Phase noise in the delta kicked rotor: from quantum to classical

In the classical δ-kicked rotor model, a pendulum is subject to a pulsed gravitational field. The pulses are modelled as instantaneous Dirac-delta functions. At other times, the pendulum is free to rotate in an environment free of any external forces. The kicked rotor exhibits chaotic behaviour for sufficiently large kick-strengths [15]. The Hamiltonian for this system (in scaled, dimensionless units) is

$$\begin{eqnarray}&&H(t)=\frac{\bar{\ }k{{p}^{2}}}{2}+k\;{\rm cos} (x)\mathop{\sum }\limits_{i=0}^{N-1}\delta (t-i),\end{eqnarray} \tag{ 1 }$$

where $\bar{\ }k$ is the pulse period, p is the angular momentum of the pendulum, k is the kick-strength, N is the number of kicks and x is the angular position.

The classical kicked rotor was extended to the quantum realm and brought to experimental realization through atom optics [16]. This allowed access to a quantum chaotic system in a direct way. In the atom optics system, the external force is not gravity, but a pulse of an optical standing wave. The standing wave is created from the interference of two counterpropagating laser beams, detuned from the atomic resonance. This results in the atoms experiencing a sinusoidal phase grating.

Interesting dynamics emerge from this system. The 'chaos' inherent in the classical system is not strictly present in the quantum system owing to the linearity of the equations of quantum mechanics [17]. Instead, the chaos manifests as dynamical localization. This curious phenomenon has strong links with Anderson localization [18]. The effect is that, after a short period where apparent classical motion occurs (termed the 'quantum break time'), diffusion in momentum space is inhibited and the system reaches a steady state where no further energy is delivered to the system.

Another specifically quantum feature of the behaviour surrounds the appearance of a 'quantum resonance'. For certain kicking frequencies (${{\omega }_{k}}=4{{\omega }_{r}}$, where ω_r is the atomic recoil frequency) the free evolution is completely negated, and subsequent kicks continue to add in phase constructively. This leads to ballistic energy growth. The origin of this behaviour is similar to the Talbot effect in near-field diffraction optics, where after certain propagation distances, the diffraction grating is re-imaged. In the quantum resonance case, the 're-imaging' occurs after a certain propagation time, known as the 'Talbot time', T_tal.

Another important concept is that of the 'anti-resonance'. At half-integer multiples of the Talbot time, the effect of the free evolution is for each successive kick to add destructively. Thus, the energy simply oscillates in time.

Our system is an extension of the classic system of (1), with the addition of a sinusoidal phase modulation of the standing wave phase grating. Our system is now described by the time-dependent Hamiltonian (again in scaled, dimensionless units)

$$\begin{eqnarray}&&H(t)=\frac{\bar{\ }k{{p}^{2}}}{2}+k{\rm cos} [x+\alpha {\rm cos} ({{\omega }_{p}}t)]\mathop{\sum }\limits_{i=0}^{N-1}\delta (t-i),\end{eqnarray} \tag{ 2 }$$

with phase modulation frequency ω_p and phase modulation amplitude α. The momentum co-ordinate p now represents the linear momentum of the atoms, while x is the linear position co-ordinate. Definitions of these units in terms of experimental parameters can be found in section 3.1.1.

In the classical system of a rotating pendulum [17], such phase modulation would correspond to the direction of gravity rotating with frequency ω_p. In our system, we modulate the phase of one of two interfering laser beams, thereby modulating the position of the optical grating.

Many of the key effects of phase modulation on the kicked rotor dynamics can be understood with the aid of a simple picture, such as that in figure 1(a). Here, we consider kicking with pulse period ${{T}_{{\rm tal}}}/2$ (T_tal is the Talbot time of 66.3 $\mu {\rm s}$ for ⁸⁷Rb). The phase modulation frequency is set to half the kicking frequency: $f=1/{{T}_{{\rm tal}}}\approx 15$ kHz. As the duration of a kick is far shorter than the timescale over which the phase changes appreciably, the phase sequence effectively reduces to a Nyquist sampling problem. It is straightforward to see that, if the modulation amplitude is set to $\pi /2$, the phase will jump by π from kick to kick. This has the effect of completely negating the quantum anti-resonance, and effectively creating a quantum resonance condition where the energy grows ballistically with kick number.

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Other pertinent points can be gathered from similar pictures. For example, phase modulation at the kicking frequency of $\approx 30$ kHz is equivalent to zero phase modulation, as the phase is the same for each kick. Even higher kicking frequencies are simply aliased to within the 0–15 kHz range (e.g. 45 kHz is equivalent to 15 kHz). These stem from the fact that the Nyquist frequency for the system is 15 kHz. Another important aspect concerns the phase at which the kick sequence begins. If, for example, kicking occurs at 15 kHz and the phase goes as ${\rm sin} ({{\omega }_{p}}t)$ rather than ${\rm cos} ({{\omega }_{p}}t)$, then the phase will be constant for each kick. It is therefore important that the initial phase is controlled for the experiment.

We can develop a more quantitative understanding of the system by considering the spectral qualities of the kick sequence. A train of δ-functions in time has a Fourier spectrum of a comb of frequencies, spaced at the kicking frequency

$$\begin{eqnarray}&&\mathcal{F}\left[ \mathop{\sum }\limits_{n=-\infty }^{\infty }\delta (t-nT) \right]=\mathop{\sum }\limits_{m=-\infty }^{\infty }\delta (\omega -m{{\omega }_{k}}).\end{eqnarray} \tag{ 3 }$$

Now we consider the phase modulated problem. The phase shift can be built in to the kick sequence, so we consider the Fourier transform

$$\begin{eqnarray}&&{{F}_{{\rm mod} }}(\omega )=\mathcal{F}\left[ \mathop{\sum }\limits_{n=-\infty }^{\infty }\delta (t-nT){{{\rm e}}^{{\rm i}\alpha {\rm cos} {{\omega }_{p}}t}} \right]\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&=\mathcal{F}\left[ \mathop{\sum }\limits_{n=-\infty }^{\infty }\delta (t-nT) \right]\;*\;\mathcal{F}\left( {{{\rm e}}^{{\rm i}\alpha {\rm cos} {{\omega }_{p}}t}} \right).\end{eqnarray} \tag{ 5 }$$

The exponential may be expanded in terms of Bessel functions [19] to give the simple expression for the Fourier transform of

$$\begin{eqnarray}&&\mathcal{F}\left( {{{\rm e}}^{{\rm i}\alpha {\rm cos} {{\omega }_{p}}t}} \right)=\mathcal{F}\left[ {{J}_{0}}(\alpha )+2\mathop{\sum }\limits_{k=1}^{\infty }{{i}^{k}}{{J}_{k}}(\alpha ){\rm cos} (k{{\omega }_{p}}t) \right]\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&=\;2\pi \mathop{\sum }\limits_{k=-\infty }^{\infty }{{i}^{k}}{{J}_{k}}(\alpha )\delta (\omega -k{{\omega }_{p}}).\end{eqnarray} \tag{ 7 }$$

The full spectrum F_mod is then the convolution of (7) with a train of δ-functions, which simply results in a continual repeating sequence of the spectrum in (7).

Based on the simple picture introduced in section 2.3, we now make the heuristic assumption that the induced resonance is due only to the frequency component at ${{\omega }_{p}}={{\omega }_{k}}/2$ and higher frequencies which alias to it. The spectrum in (7) indicates that frequency components of integer multiples of ω_p are produced, with Bessel function weightings in the modulation strength α. We then simply look at the fraction of the power spectrum concentrated at ${{\omega }_{k}}/2$ to estimate the extent of the influence of the modulation on the rotor energy.

The Fourier analysis also permits a prediction of the frequencies at which a resonance can be induced. From (7), integer multiples of ω_p are produced. Odd multiples of ${{\omega }_{k}}/2$ are aliased to ${{\omega }_{k}}/2$. This gives the simple resonance condition

$$\begin{eqnarray}&&\omega =\frac{2n+1}{m}\frac{{{\omega }_{k}}}{2},\qquad n,m\in \mathbb{N}.\end{eqnarray} \tag{ 8 }$$

We now illustrate the applicability of the Fourier spectrum in predicting the energy as a function of modulation strength. For the first order resonance, the relevant harmonics are the $1{\rm st}$ (15 kHz), $3{\rm rd}$ (45 kHz), $5{\rm th}$ (75 kHz), ..., while for the second order resonance, the relevant harmonics are the $2{\rm nd}$, $6{\rm th}$, $10{\rm th}$, ... The relevant harmonics for an mth order resonance are clearly $m(2n+1),n\in \mathbb{Z}$. We look at the fraction of the power spectrum concentrated at these harmonics, to give a general rule for the energy of an $m{\rm th}$ order resonance as

$$\begin{eqnarray}&&{{E}_{m}}(\alpha )\propto |\mathop{\sum }\limits_{n=0}^{\infty }{{i}^{m(2n+1)}}{{J}_{m(2n+1)}}(\alpha ){{|}^{2}}.\end{eqnarray} \tag{ 9 }$$

In figure 2, we compare these calculations with quantum simulations (explained in section 3.2). The simulations have been conducted directly using the Floquet operator and are independent of the Fourier analysis. Both the 7.5 kHz and the 15 kHz resonances are plotted as a function of modulation strength α. The correspondence between the two is practically perfect, indicating the validity of the Fourier treatment.

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2.4.1. Incommensurate frequency

An interesting effect occurs if ω_p is chosen to be incommensurate with the kicking frequency. As indicated in figure 1(b), the phase sequence for such a choice of frequency is pseudo-random. It is an interesting question to ask what effect this 'phase noise' will have on dynamical localization.

The original, unmodulated kicked rotor is a very well known classically chaotic system. In the quantum limit, this chaos manifests as dynamical localization for pulse periods incommensurate with the quantum resonances (see, for example, [17]). This is due to the direct mapping of the Hamiltonian onto the Anderson model [20]. In a similar vein, when the kicked rotor potential is a function of N pairwise incommensurate frequencies, the system maps to an $N-1$ dimensional Anderson problem [21]. This has been experimentally verified by Delande and co-workers, with three incommensurate frequencies controlling amplitude modulation of the kick pulses [7].

The previous works suggest that, in our system, kicking on resonance with an incommensurate phase modulation frequency would result in a one-dimensional (1D) Anderson problem, with dynamical localization eventuating after a quantum break time. Kicking off resonance, with an addition of a further incommensurate ω_p, would result in a two dimensional (2D) Anderson problem, similar to that investigated by Doron and Fishman [22]. The localization length ξ in this case should grow exponentially with the 'mean free path', which is a function of the kick strength. As suggested in the scaling theory of localization, all eigenstates are localized in a 2D system, meaning localization should eventually result with large ξ [23].

Our analysis in the vicinity of the resonances makes use of -classical theory, developed in [24]. The method revolves around making a pseudo-classical approximation to the dynamics in the vicinity of the quantum resonance. We write the pulse period $\bar{\ }k$= 2π ℓ +, where $\ell$ is the resonance order. An important parameter is , which represents the detuning from quantum resonance. $|\epsilon |$ takes the role of the effective Planckʼs constant for our pseudo-classical analysis, even though $\bar{\ }k$ is large. By writing $J=\epsilon N+\pi \ell \;+\;$ $\bar{\ }k$β and $\vartheta =x+\pi [1-{\rm sgn} (\epsilon )]/2$ (where β is the quasimomentum and N is the integer momentum), we arrive at the pseudo-classical map

$$\begin{eqnarray}&&{{J}_{t+1}}={{J}_{t}}+k|\epsilon |{\rm sin} ({{\vartheta }_{t+1}}),\qquad {{\vartheta }_{t+1}}={{\vartheta }_{t}}+{{J}_{t+1}}.\end{eqnarray} \tag{ 10 }$$

The main importance in the map lies in the resultant effective stochasticity parameter, $k|\epsilon |$: a scaling law is obeyed for kick-strength k and resonance detuning $|\epsilon |$. The map in (10) will be used for the analysis of the resonance stability in section 4.2.

We begin with a ⁸⁷Rb BEC of approximately 20 000 atoms, prepared in an all-optical BEC machine [25]. The setup is similar to that of our previous phase modulated kicked rotor work [9].

3.1.1. Kick parameters

As indicated in figure 3, an optical standing wave is created via interference of two counter-propagating laser beams overlapping the atoms, detuned by $\Delta /(2\pi )=150$ GHz from the ⁸⁷Rb D2 line. The AC Stark shift means that the standing wave functions as a phase grating. Acousto-optic modulators (AOMs) provide individual control over each of the two beams. The AOMs are driven with a dual channel Tektronix arbitrary function generator (AFG3252), running at 80 MHz, and switched with Minicircuits rf switches (ZSWA-4-30DR). The AOMs present us with the ability to control the timing and relative phase of the two beams, giving us command over the phase grating.

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A single kick results from a $\tau =300$ ns pulse of the optical lattice. The short 'on' period, with respect to the atomic velocity and the standing wave period of $\lambda /2$, means that the diffraction takes place firmly in the Raman–Nath regime. This allows us to mathematically treat each kick as a δ-function. Discrete momentum transfers of two photon recoils ($2\hbar {{k}_{L}}=2{{p}_{r}}$) are allowed in this system, as absorption occurs in one beam and stimulated emission in the other.

For convenience, we utilize the following scaled units throughout

$$\begin{eqnarray}&&x=2{{k}_{L}}X,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&p=\frac{P}{2\hbar {{k}_{L}}},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&t={{t}^{\prime }}/T,\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&k=\frac{\tau {{\Omega }^{2}}}{4\Delta },\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\bar{\ }k=\frac{4\hbar k_{L}^{2}\;T}{m},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\epsilon =\bar{\ }k-2\pi \ell ,\qquad \ell \in \mathbb{Z},|\epsilon |\lt \pi ,\end{eqnarray} \tag{ 16 }$$

where X and P are real position and momentum operators; ${{k}_{L}}=2\pi /\lambda$ is the wavenumber of the kicking laser; $T=2\pi /{{\omega }_{k}}$ is the kick period; ${{t}^{\prime }}$ is real time; m is the atomic mass; τ is the duration of a single kicking pulse; Δ is the laser detuning from resonance; and Ω is the Rabi frequency.

The pulses are spaced at intervals near either the resonant Talbot time of 66.3 $\mu {\rm s}$, or the anti-resonant time of 33.1 $\mu {\rm s}$. The timing is controlled with a programmable pulse generator. As mentioned in section 2, we require the modulation to be cosinusoidal with respect to the first kick. We therefore need to synchronize the start of the kick sequence with the phase modulation of the function generator. This is accomplished through the use of the logic output of the function generator. The pulse generator trigger signal is delayed by a D flip-flop, until the flip-flop is triggered by the rising edge of the TTL signal. A further delay of one quarter of the phase modulation period is programmed in to the start of the pulse sequence to ensure that our phase modulation is ${\rm cos} ({{\omega }_{p}}t)$, rather than ${\rm sin} ({{\omega }_{p}}t)$.

3.1.2. Analysis

We conduct numerical simulations in order to corroborate our experimental data with theoretical results. The simulations are conducted in 1D using the split-step method, which is ideally suited to the δ-kicked rotor problem. The evolution can be directly described by the Floquet operator

$$\begin{eqnarray}&&F(k,{{\omega }_{p}},\bar{\ }k,\alpha ,t)={{F}_{{\rm free}}}{{F}_{{\rm kick}}}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&=\;{\rm exp} \left( -{\rm i}\frac{\bar{\ }k{{p}^{2}}}{2} \right){\rm exp} \left[ -{\rm i}k{\rm cos} (x+\alpha {\rm cos} {{\omega }_{p}}t) \right].\end{eqnarray} \tag{ 18 }$$

The Floquet operator F describes the evolution from immediately prior to one kick to immediately prior to the next. The Floquet operator is useful as it can be separated into two distinct parts: the kick operator F_kick, and the free evolution operator F_free. The kick operator is diagonal in position space; the free evolution is diagonal in momentum space. A full kick, including free evolution, can then be completed with two multiplications and a single Fourier transform from position to momentum space.

Except where stated, we choose our initial condition to be the solution to the Gross–Pitaevskii equation for the ground state of our trap, which has been subject to 300 $\mu {\rm s}$ of expansion under mean field repulsion. This results in a wavepacket with a full-width at half-maximum of 0.4 $\hbar {{k}_{L}}$ in momentum space.

As mentioned in section 2, the most dramatic effect of the phase modulation frequency on the dynamics relates to the transformation of an anti-resonance into a resonance, and vice-versa, when the phase modulation frequency (ω_p) is half of the kicking frequency (ω_k). In this section we examine the nature of the resonance.

In figure 4 we examine the behaviour in the vicinity of the 15 kHz resonance. Data is shown for 8, 14 and 22 kicks and is overlaid with simulations. The sinc-type resonance feature in the energy, indicated in the simulation, is well replicated by the data. The Fourier-based analysis in section 2 suggests a Fourier $1/N$ scaling of the resonance width, and we observe this in both simulations and data.

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For the energy measurement, the main discrepancy between the data and simulations occurs directly on resonance. The high energies on resonance result in a number of weakly populated high momentum states, which contribute significantly to the energy. With our absorption imaging system, these are inefficiently detected, leading to lower than expected energies. Small imperfections also arise from the time-of-flight imaging method, which does not produce a perfect momentum map for short expansion times.

An interesting feature of the data is the significant peak in the zero momentum at 15 kHz, shown in figure 4(b). We include all atoms in the zeroth diffraction order in this measurement. This peak is rather counter-intuitive, as it could be expected that alongside the energy growth on resonance, fewer atoms would remain in the zero momentum state. Indeed, this prediction holds for the unmodulated kicked rotor resonances, as shown in the inset. One possible reason is the generation of a small ratchet effect. For a small number of kicks, the phase sequence may appear periodic, thereby generating small transporting island structures in phase space [9, 10]. This would lead to directed momentum transport—hence a small peak in the zero momentum state.

In addition to the main resonance at 15 kHz, we observe narrow fractional resonances. In this system, fractional resonances occur due to the overlap of higher harmonics of ω_p with the resonant frequency ${{\omega }_{k}}/2$. The second order resonance at 22.5 kHz (equivalent to the resonance at 7.5 kHz), is shown in figure 5. An interesting aspect of this resonance, confirmed by simulations, is the lack of a peak in the zero momentum population. Experimental images of the momentum distribution are shown in figure 6.

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As previously mentioned, the initial phase ϕ₀ of the modulation $\alpha {\rm cos} ({{\omega }_{p}}t+{{\phi }_{0}})$ can have a dramatic effect. In figure 7, data is shown scanning over the initial phase. The strong dependence of the initial phase on the system meant that ϕ₀ needed to be controllable: this was achieved through a flip-flop implementation (indicated in figure 3).

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Finally, it is worth noting the dependence of the resonance on even or odd numbers of kicks. The natural tendency of the unmodulated $\ell =1$ anti-resonance is an oscillating energy, where each kick undoes the effect of the previous one. Odd numbers of kicks therefore produce greater energies than even numbers. For a kick sequence with a spectral component at ω_k, an odd number of kicks will then provide increased energy on resonance. This leads to sharper resonances for odd N, as indicated in figure 8.

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One nice advantage of using phase modulation is that it allows for replicable phase 'noise' sequences to be produced. This means that the data does not need to be ensemble averaged over a number of noisy (yet random) sequences, in order for meaningful correlations between parameters and the noise level to be extracted.

In figure 9(a), data is presented with increasing levels of phase noise on the main 66.3 $\mu {\rm s}$ quantum resonance. The modulation frequency of 6495 Hz produces the same phase sequence as shown in figure 1(b). A decay in the energy is indicated with increasing α. However, the decay rate is slow, and a substantial level of noise is required before the resonance is completely negated ($\alpha \approx \pi /3$). Similar results are obtained for different incommensurate frequencies, shown in the inset, indicating that the microscopic noise details are not significant here.

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The resonance stability indicated by the data and simulations is supported by the pseudo-classical Poincaré surfaces of section in figures 9(b)–(e). The central resonance island becomes increasingly blurred by the phase noise, yet maintains its identity with very large phase noise amplitudes. This indicates that the resonance is surprisingly robust to phase noise.

The robustness of the resonance is further supported by simulations. In figure 10, energy is plotted as a function of around the $\ell =2$ resonance ($\approx 66.3\;\mu {\rm s}$ pulse period). Phase modulation is conducted at 6495 Hz. The simulations support the pseudo-classical phase space analysis in figures 9(b)–(e), indicating the persistence of the resonance peak for large levels of phase noise.

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4.2.1. Relation to previous studies

It is quite clear that if $\epsilon =0$, super-diffusion will persist for all time in the presence of amplitude noise. The phase accumulated from free evolution remains a multiple of $2\pi$, meaning each kick adds constructively. With phase noise on $\epsilon =0$, the outcome is not immediately obvious. Here we investigate the effects of noise on resonance.

Our experimental results were inconclusive, with linear energy growth observed up to 25 kicks. Our experimental setup is not designed to measure large energies due to the relatively low number of atoms in our BEC, which results in a low signal-to-noise ratio for the energy-important high momentum states. To answer the phase modulation question, we therefore decided to employ numerical simulations.

Our simulations in figure 11 indicate approximate linear energy growth for all levels of phase noise¹ . At first glance, this may appear surprising. Following the logic of Casati et al [21], the presence of two incommensurate frequencies in the system may suggest a 1D Anderson system. If this was the case, the energy would stabilize after a 'quantum break time', which is not observed. A more careful analysis reveals the reason for this behaviour.

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When $\ell =2$ and $\epsilon =0$, each momentum order accumulates a phase between kicks which is a multiple of $2\pi$: all momentum orders rephase. This means that the free evolution can, in a sense, be disregarded and the net effect is that of one 'long' kick. To find the effective phase grating, we require a sum of the individual gratings

$$\begin{eqnarray}&&{{V}_{{\rm eff}}}(x,N)=\mathop{\sum }\limits_{n=0}^{N-1}{{V}_{n}}(x)=\mathop{\sum }\limits_{n=0}^{N-1}k{\rm sin} [x+\alpha {\rm cos} (n{{\omega }_{p}}T)],\end{eqnarray} \tag{ 19 }$$

where T is the pulse period and N the number of kicks. With the individual gratings adding out of phase, the resultant amplitude is less than $Nk$. Performing this summation yields the same form of energy as a function of α as in figure 11(b). The 'steps' in the time evolution seen in figure 11(a) are also found. Fundamentally, in this case, the resonance is not broken by phase noise, but is rather diminished due to the effective noise-induced reduction of kick-strength.

The next question we can ask is the effect that being away from quantum resonance will have ($\epsilon \gt 0$). We set $\bar{\ }k$, π and ω_p to all be incommensurate. The value of $k|\epsilon |=1.2$ is chosen to be larger than 0.97, such that the classical phase space is fully chaotic. Previous analyses suggest that a 2D Anderson system would result from this system [22, 28].

In figure 12(a) we plot simulations of energy as a function of time for different modulation strengths. In the case of zero modulation, the energy simply oscillates in time with no net growth: this is simple dynamical localization. As the noise level α is increased, the energy begins to grow with power-law behaviour, $E\propto {{t}^{q}}$. The exponent q grows with α and tends to unity.

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The noise-induced destruction of dynamical localization is well illustrated in figures 12(b) and (c). The plots show the momentum space wavefunction after 70 kicks, (b) without and (c) with phase noise. The exponential localization in the absence of phase noise is clear in (b), with strong linearity in the semi-logarithmic plot. The nature of the wavefunction then changes markedly for $\alpha =\pi /6$, broadening the distribution and not giving any indication of exponential localization.

Dynamical localization relies on strict quantum correlations between momentum states. Small levels of phase noise disrupt these correlations, weakening the localization [29]. Pure diffusion results when this coherence has been destroyed, resulting in classical-like behaviour.

The contrast between the robustness of the resonance and the sensitivity of the localization highlights the fact that the transition from quantum to classical behaviour is not trivial.

It is something of an open question as to whether it is meaningful to associate this with a 2D Anderson system—the localization length would be extremely large (note clear diffusion for up to 300 kicks), and well beyond what could be experimentally observed. Physically, the relevant notion here relates to the inhibition of interference due to noise.

Based on the comparison between figures 9–10 and 12, it is clear that dynamical localization is far more sensitive to noise effects than the quantum resonance.