6] F.M Izrailev and D. L.Shepelyansky, Dok.Akad.Nauk. 249(1979) 1103 [Sov. Phys. Dok. 24 (1979) 996]; Teor. Mat. Fiz. 43(1980) 417 [Theor. Math. Phys. 43 (1980) 553] S. J. Chang and K. J. Shi, Evolution and exact eigenstates of a resonant quantum system, Phys. Rev. A, Vol 34, 1 (1986); Time Evolution and Eigenstates of a Quantum Iterative System, Phys Rev Lett. Vol. 55, 3 (1985) 269-272. [7] F.L.Moore, J.C.Robinson, C.Bharucha, P.E.Williams, M.G.Raizen, Observation of dynamical localization in atomic momentum transfer: a new testing ground for quantum chaos, Phys. Rev. Lett. 73: 2974 (1994); J.Kanem, S.Maneshi, M.Partlow, M.Spanner, A.M.Steinberg, Observation of high- order quantum resonances in the kicked rotor, P.R.L. 98: 083004 (2007); C.Ryu et al, High-order quantum resonances observed in a periodically kicked Bose- Einstein condensate, Phys. Rev. Lett. 96: 160403 (2006); A.Ullah, M.D.Hoogerland, Experimental observation of Loschmidt time reversal of a quantum chaotic system, Phys. Rev. E 83: 046218 (2012) [8] A. Bottcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices (Springer, 1999); N. Nikolski et al, Toeplitz Matrices and Operators (Cambridge Studies in Advanced Mathematics, CUP 2020) [9] A. Peres, Stability of quantum motion in chaotic and regular systems, Phys. Rev. A 30, 1610 (1984); Quantum Theory: Concepts and Methods, Springer (1995) [10] T.Gorin, T.Prosen, T.H.Seligman, M.Znidaric, Dynamics of Loschmidt echos and ﬁdelity decay, Phys. Rep. 435: 33-156 (2006); T. Prosen, T. H. Seligman, M. Znidaric, Theory of quantum Loschmidt echoes, Prog.Theo.Phys.Supp. 150 (2003), 200-228 [11] R. A. Jalabert and H. M. Pastawski, Environment-independent decoherence rate in classically chaotic systems, Phys. Rev. Lett. 86 (2001) p. 2490-2493 [12] B. Yan and W. Chemissany, Quantum Chaos on Complexity Geometry, (arXiv:2004.03501 [quant-ph]). [13] E. Wigner, On the Quantum Correction For Thermodynamic Equilibrium, Physical Review 40, 749 (1932) [14] H. J. Groenewold, Physica 12, 405 (1946); J. E. Moyal, in Mathematical Proceedings of the Cambridge Phil. Society, Vol. 45 (Cambridge Univ. Press, 1949) pp. 99-124.

15 [15] A. Kenfack and K. Zyczkowski, Negativity of the Wigner function as an indicator of non-classicality, Journal of Optics B: Quantum and Semiclassical Optics, Volume 6, Number 10 (2004). [16] M. V. Berry, Semiclassical Mechanics in Phase Space: A study of Wigner’s function, Phil. Trans. R. Soc. A. 287, 237-71 (1977). [17] M. Hillery, R. F. O Connell, M. O. Scully, and E. P. Wigner, Distribution functions in physics: Fundamentals, Phys. Rep. 106, (1984)121-167 ; C. Zachos, D. Fairlie, and T. Curtright, Quantum Mechanics in Phase Space, (World Scientific, Singapore, 2005). [18] G. P. Berman, A. R. Kolovskii, F. M. Izrailev, and A. M. Iomin, Quantum chaos in the Wigner representation, Chaos: An Interdisciplinary Journal of Nonlinear Science 1, 220 (1991). [19] A. Larkin and Y. N. Ovchinnikov, Quasiclassical method in the theory of supercon- ductivity, Sov. Phys. JETP 28 no. 6, (1969) 1200-1205. [20] J. Maldacena, S. H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 (arXiv hep-th:1503.01409); A. Kitaev, A simple model of quantum holography, parts 1 and 2, Talks at KITP, 2015; S. H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP (2014) no. 3, 67 (arXiv hep-th:1306.0622). [21] E. B. Rozenbaum, S. Ganeshan, V. Galitski, Lyapunov Exponent and Out-of-Time- Ordered Correlator’s Growth Rate in a Chaotic System, Phys. Rev. Lett. 118, 086801 (2017). [22] D. R. Grempel, S. Fishman, and R. E. Prange, Phys. Rev. Lett 49 (1982) 833-836; M. V. Berry, Physica 10D, 369 (1984). [23] L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64, 24-43 (2016), [Addendum: Fortsch.Phys. 64, 44-48 (2016)], arXiv:1403.5695 [hep-th]; D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90, 126007 (2014), arXiv:1406.2678 [hep-th]; A. Brown, D. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Holographic Complexity Equals Bulk Action?, P. R. L. 116, 191301(2016), arXiv:1509.07876 [hep-th]; D. A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04 (2017) 121, arXiv:1610.04903 [quant-ph]; J. Cotler, N. Hunter-Jones, J. Liu, and B. Yoshida, Chaos, Complexity, and Random Matrices, JHEP 11 (2017) 048, arXiv:1706.05400 [hep-th]; T. Ali, A. Bhattacharyya, S S. Haque, E. H. Kim, N. Moynihan and J. Murugan, Chaos and complexity in quantum mechanics. PhysRevD 101, 026021 (2020); F. G.S.L. Brando, W. Chemissany, N. Hunter-Jones, R. Kueng and J. Preskill, Models of quantum complexity growth, arXiv:1912.04297 [hep-th]; V. Balasubramanian, M. DeCross, A. Kar and O. Parrikar, Quantum complexity of time evolution with chaotic Hamiltonians, JHEP 134 (2020). [24] Amin A. Nizami, work in progress