Stroboscopic dynamics The dynamics of the quantum kicked rotator (with wave function | \psi(t) \rangle ) is governed by the time dependent
Schrödinger equation : i\hbar\frac{\partial }{\partial t}| \psi(t) \rangle=\left[\frac{\hat{p}^2}{2I} + K \cos \hat{\theta} \sum_{n=-\infty}^\infty \delta \left(\frac{t}{T}-n \right)\right]| \psi(t) \rangle with [ \hat{\theta},\hat{p}]=i\hbar (or equivalently \langle \theta | \hat{p} | \psi \rangle= i\hbar \frac{\partial \psi}{\partial \theta} ). As for classical dynamics, a stroboscopic point of view can be adopted by introducing the time propagator over a kicking period \hat{U} (that is the
Floquet operator) so that |\psi(t+T) \rangle = \hat{U} |\psi(t) \rangle. After a careful integration of the time-dependent Schrödinger equation, one finds that \hat{U} can be written as the product of two operators\hat{U}=\exp\left[-i\frac{\hat{p}^2 T}{2I \hbar}\right] \exp\left[-i\frac{KT}{\hbar} \cos\hat{\theta}\right]We recover the classical interpretation: the dynamics of the quantum kicked rotor between two kicks is the succession of a free propagation during a time T followed by a short kick. This simple expression of the Floquet operator \hat{U} (a product of two operators, one diagonal in momentum basis, the other one diagonal in angular position basis) allows to easily numerically solve the evolution of a given wave function using
split-step method. Because of the periodic boundary conditions at \theta=\pm \pi , any wave function | \psi \rangle can be expanded in a discrete momentum basis |l \rangle (with p=l \hbar , l integer) see
Bloch theorem), so that : \langle \theta | \psi \rangle =\sum_{l=-\infty}^{\infty} \langle l | \psi \rangle \mathrm{e}^{i l \theta} \Leftrightarrow \langle l | \psi \rangle = \int_{-\pi}^{\pi} \frac{\mathrm{d} x}{2\pi} \langle \theta | \psi \rangle \mathrm{e}^{-i l \theta} Using this relation with the above expression of \hat{U} , we find the recursion relation \langle l| \psi(t+T) \rangle = \exp\left(-i\frac{l^2 \hbar T}{2 I}\right) \sum_{m=-\infty}^\infty (-i)^{m-l} J_{m-l} \left(\frac{KT}{\hbar} \right) \langle m| \psi(t) \rangle where \textstyle {J}_n is a
Bessel function of first kind. {{hidden|Demonstration| Indeed, we have \langle l| \psi(t+T) \rangle = \langle l | \hat{U} | \psi(t) \rangle = \exp\left(-i\frac{l^2 \hbar T}{2 I}\right) \langle l | \exp\left(-i\frac{K T \cos \hat{\theta}}{\hbar}\right) | \psi(t) \rangle \langle l| \psi(t+T) \rangle =\exp\left(-i\frac{l^2 \hbar T}{2 I}\right) \int_{-\pi}^{\pi} \frac{\mathrm{d} \theta}{2\pi} \mathrm{e}^{-i l \theta} \exp\left(-i\frac{K T \cos \hat{\theta}}{\hbar}\right) \langle \theta | \psi(t) \rangle \langle l| \psi(t+T) \rangle = \exp\left(-i\frac{l^2 \hbar T}{2 I}\right) \int_{-\pi}^{\pi} \frac{\mathrm{d}\theta}{2\pi} \mathrm{e}^{-i l \theta} \sum_{n=-\infty}^\infty (-i)^n J_n \left(\frac{KT}{\hbar} \right) \mathrm{e}^{-i n \theta} \sum_{m=-\infty}^\infty \mathrm{e}^{i m \theta} \langle m | \psi(t) \rangle \langle l| \psi(t+T) \rangle = \exp\left(-i\frac{l^2 \hbar T}{2 I}\right) \sum_{n,m=-\infty}^\infty (-i)^n J_n \left(\frac{KT}{\hbar} \right) \left[ \int_{-\pi}^{\pi} \frac{\mathrm{d}\theta}{2\pi} \mathrm{e}^{i (m-l-n) \theta} \right] \langle m| \psi(t) \rangle \langle l| \psi(t+T) \rangle = \exp\left(-i\frac{l^2 \hbar T}{2 I}\right) \sum_{n,m=-\infty}^\infty (-i)^n J_n \left(\frac{KT}{\hbar} \right) \frac{\sin([m-l-n])\pi}{(m-l-n)\pi} \langle m| \psi(t) \rangle So that we recover the result, keeping only non vanishing terms m-l-n=0 in the double sum.}}
Dynamical localization It has been discovered that the classical diffusion is suppressed in the quantum kicked rotator. It was later understood that this is a manifestation of a quantum
dynamical localization effect that parallels
Anderson localization. There is a general argument that leads to the following estimate for the breaktime of the diffusive behavior : t^* \ \approx \ D_{cl}/\hbar^2 , where D_{cl} is the classical diffusion coefficient. The associated localization scale in momentum is therefore \textstyle \sqrt{D_{cl} t^*}.
Link with Anderson tight-binding model The quantum kicked rotor can actually formally be related to the Anderson tight-binding model a celebrated Hamiltonian that describes electrons in a disordered lattice with lattice site state |n \rangle, where Anderson localization takes place (in one dimension)\hat{H} = \sum_{n} \varepsilon_n |n \rangle \langle n| + \sum_{n\neq m} t_{n-m} | n \rangle \langle m |where the \varepsilon_n are random on-site energies, and the t_{n-m} are the hopping amplitudes between sites n and m. In the quantum kicked rotator it can be shown, that the plane wave | p \rangle with quantized momentum p = n \hbar play the role of the lattice sites states. The full mapping to the Anderson tight-binding model goes as follow (for a given eigenstates of the Floquet operator, with quasi-energy \omega) t_n = - \int_{-\pi}^{\pi} \frac{\mathrm{d}x}{2\pi} \tan[K \cos(x)/2] \mathrm{e}^{-i x n} \quad \text{and} \quad \varepsilon_n = \tan(\omega/2 - n^2/4) . Dynamical localization in the quantum kicked rotator then actually takes place in the momentum basis.
Effect of noise and dissipation If noise is added to the system, the dynamical localization is destroyed, and diffusion is induced. This is somewhat similar to hopping conductance. The proper analysis requires to figure out how the dynamical correlations that are responsible for the localization effect are diminished. Recall that the diffusion coefficient is D_{cl}\approx K^2/2, because the change (p(t)-p(0)) in the momentum is the sum of quasi-random kicks K\sin(x(n)). An exact expression for D_{cl} is obtained by calculating the "area" of the correlation function C(n) = \langle \sin(x(n))\sin(x(0)) \rangle , namely the sum D = K^2\sum C(n). Note that C(0)=1/2. The same calculation recipe holds also in the quantum mechanical case, and also if noise is added. In the quantum case, without the noise, the
area under C(n) is zero (due to long negative tails), while with the noise a practical approximation is C(n)\mapsto C(n) e^{-t/t_c} where the coherence time t_c is inversely proportional to the intensity of the noise. Consequently, the noise induced diffusion coefficient is : D \approx D_{cl}t^* / t_c \quad [\text{assuming }t_c \gg t^*] Also the problem of quantum kicked rotator with dissipation (due to coupling to a thermal bath) has been considered. There is an issue here how to introduce an interaction that respects the angle periodicity of the position x coordinate, and is still spatially homogeneous. In the first works a quantum-optic type interaction has been assumed that involves a momentum dependent coupling. Later a way to formulate a purely position dependent coupling, as in the Caldeira-Leggett model, has been figured out, which can be regarded as the earlier version of the
DLD model.
Experimental realization with cold atoms The first experimental realizations of the quantum kicked rotator have been achieved by
Mark G. Raizen group in 1995, later followed by the Auckland group, and have encouraged a renewed interest in the theoretical analysis. In this kind of experiment, a sample of cold atoms provided by a
magneto-optical trap interacts with a pulsed standing wave of light. The light being detuned with respect to the atomic transitions, atoms undergo a space-periodic
conservative force. Hence, the angular dependence is replaced by a dependence on position in the experimental approach. Sub-milliKelvin cooling is necessary to obtain quantum effects: because of the
Heisenberg uncertainty principle, the de Broglie wavelength, i.e. the atomic wavelength, can become comparable to the light wavelength. For further information, see. Thanks to this technique, several phenomena have been investigated, including the noticeable: • quantum Ratchets; • the Anderson transition in 3D. == See also ==