# Spontaneous-emission induced ratchet in atom-optics kicked rotor quantum
walks

Nikolai Bolik  Institute of Theoretical Physics, Heidelberg University,
Philosophenweg 16, 69120 Heidelberg, Germany  Sandro Wimberger
<EMAIL_ADDRESS>Dipartimento di Scienze Matematiche, Fisiche e
Informatiche, Università di Parma, Parco Area delle Scienze 7/A, 43124 Parma,
Italy  INFN, Sezione di Milano Bicocca, Gruppo Collegato di Parma, Parco Area
delle Scienze 7/A, 43124 Parma, Italy

###### Abstract

Quantum walks have gained significant attention over the past decades, mainly
because of their variety of implementations and applications. Atomic quantum
walks are typically subject to spontaneous emissions arising from the control
fields. We investigate spontaneous emission in an atom optics kicked rotor
quantum walk. Here, spontaneous emission occurs naturally due to the driving
by the kicks, and it is generally viewed as a nuisance in the experiment. We
find, however, that spontaneous emission may induce asymmetries in an
otherwise symmetric quantum walk. Our results underscore the utility of
spontaneous emission and the application of the asymmetric evolution in the
walker’s space, i.e. for the construction of a quantum walk ratchet or for
Parrondo-like quantum games. This highlights the potential for reinterpreting
seemingly adverse effects as beneficial under certain conditions, thus
broadening the scope of quantum walks and their applications.

Atom-optics kicked rotor, Quantum walks, Bose-Einstein condensates, Floquet-
Bloch engineering, Quantum chaos, Quantum-to-classical transition

##  I ’Introduction

Over the past two decades, quantum walks (QWs) have gained considerable
attention owing to their potential to outperform classical algorithms in a
range of computational tasks [1, 2, 3]. The distinct characteristics of QWs
stem from quantum interference phenomena. For instance, they exhibit a
ballistic expansion [1, 2] as opposed to the diffusive expansion observed in
classical random walks, which are governed by stochastic processes [4].

Much like their classical counterparts, QWs can be broadly divided into two
primary types: The discrete-time and the continuous-time QWs [1]. Notably, the
discrete-time variant introduces an additional coin-degree-of-freedom, which
dictates the direction of the walker in the subsequent step.

In preceding studies, we have investigated a distinctive model, describing
discrete-time QW, employing a Bose-Einstein condensate (BEC), consisting of
87Rb atoms, with an additional spin-1/2 degree of freedom [5, 6, 7, 8, 9].
This approach markedly differs from various other experimental investigations
[10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26],
distinguishing itself by realizing the QW to occur within quantized momentum
space, where the steps of the walk are implemented by periodic pulses applied
to the condensate. This process is meticulously described by the atom-optics
kicked rotor (AOKR) model [27, 28, 29], governed by the following Hamiltonian:

$\hat{\mathcal{H}}=\frac{\hat{p}^{2}}{2}+k\cos(\hat{\theta})\sum_{j=-\infty}^{\infty}\delta(t-j\tau).$
(1)

$\hat{p}$ and $\hat{\theta}$ are the momentum and angular position operators.
The parameters $k$ and $\tau$ represent the laser kick strength and time
interval between successive pulses. Considering that experiments occur in a
periodic potential, Bloch’s theorem may be applied [28]. In our dimensionless
variables, the momentum separates into $p=n+\beta$, with an integer part $n$
and a dimensionless quasimomentum $\beta\in[0,1)$. The experimental system’s
quasi-momentum distribution, primarily determined by the BEC’s initial
temperature, has typically a width of a few percent in the Brillouin zone.

The evolution over one time period, $\tau$, can be described by the Floquet
operator:

$\hat{\mathcal{U}}=e^{-i\tau\hat{p}^{2}/2}e^{-ik\sigma_{z}\cos(\hat{\theta})},$
(2)

that consists of a free evolution part (left factor) and a kick operator
(right factor). For AOKR QWs, the free evolution between subsequent kicks has
a duration of a full Talbot time [9, 6, 7, 30], which corresponds to
$\tau=\tau_{T}=4\pi$ in our dimensionless variables [31, 28]. Without further
nuisance, the free evolution operator is thus equal to unity.

A kick typically possesses a finite pulse-duration $\tau_{p}$ itself.
Numerically, this is handled by dividing the kick into $h$ sub-kicks [31].
Each sub-kick is characterized by $\frac{1}{h}$ of the full kicking strength
and is followed by free evolution over $\frac{1}{h}$ of the total pulse-
duration. The parameter $h$, balancing computational time and approximation
accuracy, is chosen based on numerical efficiency; $h=10$ has been deemed
sufficient for reaching convergence within our parameter regime.

$\hat{\mathcal{U}}_{\mathrm{kick}}=\prod_{h}e^{-i\frac{\hat{p}^{2}}{2}\frac{\tau_{p}}{h}}e^{-i\frac{k}{h}\mathrm{cos}(\hat{\theta})\sigma_{z}}.$
(3)

In this work, the primary concern is spontaneous emission (SE) occurring when
the kick potential is on [7, 31, 27, 28], rather than effects arising from the
finite pulse width. Since a finite pulse width effectively introduces
reflective walls in momentum space [27], we set a sufficiently small duration
for $\tau_{p}$ to ensure minimal perturbation to momentum-space distributions
from this effect.

In resonance conditions, i.e. $\tau=4\pi$, atoms exhibit ballistic movement in
momentum space. This means that momentum increases linearly with the number of
kicks, revealing the typical signature of a QW [27, 28, 31, 32]. Two internal
(spin) states of the atoms are participating in the dynamics. One is moving
left, the second one right in momentum space [30, 7, 9], mirrored by the Pauli
matrix $\sigma_{z}$ in Eqs. (2)-(3). A spin rotation by a so-called coin
matrix mixes those two states in each step of the QW [30, 7, 9]. In the
absence of external perturbations, both internal states would undergo
identical evolution due to the kicks. To put it another way, their motion
would be symmetric in the walker’s space under the influence of the AOKR’s
evolution operator. This inherent symmetry is broken by imposing a ratchet
effect on the initial state, achieved by setting specific initial conditions
in the walker’s space. To establish this initial ratchet state, we link at
least two adjacent momentum classes with a relative phase of $e^{i\pi/2}$:

$\ket{\psi_{\rm R}}=\frac{1}{\sqrt{J}}\sum_{j}e^{i\cdot j\pi/2}\ket{n=j}.$ (4)

Such a relative phase shift can be introduced experimentally by the
application of conveniently timed Bragg pulses [7, 9, 33]. The mean momentum
transfer is determined by the sign of the kick potential, which varies between
the two internal states. Notably, as the parameter $J$ in Eq. (4) increases,
the dispersion due to directed kicking decreases. With $J\geqq 3$, we observe
an evolution that most closely mirrors an ideal QW [33, 8] Unless specified
differently, our analyses will be based on an initial state with $J=3$:

$\ket{\psi_{\rm
R}}=\frac{1}{\sqrt{3}}\left(-i\ket{n=-1}+\ket{n=0}+i\ket{n=1}\right).$ (5)

The final distribution then also depends heavily on the choice of the coin.
This coin operation is executed using a Rabi coupling between the atom’s two
internal states, enabled by resonant microwave pulses, effecting a unitary
rotation of these states [7, 34, 30]. The QWs presented in Ref. [9, 7, 6] are
characterized by a series of unitary operations, as given by:

$\hat{\mathcal{U}}_{\mathrm{step}}^{t}=[\hat{\mathcal{U}}\hat{Q}_{1}]^{t}\hat{\mathcal{U}}\hat{Q}_{2}.$
(6)

Here, $t\in\mathbb{N}$ represents the steps of the walk applied to the initial
state. The final momentum distribution’s key attributes, like symmetry and
form within the walker space, hinge on the specific selection of the initial
coin $\hat{Q}_{1}$ and the coin during the walk protocol $\hat{Q}_{2}$. We
find an asymmetry that emerges due to specific combinations of the coin matrix
and a non-resonant evolution of the walk due to the effects of SE. Here, we
are especially interested in this asymmetry and in its use. SE is a quantum
effect that naturally appears in driven systems. During the application of the
laser pulse, the electron in one of the atoms is excited to the energetically
higher state and spontaneously collapses into an energetically lower state.
This effect is to some extent always present in AOKR experiments [27, 28, 29]
and is typically treated as a nuisance. In this paper, however, we will
demonstrate the utility of SE-induced asymmetry for the construction of a QW
ratchet [35].

The shape of the momentum distribution itself will primarily depend on the
choice of the coin. Since we intend to analyze the behavior of QWs under the
effect of SE, we start in Sec. II with a quick analysis of the coin
parameters. Sec. III is then devoted to the introduction of SE to the system
and identifying a candidate coin for engineering a QW ratchet. The key section
Sec. IV deals with the optimization of the QW ratchet. Experimental
implications are discussed in Sec. V, while Sec. VI summarizes our results.

##  II Global behavior of symmetry

In a recent study, we analyzed discrepancies between experimental data on QWs
in momentum space [9, 6, 7] and the corresponding theory. These discrepancies
arise from light shift-induced effects on energy levels, which effectively led
to the use of a distinct, unintended coin during the walk. This effective coin
matrix, when paired with a confined (only two momentum classes with $J=2$)
initial state in momentum space, revealed an unusual behavior, leaving a
considerable portion of the probability distribution near the origin $n=0$,
that vanishes for an initial state with $J\geqq 3$ [8]. Such observations
suggest that the phase selections within a balanced coin matrix in the
quantum-walk protocol can significantly influence key aspects of the final
momentum distribution. Here, we are specially interested in the impact of the
choice of the coin parameters on the symmetry of the walk.

###  II.1 Choice of Coin

The most general form of the balanced coin matrix for the two-state system
with a pseudo spin degree of freedom of the atom is written with three Bloch
angles $\chi$, $\gamma$ and $\alpha$ and reads

$\displaystyle\begin{aligned}
\hat{M}(\chi=\frac{\pi}{4},\gamma,\alpha)&=\begin{pmatrix}e^{i\alpha}\cos(\chi=\frac{\pi}{4})&e^{-i\gamma}\sin(\chi=\frac{\pi}{4})\\\
-e^{i\gamma}\sin(\chi=\frac{\pi}{4})&e^{-i\alpha}\cos(\chi=\frac{\pi}{4})\end{pmatrix}\
\\\ \ &=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{i\alpha}&e^{-i\gamma}\\\
-e^{i\gamma}&e^{-i\alpha}\end{pmatrix}.\end{aligned}$ (7)

Here $\chi=\frac{\pi}{4}$ determines the coin to be balanced, meaning that
both internal states are weighted equally, even though with different phases.
Since we are only interested in balanced QWs, we effectively have two free
Bloch angles $\gamma$ and $\alpha$ in the coin.

In the following analysis, all walks will be initialized by the so-called
$\hat{Y}$-coin [8], choosing in Eq. (6)

$\displaystyle\hat{Q}_{1}=\hat{Y}=\hat{M}\left(\frac{\pi}{4},\frac{\pi}{2},0\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}1&i\\\
i&1\end{pmatrix}.$ (8)

The $\hat{Q}_{2}$-coin in the walk protocol is then executed by the
$\hat{W}$-coin or the $\hat{G}_{H}$-coin [8]:

$\displaystyle\hat{W}$ $\displaystyle=$
$\displaystyle\hat{M}\left(\frac{\pi}{4},0,0\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
-1&1\end{pmatrix},$ (9) $\displaystyle\hat{G}_{H}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}\widehat{=}\frac{i}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}$ (10) $\displaystyle=$
$\displaystyle\hat{M}\left(\frac{\pi}{4},\frac{3\pi}{2},\frac{\pi}{2}\right).$
(11)

Here the equivalence in Eq. (11) is written, because both coins result in the
same walk since a global phase does not induce any change. As discussed in
detail in [8], a walk that evolves under the the $\hat{G}_{H}$-coin, without
any additional distortions, displays markedly different characteristics than a
walk evolved under the $\hat{W}$-coin. Notably, there’s a pronounced peak near
zero momentum, as illustrated in Fig. 1.

In the following, we will introduce a quantity to characterize the different
regimes of symmetry within the parameter regime spanned by the Bloch angles
$\alpha$ and $\gamma$.

Figure 1: Shown are the momentum distributions after $T=20$ steps in the
evolution. The walks are evolved under the $\hat{W}$-coin and the
$\hat{G}_{H}$-coin, respectively.

###  II.2 Mapping out parameter space

We are particularly interested in evolving asymmetries in the walk,
necessitating the definition of an asymmetry observable $S$. Similarly to
[36], the value $S$ is determined by taking the probability to the right of
the inversion point at momentum $n=0$ and subtracting the probability to the
left of it. In the context of Parrondo games two losing strategies combined
result in a winning strategy. Using the AOKR-framework for Parrondo-like
quantum games [36], the sign of the variable $S$ defines the winning or the
losing strategy, respectively. In our context, the same measure is
subsequently employed as a quantification of symmetry

$\displaystyle S=\sum_{n>0}|\psi_{n}|^{2}-\sum_{n<0}|\psi_{n}|^{2}.$ (12)

Here $|\psi_{n}|^{2}$ notes the amount of probability associated with momentum
class $n$.

In what follows, we will present arguments to shape some intuition and provide
qualitative insight into the overall behavior of the observable
$S(\gamma,\alpha)$.

Figure 2: Shown is a colormap of the symmetry indicator $S(\gamma,\alpha)$ in
dependency of the free Bloch angles $\alpha$ and $\gamma$, evaluated after
$T=15$ steps in the evolution. The blue dots reproduce the walk induced by the
$\hat{G}_{H}$-coin shown in Fig. 1, while the red dots reproduce the walk
induced by the $\hat{W}$-coin. The white lines map out the coin configurations
where an additional symmetry is found, investigated in detail in Sec. II.2.1.

####  II.2.1 Symmetric coin configurations

For special relations between $\gamma$ and $\alpha$ additional symmetries can
be found in the coin itself. Empirically, this additional symmetry in the coin
induces more symmetrical behavior in the walk. Let us discuss two cases.

For the first case consider $\gamma=\alpha$, $\gamma=\alpha\pm 2\pi$:

$\displaystyle\begin{aligned}
\hat{M}(\chi=\frac{\pi}{4},\gamma,\alpha=\gamma)&=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{i\alpha}&e^{-i\alpha}\\\
-e^{i\alpha}&e^{-i\alpha}\end{pmatrix}\ \\\ \
&\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}\phi&\rho\\\
-\phi&\rho\end{pmatrix}.\end{aligned}$ (13)

For the second case consider $\gamma=\alpha\pm\pi$:

$\displaystyle\begin{aligned}
\hat{M}(\chi=\frac{\pi}{4},\gamma,\alpha=\gamma\pm\pi)&=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{i\alpha}&-e^{-i\alpha}\\\
e^{i\alpha}&e^{-i\alpha}\end{pmatrix}\ \\\ \
&\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}{\phi}&-{\rho}\\\
{\phi}&{\rho}\end{pmatrix}.\end{aligned}$ (14)

In this setup, we consistently have two pairs of identical phases in the coin.
Generally, this condition is not fulfilled. In subsequent computations for
each timestep, the internal states are mixed in a comparable manner. It
appears plausible that when the states are mixed with similar phases along the
protocol, they also evolve in a more balanced way, however systematically in
opposite directions. This seems to induce overall a more balanced walk.

In Fig. 2, the walk evolves for $T=15$ timesteps, and the symmetry indicator
$S(\gamma,\alpha)$ is plotted as a function of the free Bloch angles. Dominant
red or blue stripes lines within the plot mark regions where the QW skews left
or right. Intersecting the white lines in-between these areas with $S=0$ align
qualitatively with the anticipated lines in parameter space $\gamma=\alpha$
and $\gamma=\alpha\pm\pi$, which possess the coin’s additional symmetry from
Eq. (13) and Eq. (14), respectively. Not all walks corresponding to $S=0$ in
Fig. 2 perfectly match predictions, seemingly distorted by high asymmetry
zones, indicating a more intricate dynamic than the rudimentary theory
suggests. Still, the theory capably predicts the qualitative behavior of where
the balanced walks are found in parameter space due to coin phase symmetry,
aiding in understanding for the system. Also the empirical lines with $S=0$
get closer to the predicted lines for a higher number of timesteps, as has
been numerically verified.

In Fig. 2, the blue and red dots are coin types that are identical up to a
global phase to $\hat{W}$ and $\hat{G}_{H}$ from Eq. (9) and Eq. (11),
respectively. Those points lie on the white lines predicted by our argument
from above.

$S$ indicates the direction of evolution for the walk, but not the walk’s
shape. This means that walks may be balanced, as quantified by $S$, but their
shape does not necessarily exhibit an inversion point, underscoring the rarity
of true symmetry. Indeed, only two classes of walks exhibit a concrete
inversion point.

One class is defined by the $\hat{G}_{H}$-coin walk, presented in Fig. 1. It
is replicated across parameter combinations on the coin-symmetry lines, with
half-integer multiples of $\pi$ for $\gamma$ and $\alpha$, indicated by blue
full dots in Fig. 2. Conversely, the $\hat{W}$-coin walk is echoed with
integer Bloch angle multiples of $\pi$, shown by red full dots in Fig. 2. All
these walks possess coins that are exclusively real or complex, respectively.
When initialized by the $\hat{Y}$-coin, they maintain the initial rotation of
$\pi$ in-between the internal states. All other cases introduce both real and
imaginary phase components, thus lifting this initial rotation which then
appears unfavorable for a mirror-symmetric walk evolution. Thus, other
configurations are either balanced without a mirror axis or are distinctly
asymmetric.

####  II.2.2 Asymmetric coin configurations

In Fig. 2, the asymmetric regions manifest as broad diagonal stripes, echoing
the observations from Sec. II.2.1. To investigate the deep asymmetric regime,
a broader relationship between $\alpha$ and $\gamma$ can be expressed as
$\gamma=\alpha+\zeta$, where $\zeta$ is arbitrary but constant, giving the
coin matrix

$\displaystyle\begin{aligned}
\hat{M}(\chi=\frac{\pi}{4},\gamma&=\alpha+\zeta,\alpha)=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{i(\alpha)}&e^{-i(\alpha+\zeta)}\\\
-e^{i\alpha+\zeta}&e^{-i\alpha}\end{pmatrix}\ \\\ \
&\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}{\phi}&{\rho}e^{-i\zeta}\\\
-{\phi}e^{i\zeta}&{\rho}\end{pmatrix}.\end{aligned}$ (15)

Empirically, we can conclude from Fig. 2 that the walk evolves asymmetrically
to the left for $0<\zeta<\pi$. Thus $e^{i\zeta}$ lives on the right half of
the complex unit circle with $\mathrm{RE}(e^{i\zeta})>0$ and $e^{-i\zeta}$
lives on the left half of the complex unit circle with
$\mathrm{RE}(e^{-i\zeta})<0$. In comparison, the walk evolves asymmetrically
to the right for $-\pi<\zeta<0$. In this case again $e^{i\zeta}$ lives on the
left part of the complex unit circle and $e^{-i\zeta}$ on the respective right
half. So if the two $\zeta$-related phase angles are effectively interchanged,
then the direction in which the walk is skewed is also switched. If this
additional $\zeta$-related phase vanishes, then we recover the scenario from
Sec. II.2.1.

Looking at the parameter scan from Fig. 2, we also find spots in the
asymmetric parameter-areas, which are exceptionally asymmetric. In the
$\mathcal{C}=(\alpha,\gamma)$ coordinate frame, we find that these spots are
located at:

$\displaystyle\mathcal{C}_{1}^{+}=(\alpha=1/2\
\pi,\gamma=0)\rightarrow\zeta=-\pi/2$ (16)
$\displaystyle\mathcal{C}_{2}^{+}=(\alpha=3/2\
\pi,\gamma=\pi)\rightarrow\zeta=-\pi/2$ (17)
$\displaystyle\mathcal{C}_{1}^{-}=(\alpha=1/2\pi,\gamma=\pi)\rightarrow\zeta=+\pi/2$
(18)
$\displaystyle\mathcal{C}_{2}^{-}=(\alpha=3/2\pi,\gamma=2\pi)\rightarrow\zeta=+\pi/2$
(19)

Here the upper index notes that the asymmetry is to the right (+) or to the
left (-), respectively. These four points have in common, that they are fully
complex on the diagonal and fully real on the off-diagonal elements. With a
respective irrelevant phase factor of $i$, this characteristic is similar to
the initialization by the coin $\hat{Y}$, which is the same for all walks.
This effectively lifts the initial rotation of $\pi$ between the internal
states provided by the $\hat{Y}$-coin and causes the walk to be highly
asymmetric.

While the results oft the section may be interesting for designing Parrondo-
like games [36], we are in the following interested in how SE affects the
symmetry of the QW.

##  III spontaneous emission

So far, our focus has been on asymmetries in the coherent QW stemming from the
selection of the coin matrix. However, noise-induced asymmetries garner
particular attention, given their potential utility in crafting a QW ratchet
[35]. The QW ratchet described in this study is particularly noteworthy as it
is induced by SE. Typically, SE is associated with merely causing decoherence
[6], but in this case it contributes to the creation of a QW ratchet. This
phenomenon represents a shift in perspective, where what is usually considered
a nuisance becomes advantageous.

###  III.1 Spontaneous emission in the AOKR

In the context of AOKR-dynamics a SE event can be interpreted as an event in
which the atom goes into an excited state during the application of the laser-
pulse and then spontaneously falls either into the energetically lower
$|1\rangle$ or $|2\rangle$ state. All excitations are assumed to be far away
from an internal electronic resonance, in contrast to [37, 38]. During the
spontaneous de-excitation a photon is emitted, inducing also a shift of the
atomic quasimomentum [31, 28]. For the AOKR we arrive at the following
effective Hamiltonian, for which a detailed derivation can be found in chapter
5 of [39] and [40] for non-resonant driving

$\displaystyle\begin{aligned}
\hat{\mathcal{H}}&=\frac{1}{1+\frac{\mu^{2}}{4\delta^{2}}}\frac{\Omega_{1}^{2}}{8\delta}\big{(}\mathrm{cos}(2k_{L}\hat{x})+1\big{)}|1\rangle\langle
1|\ \\\ \
&-\frac{1}{1+\frac{\mu^{2}}{4\Delta^{2}}}\frac{\Omega_{2}^{2}}{8\Delta}\big{(}\mathrm{cos}(2k_{L}\hat{x})+1\big{)}|2\rangle\langle
2|.\end{aligned}$ (20)

Figure 3: Level scheme of the AOKR quantum walk with the two internal levels
$\ket{1}$ and $\ket{2}$. The excited state $|e\rangle$ is used to impart the
kicks to the atoms by a off-resonant standing wave of laser light.
$\omega_{L}$ denotes the frequency of the kicking laser. $\delta$ and $\Delta$
denote the detunings of the two possible transitions with respect to the laser
frequency.

Let $\mu$ denote the SE event rate, defined as:

$\mu=\mu_{1}+\mu_{2}=\frac{k_{1}}{\tau_{p}\tau_{\rm
SE}|\delta|}+\frac{k_{2}}{\tau_{p}\tau_{\rm SE}|\Delta|}$ (21)

where $\delta$ and $\Delta$ are the respective detunings from the resonance,
shown in Fig. 3. Given that the detuning is small compared to the laser
frequency, the approximation $k_{1}=k_{2}=k$ is valid [41, 7]. Consequently we
have $\delta=-\Delta$ and thus we find $\mu_{1}=\mu_{2}$ for the SE rates in
Eq. (21). Also, $\tau_{p}$ is the pulse duration, and $\tau_{\rm SE}$
represents the lifetime of the transition.

In a numerical context, a SE event translates into a projection onto one of
the internal states:

$\displaystyle\hat{P}_{|1\rangle}$
$\displaystyle=\frac{1}{\mathcal{N}}\left(A|\psi_{|1\rangle}\rangle\otimes|1\rangle\langle
1|+B|\psi_{|2\rangle}\rangle\otimes|1\rangle\langle 2|\right),$ (22)
$\displaystyle\hat{P}_{|2\rangle}$
$\displaystyle=\frac{1}{\mathcal{N}}\left(C|\psi_{|1\rangle}\rangle\otimes|2\rangle\langle
1|+D|\psi_{|2\rangle}\rangle\otimes|2\rangle\langle 2|\right),$ (23)

accompanied by a shift in quasimomentum, $\beta_{\rm SE}$. Here, $\mathcal{N}$
serves as a normalization factor and $A,B,C,D$ are detuning-dependent weights.
For our analysis, $A,B,C,D$ can be approximated as $1/2$, see [40].

With the inclusion of SE, the kick’s evaluation becomes more complex [28, 31].
For each of the $h$ sub-kicks from Eq. (3), we determine, based on an event
probability $p_{\mathrm{event}}=\frac{\mathrm{p_{SE}}}{h}$, whether an SE
event transpires. If it does, with a 50-50 chance we decide the projection
onto either $\hat{P}_{|1\rangle}$ or $\hat{P}_{|2\rangle}$ internal states,
and the quasimomentum shift is randomly selected from $\beta_{\rm
SE}\in\mathrm{uniform}(-0.5,0.5)$. In the event’s presence, the partial kick
is expressed as:

$\displaystyle\hat{\mathcal{U}}_{\mathrm{SE}}=\hat{P}_{|i\rangle}e^{-i\frac{(\hat{p}+\beta_{\rm
SE})^{2}}{2}\frac{\tau_{p}}{h}}e^{-i\frac{k}{h}\mathrm{cos}(\hat{\theta})\sigma_{z}},$
(24)

with $|i\rangle$ being $|1\rangle$ or $|2\rangle$, with a probability of
$1/2$. The Pauli matrix $\sigma_{z}$ incodes that one internal state is moving
left, the other one right due to the kick. Our approach concurrently accounts
for the kick’s finite duration and the potential of multiple events during a
single pulse duration. For ${}^{87}\mathrm{Rb}$ quantum walks [7, 6, 9], the
Talbot time is $103\mu$s, equivalent to $4\pi$ in our units. In AOKR
experiments, the typical pulse width is of the order of $100$ns, but much
shorter pulses should be possible, see e.g. [42, 43]. We use here throughout
$\tau_{p}=0.005$, corresponding to a pulse duration of a few tens of ns. This
minimizes the impact of possible perturbations due to the finite pulse width
[44, 45, 46, 31] for longer walks of more than a few tens of steps.

The total kick is then again written as:

$\hat{\mathcal{U}}_{\mathrm{SE-
kick}}=\prod_{h}\hat{\mathcal{U}}_{\mathrm{SE}}.$ (25)

The shift in quasimomentum enters in the free evolution in-between subsequent
kicks with a duration of a full Talbot period in the same way.

$\hat{\mathcal{U}}_{\mathrm{SE-free}}=e^{-i\frac{(\hat{p}+\beta_{\rm
SE})^{2}}{2}\tau_{T}}$ (26)

Since the shift in quasimomentum comes from a uniform distribution and the
Talbot period is with a duration of $\tau_{T}=4\pi$ relatively large compared
to the duration of the pulse itself, the shift in quasimomentum is of a much
greater significance here, then during the kick pulse in Eq. (24) [28, 31].
Indeed, the free evolution between subsequent pulses with the duration
${\tau_{T}=4\pi}$ but off-resonant shift in quasimomentum due to SE, can lead
to strongly non-ballistic (nonresonant) diffusive evolution [6, 39] that
effectively destroys the QW.

has been identified as a strong nuisance due to their long duration. Mainly,
this effect limits the regime of event rates in which SE-induced effects will
be observable, which will be discussed in more detail in the following
analysis.

###  III.2 Spontaneous emissions during quantum walk

From Fig. 4, we observe that the walk under the $\hat{W}$ coin, with
increasing SE rates, becomes more classical and exhibits mild asymmetry. This
can be read off $S$ which is shown in the top right corner of each subplot.
Conversely, the walk under the $\hat{G}_{H}$ coin displays a quite pronounced
asymmetry, always a magnitude larger when compared to the $\hat{W}$ walk. For
high event probabilities, the ballistic peaks for positive momentum classes
are noticeably damped by SE. However, the peaks for negative momentum classes
are less affected. As SE rates increase, the walk becomes more classical and
the effect blurs out.

Figure 4: Two different walk protocols evaluated after $T=15$ steps in the
evolution with increasing SE. Both protocols are initialized by the
$\hat{Y}$-coin. The left row is then evolved under the $\hat{G}_{H}$-coin,
while the right row is evolved under the $\hat{W}$-coin. From top to bottom SE
rates increase. The number in the upper right corner in each plot indicates
the symmetry of the evolution $S$. The $\hat{G}_{H}$-walk develops a strong
asymmetry while the $\hat{W}$-walk becomes only slightly asymmetric. It might
be suspected that the weak asymmetry in the $\hat{W}$-protocol is due to
numerical fluctuations. However, it appears to be relatively robust. The
distributions present an average over $10000$ realizations, with the given
probabilities of $p_{\rm SE}$ per kick. Figure 5: Scan of the symmetry
$S(\gamma,\alpha)$ of the QW, evaluated after $T=15$ steps in the walk. The
walk is initialized by the $\hat{Y}$-coin and then evolved under
${\hat{M}(\chi=\frac{\pi}{4},\gamma,\alpha)}$. From top to bottom SE rates are
increased. The red dot at $(\gamma=0,\ \alpha=0)$ marks the more symmetric
walk evolved under the $\hat{W}$-coin, while the blue dot at $(\gamma=1.5\pi,\
\alpha=0.5\pi)$ marks the asymmetric walk evolved under the $\hat{G}_{H}$-coin
(compare to Fig. 2). It appears that the $\hat{G}_{H}$-walk lives in a more
unstable area in parameter space with respect to the symmetry of the evolution
under the influence of SE, causing an initially symmetric walk to evolve
asymmetrically under the introduction of SE. Each pixel is calculated as an
average over 1000 event trajectories.

With rising SE rates, the diagonal structures within the free Bloch angles
parameter domain blur, as illustrated in Fig. 5. In this parameter landscape,
the $\hat{W}$-walk (red dot) maintains a position where even at high SE-rates,
only a small deviation from the initial parameter scan from Fig. 2, resulting
in minor asymmetries, also consistent with Fig. 4. In contrast, the
$\hat{G}_{H}$-walk (blue dot) evolves strong asymmetries as it lies in a
region dominated by the pronounced asymmetry. This is due to increasing SE
rates, the asymmetric regions, governed by $\mathcal{C}_{i}^{\pm}$, shift
towards larger values of $\gamma$. This shift now situates the $\hat{G}_{H}$
coin within the asymmetric region. This finding opens a route to design an
open QW ratchet.

Within the parameter space of $\gamma$ and $\alpha$, SE introduces two key
alterations: a transition from a strong stripe to a more faint pattern,
followed by a drift of the asymmetric regions to larger values of $\gamma$.
The walk exhibiting an asymmetry then depends on the coin’s initial position
within this parameter space. The effects are counteracting each other. The
shift in $\gamma$ situates the $\hat{G}_{H}$-walk in a region of higher
asymmetry vs. the fainting amplitude of asymmetry as a consequence of the off-
resonant walk. Thus, depending on the choice of coin an asymmetric evolution
of the walk can be constructed with a corresponding SE rate. In other words,
for given windows of $p_{\rm SE}$, the initially symmetric QW can be
engineered to be maximally asymmetric.

Numerically, SE are implemented by two distinct effects: (1) the projection
onto an individual internal state with a 50:50 probability, and (2) the shift
in quasimomentum. We studied each of these effects separately for a better
understanding. Focusing solely on the projection with resonant quasimomentum,
we observe a shift in the stripe pattern in seen in Fig. 2 towards higher
values of $\gamma$. This shift becomes pronounced at SE rates of approximately
$20\%$, though it is also present at lower rates in a diminished form. The
$\hat{Y}$-Coin is located at $(0,0)$, which, within this scenario, represents
a particularity since the white line with $S=0$ always extrapolates to the
origin $(0,0)$. Hence this point is not effected by the shift. The other coin,
$\hat{G}_{H}$, is positioned at $(3/2\pi,1/2\pi)$ and finds itself in a region
with $S\neq 0$ due to the shifting of the stripes. Adding (2) the shift in
quasimomentum present in the numerical model of SE, the overall image becomes
noticeably noisier. This is attributed to the new, shifted quasimomentum being
drawn from a broad random distribution, hiding the effect of the projection.
Unfortunately, with both SE-induced effects present, the clarity of the stripe
shift in Fig. 5 is compromised. The combination of these effects leads to a
washing out of the data for $p_{\rm SE}\geq 20\%$ SE rates, which motivates
the further study of $p_{\rm SE}\leq 15\%$. Overall we can say that the SE-
induced shift of quasimomentum, making it in particular non-resonant, together
with the right choice of the coin induces the asymmetry in the QW effected by
SE.

##  IV Optimizing the QW ratchet

The SE-induced asymmetry observed for the $\hat{G}_{H}$-walk is intriguing
since it may be used for the construction of a ratchet. This section analyzes
in detail such an engineering of an AOKR-ratchet in the quantum- and as well
in the classical regime.

Previous AOKR ratchets were so-called Hamiltonian ratchets [47, 48, 49], where
the symmetry is broken by the initial conditions, while the evolution remains
coherent. This raises curiosity about the optimal parameter choice to maximize
the asymmetry of the $\hat{G}_{H}$-walk and thereby enhance the efficacy of an
SE-induced ratchet. For our analysis we will only consider the walk of the
$\hat{G}_{H}$-coin, since unperturbed it has an inversion point. Therefore, it
is a symmetric walk without SE that turns asymmetric only when perturbed by
SE. Thus our AOKR system is a suitable candidate for a purely SE-induced
ratchet.

###  IV.1 Initial state dependence

Any AOKR QW [7, 9, 33, 32, 30, 6, 8] is built on a coherent Hamiltonian
ratchet and hence on the localization of the initial conditions within the
kicking potential [47, 28, 33, 48]. The width of the initial state in momentum
space and thus the amount of momentum classes included in the initial state
$J$, as in Eq. (4), holds significant relevance for the shape of the
distribution [33]. As discussed in [8], a broader initial state considerably
diminishes the portion of the distribution proximal to $n=0$ for the
$\hat{G}_{H}$-walk. Consequently, a broader initial state implies diminished
central probability, which is instead prevalent in the tails, amplifying the
asymmetry.

The walk we are analyzing exhibits perfect symmetry for $p_{\rm SE}=0$. At
elevated SE probabilities, the walk reverts to a classical system due to
strong phase decoherence. This suggests the existence of an optimal SE
probability that maximizes the asymmetric walk evolution. The argument holds
within the region in which the walk is not yet dominated by classical
dynamics, which would be the case around $p_{\rm SE}=10\%$ and below. Our
hypothesis is supported by Fig. 6 (a) which shows the dependency of the
symmetry observable $S$ on $p_{\rm SE}$ and on $J$. We find that the curve of
the symmetry indicator $S(p_{\rm SE},J)$ improves with increasing momentum
classes $J$ in the initial state and maximizes for $T=15$ at $p_{\rm
SE}\approx 0.1$. However, from Fig. 4 we know that the regime up to which the
resonant tails of the distribution are still visible lies roughly around
$p_{\rm SE}\approx 0.05$, which would be the regime in which a ratchet that
exhibits quantum resonance [27, 28, 31] might be well observable.

Figure 6: Dependency of the symmetry observable $S(p_{\rm SE})$ on the
probability per kick $p_{\rm SE}$ for a SE event. Panel (a): $S(p_{\rm SE})$
for the walk conducted by the $\hat{G}_{H}$-coin and observed after $T=15$
steps with an initial state of increasing width, as described by Eq. (4) and
denoted by $J$. Panel (b): $S(p_{\rm SE})$ with increasing numbers of
timesteps $t$. Here the initial state is conducted by $J=3$ momentum classes.
The red curves show the system for three momentum classes in the initial state
and observed at $t=15$ steps in the evolution, as also discussed in Fig. 1.
The black curves represent $S(p_{\rm SE},t)$ for fixed $p_{\rm SE}=\ 1\%,\
5\%,\ 10\%,\ 15\%$. These curves are shown in panel (c) with their
corresponding power-law fits with exponents: $a=1$ for $p_{\rm SE}=1\%$ for
the entire time range, crossover from $a=1$ towards $a=0.5$ for $p_{\rm
SE}=5\%\dots 15\%$. With increasing SE rates, the quantum regime with $a=1$
(straight line fits) becomes shorter and shorter with respect to the crossover
part (dotted lines). The curves and their respective fits are also shown in an
inset on a double logarithmic scale. Data is obtained from averages over 10000
trajectories modeling SE [39, 40].

The biggest improvement on the asymmetry comes, when increasing from an
initial state constructed with $J=2$ to $J=3$. This observation seems to align
with the results from [8], since for $J=3$ the part of the momentum
distribution located around $n=0$ significantly decreases and is thus found in
the tails, contributing more to the asymmetry. A similar argument can be made
when increasing from $J=3$ to $J=4$. For broader initial states the
improvement becomes less significant. The better the localization in angular
$\theta$ space, the better the walks evolve with minimal dispersion. In our
simulations, we have typically chosen $J=3$ since it seems as a good tradeoff
between experimental feasibility and the amplitude of asymmetry. Also note
that the asymmetries do not fully revert to zero, even far in the classical
regime.

###  IV.2 Time dependence of the asymmetry

The temporal evolution of asymmetry offers insights into reasonable timescales
for observing this effect experimentally. In Fig. 6 (b), $S(p_{\rm SE},t)$
reveals how this unfolds. Notably, the red curve represents the state at
$t=15$, as previously elaborated. The black curves delineate the asymmetry’s
temporal development, $S(p_{\rm SE},t)$, at specific $p_{\rm SE}$ values. The
measurement of $S$ is connected to the speed of the expansion of the walks,
which clearly depends on $p_{\rm SE}$. The signature of QWs is their ballistic
expansion, which would correspond to an exponent of $a=1$ for a power-law fit
of $S(t)\propto t^{a}$. Conversely, a classical walk showcases a diffusive
expansion, signified by $a=1/2$. As Fig. 6 (c) suggests, at a marginal $p_{\rm
SE}=1\%$, the expansion remains entirely within the quantum regime, exhibiting
$a=1$. As $p_{\rm SE}$ increases, we discern a progression from a purely
quantum expansion ($a=1$) to one that is more classical with a diminished
exponent $a$. Notably, at elevated $p_{\rm SE}$ values, the trend
predominantly aligns with classical behavior, approaching an exponent of
$1/2$. Also note that for progressing timesteps $t$ the maximal observable
asymmetry is visible at smaller values of $p_{\rm SE}$. This effect is mainly
caused by the off-resonant quasimomenta in the free evolution between
subsequent pulses.

For event probabilities far in the classical regime the walk exhibits an
asymmetry that initially grows fast and then reverts due to strong decoherence
as time proceeds. This is especially present in the black curve in Fig. 6,
corresponding to an event probability of $p_{\rm SE}=15\%$. In this case, not
discussed in further detail, the off-resonant quasimomenta dominate the
dynamics and asymmetries effectively decline as time proceeds.

Essentially, all this demonstrates a transition from quantum to classical
behavior. Augmented decoherence — either due to pronounced $p_{SE}$ values or
a combination of moderate $p_{\rm SE}$ with extended durations — renders the
walk diffusive. Three distinct regimes emerge, identified by the best
parameters from a powerlaw fitting:
(i) The pure quantum regime, evidenced at $p_{\rm SE}=1\%$.
(ii) An intermediate regime exhibiting a quantum-to-classical transition as
$t$ increases, seen at ${p_{\rm SE}=5\%,10\%}$.
(iii) A predominantly classical regime, as evident for $p_{\rm SE}=15\%$ and
above. This observation is in agreement with Fig. 4, where we identified the
diminishing resonant tails in the momentum distribution for $p_{\rm SE}=15\%$.

##  V Experimental constraints

Typical experiments start with a Bose-Einstein condensate centered around
$n=0$, but with a distribution in quasimomentum of finite width [9, 7, 42].
In-between the pulses and during the pulse itself, this nonzero quasimomentum
causes non-resonant phase effects [28, 31]. Thus, similar to the influence of
SE, quantum resonances are perturbed by non-resonant $\beta$´s. While up to
now, the initial quasimomentum was chosen to be vanishing
$\beta_{\mathrm{init}}=0$, now we consider for our simulations a Gaussian
distribution of width $\beta_{\rm FWHM}$ modeling experimental reality [9, 7,
28, 42].

Figure 7: $S$ as a function of the free Bloch angles $\alpha$ and $\gamma$
under increasing width of the initial quasimomentum distribution. $S$ is
evaluated after $T=15$ steps in the evolution of the walks. The quasimomentum
is drawn from a Gaussian distribution with width $\beta_{\mathrm{FWHM}}$. Each
point in the parameter scan is averaged over 1000 trajectories. The red dot at
$(\gamma=0,\ \alpha=0)$ and the blue dot at $(\gamma=1.5\pi,\ \alpha=0.5\pi)$
mark the parameters corresponding to the $\hat{W}$-walk and the
$\hat{G}_{H}$-walk, respectively.

Similarly to the introduction of SE from Sec. III.2, a finite quasimomentum
distribution causes the asymmetric areas to bulk around the points of high
asymmetry $\mathcal{C}_{i}^{\pm}$, as demonstrated in Fig. 7 (a) and (b). A
similar deviation from the original stripe pattern has been noted In Sec.
III.2. For broader quasimomentum distributions, see Fig. 7 (b), the
asymmetries become weaker and the walks become more diffusive. However, in
contrast to the perturbations caused by SE, there is no additional shift of
the asymmetric regions to higher $\gamma$. Thus the walks calculated by the
$\hat{G}_{H}$-coin and the $\hat{W}$-coin do not evolve new strong
asymmetries. Overall they just become more "classical"-like in the sense that
they show a diffusive instead of a ballistic evolution. Thus the shift noticed
in the asymmetric regions of Fig. 5 seems to be an effect unknown to other
typical forms of perturbations in the AOKR-framework.

However, the qualitative behavior of $S(\gamma,\alpha)$ under the influence of
a non-resonant quasimomentum distribution is similar to the non-resonant
evolution found under the influence of SE.

Figure 8: Shown is the time dependence of the symmetry observable $S$.
Considered are the combined effects of finite quasimomentum with $\beta_{\rm
FWHM}=0.01$ and SE. The individual curves showcase several different values of
${p_{\rm SE}=1\%,\ 5\%,\ 10\%,\ 15\%}$. The straight lines mark out the regime
that results with an exponent of $\alpha=1$ in a power-law fit, while the
dotted areas mark the regime with $\alpha<1$. The curves are evaluated as an
average of 200 initial quasimomenta, each realized with 1000 SE-trajectories.

It has been numerically verified that the walk does indeed demonstrate no
significant asymmetries over the range of 20 timesteps under the influence of
solely finite quasimomentum and without SE. In Fig. 8, we see the time
dependence of $S$ for several different values of $p_{\rm SE}$, similar to
Fig. 6 (c), however with additional quasimomentum of $\beta_{\rm FWHM}=0.01$.
The figure indicates that the effect survives also with an additional
quasimomentum, which would be present in an experimental setup. Again we
observe qualitatively similar to Fig. 6 (c) the quantum to classical
transition with three different regimes, corresponding (i) to a pure quantum
mechanical regime, (ii) a regime that starts ballistic and then evolves a
diffusive motion as decoherence increases over time, and (iii) the
predominantly classical regime. The finite initial quasimomentum introduces
additional dispersion, which sets the expectation that the transition should
settle in earlier. However, this is only partially true. For higher event
probabilities $p_{\rm SE}=10\%$, we find that the transition settles in for
shorter times, which would also be expected due to higher dispersion in the
system. This additional dispersion slightly shifts the borders of the quantum
mechanical regime (ii) with respect to the case of Fig. 6. It is worth
mentioning that for a hypothetical future experiment, too large pulse
durations for $\tau_{p}$ would effectively induce barriers in momentum space
[27, 44, 50]. However, a QW conducted with a ${}^{87}\mathrm{Rb}$ BEC has
already been found to be observable on the discussed timescales of up to 20
timesteps [9, 6, 7, 8].

##  VI Conclusion

Our study has elucidated insights into the intricate interplay between
symmetry and the pivotal role of the choice of the coin and the impact it has
on the system’s overall dynamics. The introduction of SE has proven
instrumental as a mechanism that introduces asymmetry into an initially
symmetric QW for the construction of a QW ratchet.

Typically, SE-induced effects are detrimental. However, SE manifests indeed
beneficially under certain conditions [51, 37, 31]. Our observations on the
temporal behavior of the symmetry observable $S$ chart a distinct transition
from quantum to classical behavior depending on the strength of the introduced
SE. Evidence for this is found by an power-law fit to $S(t)$ thereby observing
a transition from a ballistic to a sub-ballistic expansion, corresponding to
the quantum to classical transition in the walk. The origin of the observed
asymmetry is an interference effect arising from specific choices of the coin
matrix and the non-resonant free evolution due to the shifts in quasimomentum
induced by SE.

Our findings have proven to be robust under the influence of finite initial
quasimomentum, which would be present in an experimental setup due to finite
temperature. The resilience of the observed effects under this perturbation
underscores the feasibility for the experimental realization of an SE-induced
QW ratchet.

Overall, this investigation offers a deeper, more nuanced understanding of the
dynamics of AOKR QWs and the influence of SE on its dynamics. By delineating
the unexpected utility of SE for an AOKR ratchet and highlighting the
robustness of our system against quasimomentum, our work provides insight for
potential future experiments along the line of [7, 6, 9, 36].

###### Acknowledgements.

We are grateful to Caspar Groiseau for help on the implementation of SE
simulations. S.W. acknowledges funding by Q-DYNAMO (EU HORIZON-
MSCA-2022-SE-01) with project No. 101131418 and by the National Recovery and
Resilience Plan, Mission 4 Component 2 Investment 1.3 – Call for tender No.
341 of 15/03/2022 of Italian MUR funded by NextGenerationEU, with project No.
PE0000023, Concession Decree No. 1564 of 11/10/2022 adopted by MUR, CUP
D93C22000940001, Project title “National Quantum Science and Technology
Institute".

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