# Theory of dissipative chaotic atomic transport in an optical lattice

V.Yu. Argonov and S.V. Prants  Laboratory of Nonlinear Dynamical Systems,
Pacific Oceanological Institute of the Russian Academy of Sciences,
43 Baltiiskaya st., 690041 Vladivostok, Russia

###### Abstract

We study dissipative transport of spontaneously emitting atoms in a 1D
standing-wave laser field in the regimes where the underlying deterministic
Hamiltonian dynamics is regular and chaotic. A Monte Carlo stochastic
wavefunction method is applied to simulate semiclassically the atomic dynamics
with coupled internal and translational degrees of freedom. It is shown in
numerical experiments and confirmed analytically that chaotic atomic transport
can take the form either of ballistic motion or a random walking with specific
statistical properties. The character of spatial and momentum diffusion in the
ballistic atomic transport is shown to change abruptly in the atom-laser
detuning regime where the Hamiltonian dynamics is irregular in the sense of
dynamical chaos. We find a clear correlation between the behavior of the
momentum diffusion coefficient and Hamiltonian chaos probability which is a
manifestation of chaoticity of the fundamental atom-light interaction in the
diffusive-like dissipative atomic transport. We propose to measure a linear
extent of atomic clouds in a 1D optical lattice and predict that, beginning
with those values of the mean cloud’s momentum for which the probability of
Hamiltonian chaos is close to 1, the linear extent of the corresponding clouds
should increase sharply. A sensitive dependence of statistical characteristics
of dissipative transport on the values of the detuning allows to manipulate
the atomic transport by changing the laser frequency.

###### pacs:

37.10.Vz, 05.45.Mt, 05.45.-a

##  I Introduction

An atom placed in a laser standing wave is acted upon by two radiation forces,
deterministic dipole and stochastic dissipative ones Minogin ; Kaz ; Meystre .
The mechanical action of light upon neutral atoms is at the heart of laser
cooling, trapping, and Bose-Einstein condensation. Numerous applications of
the mechanical action of light include isotope separation, atomic litography
and epitaxy, atomic-beam deflection and splitting, manipulating translational
and internal atomic states, measurement of atomic positions, and many others.
Atoms and ions in an optical lattice, formed by a laser standing wave, are
perspective objects for implementation of quantum information processing and
quantum computing. Advances in cooling and trapping of atoms, tailoring
optical potentials of a desired form and dimension (including one-dimensional
optical lattices), controlling the level of dissipation and noise are now
enabling the direct experiments with single atoms to study fundamental
principles of quantum physics, quantum chaos, decoherence, and quantum-
classical correspondence (for recent reviews on cold atoms in optical lattices
see Ref. GR01 ; MO06 ).

Experimental study of quantum chaos has been carried out with ultracold atoms
interacting in $\delta$-kicked optical lattices MR94 ; RB95 ; DM95 ; Amm98 ;
Rin00 ; From00 ; Hens03 ; JS04 . To suppress spontaneous emission (SE) and
provide a coherent quantum dynamics atoms in those experiments were detuned
far from the optical resonance. Adiabatic elimination of the excited state
amplitude leads to an effective Hamiltonian for the center-of-mass (CM) motion
GSZ92 , whose 3/2 degree-of-freedom classical analogue has a mixed phase space
with regular islands embedded in a chaotic sea. De Brogile waves of
$\delta$-kicked ultracold atoms have been shown to demonstrate under
appropriate conditions the effect of dynamical localization in momentum
distributions which means the quantum suppression of chaotic diffusion MR94 ;
RB95 ; DM95 ; Amm98 ; Rin00 ; From00 ; Hens03 ; JS04 ; GSZ92 . Decoherence due
to SE or noise tends to suppress this quantum effect and restore classical-
like dynamics KO98 ; Ball99 ; SM00 ; Arcy01 . Another important quantum
chaotic phenomenon with cold atoms in far detuned optical lattices is a chaos
assisted tunneling. In experiments Steck01 ; HH01 ; Steck02  ultracold atoms
have been demonstrated to oscillate coherently between two regular regions in
mixed phase space even though the classical transport between these regions is
forbidden by a constant of motion (other than energy).

The transport of cold atoms in optical lattices has been observed to take the
form of ballistic motion, oscillations in wells of the optical potential,
Brownian motion Chu85 , anomalous diffusion and Lévy flights BB02 ; BB94 ;
JD96 ; KS97 ; ME96 . The Lévy flights have been found in the context of
subrecoil laser cooling BB02 ; BB94  in the distributions of escape times for
ultracold atoms trapped in the potential wells with momentum states close to
the dark state. In those experiments the variance and the mean time for atoms
to leave the trap have been shown to be infinite.

A new arena of quantum nonlinear dynamics with atoms in optical lattices is
opened if we work near the optical resonance and take the dynamics of internal
atomic states into account. A single atom in a standing-wave laser field may
be treated as a nonlinear dynamical system with coupled internal
(electronical) and external (mechanical) degrees of freedom PRA01 ; JETPL01 ;
JETPL02 . In the semiclassical and Hamiltonian limits (when one treats atoms
as point-like particles and neglects SE and other losses of energy), a number
of nonlinear dynamical effects have been analytically and numerically
demonstrated with this system: chaotic Rabi oscillations PRA01 ; JETPL01 ;
JETPL02 , Hamiltonian chaotic atomic transport and dynamical fractals JETP03 ;
PLA03 ; pra07 , Lévy flights and anomalous diffusion PRA02 ; JETPL02 ; JRLR06
. These effects are caused by local instability of the CM motion in a laser
field. A set of atomic trajectories under certain conditions becomes
exponentially sensitive to small variations in initial quantum internal and
classical external states or/and in the control parameters, mainly, the atom-
laser detuning. Hamiltonian evolution is a smooth process that is well
described in a semiclassical approximation by the coupled Hamilton-Schrödinger
equations. A detailed theory of Hamiltonian chaotic transport of atoms in a
laser standing wave has been developed in our recent paper pra07 .

The aim of the present paper is to study dissipative chaotic transport of
atoms in a one-dimensional optical lattice in the presence of SE events which
interrupt coherent Hamiltonian evolution at random time instants. Generally
speaking, deterministic (dynamical) chaos is practically indistinguishable in
some manifestations from a random (stochastic) process. The problem becomes
much more complicated when noise acts on a dynamical system which is chaotic
in the absence of noise. Such systems are of a great practical interest.
Comparatively weak noise may be treated as a small perturbation to
deterministic equations of motion, and one can study in which way the noise
modifies deterministic evolution on different time scales. However, SE is a
specific shot quantum noise that cannot be treated as a weak one because the
internal state may change significantly after the emission of a spontaneous
photon. Special methods are needed to describe correctly the dynamics of a
spontaneously emitting single atom in an optical lattice. The purpose of this
paper is twofold. Our first goal is to give a description of possible regimes
of dissipative atomic transport in the presence of SE and to quantify their
statistical properties. Our secondary intent is a search for manifestations of
the fundamental dynamical instability and Hamiltonian atomic chaos in the
diffusive-like CM motion of spontaneously emitting atoms in a laser standing
wave which can be observed in real experiments.

The paper is organized as follows. In Sec. II we formulate a Monte Carlo
stochastic wavefunction approach to solving semiclassical Hamilton-Schrödinger
equations of motion for a two-level atom in a one-dimensional monochromatic
standing light wave. This approach allows to get the most probabilistic
outcome that can be compared directly with corresponding experimental output
with single atoms. In Sec. III we review briefly our previous results on
Hamiltonian chaotic CM motion which are necessary to quantify and interpret
dissipative dynamics. Sec. IV is devoted to description of possible regimes of
dissipative CM motion of spontaneously emitting atoms in a standing wave.
Monte Carlo simulations of the well-known effects of acceleration,
deceleration, and velocity grouping, and of a novel effect of dissipative
chaotic walking of atoms are presented in this section. Anomalous statistical
properties of dissipative chaotic walking are quantified and discussed in Sec.
V. Whereas Secs. IV and V are devoted to solving the first task of this paper,
in Sec. VI we consider the problem of manifestations of dynamical instability
and Hamiltonian chaos in dissipative atomic transport. We demonstrate
analytically and numerically that character of diffusion of spontaneously
emitting atoms changes qualitatively in the detuning regime where the
underlying Hamiltonian dynamics is chaotic. To observe this effect in a real
experiment with cold atoms in a one-dimensional optical lattice we propose to
measure the linear extent of atomic clouds with different values of their mean
momentum and predict that the extent should increase significantly with those
values of the mean momentum for which the underlying Hamiltonian evolution is
chaotic.

##  II Monte Carlo wavefunction modeling of the atomic dynamics

In the frame rotating with the laser frequency $\omega_{f}$, the standard
Hamiltonian of a two-level atom in a strong standing-wave 1D laser field has
the form

$\displaystyle\hat{H}=$
$\displaystyle\frac{\hat{P}^{2}}{2m_{a}}+\frac{1}{2}\hbar(\omega_{a}-\omega_{f})\hat{\sigma}_{z}-$
(1)
$\displaystyle-\hbar\Omega\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)\cos{k_{f}\hat{X}}-i\hbar\frac{\Gamma}{2}\hat{\sigma}_{+}\hat{\sigma}_{-},$

where $\hat{\sigma}_{\pm,z}$ are the Pauli operators for the internal atomic
degrees of freedom, $\hat{X}$ and $\hat{P}$ are the atomic position and
momentum operators, $\omega_{a}$, $\omega_{f}$, and $\Omega$ are the atomic
transition, laser, and Rabi frequencies, respectively, and $\Gamma$ is the
spontaneous decay rate. Internal atomic states are described by the
wavefunction ${|\Psi(t)\closeket}=a(t){|2\closeket}+b(t){|1\closeket}$, with
$a$ and $b$ being the complex-valued probability amplitudes to find an atom in
the excited ${|2\closeket}$ and ground ${|1\closeket}$ states. Note that the
norm of the wavefunction, $|a|^{2}+|b|^{2}$, is not conserved due to non-
Hermitean term in the Hamiltonian.

We use the standard Monte Carlo wavefunction technique Carmichael  to simulate
the atomic dynamics with the coupled internal and external degrees of freedom
in an optical lattice. The evolution of an atomic state ${|\Psi(t)\closeket}$
consists of two parts: (i) jumps to the ground state ($a=0$, $|b|^{2}=1$) each
of which is accompanied by the emission of an observable photon at random time
moments with the mean time $2/\Gamma$ (actually, the probability of SE depends
on the atomic population inversion) and (ii) coherent evolution with
continuously decaying norm of the atomic state vector without the emission of
an observable photon. The decaying norm of the state vector is equal to the
probability of spontaneous emission of the next photon. It is convenient to
introduce the new real-valued variables (normalized all the time) instead of
the amplitudes $a$ and $b$ (renormalized after SE events only)

$\displaystyle
u\equiv\frac{2\operatorname{Re}\left(ab^{*}\right)}{\left|a\right|^{2}+\left|b\right|^{2}},\quad
v\equiv\frac{-2\operatorname{Im}\left(ab^{*}\right)}{\left|a\right|^{2}+\left|b\right|^{2}},\quad
z\equiv\frac{\left|a\right|^{2}-\left|b\right|^{2}}{\left|a\right|^{2}+\left|b\right|^{2}},$
(2)

which have the meaning of synphase and quadrature components of the atomic
electric dipole moment and the population inversion, respectively. We stress
that the length of the Bloch vector, $u^{2}+v^{2}+z^{2}=1$, is conserved.

Since we study manifestation of quantum nonlinear effects in ballistic
transport of atoms, when the average atomic momentum is very large as compared
with the photon momentum $\hbar k_{f}$, the translational motion is described
classically by Hamilton equations. The whole atomic dynamics is governed by
the following Hamilton-Schrödinger equations Acta06 ; epl08

$\displaystyle\dot{x}$ $\displaystyle=\omega_{r}p,\quad\dot{p}=-u\sin
x+\sum\limits_{j=1}^{\infty}\rho_{j}\delta(\tau-\tau_{j}),$ (3)
$\displaystyle\dot{u}$ $\displaystyle=\Delta
v+\frac{\gamma}{2}uz-u\sum\limits_{j=1}^{\infty}\delta(\tau-\tau_{j}),$
$\displaystyle\dot{v}$ $\displaystyle=-\Delta u+2z\cos
x+\frac{\gamma}{2}vz-v\sum\limits_{j=1}^{\infty}\delta(\tau-\tau_{j}),$
$\displaystyle\dot{z}$ $\displaystyle=-2v\cos
x-\frac{\gamma}{2}(u^{2}+v^{2})-(z+1)\sum\limits_{j=1}^{\infty}\delta(\tau-\tau_{j}),$

where $x\equiv k_{f}{\lang\hat{X}\rang}$ and $p\equiv{\lang\hat{P}\rang}/\hbar
k_{f}$ are normalized atomic CM position and momentum. The dot denotes
differentiation with respect to the normalized time $\tau\equiv\Omega t$.
Throughout the paper we fix the values of the normalized decay rate
$\gamma\equiv\Gamma/\Omega$ and the recoil frequency $\omega_{r}\equiv\hbar
k_{f}^{2}/m_{a}\Omega$ to be $\gamma=3.3\cdot 10^{-3}$ and
$\omega_{r}=10^{-5}$. This values are similar to those used in experiments
with Na MR94 ; RB95 , Cs Amm98 ; Hood00  and Rb Hens03  cold atoms in a
standing-wave laser field with the maximal Rabi frequency of the order of
$1\div 5$ GHz. So, the normalized detuning between the field and atomic
frequencies, $\Delta\equiv(\omega_{f}-\omega_{a})/\Omega$, is a single
variable parameter. Also we fix the initial conditions as follows:
$x_{0}=v_{0}=u_{0}=0,z_{0}=-1$, and vary the initial momentum $p_{0}$ only. In
Eqs. (3) $\tau_{j}$ are random time moments of SE events and $\rho_{j}$ are
random recoil momenta with the values between $\pm 1$ (1D case). In terms of
the normalized time $\tau$ the rate of occurrence of SE events is
$\gamma(z+1)/2$. At $\tau=\tau_{j}$, the atomic variables change as follows:

$\tau=\tau_{j}\Rightarrow u\to 0,\ v\to 0,\ z\to-1,\ p\to p+\rho_{j},\ -1\leq
p_{j}\leq 1.$ (4)

##  III A brief review of Hamiltonian atomic dynamics

In this section we review briefly our main results on Hamiltonian atomic
dynamics (see Refs. PRA01 ; JETPL01 ; JETPL02 ; JETP03 ; JRLR06 ; pra07 )
which will be used in the next sections. In the absence of any losses
($\gamma=0$) the total atomic energy is conserved,

$H\equiv\frac{\omega_{r}p^{2}}{2}+U,\quad U\equiv-u\cos x-\frac{\Delta}{2}z.$
(5)

The corresponding lossless equations of motion with two independent integrals
of motion, the energy $H$ and the length of the Bloch vector, have been shown
JETPL01 ; PRA01  to be chaotic in the sense of an exponential sensitivity to
small variations in initial conditions and/or the control parameters. The CM
motion is governed by the simple equation for a nonlinear physical pendulum
with the frequency modulation JETPLP02

$\ddot{x}+\omega_{r}u(\tau)\sin x=0,$ (6)

where the synchronized component of the atomic dipole $u$ is a function of all
the other atomic variables including the translational ones. Besides the
regular CM motion, namely, oscillations in a well of the optical potential and
a ballistic motion over its hills, we have found and quantified chaotic CM
motion JETPL01 ; PRA01 ; JETPLP02 . On the exact atom-laser resonance with
$\Delta=0$, $u$ is a constant, and the CM performs either regular
oscillations, if $H<|u|$, or moves ballistically, if $H>|u|$.

At $\Delta\neq 0$, the depth of the potential wells changes in course of time,
and atoms may wander in a rigid optical lattice (without any modulations of
its parameters) in a chaotic way with alternating trappings in the wells and
flights of different lengths and directions over the hills. At small
detunings, $|\Delta|\ll 1$, the second term of the potential energy $U$ in Eq.
(5) is small, and $U$ can be approximated by a function of only one internal
variable $u$. In this case we have approximate solutions for $v$ and $z$

$\displaystyle v(\tau)=\pm\sqrt{1-u^{2}}\
\cos\left(2\int\limits_{0}^{\tau}\cos xd\tau^{\prime}+\chi_{0}\right),$ (7)
$\displaystyle z(\tau)=\mp\sqrt{1-u^{2}}\
\sin\left(2\int\limits_{0}^{\tau}\cos xd\tau^{\prime}+\chi_{0}\right),$

where $\chi_{0}$ is an integration constant which is a function of initial
values of $z$ and $u$. Using these solutions one can prove that at
$|\Delta|\ll 1$, $u$ performs shallow oscillations when the atom moves between
the nodes (recall that $u=const$ at $\Delta=0$). These oscillations are
synchronized with the oscillations of $z$, and when an atom approaches any
node with $\cos x=0$, where the strength of the laser field changes the sign,
they slow down (see Eq. (7)). The swing of oscillations of $u$ gradually
increases, and exactly at the node $u$ changes abruptly its value (see Fig.
1). Thus, $u$ is practically a constant between the nodes and it performs a
sudden jump at every node.

In the Raman-Nath approximation, where $x=\omega_{r}p\tau$ and $p=const$, we
have managed to derive the deterministic mapping allowing to compute the value
$u_{m}$ just after crossing the $m$th node

$\displaystyle
u_{m}\simeq\sin\left\\{\frac{\Delta}{\sqrt{1-u^{2}_{m-1}}}\left[\sqrt{\frac{\pi}{\omega_{r}p}}\left(v_{0}\cos\left(\frac{2}{\omega_{r}p}-\frac{\pi}{4}\right)+\right.\right.\right.$
(8)
$\displaystyle\left.\left.\left.+(-1)^{m}z_{0}\sin\left(\frac{2}{\omega_{r}p}-\frac{\pi}{4}\right)\right)+(-1)^{m}z_{0}\right]+\arcsin
u_{m-1}\right\\},$

where $v_{0}$ and $z_{0}$ are the values of $v$ and $z$ at the antinodes of
the standing wave at $x=\pi k$, $k=0,1,2,...$. They are the same at all the
antinodes because in the Raman-Nath approximation $v$ and $z$ are periodic
functions of $x$ (see solution (7)). Formula (8) describes the series of jumps
of two alternating magnitudes (for odd and even $m$). Strictly speaking, (8)
is valid with fast ballistic atoms and not on a very long time scale.
Deviation of the analytic calculations with Eq. (8) from the exact numerical
results is demonstrated in Fig. 1a where we plot the function $u(\tau)$ for a
fast atom with $p_{0}=1900$. It is obvious that the signal is rather regular
but the magnitude of the jumps changes slowly because the Bloch components $v$
and $z$ are not strictly periodic functions of time.

Figure 1b plots $u(\tau)$ in the regime of Hamiltonian chaotic walking. To
quantify chaotic jumps of $u$ we proposed in Ref. pra07  the stochastic map

Figure 1: Time evolution ($\tau$ is in units of $\Omega^{-1}$) of the synphase
component of the electric dipole moment $u$ and the atomic energy $H$ (in
units of $\hbar\Omega$). (a) Regular Hamiltonian dynamics ($p_{0}=1900$,
$\gamma=0$), (b) chaotic Hamiltonian dynamics ($p_{0}=700$, $\gamma=0$), (c)
regular dissipative dynamics ($p_{0}=1900$, $\gamma=0.0033$), (d) chaotic
dissipative dynamics ($p_{0}=700$, $\gamma=0.0033$). In all the panels,
$\Delta=-0.0005$. The initial part of (a) agrees with approximate solution (8)
with $v_{0}=0$, $z_{0}=-1$.

$\displaystyle
u_{m}\simeq\sin\left(|\Delta|\sqrt{\frac{\pi}{\omega_{r}p_{\rm{node}}}}\sin\phi_{m}+\arcsin
u_{m-1}\right),$ (9)

which was derived from the deterministic map (8) by introducing random phases
$\phi_{m}$ $(0\leq\phi\leq 2\pi)$ instead of arguments of trigonometric
functions which may differ significantly from node to node due to strong
variations in the atomic momentum beyond the Raman-Nath approximation. Note
that the value of the momentum at the instant when the atom crosses a node,
$p_{\rm node}=\sqrt{2H/\omega_{r}}$, is approximately the same for all nodes.

The map (9) describes a random Markov process in the $u$ space with $u_{m}$
varying in the range $-1\leq u_{m}\leq 1$. This quantity may be smaller or
larger than the atomic energy $H$ (which is a constant in the Hamiltonian
limit). Since the values of $u$ define the atomic potential energy, its random
walking governs a random walking of atoms in the lattice. The possible regimes
of the Hamiltonian CM motion can be summarized as follows pra07 : At $|u|>H$,
an atom oscillates in one of the potential wells, at $|u|<H$, it moves
ballistically. It can walk chaotically if $0<H<1$. In the process of
Hamiltonian chaotic walking the atom wanders in an optical lattice with
alternating trappings in wells of the optical potential and flights over its
hills changing the direction of motion many times. “A flight” is an event when
the atom passes, at least, three nodes. CM oscillations in a well of the
optical potential is called “a trapping”. The number of node crossings $l$
during a single flight or a single trapping event is a discrete measure of the
length and durations of those events. We have derived in Ref. pra07  the
following formulas for the probability density functions (PDFs) for the flight
and trapping events in the diffusive approximation:

$\displaystyle P_{fl}(l)\simeq\frac{Q(D_{u})}{\arcsin^{3}H}$
$\displaystyle\sum\limits_{j=0}^{\infty}(j+1/2)^{2}*$ (10)
$\displaystyle*\exp\frac{-(j+1/2)^{2}\pi^{2}D_{u}l}{\arcsin^{2}H},$
$\displaystyle P_{tr}(l)\simeq\frac{Q(D_{u})}{\arccos^{3}H}$
$\displaystyle\sum\limits_{j=0}^{\infty}(j+1/2)^{2}*$
$\displaystyle*\exp\frac{-(j+1/2)^{2}\pi^{2}D_{u}l}{\arccos^{2}H}.$

Here $Q$ is a constant, $D_{u}=\Delta^{2}\pi/4\omega_{r}p_{\rm{node}}$ is a
diffusion coefficient in the $u$ space. For comparatively small values of $l$
(i. e., with short flights and trappings), we get from Eq. (10) the power
decay

$P_{fl}\propto P_{tr}\propto l^{-1.5},$ (11)

whereas for large $l$ the decay is exponential. Numerical simulation of the
Hamiltonian equations of motion agrees well with the analytical results (10)
in different ranges of the detunings. A typical PDF for the flight and
trapping events decays initially algebraically and has an exponential tail.
The length of the initial power-law segment is inversely proportional to the
value of the detuning $\Delta$ and can be rather large.

In which way SE changes the statistical properties of the Hamiltonian motion?
Can we find fingerprints of Hamiltonian instability and chaos in the motion of
spontaneously emitting atoms or SE totally suppresses any manifestations of
coherent (but chaotic!) Hamiltonian dynamics? These questions will be
addressed in the next sections.

##  IV Dissipative atomic transport in a laser standing wave

The emission of a photon into the continuum of modes of the electromagnetic
field is accompanied by an atomic recoil. The dissipative (friction) force
$F\equiv{\lang\dot{p}\rang}$ (which does not exist in the Hamiltonian system)
depends on the atomic momentum $p$ and the sign of the detuning in a
complicated way Kaz ; PRA05 . The effects of acceleration, deceleration, and
velocity grouping (at $\Delta<0$) are well-known in the literature Kaz ;
Meystre . A novel effect we report in this section is dissipative chaotic
walking. It appears under appropriate conditions that are different from those
specified for Hamiltonian chaotic walking in the preceding section.

To illustrate the possible regimes of dissipative atomic transport in a
standing wave we integrate by the Monte Carlo method dissipative equations of
motion (3) with 2000 atoms whose positions and momenta are distributed in a
quasi-Gaussian manner (Fig. 2a). In Fig. 2b we demonstrate the

Figure 2: Atomic momentum and position distribution illustrating the effects
of atomic acceleration, deceleration, and the velocity grouping: (a) $\tau=0$,
(b) $\tau=10^{5}$, $\Delta=-0.2$, (c) $\tau=10^{5}$, $\Delta=0.1$, (d)
$\tau=10^{5}$, $\Delta=-0.05$. Momentum $p$ is given in units of $\hbar
k_{f}$, the position in units of $k_{f}^{-1}$.

velocity grouping effect at $\Delta=-0.2$ and $\tau=10^{5}$. A large number of
atoms is grouped around two values of the capture momentum $p_{\rm
cap}\simeq\pm 1300$ because of acceleration of slow atoms and deceleration of
the fast ones in the initial ensemble. The slower the atoms are the longer is
the process of the velocity grouping. Note that atoms with $|p|\lesssim 100$,
trapped initially in a well of the optical potential, could not quit the well
up to $\tau=10^{5}$. Contrary to that, at positive values of the detuning fast
atoms are accelerated and slow ones are decelerated. As a result, we observe a
pronounced peak around $x\simeq p\simeq 0$ shown in Fig. 2c at $\Delta=0.1$
and $\tau=10^{5}$.

Dependence of the friction force $F$ on the current atomic momentum $p$ is
shown in Fig. 3 at $\Delta=-0.2$. It has been computed with our main equations
(3) when averaging over seven thousands atoms with different initial momentum.
The function $F(p)$ resembles the behavior of the friction force computed with
another methods

Figure 3: Dependence of the friction force $F\equiv{\lang\dot{p}\rang}$ on the
current atomic momentum $p$ at the detuning $\Delta=-0.2$.

(see Kaz  and Fig. 1a in PRA05 ). The friction force decreases up to zero
value and then begin to oscillate with increasing $p$. It has a number of
zeroes (the detailed view is shown in Fig. 3b) like the corresponding
functions in Refs. Kaz ; PRA05 . Zero values of $F$ correspond to
quasistationary values of the momentum which depend on $\Delta$. Some of them
are attractors and atoms with close values of the momentum tend to $p_{\rm
cap}$, another ones are repellors. The attractors and repellors are not
deterministic because of a random nature of SE. Thus, an atoms walks randomly
in the momentum space between different values of the capture momentum $p_{\rm
cap}$. When it reaches a certain value of the capture momentum the atom does
not stop in the momentum space and goes on to fluctuate because of both the
Hamiltonian instability and SE effect.

In the preceding section we described the Hamiltonian chaotic walking that may
occur in the absence of any losses. Dissipation causes additional strong
fluctuations of the momentum. If $\Delta>0$ or if it is negative but
comparatively large, nothing principally new happens to atoms as compared with
the Hamiltonian limit. However, at negative small values of $\Delta$, a
characteristic capture momentum becomes smaller than a typical range of
momentum fluctuations due to atomic recoils. As a result, atoms may change
their direction of motion in an irregular way. Such a dissipative chaotic
atomic walking is illustrated in Fig. 2d at $\Delta=-0.05$ and $\tau=10^{5}$
with the atoms distributed widely in the phase plane. Typical atomic
trajectories are shown in Fig. 4 in the momentum space. Figures 4a and b
illustrate how the friction force near the resonance ($\Delta=-0.001$ and
$\Delta=-0.01$) decelerates atoms with large values of the initial momentum
down to so small values of the capture momentum when the dissipative chaotic
walking becomes possible. With increasing the absolute value of the negative
$\Delta$, the capture momentum increases and the atom changes rarely the
direction of motion (Fig. 4c with $\Delta=-0.1$). Panels d and e in Fig. 4
illustrate the velocity grouping effect at $\Delta=-0.15$ with different
values of the initial momentum.

Figure 4: Typical atomic trajectories in the momentum space: (a)-(c)
dissipative chaotic walking with different statistics of atomic flights,
(d)-(e) the effect of velocity grouping. Note that atoms with very different
initial momentum acquire a close value of the capture momentum.

##  V Statistical properties of dissipative chaotic walking

Statistics of Hamiltonian chaotic walking is quantified by the flight and
trapping PDFs (10) with algebraically decaying head segments and exponential
tails whose lengths strongly depend on $\Delta$. We will show in this section
that PDFs for dissipative chaotic walking are even more sensitive to
variations in $\Delta$. Figures 4a and b clearly demonstrate that at very
small detuning $\Delta=-0.001$ long flights dominate, whereas there appears a
number of short flights with larger value of $\Delta=-0.01$. In Fig. 5 we plot
the PDFs $P_{fl}$ for the duration of atomic flights $T$ with different values
of the detuning $\Delta$. At small detunings (Fig. 5a), the length of the
power-law segments depends on $\Delta$ in a similar way as in the Hamiltonian
case (compare this figure with Figs. 5a, 6a, and 7a in Ref. pra07 ). However,
the slope slightly differs from the Hamiltonian slope which is equal to
$-1.5$. The difference in the statistics of dissipative and Hamiltonian
walkings is more evident with larger values of the detuning (Fig. 5b). The
length of the power-law segments increases drastically with increasing
$\Delta$. This effect is absent in Hamiltonian dynamics. The corresponding
slope $\alpha$ decreases with changing the detuning from $\Delta=-0.09$ to
$\Delta=-0.12$ because of the corresponding increase in the length of atomic
flights (see Figs. 4a, b, and c). In Fig. 6 we plot the dependencies of the
mean duration of atomic flights ${\lang T\rang}$ and the slope of the PDF
powerlaw fragments $\alpha$ on the detuning $\Delta$ in the range of its
medium values $-0.12\leq\Delta\leq-0.06$. Both the quantities correlate well
with each other. It means that, changing the value of $\Delta$, one can
manipulate statistical properties of dissipative atomic transport in an
optical lattice. This control is nonlinear in the sense that slightly changing
$\Delta$, say, from $-0.08$ to $-0.12$, we increase the mean duration of
flights in a few orders of magnitude. This effect may be qualitatively
explained as follows. When increasing the absolute value of the negative
detuning, the capture momentum increases but fluctuations of the current
momentum $p$ decrease providing long atomic flights epl08 .

To explain the statistical properties of the dissipative chaotic walking let
us consider the behavior of the quasienergy

Figure 5: The PDFs $P_{\rm fl}$ for the duration of atomic flights $T$ with
(a) small detunings (crosses $\Delta=-0.01$, stars $\Delta=-0.001$, circles
$\Delta=-0.0001$, squares $\Delta=-0.00001$) and (b) medium detunings (stars
$\Delta=-0.09$, $\alpha=-0.77$; circles $\Delta=-0.1$, $\alpha=-0.27$; squares
$\Delta=-0.12$, $\alpha=-0.05$). Straight lines show slopes $\alpha$ of the
power-law fragments of the PDFs in log-log scale.  Figure 6: Dependencies of
the logarithms of the mean duration of atomic flights $\langle T\rangle$
(solid line) and of the slope $\alpha$ of the PDF power-law fragments
(squares) on the detuning $\Delta$ (in units of $\Omega$).

$\displaystyle\tilde{H}_{j}$
$\displaystyle\equiv\frac{\omega_{r}}{2}p^{2}-u\cos
x-\frac{\Delta}{2}z-\frac{\Delta\gamma}{4}{\lang
1-z^{2}\rang}(\tau-\tau_{j})=$ (12)
$\displaystyle=H-\frac{\Delta\gamma}{4}{\lang
1-z^{2}\rang}(\tau-\tau_{j}),\quad\tau_{j}<\tau<\tau_{j-1},$

which is equal to the total atomic energy (5) in the absence of relaxation.
Near the resonance, $|\Delta|\ll 1$, $\tilde{H}_{j}$ is almost conserved
between SE events, i. e., in the interval $\tau_{j}<\tau<\tau_{j-1}$. The real
energy $H$ (see Fig. 1d) decreases a little in between in a linear way. The
rate of this decrease is defined by the coefficients of spontaneous emission
$\gamma$, the detuning $\Delta$, and the average probability to find the atom
in the excited state. Both the quantities, $H$ and $\tilde{H}$, changes
abruptly just after SE (because of the corresponding changes in the atomic
variables (4)). Just after emitting a $j$th spontaneous photon at
$\tau=\tau_{j}$, they have the same values. So, we will model the evolution of
the energy as a map $H_{j}\equiv H(\tau_{j}^{+})$ taken at the moments
$\tau_{j}^{+}$ just after SE

$\displaystyle
H_{j}=\tilde{H}_{j}-\tilde{H}_{j-1}+H_{j-1}=H_{j-1}+\omega_{r}p(\tau_{j}^{-})\rho_{j}+$
(13)
$\displaystyle+\frac{\omega_{r}}{2}\rho_{j}^{2}+\frac{\Delta}{2}+u(\tau_{j}^{-})\cos
x(\tau_{j})+$
$\displaystyle+\frac{\Delta}{2}z(\tau_{j}^{-})+\frac{\Delta\gamma}{4}{\lang
1-z^{2}\rang}(\tau_{j}-\tau_{j-1}),$

where the values of the atomic variables $p(\tau_{j}^{-}),u(\tau_{j}^{-})$,
and $z(\tau_{j}^{-})$ are taken at the moments $\tau_{j}^{-}$ just before SE.
They are, in turn, determined by the coherent evolution between SE.

The stochastic map for the atomic energy (13) provides an important
information about the CM motion. It has been shown in Ref. pra07  that atoms
move ballistically, if the atomic energy satisfies to the condition
$H\gtrsim|u|$, whereas at $H\lesssim|u|$ they may change the direction of
motion. The dissipative chaotic walking takes place when the atomic energy $H$
alternatively takes the values larger and smaller than a critical value
$H=|u|$. In the Hamiltonian limit, where the energy is conserved, the problem
of the CM chaotic walking has been reduced to the task of random walking of
the Bloch component $u$ (see Sec. III and Ref. pra07 ). The energy is not
conserved in the presence of relaxation, but the values of $u$ are always
small (see Appendix and Fig. 1c and d). Thus, atoms oscillate in the wells of
the optical potential if $H\lesssim 0$ and move ballistically if $H\gtrsim 0$.

On a time scale exceeding the mean time between SE events $2/\gamma$, the
evolution of energy can be treated as a diffusion process with a drift in the
energetic space. The probability to have the energy $H$ at time $\tau$ is
governed by the Fokker-Planck equation

$\dot{P}(H,\tau)=-2c_{H}\frac{\partial P}{\partial
H}+D_{H}\frac{\partial^{2}P}{\partial H^{2}},$ (14)

where $D_{H}$ is an energy diffusion coefficient and $c_{H}$ is an energy
drift coefficient which can be estimated with the help of Eq. (13) as follows:

$c_{H}\equiv\frac{{\lang
H_{j}-H_{j-1}\rang}}{{\lang\tau_{j}-\tau_{j-}\rang}}={\lang\dot{H}\rang}\simeq\frac{\omega_{r}\gamma}{12}+\frac{\Delta\gamma}{2}.$
(15)

In deriving this formula we adopt the average value of the squared recoil
momentum ${\lang\rho_{j}^{2}\rang}=1/3$ (the projections of the recoil momenta
on the standing-wave axis $x$, $\rho_{j}$, are assumed to be distributed in
the range $\pm 1$ with the same probability), the average value of the
population inversion just before a SE event to be $z(\tau_{j}^{-})=1/2$, the
average value of $z$ over the whole time scale and its mean squared deviation
from 1 to be ${\lang z\rang}=0$ and ${\lang 1-z^{2}\rang}=1/2$, respectively
(see the solution (33) in Appendix). Moreover, neglecting the correlation, we
put ${\lang u\cos x\rang}\simeq{\lang u\rang}{\lang\cos x\rang}=0$, which is
valid if $|\omega_{r}p|\gtrsim\gamma/2$, i. e., when $p\gtrsim 100$ with our
choice of the parameters. Since the first term in (15) is small and may be
neglected, the drift velocity of an atom in the energetic space is
approximately proportional to the detuning $\Delta$, and, therefore in average
atoms accelerate and decelerate at $\Delta>0$ and $\Delta<0$, respectively, as
it should be for $|\Delta|\ll 1$. In the weak Raman-Nath approximation, (32)
and (34), the drift coefficient in the energetic space is simply related to
the friction force $F$ acting upon atoms

$F\equiv{\lang\dot{p}\rang}\simeq\frac{{\lang\dot{H}\rang}}{\omega_{r}p}.$
(16)

The friction force plays the role of a drift coefficient in the momentum
space. Strictly speaking, the weak Raman-Nath approximation is not valid near
the turning points when the atomic velocity is comparatively small. However,
most of the flight time it is valid.

The diffusion coefficient in the energetic space is given by the formula

$\displaystyle\ D_{H}\equiv\frac{{\lang(H_{j}-H_{j-1})^{2}\rang}-{\lang
H_{j}-H_{j-1}\rang}^{2}}{2{\lang\tau_{j}-\tau_{j-1}\rang}},$ (17)

which can be rewritten with the help of (13) as follows:

$\displaystyle
D_{H}\simeq\frac{\gamma\omega_{r}^{2}p^{2}(\tau_{j}^{-})}{{12}}+\frac{{\lang
u^{2}(\tau_{j}^{-})\rang}\gamma}{8}.$ (18)

Using weak Raman-Nath approximation, (32) and (34), the first term can be
replaced by $\gamma\omega_{r}H/6$. Using the estimation (38) for ${\lang
u^{2}(\tau_{j}^{-})\rang}$ in the irregular CM motion regime (see Appendix),
we get the following expression for the energy diffusion coefficient:

$D_{H}^{ch}\simeq\frac{\gamma\omega_{r}H}{{6}}+\frac{\Delta^{2}}{8}.$ (19)

This expression is valid for moderately small momentums ($p\lesssim 1000$)
when the strong Raman-Nath approximation cannot be applied. In the process of
dissipative chaotic walking, the probability to get higher values of the
momentums is almost zero.

Now we will try to derive analytically a distribution of the durations $T$ of
atomic flights in the process of dissipative chaotic walking. In fact, it is a
problem of the first passage time for the atomic energy $H$ to return to its
zero value. Recall that at small detunings we have $H\simeq 0$ in the very
beginning of every flight. In course of time $H$ can reach rather large
values, and it returns to zero at the end of the flight. If the random jumps
of the energy would be symmetric ($c_{H}=0$), the probability to find a flight
with duration $T$ would be proportional to $T^{-1.5}$, where the exponent
$-1.5$ does not depend on the diffusion coefficient. This conjecture follows
from the known theorem in probability theory. More general result (see chapter
XIV in Ref. Feller ) proves that in the case of an asymmetric random walking
in the energetic space ($c_{H}\neq 0$) the PDF for the flight durations in
configuration space is

$P_{\rm{fl}}\propto e^{-c_{H}^{2}T/D_{H}}T^{-1.5},$ (20)

if the drift and diffusion coefficients in the Fokker-Planck equation for the
random walking are assumed to be constants. This formula gives a distribution
of the flight durations with a power-law fragment followed by an exponential
tail and agrees qualitatively with the exact numerical computations of
$P_{fl}$ shown in Fig. 5a for a few values of the detuning $\Delta$. The main
disadvantage of this formula is that (20) does not depend on $\Delta$ as the
exact PDFs in Fig. 5a. At very small $\Delta=-10^{-5}$, the PDF, shown by
squares in Fig. 5a, decays mostly algebraically, whereas at larger values of
the detuning the power-law fragments are much shorter. A discrepancy between
the analytical and numerical PDFs arises because we assumed in deriving (20)
that $D_{H}$ and $c_{H}$ do not depend on the energy $H$. In fact, it is not
the case for small values of the momentum $p$, and a more accurate formula for
$P_{fl}(T)$ is required.

The PDFs for Hamiltonian (10) and dissipative (20) transport are similar in
the sense that both $P_{fl}$ contain power-law fragments followed by
exponential tails, but the origin of each statistics is different. In the
Hamiltonian limit the statistics is governed by the behavior of $u$, not the
energy, as in the dissipative system. A turnover from a power law to an
exponential decay in the Hamiltonian case is explained by a boundedness of
$|u|\lesssim 1$, whereas in the dissipative system it is explained by a
negative drift of the energy $H$. Each of the factors prevents the
corresponding randomly walking quantity to go far away from its critical value
(at which the atoms can change the directions of motion) decreasing the
probability of long flights in at exponential way.

##  VI Manifestation of Hamiltonian chaos in dissipative atomic transport

In the absence of SE the atomic dynamics can be regular or chaotic depending
on the initial conditions and/or the detuning. In experiments one measures
statistical characteristics of spontaneously emitting atoms. Is there a
correlation between those characteristics and the underlying Hamiltonian
dynamics? Can we find any manifestations of Hamiltonian instability, chaos,
and order, in the diffusive-like dissipative atomic transport? These questions
will be addressed in the present section.

The common quantitative criterion of deterministic chaos, the maximal Lyapunov
exponent $\lambda$, is a measure of a divergence of two trajectories in the
phase space with close initial conditions LL83 . To quantify probability of
chaos in the mixed Hamiltonian dynamics, when $\lambda=0$ with some values of
$p_{0}$ and $\lambda>0$ with another values of $p_{0}$, we introduce a
probabilistic measure of Hamiltonian chaos

$\Lambda\equiv{\lang 2\Theta(\lambda)-1\rang},$ (21)

where $\Theta(\lambda)$ is a Heaviside function ($\Theta=0$ if $\lambda<0$,
$\Theta=1/2$ if $\lambda=0$, and $\Theta=1$ if $\lambda>0$). The probability
of Hamiltonian chaos $\Lambda$ is computed by averaging over a large number of
atomic trajectories with different values of $p_{0}$. If all the trajectories
in the set turn out to be stable, one gets $\Lambda=0$, and if all they are
exponentially unstable, then $\Lambda=1$. One gets $0<\Lambda<1$, if some
trajectories in the set are stable but the other ones are not. The magnitude
of $\Lambda$ is proportional to the fraction of trajectories with positive
$\lambda$s.

To examine manifestations of the underlying Hamiltonian dynamics in
dissipative transport in is convenient to consider atomic diffusion not in the
energetic but in the momentum space. The momentum diffusion coefficient, which
is a measure of momentum fluctuations, can be written with the help of (17)
and (32) as follows:

Figure 7: Correlation between the average momentum diffusion coefficient in a
log-log scale $D_{p}$ (in units of $\hbar^{2}k^{2}_{f}\Omega$) and probability
of Hamiltonian chaos $\Lambda$ in their dependencies on current atomic
momentum $p$ (in units of $\hbar k_{f}$) at $\Delta=-0.01$ and
$\Delta=-0.0005$. Dashed lines with the slopes $~{}p^{-2}+{\rm const}$ and
$~{}p^{-1}+{\rm const}$ are theoretical curves (23) and (24) valid in the
regimes of Hamiltonian chaos and order, respectively. Solid line is a
theoretical curve (25) derived to fit the exact numerical results. An abrupt
change in the decay laws for $D_{p}$ occurs for those values of $p$ where the
transition from order to chaos takes place in the underlying Hamiltonian
dynamics.

$\displaystyle
D_{p}\simeq\frac{D_{H}}{\omega_{r}^{2}p^{2}}\simeq\frac{\gamma}{{12}}+\frac{{\lang
u^{2}(\tau_{j}^{-})\rang}\gamma}{8\omega_{r}^{2}p^{2}}.$ (22)

Using the formula (19), we get $D_{p}$ in the regime of chaotic oscillations
of the Bloch component $u$

$D_{p}^{ch}\simeq\frac{\gamma}{{12}}+\frac{\Delta^{2}}{8\omega_{r}^{2}p^{2}}.$
(23)

The momentum diffusion coefficient $D_{p}$ is computed with the main equations
(3) in the following way. The range of all possible values of the atomic
energy $H$ (5) is partitioned in a large number of bins. For a large number of
initial conditions (in fact, we change only the initial momentum $p_{0}$
keeping the other conditions to be fixed), after any $j$th SE event we compute
the difference $H_{j}-H_{j-1}$ and the squared difference
$(H_{j}-H_{j-1})^{2}$. They are random values, but their statistics depend on
the preceding energy value $H_{j-1}$. So we calculate the histograms of the
average values of ${\lang H_{j}-H_{j-1}\rang}$ and
${\lang(H_{j}-H_{j-1})^{2}\rang}$ as functions of energy $H_{j-1}$. After that
we can compute the energy diffusion coefficient $D_{H}$ (17) which, being
divided by $\omega_{r}^{2}p^{2}$, yields the momentum diffusion coefficient
$D_{p}$ which is better to present as a function of the current momentum
$p\simeq\sqrt{2H_{j-1}/\omega_{r}}$.

The main result in this section is illustrated with Fig. 7. In the upper left
and right panels the dependencies of the momentum diffusion coefficient
$D_{p}$ on the current momentum $p$ are plotted in a log-log scale for
$\Delta=-0.01$ and $\Delta=-0.0005$, respectively. In both the cases, we put
$\gamma=0.0033$. These plots should be compared with the corresponding lower
panels where the probability of the Hamiltonian chaos $\Lambda$ is plotted
against $p$ with $\gamma=0$ (i. e. in the Hamiltonian limit of Eqs. (3)). It
is evident that the character of the momentum diffusion changes abruptly at
those values of the current momentum where a transition from chaos to order
occurs in the underlying Hamiltonian dynamics. Such a turnover takes place in
a range of small negative detunings and is a manifestation of the
peculiarities of the underlying Hamiltonian evolution in the diffusive-like
dissipative transport of atoms in a standing-wave laser field. We may conclude
that in spite of random atomic recoils due to SE the chaotic (regular)
dynamics between the acts of SE clearly manifests itself in the behavior of
the measurable characteristic of the atomic transport, the momentum diffusion
coefficient $D_{p}$. The behavior of $D_{p}$ in the range of $p$, where the
underlying Hamiltonian evolution is chaotic, is well described by the formula
(23) with $D_{p}^{ch}\sim p^{2}+const$ (see both the upper panels in Fig. 7
where this dependence is shown by dashed lines).

However, the formula (23) does not work in the regimes when the underlying
Hamiltonian dynamics is mixed or regular because in deriving it we supposed
fully chaotic behavior of $u$. We have managed to estimate analytically
$D_{p}$ in the Hamiltonian regular regime at extremely small values of the
detuning $|\Delta|\lll 1$ and for atoms whose momentum is so large that we can
neglect its fluctuations between SE events (the exact Raman-Nath approximation
with $x=\omega_{r}p\tau$). Figure 1a illustrates the ladder-like behavior of
$u$ which is descibed by the deterministic mapping (8) on a comparatively
short timescale. To get $D_{p}^{RN}$ from Eq. (22) we use the expression (37)
for $u^{2}(\tau_{j}^{-})$ derived in Appendix

$D_{p}^{RN}\simeq\frac{\gamma}{{12}}+\frac{\Delta^{2}}{8\omega_{r}p\gamma\pi}.$
(24)

Thus, we derived the formulas for the momentum diffusion coefficient $D_{p}$
in the regimes of Hamiltonian chaos (23) with $\Lambda=1$ and Hamiltonian
order (24) with $\Lambda=0$. In a general case with $0\leq\Lambda\leq 1$, we
will assume a linear combination

$\displaystyle D_{p}\simeq(1-\Lambda)D_{p}^{RN}+\Lambda D_{p}^{ch}\simeq$ (25)
$\displaystyle\simeq\frac{\gamma}{{12}}+\frac{\Delta^{2}}{8\omega_{r}p}\left(\frac{1-\Lambda}{\gamma\pi}+\frac{\Lambda}{\omega_{r}p}\right).$

This function, shown by the solid line in the right upper panel in Fig. 7,
fits rather well exact numerical results.

We would like to end this section with the proposal of a simple experimental
scheme to observe our main theoretical and numerical result on an abrupt
change in the character of atomic diffusion in a laser standing wave under
conditions corresponding to two different regimes of the underlying
Hamiltonian evolution, chaotic and regular ones. Let us consider a small
atomic cloud moving in one direction with an average atomic momentum ${\lang
p_{c}\rang}$. Initial position and momentum distribution are assumed to be a
Gaussian with the standard deviations $\sigma^{2}_{x}\equiv{\lang(x-{\lang
x_{c}\rang})^{2}\rang}$ and $\sigma^{2}_{p}\equiv{\lang(p-{\lang
p_{c}\rang})^{2}\rang}$. The momentum diffusion coefficient is

$D_{p}=\frac{d(\sigma^{2}_{p})}{2d\tau}.$ (26)

The temperature of gas and its rate of heating in Kelvins per second are

$T\equiv\frac{2{\lang
E_{k}\rang}}{k_{B}}=\frac{\hbar^{2}k^{2}_{f}\sigma^{2}_{p}}{m_{a}k_{B}},\quad\frac{dT}{dt}=\frac{2\hbar^{2}k^{2}_{f}\Omega
D_{p}}{m_{a}k_{B}},$ (27)

where $E_{k}$ is a kinetic energy of atoms (in Joules) in the reference frame
moving with the CM of the cloud. It follows from (27) that the rate of heating
is proportional to $D_{p}$ whose behavior is different in the regimes of
regular and chaotic underlying Hamiltonian dynamics.

The linear extent of the cloud in meters is $L_{X}\equiv 2\sigma_{x}/k_{f}$.
On a comparatively short time scale, $\tau\ll|({\lang p_{c}\rang}/F)|$, and
low temperatures $\sigma_{p}\ll|{\lang p_{c}\rang}|$, $D_{p}$ is approximately
the same for all the atoms in the cloud because the CM velocity could not
change significantly under the action of the friction force $F$ during the
observation time. Using the first equation in the set (3) and the Eq.(26), we
obtain

$\sigma^{2}_{x}\simeq\sigma^{2}_{x}(0)+\frac{1}{2}\omega^{2}_{r}\sigma^{2}_{p}(0)\tau^{2}+\frac{2}{3}D_{p}\omega^{2}_{r}\tau^{3}.$
(28)

Figure 8: A log-log dependence of the cloud’s linear extent $L_{X}$ (in
microns) on the average initial momentum ${\lang p_{c}\rang}$ of atoms in a
cloud at two moments of time. The analytic dashed lines were computed with the
formula (28) with $D_{p}=D_{p}^{ch}$ valid in the chaotic regime on a short
time scale. Note an abrupt change in the decay of $L_{X}({\lang p_{c}\rang})$
in the range of ${\lang p_{c}\rang}\simeq 1200$ where a chaos-order transition
takes place in the underlying Hamiltonian motion. $\Delta=-0.01$,
$2\sigma_{x}(0)=0.5$, $2\sigma_{p}(0)=5$, wavelength
$\lambda_{f}=2\pi/k_{f}=850$ nm.

We have computed $L_{X}$ with that formula with $D_{p}$ given by (23) and
compare the result with numerical simulation of Eqs. (3) for a number of
atomic clouds with different initial values of ${\lang p_{c}\rang}$. In Fig. 8
the dependence $L_{X}({\lang p_{c}\rang})$ is plotted for $\Delta=-0.01$ at
two moments of time. The analytic dashed lines fit well the exact numerical
results in the range ${\lang p_{c}\rang}\lesssim 1200$ where the underlying
Hamiltonian dynamics is chaotic (see the left column in Fig. 7). Note an
abrupt change in the decay of $L_{X}({\lang p_{c}\rang})$ beginning with those
values of ${\lang p_{c}\rang}\simeq 1200$ where the Hamiltonian dynamics
becomes more regular. Since the linear extent of the atomic clouds changes
abruptly at the chaos-order border one may conclude that in real experiments
the value of $L_{X}$ should increase sharply with those values of the average
cloud momentum ${\lang p\rang}$ for which the underlying Hamiltonian evolution
is chaotic.

##  VII Conclusion

Coherent evolution of the atomic state in a near-resonant standing-wave laser
field is interrupted by SE events at random moments of times. The Hamiltonian
evolution between these events has been shown previously (for a summary of
Hamiltonian theory for cold atoms in a 1D optical lattice see Ref. pra07 ) to
be regular or chaotic depending on the values of the detuning $\Delta$ and the
initial momentum $p_{0}$. We stress that dynamical chaos may happen without
any noise and any modulation of the lattice parameters. It is a specific kind
of dynamical instability in the fundamental interaction between the matter and
radiation.

In reality Hamiltonian chaos is masked by random events of SE. The behavior of
spontaneously emitting atoms in the detuning and momentum regimes where the
underlying Hamiltonian dynamics is chaotic may be called stochastic chaos. We
have specified and quantified two regimes of the stochastic chaos, namely,
random walking and dissipative ballistic transport. In the first regime, atoms
wander in an optical lattice in a random way performing flights in both the
directions with the PDFs strongly depending on the detuning (see Figs. 5 and
6). In the ballistic regime, atoms move in the same direction with momentum
fluctuations caused both by the Hamiltonian instability as well as SE events.
It has been shown in our numerical experiments and confirmed analytically that
the character of momentum diffusion changed abruptly in the regime where the
underlying Hamiltonian dynamics proved to be chaotic. A clear correlation
between the decay of the momentum diffusion coefficient $D_{p}$ and
probability of Hamiltonian chaos $\Lambda$ has been found (Fig. 7). In order
to observe the manifestation of Hamiltonian chaos in real experiments we
proposed to measure a linear extent of atomic clouds $L_{X}$ in a 1D optical
lattice and predicted a significant increase in $L_{X}$ for the atomic clouds
with $\Lambda\simeq 1$.

In conclusion we would like to discuss some possible applications of the
theory developed and the results obtained. A sensitive dependence of
statistical properties of dissipative chaotic walking and ballistic transport
on the values of the detuning $\Delta$ provides a possibility to manipulate
atomic CM motion by changing $\Delta$. For example, one can increase the mean
duration of atomic flights in three orders of magnitude by changing $\Delta$
only by thirty percents (see Fig. 6).

Cold atoms in optical lattices is an ideal system to study different phenomena
in statistical physics. Besides dynamical chaos, the phenomena of stochastic
resonance has been observed in a near-resonant optical lattice SC02 . Another
phenomenon of considerable current interest is cold atom ratchets ML99 ; SS03
; JG04 ; GB07 ; GP07 . A ratchet is a spatially periodic device which is able
to produce a directed transport of particles in the absence of a net bias (i.
e., when the time- and space-averaged forces are zero). In order to realize
the ratchet effect it is necessary to break time or/and spatial symmetries
which generate a countermoving partner to each trajectory FY00 . Different
classes of the ratchets have been experimentally realized with cold atoms in
optical lattices ML99 ; SS03 ; JG04 . The interrelation of Hamiltonian chaos
and SE noise, found in this paper, provides an additional possibility to
create and manipulate directed transport of atoms in rigid optical lattices.

## Appendix

We work in the regime of small detunings

$|\Delta|\ll 1,$ (29)

moderate mean atomic velocities

${\lang|\omega_{r}p|\rang}\sim\gamma/2\ll 1,$ (30)

and diffusive motion

$\tau\gg 2/\gamma.$ (31)

Due to (29), we may neglect the last term in the potential energy (5) and
adopt the Hamiltonian solutions for (7) between any two acts of SE. The
evolution of $u$ is described by the approximate solutions (8) for the regular
Raman-Nath motion and (9) for the chaotic motion. It follows from the
condition of moderate atomic velocity (30) and solutions (8), (9) that $|u|\ll
1$. In other words, $u$ cannot go far from zero between acts of SE after each
of which $u=0$. Under the conditions (29) and $|u|\ll 1$, the mean kinetic
atomic energy is much greater than the potential one and in this weak Raman-
Nath approximation

$|u\sin x+\frac{\Delta z}{2}|\ll\frac{\omega_{r}p^{2}}{2},$ (32)

the solution (7) for $z$ is simplified

$\displaystyle z(\tau)\simeq\mp\sin\left(2\int\limits_{0}^{\tau}\cos
xd\tau^{\prime}+\chi_{0}\right).$ (33)

In the diffusion regime (31) and in the weak Raman-Nath approximation (32) the
momentum fluctuations between two SE are small in comparison with its
fluctuations over the time scale of atomic transport. So, the atomic momentum
just before SE is equal to its value at the node

$p\simeq p(\tau_{j}^{-})\simeq p_{\rm node}.$ (34)

Now we can get simplified solutions for $u$. In the exact Raman-Nath
condition, $x=\omega_{r}p\tau$, we have from (8) the approximate deterministic
map written in a two-step form

$\displaystyle u_{m}^{RN}\approx 2\Delta\sqrt{\frac{\pi}{\omega_{r}p}}\
v_{0}\cos\left(\frac{2}{\omega_{r}p}-\frac{\pi}{4}\right)+u_{m-2}^{RN}.$ (35)

In the chaotic regime we have from $(\ref{u_m})$ the stochastic map

$\displaystyle
u_{m}^{ch}=|\Delta|\sqrt{\frac{\pi}{\omega_{r}p}}\sin\phi_{m}+u_{m-1}^{ch}.$
(36)

The maps derived enable us to estimate the values of $u(\tau_{j}^{-})$ just
before $j$th act of SE. After the $(j-1)$th SE $u_{0}=u(\tau_{j-1}^{+})=0$,
$u(\tau_{j}^{-})=u_{M}$ is an accumulated value of $u$ after passing $M$ nodes
in the interval $\tau_{j-1}<\tau<\tau_{j}$. The average number of node
crossings can be estimated to be ${\lang M\rang}\simeq
2|\omega_{r}p|/\gamma\pi$. In the exact Raman-Nath limit,
$u_{M}=M(u_{m}-u_{m-2})/2$, and the mean squared value in the regular regime

${\lang u^{2}(\tau_{j}^{-})\rang}_{RN}={\lang
u^{2}_{M}\rang}_{RN}\simeq\frac{4\Delta^{2}\omega_{r}p}{\pi\gamma^{2}}\ {\lang
v^{2}_{0}\rang}\cos^{2}\left(\frac{2}{\omega_{r}p}-\frac{\pi}{4}\right).$ (37)

Beyond the strong Raman-Nath limit, $u(\tau_{j}^{-})=u_{M}$ is a sum of $M$
random numbers which are proportional to $\sin\phi_{m}$. From the probability
theory we get

${\lang u^{2}(\tau_{j}^{-})\rang}_{ch}={\lang u^{2}_{M}\rang}_{ch}\simeq{\lang
M\rang}\Delta^{2}\frac{\pi}{2\omega_{r}p}\simeq\frac{\Delta^{2}}{\gamma}.$
(38)

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# TOP detector for particle identification at Belle II

Ezio Torassa
( On behalf of the Belle II TOP group)  INFN Sezione di Padova
35131 Padova, Italy
<EMAIL_ADDRESS>
(Day Month Year; Day Month Year)

###### Abstract

The Time-Of-Propagation (TOP) counter is a ring-imaging Cherenkov detector
designed to identify the charged hadrons in the barrel region of the Belle II
detector. The Belle II experiment collected data delivered by the SuperKEKB
accelerator from March 2019 to June 2022. After the first run period (Run1) a
long shutdown (LS1) was dedicated to implement several accelerator and
detector upgrades. The most important upgrade was the completion of the vertex
detector. The second collision period (Run2), started in January 2024. The
components of the TOP detector and the performance during the Run1 will be
reported, the detector upgrade during LS1 and the future upgrades will be
described.

###### keywords:

TOP; particle identification; Belle II.

PACS numbers:

##  1 Introduction

The TOP detector replaced the Aerogel Cherenkov Counter of the Belle
experiment, it consists of 16 modules made by quartz bar radiators read-out by
Micro-Channnel Plate Photomultipliers (MCP-PMTs). The TOP modules are arranged
in the central region of the Belle II detector [1] between the Central Drift
Chamber (CDC) and the Electromagnetic Calorimeter (ECL) (Fig. 1, left) . The
quartz bars act as Cherenkov radiator and photon collectors. Thanks to the
high average refractive index, the Cherenkov radiation remains trapped inside
the bars and propagates to the photodetectors through internal reflection.
Different hadrons crossing the quartz bar with the same angle have Cherenkov
photons emitted at different angles, they arrive at the photodetector plane in
different channels and at different times (Fig. 1, right). The time of arrival
is measured relative to the $e^{+}e^{-}$ collision time, it includes the time-
of-flight (Tof) of the particle and the time-of-propagation (Top) of photons:

$K/\pi~{}\mathrm{Tof~{}difference}~{}\sim
50~{}ps/m~{}~{}~{}~{}~{}~{}~{}~{}~{}K/\pi~{}\mathrm{Tof~{}difference}~{}\sim
75~{}ps/m$
$\mathrm{PID~{}sensitivity}\propto\frac{\Delta~{}Tof~{}+\Delta~{}Top}{\sigma_{Time}}\sqrt{N_{\gamma}}$

The number of photons $N_{\gamma}$ is 20-30 depending to the track angle.
Radius of the TOP cylinder is $1.2~{}m$ therefore
$\Delta~{}Tof~{}+\Delta~{}Top$ is on the order of $100~{}ps$, it follows the
requirement of the time resolution $\sigma_{Time}\leq~{}100~{}ps$ to keep a
good PID sensitivity. The arrival position and time of photons are measured
with a 2D segmentation of 16 channels per $in^{2}$ and a time resolution
better than 100 ps. For every track the probability to be a specific particle
is estimated by comparing the resulting space-time distribution of the
arriving photons with the probability density function expected for this
particle hypothesis. The particle identification capability has been tested by
selecting pure samples of pions, kaons and protons tagged reconstructing
$D^{*}$, $K_{S}$ and $\Lambda$ particles decays.

Figure 1: The Belle II and the TOP detectors (left). Cherenkov photons emitted
at different angles by pion and kaon hadrons (right).

The TOP detector was constructed within the original time schedule, the
installation was completed in May 2016. The drawback was the use of a small
fraction of the latest generation of photomultipliers having long lifetime.
Half of the MCP-PMTs have been replaced during LS1, the others will replaced
during the next long shutdown.

##  2 TOP detector

The key elements of the TOP detector are described in the following.

###  2.1 Cherenkov radiator and light guide

The Cherenkov radiator and light guide elements are shown in Fig. 2 for a
single TOP module.

Figure 2: Cherenkov radiator and light guide elements for a single TOP module.

Four parts are glued together: two fused silica bars each of dimension (1250 x
450 x 20) $mm^{3}$ acting as the Cherenkov radiator, a mirror located in the
forward region with dimension (100 x 450 x 20) $mm^{3}$ and a prism that
couples the bar to an array of MCP-PMTs in the backward region with dimension
(100 x 456 x 20-51) $mm^{3}$. The high refractive index of the radiator
(n=1.44 for $\lambda=405~{}nm$) with high geometrical precision, bulk
transmittance and surface reflectance (Table 2.1) minimize the loss of photons
during propagation [2].

Manifacturing specifications of the quarts bars.  Quartz properties
Requirements Flatness $<6.3~{}\mu m$ Perpendicularity $<20~{}arcsec$
Parallelism $<4~{}arcsec$ Roughness $<0.5~{}nm$ (RMS) Bulk transmittance
$>98\%/~{}m$ Surface reflectance $>99.9\%$ / reflection

###  2.2 MCP-PMT photodetectors

The expansion volume of the quartz radiator bar is coupled in the back side
with an array of two rows of MCP-PMTs, in total 32 photodetectors per module.
Each MCP-PMT is square-shaped, has an effective area of (23 x 23) $mm^{2}$ and
is segmented into 16 channels (Fig. 3, left). The 16 TOP modules have in total
8192 channels. The photoelectrons are multiplied inside two microchannel
planes, each plane is 400 $\mu m$ thick, each channel has 10 $\mu m$ diameter.
Channels are tilted to prevent electrons from passing through the holes
without multiplication process (Fig. 3, right). The multiplication in the
short path makes the transit time spread less than 50 $ps$.

Figure 3: The Hamamatsu R10754-07-M16(N) microchannel plate photomultiplier
tube (left). Two microchannel plates with angled channels (right).

During the mass production, three types of MCP-PMTs have been developed to
improve the lifetime of the photocathode [3].

  1. 1.

Conventional MCP-PMT
In the manufacturing process of conventional MCP-PMT a lead glass disk with
holes is built with thermal and etching processes [4]. This lead glass
capillary array is then heated in presence of of hydrogen, which chemically
reduces the surfaces of holes, leaving a resistive and emissive surface that
is effective for electron amplification.

  2. 2.

Atomic layer deposition ALD MCP-PMT
This new technology does not use material containing lead which is a harmful
element restricted by the RoHS (restriction of hazardous substances) directive
of the European Union. The new glass capillary array is coated using the ALD
technology with a resistive film and with a secondary electron multiplier film
with high uniformity in each micro channel.

  3. 3.

Life-extended ALD MCP-PMT
Each components of of the MCP-PMT get an optimized backing to reduce the
trapped ions.The residual ions can be released during the development of the
electron avalanche and can reduce the lifetime of the photocathode.

Based on the development of the MCP-PMT technology during the TOP detector
construction, $44\%$ of the installed PMTs were conventional MCP-PMTs, $43\%$
were ALD MCP-PMTs and $13\%$ were life-extended ALD MCP-PMTs.

###  2.3 Front-end readout electronics

The TOP front-end electronics are required to read out the signals of all 8192
MCP-PMT channels in the whole TOP system with a global timing resolution of
better than 100 ps at a nominal trigger rate of up to 30 kHz. This is achieved
by employing specially designed $2.7~{}GHz$ wave-form sampling electronics.
The TOP readout is organized as an ensemble of 4 boardstacks per module (Fig.
4). Every boardstack contains 4 ASIC carrier boards and a single Standard
Control Read-Out data (SCROD). Every carrier board contains 4 Ice Ray Sampler
version X (IRSX) chips with 8 channels/chip [5].

Figure 4: TOP boardstack consisting of one SCROD board, four AISC boards and
one HV supply module.

For every channel an internal trigger system based on a fixed threshold marks
regions of interest. Every region of interest has a length of 32 samples,
about 12 ns. The SCROD extracts the timing of photon pulses for every channel
and transfer the data to the Belle II DAQ system.

###  2.4 Laser calibration system

The laser calibration system was designed to calibrate the time differences
between the 8192 channels of the TOP detector with a resolution better than
$100~{}ps$ [6]. To achieve this, it is necessary to reach all channels of all
photodetectors with a single photon source while maintaining a time resolution
of few tens of picoseconds in each step. To inject the calibration light in
the box containing the quartz prism, the number and size of holes should be as
minimal as possible to avoid loss of Cherenkov light. Simulation studies shown
the best optical and mechanical solution was to inject the light in 9 holes
with steps of $50.7~{}mm$ with a vertical angle of $15^{o}$ and light
numerical aperture $0.5<NA<0.6$ (Fig. 5, right).

Figure 5: Scheme of the laser calibration system (left). Picture of the prism
box with the laser calibration fibers installed.

The calibration light source is provided by a PiL040X pulsed laser diode with
$405~{}nm$ of wavelength, pulse width less than $27~{}ps$, and peak power of
$300~{}mW$. A Planar Lightwave Circuit (PLC High Power Splitter from PPI Inc.)
followed by $30~{}m$ long single-mode (SM) fibers, demultiplex the light for
the 16 TOP modules. For every module a multi-mode (MM) fiber bundle split the
light to the 9 light entries. For every bundle the end side of every fiber is
coupled with a grin lens (Selfoc-SLH) with a nominal $NA=0.6$ (Fig. 5, left).

##  3 Time calibration

The time calibration is performed in four steps [7]:

  1. 1.

Time Base Calibration
Each channel of the IRSX ASIC consists 128 sampling windows, the analog
information are stored in 32k cells and readout by 64 Wilkinson ADCs. The
different sampling windows have different delays to be precisely calibrated. A
double peak pulse with fixed time difference of $22~{}ns$ is injected in the
electronics. In order to cover all the 128 elements the double pulse is not
synchronized with the accelerator clock. Figure 6 shows the measurement of the
time difference as a function of the sample number. Time fluctuations up of
$1~{}ns$ have been reduced after the time base calibration to $42~{}ps$ of
r.m.s.

Figure 6: Time difference of two calibration pulses as a function of sample
number before (left) and after (right) the time base calibration.

  2. 2.

Time assignment of channels within a module
The time alignment of channels within a module is performed with the laser
calibration system. The small time jitter of the laser allows the different
time response of the 512 channels of each module to be measured. However
single channels can receive light directly or after reflections inside the
prism, they can also be illuminated by two fibers. A simple method is to align
the average of the time distribution in each channel. Figure 7 shows the times
for the 128 channel in a single boardstack before and after the alignment,
with respect to a common time related to the laser trigger. A more precise
method with time offset fit was implemented, it takes into account the
different intensities after reflections and the possibile propagation times
taken from Monte Carlo.

Figure 7: Times for the 128 channel in a single boardstack before (left) and
after (right) alignment.

  3. 3.

Time alignment of modules
The 16 TOP modules are aligned together within few tens of ps using cosmic
rays and dimuons from collision data. The same kind of data can be used also
for the geometrical alignment after the summer and winter stops or after an
earthquake.

  4. 4.

Alignment relative to collision time
The global $T_{0}$ with respect to the RF accelerator clock is measured using
dimuons from collision data.

##  4 TOP detector performance

Particle identification performance is estimated with Likelihood ratio by
comparing the assumed particle with respect to the sum of alternative options.
Different choices are possibile for this ratio:

$R[K/\pi]=\frac{\mathcal{L}_{K}}{{\mathcal{L}_{K}}{+\mathcal{L}_{\pi}}}~{}~{}~{}~{}~{}~{}p[h]=\frac{\mathcal{L}_{h}}{\sum_{h^{\prime}}}{\mathcal{L}_{h^{\prime}}}~{}~{}~{}~{}~{}~{}~{}\mathcal{L}_{h}=\mathcal{L}_{h}^{TOP}~{}~{}~{}~{}$

(binary PID) (global PID) (TOP only)

The PID performance of the TOP detector shown in Figure 8 was obtained by
varying the $K-\pi$ binary ratio. Two data-set periods are reported and
compared with the Monte Carlo expectation. The performance is stable in time,
the difference with respect to Monte Carlo shows the simulation of the
detector can still be improved.

Figure 8: K identification efficiency as a function of the pion mis-
identification rate for two data-set periods (blue and grey) and for the Monte
Carlo (red).

The identification efficiency for Kaons selected with $R(K/\pi)>0.5$ and the
mis-identification of pions are shown in Figure 9 in bins of momentum using
all sub-detectors (left) or TOP only (right) information. Kaons and pions have
been tagged from the $D^{*+}\to D^{0}[K^{-}\pi^{+}]\pi^{+}$ decay. The
efficiency is about $85\%$ and almost uniform in all the TOP slots. Machine
learning approach is under study where weights used to combine PID information
are not static but function of charge of the track and its kinematics.

Figure 9: Kaon efficiency as a function of momentum in the laboratory frame
for the PID criterion $R(K/\pi)>0.5$ using all sub-detectors (left) or TOP
only (right).

##  5 TOP detector upgrade

After the first run period 4 boardstacks were dead, 10 boardstacks were
showing small issues during operation. The TOP detector had in total about
600-700 dead channels out of 8192. All the boardstacks and electronics boards
showing problems have been replaced during LS1. The accumulated charge of the
photodetectors was between $0.2~{}C/cm^{2}$ and $0.3~{}C/cm^{2}$ depending to
the slot and to the photodetector type. The lifetime of MCP-PMTs is defined as
the accumulated charge which has the effect of reducing the photocathode
quantum efficiency (QE) to $80\%$. The lifetime measured in laboratory is
$1.1~{}C/cm^{2}$ in average for the conventional MCP-PMTs, $10.4~{}C/cm^{2}$
in average for the ALD MCP-PMTs and $>10.4~{}C/cm^{2}$ for the extended-
lifetime ALD MCP-PMTs. Considering the large fluctuation of these measurement
(from 0.3 to 1.7 $C/cm^{2}$ for the conventional type) and the expected
increase of luminosity and background, the conventional MCP-PMTs have been
replaced even though the QE was not expect to be significantly degraded in
average. The removed MCP-PMTs have been tested in laboratory, the measured QE
shown a faster degradation than expected. The possibile explanation is the
heating of the photodetectors due to electronics. The hypothesis was verified
by measuring the quantum efficiency degradation at different environmental
temperatures, a faster degradation was found for higher temperatures.

Figure 10: Distribution of conventional MCP-PMTs (blue), ALD MCP-PMTs (green)
and life-extended ALD MCP-PMTs (red) inside the 16 modules of the TOP detector
before LS1, after LS1, and after the next long shutdown.

Figure 10 shows the distribution of the different type of MCP-PMT before LS1,
after LS1 and the plan for the next long shutdown LS2. The production of the
missing life-extended MCP-PMTs is ongoing, 150 new MCP-PMTs have been already
delivered or ordered out of 220. The ALD MCP-PMTs have been moved from the top
to the bottom slots to have a backup option of early replacement during the
annual summer break in case of faster-than-expected QE degradation. Upgraded
electronics with less power consumption and more compact design is being
considered as well as further lifetime improvements for MCP-PMTs or new type
of photodetectors like SiPMs.

##  6 Conclusions

TOP is a new concept of a compact Cherenkov detector for particle
identification which relies on multichannel long-lifetime MCP-PMTs for the
precise measurement of the arrival position and time of individual photons.
The installation of the TOP detector has been completed in May 2016, it is
successfully operating since the start of physics collisions in March 2019.
TOP only binary PID gives $85\%$ of Kaon identification efficiency with $10\%$
of pion mis-identification rate. After LS1 the fraction of active channels
increased from $91-93\%$ to $99.5\%$. The TOP upgrade program for the LS2
shutdown is well underway with $68\%$ of missing life-extended MCP-PMTs
already delivered or ordered.

## Acknowledgements

This work was supported by the following European Union’s funding sources:
Horizon 2020 Marie Sklodowska-Curie RISE project JENNIFER2 grant agreement No.
82207, Horizon 2020 Research and Innovation project AIDAinnova grant agreement
No. 101004761.

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