Kicked top
| Fritz Haake and Marek Kuś (2010), Scholarpedia, 5(11):10242. | doi:10.4249/scholarpedia.10242 | revision #137061 [link to/cite this article] |
The kicked top is a simple dynamical system, Hamiltonian in character and capable of chaotic motion. It has a single degree of freedom and its two dimensional classical phase space is a sphere. Chaos is possible due to periodic driving, for simplicity chosen as a sequence of (delta function) kicks. Due to the compactness of the classical phase space and the finiteness of the quantum mechanical Hilbert space the dynamics is relatively easy to treat theoretically and numerically. Experimental realizations of both the classical and the quantum cases are available
Contents |
Classical top
The classical Hamiltonian can be expressed in terms of the components of an angular momentum \(\vec{J}=(J_x,J_y,J_z)\) as the only dynamical variables. Consequently, the squared angular momentum is conserved, \(\vec{J}^2={\rm const}>0\ ,\) and that conservation law reveals the manifold accessible to the vector \(\vec{J}\) as a sphere. Equivalently, one may express the vector \(\vec{J}\) through a canonical pair with coordinate \(q\equiv \phi\) and momentum \(p\equiv \cos \theta\) where \(\theta\) and \(\phi\) are the polar and azimuthal angles in the familiar polar-coordinate representation of the sphere, \(J_x=\sqrt{\rm const}\sin \theta \cos \phi,\, J_y=\sqrt{\rm const}\sin \theta \sin \phi,\, J_z=\sqrt{\rm const}\cos \theta\ .\) The canonical Poisson brackets \(\{p,q\}=1\) imply the angular-momentum algebra given by the Poisson bracket \(\{J_x,J_y\}=J_z\) and its cyclic permutations.
A Hamiltonian generating chaotic motion arises if two polynomials of \(\vec{J}\) are summed, one of which carries the driving,
\[\tag{1} H(\vec{J},t)=H_0(\vec{J})+H_1(\vec{J})\sum_{n=0,\pm 1,\pm 2,\ldots}T\delta(t-nT) \]
where \(T\) is the period of the driving. The two polynomials
\(H_0\) and \(H_1\) must not Poisson commute,
\(\{H_0,H_1\}\neq 0\ ,\) and at least one of them has to have an
order exceeding one. A simple example is
\[\tag{2} H_0=\alpha J_x\,,\quad H_1=\tau J_z^2\,; \]
the first term generates rotation about the \(x\) direction with a velocity
\(\alpha\) while the second one may be said to generate torsion of strength
$\tau$ about the $z$ direction, i.e. a state dependent rotation by an angle
proportional to \(J_z\ .\)
A period-to-period stroboscopic description is convenient in terms of a map \(\vec{J}^{(n)}\to\vec{J}^{(n+1)}\) for which we may imagine an initial condition \(\vec{J}^{(0)}\) set at some arbitrary moment \(t_0\ ;\) the integer \(n\) then counts the number of subsequent periods and corresponds to the time \(t_n=t_0+nT\) [Haake (2010)]. If we scale the period \(T\) to unity and choose the initial time \(t_0\) immediately after a kick the map takes the form \[\tag{3} J_x^{(n+1)}=J_x^{(n)}\cos[\tau(J_y^{(n)}\cos\alpha+J_z^{(n)}\sin\alpha)] -(J_y^{(n)}\cos\alpha-J_z^{(n)}\sin\alpha)\, \sin[\tau(J_y^{(n)}\cos\alpha+J_z^{(n)}\sin\alpha)] \]
\[\tag{4} J_y^{(n+1)}=J_x^{(n)}\sin[\tau(J_y^{(n)}\cos\alpha+J_z^{(n)}\sin\alpha)] +(J_y^{(n)}\cos\alpha-J_z^{(n)}\sin\alpha)\, \cos[\tau(J_y^{(n)}\cos\alpha+J_z^{(n)}\sin\alpha)] \]
\[\tag{5} J_z^{(n+1)}=J_y^{(n)}\cos\alpha+J_z^{(n)}\sin\alpha \]
A top Hamiltonian like (2) was experimentally realized in 1985
by F. Waldner and coworkers [Waldner (1985)], who employed a magnetized
crystallite where a large number nuclear spins collectively build a
magnetization representable by a classical angular momentum. Precession of
the magnetization about an external magnetic field made for the rotation term
\(H_0\ ;\) the torsion term, \(H_1\ ,\) was due to a
nonlinear crystal anisotropy. In that experiment, the magnetic field was
sinusoidally modulated (instead of by a train of kicks) in time such that
\(H_0\) instead of \(H_1\) carried the periodic driving.
Typical stroboscopic phase-space portraits of such a top are shown in Figures 1-3. The extensions of regular and chaotic motion are varied from one portrait to the next by different choices for the strengths of rotation and torsion. Figure 1 corresponds to a nearly integrable top - both precession and torsion are present, albeit the torsion strength is close to zero. Figure 2 displays some islands of regular motion surrounded by a large chaotic sea. Finally, chaos is made predominant in Figure 3.
Quantum top
If the length of the angular momentum is sufficiently small quantum effects become important. The angular momentum \(\vec{J}\) must then be taken as a triple of non-commuting operators; their classical Poisson brackets are replaced by the commutator \([J_x,J_y]=J_xJ_y-J_yJ_x={\rm i}\hbar J_z\) and its cyclic permutations. The Hamiltonian operator looks like the above classical Hamiltonian with the component of angular momentum replaced by their quantum counterparts. With the choice of \(H_0\) and \(H_1\) as in Eq.(2) \[\tag{6} \hat{H}=\alpha J_y+\frac{\tau}{2j+1}\,J_z^2 \sum_{n=0,\pm 1,\pm 2,\ldots}T\delta(t-nT)\,. \]
The factor \((2j+1)^{-1}\) appears in the second component in order to give the same weight to both parts in the limit of large \(j\ ,\) if both \(\alpha\) and $\tau$ are taken independent of \(j\ .\) The squared angular momentum is again conserved, but the square is not capable of taking on arbitrary positive values any longer; instead, we face \(\vec{J}^2=\hbar^2 j(j+1)\) with the dimensionless quantum number \(j\) restricted to either integer or half-integer values. The classical limit is attained for large \(j\ ;\) formally, we may take \(j\to\infty,\,\hbar\to 0\) with the fixed product \(\hbar j\) specifying the desired classical length. Besides determining the squared angular momentum the quantum number \(j\) fixes the dimension of the Hilbert space as \(2j+1\ .\)
A stroboscopic description of the quantum evolution is provided by a period-to-period map for the wave function of the top, \(\psi_n\to\psi_{n+1}=F\psi_n\) in terms of a unitary time evolution operator \(F\) to be called Floquet operator. The state \(\psi_n=F^n\psi_0\) arises at the time \(t_n=t_0+nT\ ,\) out of the intimal state \(\psi_0\) specified at the time \(t_0\ .\) For the Hamiltonian (1),(2) the Floquet operator reads
\[\tag{7} F=\exp\left\{-\frac{{\rm i}\tau J_z^2}{\hbar^2(2j+1)}\right\}\, \exp\left\{-\frac{{\rm i}\alpha J_x}{\hbar}\right\} \]
where, again, \(T=1\ .\) It is well to
note that the product structure of the model Floquet operator (7)
is owed to the assumed delta-shaped kicks: loosely speaking, while
momentarily on the delta kick overwhelms the time independent precession,
and while off it lets the precession act exclusively. The initial time
\(t_0\) is chosen as immediately after a kick, for \(F\) to
consist of just two factors, one for precession (on the right) and the other
for the kick (on the left).
The Floquet operator is representable by a unitary \((2j+1)\times (2j+1)\) matrix. The \((2j+1)\) eigenvalues and eigenvectors display interestingly different properties depending on whether the coupling constants \(\tau\) and \(\alpha\) are chosen so as to have chaos or regular motion predominant in the classical limit, see further below.
An angular momentum with a Hamiltonian like (1),(2) was experimentally realized by P. Jessen and coworkers [Jessen (2009)]. The angular momentum there was the sum of nuclear and electron spins of \(^{133}\) Cs atoms, with \(j=3\ .\) Even though that latter value of \(j\) places the top deeply within the quantum realm, the experiments clearly establish, on average over many runs, a surprisingly close correspondence to the classical behavior, including a resolution of the partition of phase space into regular and chaotic parts.
Symmetries
The particular example of a kicked top specified by the Floquet operator (7) enjoys two symmetries worthy of discussion. First, that Floquet operator is invariant under rotation by the angle \(\pi\) about the \(J_x\) -axis which leaves both \(J_x\) and \(J_z^2\) unchanged. The pertinent rotation operator reads \(R={\rm e}^{{\rm i}\pi J_x/\hbar}\ ;\) it obeys \(R^2=+1\) for integer \(j\) and \(R^2=-1\) for half integer \(j\ .\) The eigenvalues of \(R\) are respectively \(\pm 1\) or \(\pm{\rm i}\) for \(j\) integer or half integer; in both cases, the sign suggests to speak of positive or negative parity. Since \(R\) and the Floquet operator \(F\) commute the eigenstates of \(F\) have definite parity as well, and the spectrum of \(F\ ,\) consisting of \(2j+1\) unimodular eigenvalues, falls into two separate subspectra, one for each parity.
Second, the Floquet operator under discussion is time reversal covariant,
\[ TFT^{-1}=F^\dagger\,. \]
The anti-unitary time reversal operator reads
\[\tag{8} T={\rm e}^{{\rm i}\alpha J_x/\hbar}K \]
where \(K\) denotes the operation of complex conjugation with
respect to some standard representation wherein \(J_z\) and
\(J_x\) become real matrices while \(J_y\) is an imaginary
matrix; the most convenient such representation is provided by the joint
eigenstates \(|jm\rangle\) of \(\vec{J}^2\) and
\(J_z\) with the respective eigenvalues \(\hbar^2j(j+1)\)
and \(\hbar m\ .\) The time reversal operator (8) squares
to unity, \(T^2=1\ ,\) both for integer and half integer
\(j\) since \(J_x\) and \(J_z\) are real matrices.
Extension to different symmetry classes
Time reversal invariance can be broken by imposing two separate kicks per period of the driving such that the Floquet operator becomes a product of three non-commuting unitary factors. The polynomials in \(\vec{J}\) appearing in the respective exponents can be chosen such that no geometric symmetry reigns [Kuś(1987)]. An example is
\[\tag{9} F=F_xF_yF_z\,,\qquad F_i=\exp\left\{-\frac{{\rm i}\alpha_i J_i}{\hbar}- \frac{{\rm i}\tau_i J_i^2}{\hbar^2(2j+1)}\right\}\,,\; i=x,y,z\,. \]
If the six coupling constants \(\alpha_i,\tau_i\) are given
different values of order unity there are no unitary or antiunitary
symmetries. Such Floquet operators describe tops from what is called the
unitary symmetry class.
Upon setting one of the three factors $F_i$ in the foregoing $F$ equal to unity (by letting the pertinent rotation angle and torsion constant vanish) we recover a time reversal symmetry with $T=F_jK$ and $F_j$ one of the two remaining factors in $F$. The time reversal operator $T$ then squares to unity since two angular momentum components can always be given real representations. The symmetry class thus accessed is called the orthogonal one. In general no geometric (unitary) symmetry persists but of course special choices of the coupling constants will entail such symmetries, like the one encountered in the preceding section.
Upon setting one of the three factors \(F_i\) in the foregoing \(F\) equal to unity (by letting the pertinent rotation angle and torsion constant vanish) we recover a time reversal symmetry with \(T=F_jK\) and \(F_j\) one of the two remaining factors in \(F\ .\) The time reversal operator \(T\) then squares to unity since two angular momentum components can always be given real representations. The symmetry class thus accessed is called the orthogonal one. In general no geometric (unitary) symmetry persists but of course special choices of the coupling constants will entail such symmetries, like the one encountered in the preceding section.
Tops from a third symmetry class, known as the symplectic one, are attained by choosing a representation with half integer \(j\) for the Floquet operator [Scharf(1988)]
\[ F=F_1F_2\,,\qquad F_1=\exp\left\{-\frac{{\rm i}\tau_1 J_z^2}{\hbar^2(2j+1)}\right\} \] \[\tag{10} F_2=\exp\left\{-\frac{\rm i}{\hbar^2(2j+1)}\left[\tau_2 J_z^2+ \tau_3(J_xJ_z+J_zJ_x) +\tau_4(J_xJ_y+J_yJ_x)\right]\right\}\,. \]
Time reversal covariance holds, \(TFT^{-1}=F^\dagger\ ,\) with
\(T=F_1{\rm e}^{{\rm i}\pi J_y/\hbar}K\ ,\) and that time reversal
operator squares to minus unity, \(T^2=-1\ ,\) due to the restriction
to half integer values of \(j\ .\) For integer values of
\(j\ ,\) on the other hand, the present \(F\) belongs to the
orthogonal class since in that case \(T^2=+1\ .\)
Worthy of mention is the integrable behavior resulting when the Floquet operator is taken as just one of the foregoing \(F_i\) in (9),(10). Such a choice corresponds to a time independent Hamiltonian \(\tilde{H}\) identified from \(F_i=\exp(i\tilde{H}T)\ ,\) for which the conservation of both the energy, \(\tilde{H}=E\ ,\) and the squared angular momentum, \(\vec{J}^2=\hbar^2 j(j+1)\) entail classical integrability.
The altogether four symmetry classes just gone through entail observably different behaviors, for instance in the fluctuations in the sequence of the eigenphases (quasienergies) of their Floquet operators. The integrable case is characterized by Poissonian spectral fluctuations, as if the quasienergies bore no mutual correlations at all. The probability density of nearest-neighbor spacings then is the exponential \(P(S)={\rm e}^{-S}\) if the unit is chosen such that the mean spacing is unity, \(\langle S\rangle=1\ ;\) one speaks of level clustering since the most probable spacing is zero.
In contrast, the three symmetry classes corresponding to predominance of
classical chaos display level repulsion inasmuch as the pertinent spacing
distributions vanish at zero spacing. Random-matrix theory provides the so
called Wigner surmise \(P(S)=AS^\beta{\rm e}^{-BS^2}\) with the
exponent \(\beta\) respectively taking the values \(1, 2,\)
and 4 for the orthogonal, unitary, and symplectic symmetry class. One
therefore speaks of linear, quadratic, and quartic level repulsion. The two
constants \(A\) and \(B\) in the Wigner surmise can be
fixed by normalization, \(\int_0^\infty dSP(S)=1\) and \(\langle
S\rangle=1\ .\)
As revealed in Figure 4, kicked tops have spacing distributions in impressive agreement with the theoretical predictions for the pertinent symmetry classes.
The rotator limit and quantum localization
An interesting limit of the top (1),(2) arises when torsion so strongly overwhelms rotation that almost all classical phase-space trajectories remain confined to a narrow equatorial waistband, rather than exploring all of the sphere [Haake (1988)]. For that limit to arise, the torsion constant and the rotation angle must scale with \(j\) as \(\tau\propto j\) and \(\alpha\propto {1}/{j}\ .\) The component \(J_z/\hbar\) then remains of order unity if so chosen initially while the much larger transverse components of the angular momentum can be written as \(J_x=\hbar j\cos\phi,\, J_y=\hbar j\sin\phi\ .\) The classical phase space then effectively looks like a cylinder (with \(\phi\) an angular coordinate and \(J_z/\hbar\) a dimensionless momentum. The dynamics becomes equivalent to that of the kicked rotator, another standard model of nonlinear dynamics. In particular, if we choose \(\tau=j\) and \(\alpha=K/j\) we recover in the described limit the kicked rotator (the Chirikov standard map) with the parameter \(K\ .\) The most prominent classical feature of that kicked rotator is rapid motion (of the angle \(\phi\)) around the cylinder and slow diffusion (of the momentum) along the cylinder.
The quantum version of the rotator displays, for generic values of the two coupling constants, the phenomenon of dynamical localization: the eigenfunctions of \(F\) are exponentially localized in the eigenbasis of the momentum \(J_z/\hbar\ .\) A dimensionless localization length can be defined as \(l=\sqrt{{\rm var}(J_z/\hbar)}=\sqrt{\langle (J_z/\hbar)^2\rangle-\langle J_z/\hbar\rangle^2}\) and must obviously be restricted as \(l\ll j\ .\) The above outlined equivalence between the rotator and the top in the described limit allows to estimate the localization length as \[\tag{11} l\sim\frac{K^2}{2\hbar^2}, \quad \textrm{for\ }K\gg 1. \]
Other applications
The kicked top naturally appears in a description of dynamics of resonant tunneling diode (see [Shepelyansky & Stone (1995)]). It was also used for studies of the Loschmidt echo behavior (see [Jacqquod et al. (2001)]).
References
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- Waldner, F.; Barberis, D R and Yamazaki, H (1985). Route to chaos by irregular periods: Simulations of parallel pumping in ferromagnets. Phys. Rev.A 31: 420. doi:10.1103/physreva.31.420.
- Chaudhury, S; Smith, A; Anderson, B E; Ghose, S and Jessen, P S (2009). Quantum signatures of chaos in a kicked top. Nature 461: 768. doi:10.1038/nature08396.
- Kuś, M.; Scharf, R. and Haake, F (1987). Symmetry versus degree of level repulsion for kicked quantum systems. Z. Phys. B 66: 129. doi:10.1007/bf01312770.
- Scharf, R; Dietz, B; Kuś, M; Haake, F and Berry, M V (1988). Kramers degeneracy and quartic level repulsion. Europh. Lett. 5: 383. doi:10.1209/0295-5075/5/5/001.
- Mehta, M L (2004). Random Matrices. 3rd ed. Elsevier, New York.
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- Shepelyansky(1995). Chaotic Landau level mixing in classical and quantum wells. Phys.Rev.Lett. 74: 2098. doi:10.1103/physrevlett.74.2098.
- Jacquod, Ph.; Silvestrov, P.G. and Beenakker, C.W.J. (2001). Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo. Phys.Rev.E 64: 055203(R). doi:10.1103/physreve.64.055203.
