Kicked Rotor Preview

The kicked rotor is a particle on a circle that rotates freely between short, periodic impulses. In the ideal delta-kick limit, one period reduces exactly to a two-step map. The same model therefore provides a compact meeting point for Hamiltonian chaos, Floquet operators, quantum maps, quantum resonance, and interference-induced suppression of classical diffusion.

This page owns that model and its basic comparison between classical and quantum stroboscopic motion. Particle on a Ring owns the Hilbert space and angular-momentum basis; Floquet Operators owns general one-period spectral theory; Quantum Maps and Discrete-Time Evolution owns the abstract map viewpoint; and Quantum Chaos Preview owns the broader diagnostic map. Detailed scrambling and many-body localization belong to later specialist treatments.

Classical Kicked Rotor

Let $\theta$ be an angle, $L$ its conjugate angular momentum, $I$ the moment of inertia, and $T$ the time between kicks. A convenient physical Hamiltonian is

$$H(\theta,L,t) = \frac{L^2}{2I} + \kappa\cos\theta \sum_{n\in\mathbb Z} \delta(t-nT).$$

Because the delta distribution has units of inverse time, $\kappa$ has units of action, or angular momentum. The sign of the kick is conventional: changing $\kappa\cos\theta$ to $-\kappa\cos\theta$ is equivalent to shifting $\theta$ by $\pi$.

Sample the motion immediately before each kick:

$$t_n=nT^-.$$

Hamilton’s equations are

$$\dot\theta=\frac{L}{I}, \qquad \dot L = \kappa\sin\theta \sum_{n\in\mathbb Z} \delta(t-nT).$$

The angle is continuous through an ideal kick, while the angular momentum jumps:

$$L_n^+ = L_n^- + \kappa\sin\theta_n.$$

Free rotation during the following interval gives

$$\theta_{n+1} = \theta_n + \frac{T}{I}L_n^+ \quad \text{mod }2\pi.$$

Introduce the dimensionless momentum and kick strength

$$P_n = \frac{T}{I}L_n^-, \qquad K = \frac{T}{I}\kappa.$$

The stroboscopic dynamics is then the standard map:

$$\begin{aligned} P_{n+1} &= P_n+K\sin\theta_n, \\ \theta_{n+1} &= \theta_n+P_{n+1} \quad \text{mod }2\pi. \end{aligned}$$

The angle is periodic, but $P$ is normally left unbounded. Classical phase space is therefore a cylinder. Reducing $P$ modulo $2\pi$ produces a torus map useful for phase portraits, but it discards information about unbounded momentum transport.

Area Preservation and Classical Chaos

The Jacobian of one step is

$$J_n = \begin{pmatrix} 1+K\cos\theta_n & 1 \\ K\cos\theta_n & 1 \end{pmatrix},$$

where the first row differentiates $\theta_{n+1}$ and the second differentiates $P_{n+1}$ with respect to $(\theta_n,P_n)$. Its determinant is

$$\det J_n=1.$$

Thus the standard map preserves phase-space area, as a stroboscopic Hamiltonian map must. Chaos here does not arise from dissipation or attraction to a strange attractor.

Its qualitative behavior depends strongly on $K$:

• At $K=0$, momentum is constant and the rotor is integrable.
• For small nonzero $K$, invariant curves, resonant islands, and chaotic layers coexist.
• As $K$ grows, large chaotic regions develop, although stable islands can survive.
• For many strongly chaotic trajectories, momentum behaves approximately diffusively over suitable time ranges.

Under a random-phase approximation,

$$\left\langle (P_n-P_0)^2 \right\rangle \approx D_{\rm cl}n,$$

with

$$D_{\rm cl} \approx \frac{K^2}{2}.$$

The last formula is not exact for every $K$. Correlations between kicks, accelerator modes, islands, and anomalous transport can produce substantial corrections. The kicked rotor is valuable precisely because one elementary map already displays both regular and chaotic classical regimes.

Quantization on the Circle

The quantum Hilbert space is

$$\mathcal H=L^2(S^1,d\theta).$$

Rather than relying on a globally defined canonical angle operator, use multiplication by the periodic function $\cos\theta$ and the self-adjoint angular momentum

$$\hat L = -i\hbar\frac{d}{d\theta}$$

on its periodic domain. The normalized momentum states are

$$\langle\theta\vert m\rangle = \frac{e^{im\theta}}{\sqrt{2\pi}}, \qquad \hat L\lvert m\rangle = m\hbar\lvert m\rangle, \qquad m\in\mathbb Z.$$

The ideal delta pulse produces the kick unitary

$$U_K = \exp\left( -\frac{i\kappa}{\hbar} \cos\theta \right),$$

while free rotation for one period produces

$$U_0 = \exp\left( -\frac{iT\hat L^2}{2I\hbar} \right).$$

With snapshots taken immediately before each kick, the kick acts first and the free rotation second. The Floquet operator is therefore

$$U_F=U_0U_K.$$

The rightmost factor acts first. Reversing the sampling convention reverses the displayed order.

Dimensionless Quantum Map

Define

$$\hat P = \frac{T}{I}\hat L, \qquad \hbar_{\rm eff} = \frac{\hbar T}{I}.$$

Then

$$\hat P\lvert m\rangle = m\hbar_{\rm eff}\lvert m\rangle,$$

and the one-period unitary becomes

$$U_F = \exp\left( -\frac{i\hat P^2}{2\hbar_{\rm eff}} \right) \exp\left( -\frac{iK\cos\theta}{\hbar_{\rm eff}} \right).$$

Two independent dimensionless parameters now appear:

• $K$ controls the corresponding classical standard map;
• $\hbar_{\rm eff}$ controls quantum phase resolution.

Many papers instead denote the kick phase by $k=K/\hbar_{\rm eff}=\kappa/\hbar$. Comparing formulas without checking which quantity is called the “kick strength” is a common source of apparent disagreement.

Using the Jacobi–Anger expansion,

$$e^{-iz\cos\theta} = \sum_{r\in\mathbb Z} (-i)^rJ_r(z)e^{ir\theta},$$

the momentum-basis matrix elements are

$$\begin{aligned} \langle m\vert U_F\vert m'\rangle &= e^{-i\hbar_{\rm eff}m^2/2} (-i)^{m-m'} \\ &\quad\times J_{m-m'}\left( \frac{K}{\hbar_{\rm eff}} \right). \end{aligned}$$

The kick couples many angular-momentum states through Bessel amplitudes. The free step leaves their probabilities unchanged but assigns the quadratic phases that govern interference between later kicks.

Worked Example: One Kick from Zero Momentum

Start in $\lvert0\rangle$. After one period,

$$\langle m\vert\psi_1\rangle = e^{-i\hbar_{\rm eff}m^2/2} (-i)^m J_m\left( \frac{K}{\hbar_{\rm eff}} \right).$$

Therefore

$$P_m(1) = \left\lvert J_m\left( \frac{K}{\hbar_{\rm eff}} \right) \right\rvert^2.$$

The Bessel identities

$$\sum_{m\in\mathbb Z}J_m(z)^2=1$$

and

$$\sum_{m\in\mathbb Z}m^2J_m(z)^2 = \frac{z^2}{2}$$

give

$$\langle\hat P\rangle_1=0, \qquad \langle\hat P^2\rangle_1 = \frac{K^2}{2}.$$

The first kick thus has the same second-moment increment suggested by the classical random-phase estimate. The long-time behavior is different because later amplitudes interfere coherently.

Floquet Spectrum and Sampling Phase

Floquet eigenstates satisfy

$$U_F\lvert\phi_\alpha\rangle = e^{-i\omega_\alpha} \lvert\phi_\alpha\rangle,$$

where $\omega_\alpha$ is an eigenphase modulo $2\pi$. If one assigns a period $T$, the associated quasienergy is

$$\varepsilon_\alpha = \frac{\hbar\omega_\alpha}{T} \quad \text{mod }\hbar\Omega, \qquad \Omega=\frac{2\pi}{T}.$$

Sampling immediately after each kick gives

$$U_F'=U_KU_0.$$

The two maps are unitarily conjugate:

$$U_F' = U_KU_FU_K^\dagger.$$

They have the same eigenphases but different eigenvectors at the chosen drive phase. This is a concrete instance of the general reference-phase dependence explained in Stroboscopic Dynamics.

Dynamical Localization Preview

Suppose the classical map lies in a regime where an ensemble spreads diffusively in momentum. A nonresonant quantum wave packet can initially show similar growth, but coherent interference eventually suppresses further diffusion. In the standard localization regime,

$$\langle\hat P^2\rangle_n$$

approaches a bounded, fluctuating scale rather than continuing to grow linearly. Corresponding Floquet eigenstates are exponentially localized in the momentum index:

$$\left\lvert \langle m\vert\phi_\alpha\rangle \right\rvert^2 \sim \exp\left( -\frac{\lvert m-m_\alpha\rvert}{\ell} \right),$$

where the localization length $\ell$ depends on the parameters and on convention-dependent definitions.

This phenomenon is called dynamical localization because localization appears in the evolution of a deterministic, periodically driven system. It is not friction: the evolution remains unitary, purity is preserved, and energy is not carried into an environment.

Fishman, Grempel, and Prange related the kicked-rotor Floquet eigenvalue problem to a one-dimensional Anderson-like tight-binding equation in momentum space. The quadratic free-rotation phases act as deterministic pseudodisorder when their arithmetic is sufficiently incommensurate. This mapping explains why interference can halt classical diffusion, but it should not be paraphrased as a literal random potential in the laboratory Hamiltonian.

Quantum Resonances

Dynamical localization is not universal for every $\hbar_{\rm eff}$. When

$$\frac{\hbar_{\rm eff}}{4\pi} \in \mathbb Q,$$

the free phases become periodic in the integer momentum label. Translation symmetries in momentum space can then produce quantum resonances. For suitable initial states and boundary phases, the energy grows ballistically:

$$\langle\hat P^2\rangle_n \propto n^2,$$

rather than diffusing and then localizing.

At the primary resonance $\hbar_{\rm eff}=4\pi$,

$$e^{-i\hbar_{\rm eff}m^2/2} = e^{-i2\pi m^2} = 1$$

for every $m\in\mathbb Z$. The free-rotation factor is exactly the identity on the periodic rotor Hilbert space, so successive kick phases add coherently. This exceptional arithmetic behavior is why a statement about localization must always declare the effective Planck constant and boundary conditions.

What the Model Says About Quantum Chaos

The classical standard map can have positive Lyapunov exponents and exponentially separating nearby trajectories. Exact quantum evolution is linear and unitary, so the Hilbert-space distance between two state vectors is preserved. Quantum chaos is therefore not defined by copying classical trajectory separation verbatim.

Instead, one asks how classically chaotic structures appear in quantum observables, Floquet spectra, eigenstates, semiclassical propagation, and correlation functions. The kicked rotor offers several such signatures:

• early-time correspondence with classical momentum diffusion;
• quasienergy and eigenvector statistics;
• dynamical localization caused by long-time phase coherence;
• quantum resonances controlled by number-theoretic phase relations;
• sensitivity of transport to noise, decoherence, and finite pulse duration.

These are entry points, not a complete definition of quantum chaos. Quantum Chaos Preview compares spectral, semiclassical, transport, and operator-growth diagnostics. Why Phase Space in Quantum Mechanics? explains the operator-to-phase-space bridge without assigning forbidden simultaneous sharp values to angle and momentum.

Experimental Realizations

In atom-optics realizations, ultracold atoms move through a pulsed optical standing wave. Position modulo the lattice period plays the role of the rotor angle, atomic momentum supplies the momentum ladder, and short light pulses approximate delta kicks. This is a simulator of the rotor map rather than a literal mechanical rod rotating in space.

Experiments reported suppression of classical momentum diffusion and later implemented a direct atom-optics quantum delta-kicked rotor. Real experiments also expose assumptions hidden by the ideal map:

• pulses have finite duration;
• an atomic cloud can contain a distribution of quasimomenta;
• spontaneous emission and technical noise reduce coherence;
• the measured momentum range is finite;
• interactions may need to be negligible or modeled explicitly.

These effects can weaken localization, broaden resonances, or change the effective Floquet operator. Agreement with the ideal model therefore requires a parameter and convergence analysis, not only a visually similar momentum profile.

Numerical Propagation

The factorized map supports an efficient split-operator algorithm that is exact for the ideal delta-kicked model:

1. Represent the state by momentum amplitudes $c_m$.
2. Transform to an angular grid by a discrete Fourier transform.
3. Multiply by $\exp[-iK\cos\theta/\hbar_{\rm eff}]$.
4. Transform back to momentum space.
5. Multiply by $\exp[-i\hbar_{\rm eff}m^2/2]$.

Unlike a generic Trotter approximation, these two factors are the exact one-period map under the declared sampling convention. Numerical errors instead come from finite momentum cutoffs, angular-grid aliasing, floating-point accumulation, and inadequate resolution of long localization tails.

A reliable simulation should monitor

$$\sum_m\lvert c_m\rvert^2,$$

population near the momentum-grid edges, convergence under grid enlargement, and sensitivity to small changes in $\hbar_{\rm eff}$. A finite grid can create artificial recurrences or false saturation that resembles localization.

Scope and Limitations

The delta-kicked rotor is an idealized single-particle Floquet model. It cleanly separates free and impulsive evolution, but that solvability should not be mistaken for universality.

• Finite-width pulses can alter the classical map and quantum phases.
• Mixed classical phase space invalidates a single global diffusion picture.
• Quantum resonance competes with localization at special arithmetic parameters.
• Decoherence can restore diffusive transport by destroying the interference responsible for localization.
• Interacting kicked systems introduce genuinely many-body questions not contained in this page.

The standard map and classical phase-space geometry continue in Phase Space and Hamiltonian Mechanics Review. General quasienergy structure belongs to Quasienergies. Inverse-frequency methods for smooth periodic drives belong to High-Frequency Expansions; a singular delta-kick sequence requires separate attention to convergence and operator domains.

Common Mistakes

• Forgetting to state whether snapshots are taken before or after each kick.
• Reading $U_0U_K$ from left to right in time; the rightmost factor acts first.
• Treating $K$ and $K/\hbar_{\rm eff}$ as the same parameter.
• Reducing momentum modulo $2\pi$ and then claiming to have measured unbounded diffusion.
• Assuming $D_{\rm cl}=K^2/2$ is exact for all kick strengths.
• Claiming dynamical localization at a quantum-resonant value of $\hbar_{\rm eff}$.
• Interpreting localization as dissipation or loss of purity.
• Trusting saturation on a finite numerical grid without checking edge population.
• Calling unitary Hilbert-space evolution chaotic because classical trajectories separate.

References

• B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Physics Reports 52, 263–379, 1979, doi:10.1016/0370-1573(79)90023-1.
• G. Casati, B. V. Chirikov, F. M. Izrailev, and J. Ford, “Stochastic behavior of a quantum pendulum under a periodic perturbation,” in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Lecture Notes in Physics 93, 334–352, Springer, 1979.
• F. M. Izrailev and D. L. Shepelyanskii, “Quantum resonance for a rotator in a nonlinear periodic field,” Theoretical and Mathematical Physics 43, 553–561, 1980, doi:10.1007/BF01029131.
• S. Fishman, D. R. Grempel, and R. E. Prange, “Chaos, quantum recurrences, and Anderson localization,” Physical Review Letters 49, 509–512, 1982, doi:10.1103/PhysRevLett.49.509.
• D. R. Grempel, R. E. Prange, and S. Fishman, “Quantum dynamics of a nonintegrable system,” Physical Review A 29, 1639–1647, 1984, doi:10.1103/PhysRevA.29.1639.
• F. M. Izrailev, “Simple models of quantum chaos: spectrum and eigenfunctions,” Physics Reports 196, 299–392, 1990, doi:10.1016/0370-1573(90)90067-C.
• F. L. Moore, J. C. Robinson, C. Bharucha, P. E. Williams, and M. G. Raizen, “Observation of dynamical localization in atomic momentum transfer: a new testing ground for quantum chaos,” Physical Review Letters 73, 2974–2977, 1994, doi:10.1103/PhysRevLett.73.2974.
• F. L. Moore, J. C. Robinson, C. F. Bharucha, B. Sundaram, and M. G. Raizen, “Atom optics realization of the quantum delta-kicked rotor,” Physical Review Letters 75, 4598–4601, 1995, doi:10.1103/PhysRevLett.75.4598.
• F. Haake, Quantum Signatures of Chaos, 3rd ed., Springer, 2010.

Exercises

1. Derive the standard map from the physical Hamiltonian and verify the dimensions of $\kappa$, $P$, and $K$.

Solution

Integrating Hamilton’s equation for $L$ across a kick gives

$$L_n^+-L_n^- = \kappa\sin\theta_n.$$

Free evolution then gives

$$\theta_{n+1} = \theta_n + \frac{T}{I}L_n^+ \quad \text{mod }2\pi.$$

Multiplying the momentum equation by $T/I$ and defining

$$P_n=\frac{T}{I}L_n^-, \qquad K=\frac{T}{I}\kappa$$

produces

$$P_{n+1}=P_n+K\sin\theta_n,$$

followed by

$$\theta_{n+1} = \theta_n+P_{n+1} \quad \text{mod }2\pi.$$

The delta function has units $1/\text{time}$, so $\kappa$ has units of energy times time, equal to angular momentum. Since $T/I$ has units of inverse angular momentum, both $P$ and $K$ are dimensionless.

2. Show directly that the standard map preserves area.

Solution

Differentiate

$$P'=P+K\sin\theta, \qquad \theta'=\theta+P'.$$

The Jacobian is

$$\frac{\partial(\theta',P')} {\partial(\theta,P)} = \begin{pmatrix} 1+K\cos\theta & 1 \\ K\cos\theta & 1 \end{pmatrix}.$$

Therefore

$$\det J = (1+K\cos\theta)-K\cos\theta = 1.$$

The map preserves the area element $d\theta\,dP$.

3. Derive the momentum-basis matrix elements of the Floquet operator.

Solution

Set

$$z=\frac{K}{\hbar_{\rm eff}}.$$

The Jacobi–Anger expansion gives

$$e^{-iz\cos\theta} = \sum_r(-i)^rJ_r(z)e^{ir\theta}.$$

Because

$$\langle m\vert e^{ir\theta}\vert m'\rangle = \delta_{m,m'+r},$$

the kick matrix element is

$$\langle m\vert U_K\vert m'\rangle = (-i)^{m-m'}J_{m-m'}(z).$$

The free unitary is diagonal:

$$\langle m\vert U_0\vert r\rangle = e^{-i\hbar_{\rm eff}m^2/2} \delta_{mr}.$$

Multiplying $U_F=U_0U_K$ yields

$$\begin{aligned} \langle m\vert U_F\vert m'\rangle &= e^{-i\hbar_{\rm eff}m^2/2} (-i)^{m-m'} \\ &\quad\times J_{m-m'}\left( \frac{K}{\hbar_{\rm eff}} \right). \end{aligned}$$

4. For an initial state $\lvert0\rangle$, compute $\langle\hat P^2\rangle$ after one kick.

Solution

The momentum probability is

$$P_m(1) = J_m\left( \frac{K}{\hbar_{\rm eff}} \right)^2.$$

Since $\hat P\lvert m\rangle=m\hbar_{\rm eff}\lvert m\rangle$,

$$\begin{aligned} \langle\hat P^2\rangle_1 &= \hbar_{\rm eff}^2 \sum_m m^2 J_m\left( \frac{K}{\hbar_{\rm eff}} \right)^2 \\ &= \hbar_{\rm eff}^2 \frac{1}{2} \left( \frac{K}{\hbar_{\rm eff}} \right)^2 \\ &= \frac{K^2}{2}. \end{aligned}$$

5. At the primary resonance $\hbar_{\rm eff}=4\pi$, show that an initial $\lvert0\rangle$ state has ballistic second-moment growth.

Solution

At $\hbar_{\rm eff}=4\pi$, the free phase is unity for every integer $m$, so

$$U_F=U_K.$$

After $n$ periods,

$$U_F^n = \exp\left( -\frac{inK\cos\theta}{\hbar_{\rm eff}} \right).$$

The momentum probabilities are therefore

$$P_m(n) = J_m\left( \frac{nK}{\hbar_{\rm eff}} \right)^2.$$

Using the same Bessel second-moment identity as in the previous exercise gives

$$\langle\hat P^2\rangle_n = \frac{n^2K^2}{2}.$$

Quadratic growth is ballistic and is incompatible with long-time dynamical localization.

6. Explain why saturation of $\langle\hat P^2\rangle_n$ does not by itself prove dynamical localization in a numerical calculation.

Solution

A finite momentum grid has edges or periodic wraparound. Once the wave packet reaches those boundaries, the second moment can stop growing or recur even when the infinite-system dynamics would continue to spread. A convincing localization test should enlarge the grid, confirm negligible edge population, resolve an exponential momentum tail over a stable range, preserve norm, and check that the result survives longer propagation. One should also verify that $\hbar_{\rm eff}$ is not accidentally near a resonance whose long transient is being truncated.