# On Base Station Localization for State Estimation over Lossy Networks

Ufuk Topcu, Kenneth Hsu, and Kameshwar Poolla The authors contributed equally
to this work.This work was supported in part by the NSF under Grant ECS
03-02554.Authors are with the Department of Mechanical Engineering at the
University of California at Berkeley.

###### Abstract

We consider a state estimation problem where observations are made by multiple
sensors. These observations are communicated over a lossy wireless network to
a central base station that computes estimates via a Kalman filter. The goal
is to determine the optimal location of the base station under a certain class
of packet loss probability models. It is shown in the two sensor case that the
base station is optimally located at one of the sensor locations. Empirical
evidence suggests that the result holds in some generality.

##  1 Introduction

The recent confluence of low-cost sensing, communication, and computation
technologies has led to much interest in the development of wireless sensor
networks for estimation and control. These wireless sensor networks provide
for an economical means of extracting greater performance and efficiency in a
variety of applications, such as manufacturing and chemical plant processes
[Moyne], indoor climate control (HVAC) [Pakzad], environment monitoring
[Szewczyk], electrical power distribution [Chong], and automatic traffic flow
[Giridhar].

There are many advantages that wireless sensor networks hold over their wired
counterparts. For example, the fact that communication is performed wirelessly
eliminates any physical connection between the various nodes in the network.
This enables wireless sensor networks to be implemented without any major
infrastructure overhauls. This also allows for the design of networks that can
gracefully accept dynamic changes to its structure, such as the addition or
subtraction of nodes, or in the case of mobile sensor networks, changing
communication topologies.

While the physical disconnection within the network allows for many practical
and performance related advantages, it also introduces many fundamental issues
(e.g. loss/delay of information, limits on communication bandwidth, power
constraints [Culler]) that are of lesser concern in wired configurations.
Issues such as packet loss/delay are crucial in estimation and control
problems where information flow is assumed to be uninterrupted, and much work
has been done to examine performance in the presence of these complications
[Sinopoli, Schenato, Hespanha].

In this paper, we consider the problem where (noisy) observations of a dynamic
process are made by multiple sensors at fixed locations. These sensors
communicate over a lossy wireless network to a base station that computes the
optimal state estimate by processing the data via a Kalman filter. Since the
physical distance between the sensors and the base station affects the packet
loss probability (which in turn affects the quality of the estimates), our
goal is to determine the optimal position of the base station. A similar
problem of locating control logic has been studied in [Kumar].

There has been much recent work on decentralized estimation problems where no
single node is burdened with the majority of the computational tasks [Grime,
Spanos]. However, implementation of these ideas require “smart” sensors that
have local computational abilities. While these approaches have been shown to
possess various performance advantages, they are not always realizable. In
order to examine more fundamental issues regarding the implementation of
wireless sensor networks, we restrict our attention to the case where
stationary sensors merely _sense_ and _communicate_ , and possess no
computational power. We leave the more general problem for future work.

The remainder of the paper is organized as follows. In Section 2, we give a
brief overview of Kalman filtering. Section 3 contains our problem formulation
and main results. Some discussion regarding more general cases can be found in
Section 4.

##  2 Preliminaries

###  2.1 Kalman Filtering

We begin by giving a brief overview of the canonical state estimation problem.
Consider a process whose dynamics are modeled by the (possibly time-varying)
state space equation

$x_{k+1}=Ax_{k}+w_{k},$ (1)

where $x_{k}\in\mathds{R}^{n}$ is the state vector at time $k$, and $w$ is a
sequence of independent Gaussian random vectors with zero mean and covariance
matrix $Q$. The initial condition $x_{0}$ is unknown and modeled as a Gaussian
random variable with zero mean and covariance matrix $P_{0}$.

We monitor this process via measurements of the form

$y_{k}=Cx_{k}+v_{k},$

where $C$ is possibly time-varying, and $v$ is a sequence of independent
Gaussian random vectors with zero mean and covariance matrix $R$. The
sequences $w$ and $v$ are assumed to be uncorrelated. The minimum variance
estimate of the sequence $x$ is then given recursively by the Kalman filter
[Kalman],

$\displaystyle K_{k}$ $\displaystyle=$ $\displaystyle
AP_{k}C^{*}(CP_{k}C^{*}+R)^{-1}$ (2a) $\displaystyle\hat{x}_{k+1}$
$\displaystyle=$ $\displaystyle A\hat{x}_{k}+K_{k}(y_{k}-C\hat{x}_{k})$ (2b)
$\displaystyle P_{k+1}$ $\displaystyle=$ $\displaystyle
AP_{k}A^{*}+Q-K_{k}CP_{k}A^{*}.$ (2c)

Here, $M^{*}$ denotes the complex conjugate transpose of a matrix $M$.

Due to the increasing availability of high quality, low-cost sensors, we are
often confronted with the situation where we have access to measurements from
multiple sensors. That is, suppose that we monitor the process (1) via $N$
sensors, whose measurements are represented as

$\displaystyle y_{k}^{1}$ $\displaystyle=$ $\displaystyle
C_{1}x_{k}+v_{k}^{1}$ $\displaystyle\vdots$ $\displaystyle y_{k}^{N}$
$\displaystyle=$ $\displaystyle C_{N}x_{k}+v_{k}^{N}.$

Here, $v^{1},\dots,v^{2}$ are sequences of independent Gaussian random vectors
with zero mean and covariance matrices $R_{1},\dots,R_{N}$, respectively. The
sequences $v^{1},\dots,v^{N}$ and $w$ are assumed to be uncorrelated. The
resulting state estimates are again given by (2), where we now define

$\displaystyle C$ $\displaystyle=$ $\displaystyle\mbox{blkcol}(C_{j}),\ \
j=1,\dots,N$ $\displaystyle R$ $\displaystyle=$
$\displaystyle\mbox{blkdiag}(R_{j}),\ \ j=1,\dots,N.$

Here, blkcol denotes the vertical stacking of its arguments, and blkdiag
denotes a block diagonal matrix of its arguments.

###  2.2 Kalman Filtering with Packet Losses

We now discuss the minimum variance state estimation problem in the presence
of packet losses. Let $\Omega$ be the power set of $\\{1,\dots,N\\}$ and let
each subset $\omega\in\Omega$ be given the standard simple order. Since the
Kalman filtering equations (2) yield the minimum variance state estimate for
linear time-varying systems, we obtain the following optimal filter for state
estimation with packet losses

$\left.\begin{array}[]{l}K_{k}=AP_{k}F_{\omega_{k}}^{*}(F_{\omega_{k}}P_{k}F_{\omega_{k}}^{*}+G_{\omega_{k}})^{-1}\\\
\hat{x}_{k+1}=A\hat{x}_{k}+K_{k}(y_{k}-F_{\omega_{k}}\hat{x}_{k})\\\
P_{k+1}=AP_{k}A^{*}+Q-K_{k}F_{\omega_{k}}P_{k}A^{*}\end{array}\right\\}\mbox{
for }\omega_{k}\in\Omega\backslash\phi,$

and

$\left.\begin{array}[]{l}\hat{x}_{k+1}=A\hat{x}_{k}\\\
P_{k+1}=AP_{k}A^{*}+Q\end{array}\right\\}\mbox{ for }\omega_{k}=\phi,$

where

$\displaystyle F_{\omega_{k}}$ $\displaystyle=$
$\displaystyle\mbox{blkcol}(C_{j}),\ \ j\in\omega_{k}$ $\displaystyle
G_{\omega_{k}}$ $\displaystyle=$ $\displaystyle\mbox{blkdiag}(R_{j}),\ \
j\in\omega_{k}.$

For a similar treatment, see [Liu, Hounkpevi].

##  3 Base Station Placement

###  3.1 Problem Formulation

We will model the packet arrival process for each sensor as a sequence of
independent binomial random variables. More specifically, for $i=1,\dots,N$,
let $\lambda_{i}$ be the probability that the observation from sensor $i$ is
received. Since the sensor locations are fixed, a given base station location
gives rise to distances $d_{1},\dots,d_{N}$. We will assume that the
probability of packet arrival decreases as the distance $d_{i}$ between a
sensor and the base station increases.

Note that the sequences $(F_{\omega})_{k}$ and $(G_{\omega})_{k}$ both depend
on whether packets have been received or lost, and hence the sequence
$(P_{k})_{k}$ is itself random. We will then aim to minimize (with respect to
the Loewner ordering [Horn], i.e., $X\succeq Y$ if $X-Y$ is positive semi-
definite). the expected estimation error covariance. That is, we consider the
problem

$\min{\bf E}\left[P_{k+1}|P_{k}\right],$

where the minimization is performed over the set of possible base station
locations. If we define for each nonempty $\neq\omega\in\Omega$

$\displaystyle\alpha_{\omega}$ $\displaystyle=$
$\displaystyle\prod_{j\in\omega}\lambda_{j}\prod_{j\in\\{1,\dots,N\\}\backslash\omega}(1-\lambda_{j})$
$\displaystyle K_{k}$ $\displaystyle=$ $\displaystyle
AP_{k}F_{\omega}^{*}(F_{\omega}P_{k}F_{\omega}^{*}+G_{\omega})^{-1},$

and $\alpha_{\omega}=0$ for $\omega=\phi$, we can write

${\bf
E}\left[P_{k+1}|P_{k}\right]=AP_{k}A^{*}+Q-\sum_{\omega\in\Omega}\alpha_{\omega}K_{k}F_{\omega}P_{k}A^{*}.$

As stated, the above optimization problem is a difficult nonlinear programming
problem that can be attacked by the appropriate solvers. Since it is not
feasible for the location of the base station to change over time, we will
insist that the solution to the above problem minimizes ${\bf
E}\left[P_{k+1}|P_{k}\right]$ for any $P_{k}$. Consequently, we shall
hereafter restrict our attention to an illuminating instance of the above
problem.

###  3.2 The 2 Sensor Case

In this section, we consider the case where measurements are obtained from $2$
homogeneous sensors, $s_{1}$ and $s_{2}$. Without loss of generality, these
sensors are located at $\xi_{1}=0$ and $\xi_{2}=1$ on the real line. The goal
is to find the best location $d\in[0,1]$ for the base station. Note that
optimal solution must lie in the interval $[0,1]$. The situation $d=0$ (or
$d=1)$ corresponds to the base station being physically wired to one of the
sensors, and wireless communication is performed only by the other sensor.

Although there may be many factors that influence the packet loss probability,
we shall consider only the effects due to the physical distance between the
communicating nodes. Since the sensors have identical broadcasting
capabilities, it is then natural to model the probability of packet loss in
the following manner. Let $f:[0,1]\rightarrow[0,1]$ be convex and decreasing,
with $f(0)=1$. When the base station is located at some $d\in[0,1]$, the
probability that a packet is received from sensor $s_{1}$ is $f(d)$, and the
probability that a packet is received from sensor $s_{2}$ is $f(1-d)$. The
constraint that $f(0)=1$ captures the implicit assumption that no packet is
lost when a sensor is affixed to the base station and communication is not
performed wirelessly.

####  3.2.1 Vector-valued Measurements, Same Covariances

We will first restrict our attention to the case where

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle
C_{2}=C\in\mathds{R}^{m\times n}$ $\displaystyle R_{1}$ $\displaystyle=$
$\displaystyle R_{2}=R\in\mathds{R}^{m\times m}.$

If we model the probability of packet loss as described above, we can examine
the estimation error covariance as a function of $d$. For ease of notation,
let us define

$\displaystyle M_{1}$ $\displaystyle=$ $\displaystyle
AP_{k}C^{*}(CP_{k}C^{*}+R)^{-1}CP_{k}A^{*}$ $\displaystyle M_{2}$
$\displaystyle=$ $\displaystyle AP_{k}C^{*}(CP_{k}C^{*}+R/2)^{-1}CP_{k}A^{*}$
$\displaystyle S$ $\displaystyle=$ $\displaystyle AP_{k}A^{*}+Q.$

The estimation error covariance $P_{k+1}$ then assumes values as follows:

$\begin{array}[]{|c|c|}\hline\cr P_{k+1}&\mbox{Probability}\\\ \hline\cr
S-M_{2}&\ \ f(d)f(1-d)\\\ \hline\cr\ \ S-M_{1}&\ \ f(d)-2f(d)f(1-d))+f(1-d)\\\
\hline\cr S&(1-f(d))(1-f(1-d))\\\ \hline\cr\end{array}$

Note that in the case where packets from both sensors are lost, the resulting
covariance corresponds to that of propagating the previous state estimate. The
expected error covariance can then be computed as

$\displaystyle{\bf E}\left[P_{k+1}|P_{k}\right]$ $\displaystyle=$
$\displaystyle S-f(d)f(1-d)M_{1}$ $\displaystyle\hskip
3.61371pt-(f(d)-2f(d)f(1-d)+f(1-d))M_{2}.$

We have the following result.

###### Proposition 1

Let $C_{1}=C_{2}=C\in\mathds{R}^{m\times n}$ and
$R_{1}=R_{2}=R\in\mathds{R}^{m\times m}$. Let $f:[0,1]\rightarrow[0,1]$ be
twice differentiable, convex, and decreasing, with $f(0)=1$. Then, ${\bf
E}\left[P_{k+1}|P_{k}\right]$ is matrix concave in $d$ on $[0,1]$.

Proof Define

$\displaystyle J_{k+1}(d)$ $\displaystyle=$ $\displaystyle{\bf
E}\left[P_{k+1}|P_{k}\right]$ $\displaystyle=$ $\displaystyle
AP_{k}A^{*}+Q-f(d)f(1-d)M_{2}$ $\displaystyle\hskip
10.84006pt-(f(d)-2f(d)f(1-d)+f(1-d))M_{1}.$

Taking the second derivative yields

$\displaystyle J_{k+1}^{\prime\prime}(d)$ $\displaystyle=$ $\displaystyle
f^{\prime\prime}(d)(f(1-d)-1)M_{1}$
$\displaystyle+f^{\prime\prime}(1-d)(f(d)-1)M_{1}$
$\displaystyle-2f^{\prime}(d)f^{\prime}(1-d)(2M_{1}-M_{2})$
$\displaystyle+(f^{\prime\prime}(d)f(1-d)+f(d)f^{\prime\prime}(1-d))(M_{1}-M_{2}).$

Since $f$ is convex and decreasing with $f(0)=1$, we have $f^{\prime}(x)\leq
0$ and $f^{\prime\prime}(x)\geq 0$ for $x\in[0,1]$. Consequently,
$J^{\prime\prime}(d)\preceq 0$, and hence ${\bf E}\left[P_{k+1}|P_{k}\right]$
is matrix concave on $[0,1]$ (see [Boyd], p.110). $\blacksquare$

###### Corollary 1

Let $C_{1}=C_{2}=C\in\mathds{R}^{m\times n}$ and
$R_{1}=R_{2}=R\in\mathds{R}^{m\times m}$. Let $f:[0,1]\rightarrow[0,1]$ be
twice differentiable, convex, and decreasing, with $f(0)=1$. Then, the base
station is optimally located at one of the sensor positions.

####  3.2.2 Scalar-valued Measurements, Same Covariances

A similar result may be obtained in the case where
$C_{1}=C_{2}=C\in\mathds{R}^{1\times n}$ and the noise covariances $R_{1}$ and
$R_{2}$ are not necessarily equal. We will need the following preliminary
result.

###### Lemma 1

Let $C_{1}=C_{2}=C\in\mathds{R}^{1\times n}$ and define

$\displaystyle T$ $\displaystyle=$ $\displaystyle CP_{k}C^{*}$ $\displaystyle
M_{0}$ $\displaystyle=$ $\displaystyle AP\begin{bmatrix}C\\\
C\end{bmatrix}^{*}\begin{bmatrix}T+R_{1}&T\\\
T&T+R_{2}\end{bmatrix}^{-1}\begin{bmatrix}C\\\ C\end{bmatrix}PA^{*}$
$\displaystyle M_{1}$ $\displaystyle=$ $\displaystyle
APC^{*}(T+R_{1})^{-1}CPA^{*}$ $\displaystyle M_{2}$ $\displaystyle=$
$\displaystyle APC^{*}(T+R_{2})^{-1}CPA^{*}.$

Then, $M_{0}-M_{1}-M_{2}\succeq 0$.

Proof Let us define

$\begin{bmatrix}\alpha&\beta\\\
\beta&\gamma\end{bmatrix}=\begin{bmatrix}T+R_{1}&T\\\ T&T+R_{2}\end{bmatrix},\
\mbox{ and }\ U=\begin{bmatrix}C\\\ C\end{bmatrix}P_{k}A^{*}.$

Then, $M_{0}-M_{1}-M_{2}$ can be written as

$\displaystyle M_{0}-$ $\displaystyle M_{1}$ $\displaystyle-M_{2}$
$\displaystyle=$ $\displaystyle U^{*}\left(\begin{bmatrix}\alpha&\beta\\\
\beta&\gamma\end{bmatrix}^{-1}-\begin{bmatrix}\alpha&0\\\
0&\gamma\end{bmatrix}^{-1}\right)U$ $\displaystyle=$ $\displaystyle
U^{*}\begin{bmatrix}\frac{1}{\alpha-\beta^{2}\gamma^{-1}}-\alpha^{-1}&-\frac{\beta\gamma^{-1}}{\alpha-\beta^{2}\gamma^{-1}}\\\
-\frac{\gamma^{-1}\beta}{\alpha-\beta^{2}\gamma^{-1}}&\frac{\gamma^{-2}\beta^{2}}{\alpha-\beta^{2}\gamma^{-1}}\end{bmatrix}U$
$\displaystyle=$ $\displaystyle
U^{*}\begin{bmatrix}\frac{1}{\alpha-\beta^{2}\gamma^{-1}}\\\
-\frac{\gamma^{-1}\beta}{\alpha-\beta^{2}\gamma^{-1}}\end{bmatrix}\begin{bmatrix}\frac{1}{\alpha-\beta^{2}\gamma^{-1}}&-\frac{\beta\gamma^{-1}}{\alpha-\beta^{2}\gamma^{-1}}\end{bmatrix}U$
$\displaystyle\hskip 10.84006pt-U^{*}\begin{bmatrix}-\alpha^{-1}&0\\\
0&0\end{bmatrix}U$ $\displaystyle=$ $\displaystyle
AP_{k}C^{*}\left(\frac{(1-\gamma^{-1}\beta)^{2}}{\alpha-\beta^{2}\gamma^{-1}}-\alpha^{-1}\right)CP_{k}A^{*},$

where the second equality follows from the matrix inversion lemma [Horn]. Now,
let

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{(1-\gamma^{-1}\beta)^{2}}{\alpha-\beta^{2}\gamma^{-1}}-\alpha^{-1}$
$\displaystyle=$
$\displaystyle\frac{(1-\gamma^{-1}\beta)^{2}-\alpha^{-1}(\alpha-\beta^{2}\gamma^{-1})}{\alpha-\beta^{2}\gamma^{-1}}.$

The desired result follows from noticing that the denominator is positive and
the numerator satisfies

$\displaystyle(1-\gamma^{-1}\beta)^{2}-\alpha^{-1}(\alpha-\beta^{2}\gamma^{-1})$
$\displaystyle\hskip
25.29494pt=-\beta\gamma^{-1}\left(2-\frac{CP_{k}C^{*}}{CP_{k}C^{*}+R_{2}}-\frac{CP_{k}C^{*}}{CP_{k}C^{*}+R_{1}}\right)$
$\displaystyle\hskip 25.29494pt\leq 0.$

$\blacksquare$

###### Proposition 2

Let $C_{1}=C_{2}=C\in\mathds{R}^{1\times n}$ and suppose that $R_{1}$ and
$R_{2}$ are not necessarily equal. Let $f:[0,1]\rightarrow[0,1]$ be twice
differentiable, convex, and decreasing, with $f(0)=1$. Then, ${\bf
E}\left[P_{k+1}|P_{k}\right]$ is matrix concave in $d$ on $[0,1]$.

Proof Define

$\displaystyle T$ $\displaystyle=$ $\displaystyle CP_{k}C^{*}$ $\displaystyle
M_{0}$ $\displaystyle=$ $\displaystyle AP\begin{bmatrix}C\\\
C\end{bmatrix}^{*}\begin{bmatrix}T+R_{1}&T\\\
T&T+R_{2}\end{bmatrix}^{-1}\begin{bmatrix}C\\\ C\end{bmatrix}PA^{*}$
$\displaystyle M_{1}$ $\displaystyle=$ $\displaystyle
APC^{*}(T+R_{1})^{-1}CPA^{*}$ $\displaystyle M_{2}$ $\displaystyle=$
$\displaystyle APC^{*}(T+R_{2})^{-1}CPA^{*}.$

We can then write

$\displaystyle J_{k+1}(d)$ $\displaystyle=$ $\displaystyle{\bf
E}\left[P_{k+1}|P_{k}\right]$ $\displaystyle=$ $\displaystyle AP_{k}A^{*}+Q$
$\displaystyle\hskip 10.84006pt-f(d)f(1-d)M_{0}-f(d)(1-f(1-d))M_{1}$
$\displaystyle\hskip 10.84006pt-f(1-d)(1-f(d))M_{2}$

Taking the second derivative yields

$\displaystyle J_{k+1}^{\prime\prime}(d)$ $\displaystyle=$ $\displaystyle
f^{\prime\prime}(d)(f(1-d)-1)M_{1}$
$\displaystyle+f^{\prime\prime}(1-d)(f(d)-1)M_{2}$
$\displaystyle-2f^{\prime}(d)f^{\prime}(1-d)(M_{1}+M_{2}-M_{0})$
$\displaystyle+f^{\prime\prime}(d)f(1-d)(M_{2}-M_{0})$
$\displaystyle+f(d)f^{\prime\prime}(1-d)(M_{1}-M_{0}).$

Since $f$ is convex and decreasing with $f(0)=1$, we have $f^{\prime}(x)\leq
0$ and $f^{\prime\prime}(x)\geq 0$ for $x\in[0,1]$. From Lemma 1 we have
$M_{1}+M_{2}-M_{0}\preceq 0$. As a result, $J^{\prime\prime}(d)\preceq 0$, so
${\bf E}\left[P_{k+1}|P_{k}\right]$ is matrix concave on $[0,1]$ (see [Boyd],
p.110). $\blacksquare$

###### Corollary 2

Let $C_{1}=C_{2}=C\in\mathds{R}^{1\times n}$ and suppose that $R_{1}$ and
$R_{2}$ are not necessarily equal. Let $f:[0,1]\rightarrow[0,1]$ be twice
differentiable, convex, and decreasing, with $f(0)=1$. Then, the base station
is optimally located at one of the sensor positions. If $R_{1}\succeq R_{2}$,
then the base station is optimally located at the second sensor.

####  3.2.3 An Example

We now offer an illustrative example to complement our main results.

Suppose that there are two sensors, located respectively at the points $0$ and
$1$ on the real line. Let us model our packet arrival probability function as

$f(d)=e^{-d}.$

Note that $f$ is convex and decreasing on $[0,1]$ with $f(0)=1$. The process
being monitored is described by the state space equations in (1), where all
the eigenvalues of $A\in\mathds{R}^{2\times 2}$ lie in the unit circle and
$C\in\mathds{R}^{1\times 2}$. The trace of the estimation error covariance
matrices is plotted in Figure 1 for the case where the base station is
situated at the point $0$ and the case where it is situated at the point
$0.5$.

Figure 1: Covariances for $d=0$ (dashed) and $d=0.5$ (solid).

##  4 General Cases

We now offer some commentary on extending our main results.

Although the previous results were confined to the case $C_{1}=C_{2}$,
simulations suggest that they also hold in more generality. Consider the case
where $N$ sensors located colinearly are available to make observations. In
general, we may have different values for $C_{1},\dots,C_{N}$ and different
values for $R_{1},\dots,R_{N}$. In this case, empirical evidence supports the
extension of our main result in the sense that the expected error covariance
is piecewise concave over the intervals defined by the location of the
sensors. Figure 2 illustrates this situation for the case $N=3$.

Figure 2: Piecewise concavity for multiple colinear sensors.

In the case where the convex hull of the sensor locations has a nonempty
interior in $\mathds{R}^{2}$, it can be shown that the optimal base station
location may lie at a point other than the sensor locations. Consider 3
sensors, located at the vertices of an equilateral triangle with side length
$0.5$. For $P_{k}=I$, $f(d)=e^{-d}$, $C_{1}=C_{2}=C_{3}=0.6$, and unit noise
covariances, the expected error covariance (as a function of base station
location) has a local minimum at the centroid, as illustrated in Figure 3.

Figure 3: Local minimum in the interior of the convex hull.

##  5 Conclusion

We considered the problem of positioning a central computational node within a
wireless sensor network for the purpose of state estimation. In the presence
of potential packet losses, the optimal estimate is given by a time-varying
Kalman filter. We then strived to determine the optimal location of the base
station so as to minimize the expected conditional estimation error
covariance. It was shown for the case of a 2-sensor configuration that under a
certain class of packet loss probability models, the base station is optimally
located at one of the sensor positions.


#  Persistent coherent many-body oscillations in a non-integrable quantum
Ising chain

Francis A. Bayocboc Jr  Jagiellonian University, Institute of Theoretical
Physics, Łojasiewicza 11, PL-30348 Kraków, Poland  Jacek Dziarmaga
Jagiellonian University, Institute of Theoretical Physics, Łojasiewicza 11,
PL-30348 Kraków, Poland  Marek M. Rams  Jagiellonian University, Institute of
Theoretical Physics, Łojasiewicza 11, PL-30348 Kraków, Poland  Wojciech H.
Zurek  Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA

(May 22, 2024)

###### Abstract

We identify persistent oscillations in a nonintegrable quantum Ising chain
left behind by a rapid transition into a ferromagnetic phase. In the
integrable chain with nearest-neighbor (NN) interactions, the nature, origin,
and decay of post-transition oscillations are tied to the Kibble-Zurek
mechanism (KZM). However, when coupling to the next nearest neighbor (NNN) is
added, the resulting nonintegrable Ising chain (still in the quantum Ising
chain universality class) supports persistent post-transition oscillation:
KZM-like oscillations turn into persistent oscillations of transverse
magnetization. Their longevity in our simulations is likely limited only by
the numerical accuracy. Their period differs from the decaying KZM oscillation
but their amplitude depends on quench rate. Moreover, they can be excited by
driving in resonance with the excitations’ energy gap. Thus, while one might
have expected that the integrability-breaking NNN coupling would facilitate
relaxation, the oscillations we identify are persistent. At low to medium
transverse fields, they are associated with Cooper pairs of Bogoliubov
quasiparticles – kinks. This oscillation of the pair condensate is a
manifestation of quantum coherence.

In a recent paper [1], we have identified and characterized oscillations
excited by the quench across the critical point of phase transitions in
quantum Ising spin systems. These coherent many-body oscillations of the
transverse magnetization are caused by quantum superpositions of broken
symmetry states left behind in the wake of a nonequilibrium (nonadiabatic)
quantum phase transition. They can be understood as a consequence of the
Kibble-Zurek mechanism (KZM) [2, *K-b, *K-c, 5, *Z-b, *Z-c, *Z-d] which is
also responsible for the generation of topological defects in both classical
[9, 10, 11, 12, 13, 14, 15, *KZnum-h, *KZnum-i, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48] and quantum [49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78,
79, 80, 81, 82, 83, 84, 79, 85, 86, 87, 1, 30, 88, 89, 90, 91, 92, 93, 94, 95,
96, 97, 98, 99, 100, 101] phase transitions. In particular, frequencies and
amplitudes of these post-critical oscillations follow the KZM scalings.

Figure 1: Oscillations during KZ ramp. Top: the KZ ramp crossing the critical
point at $g_{c}=2.47725$ to end in the ferromagnetic phase at non-zero
$g=0.5$. Middle: Rescaled transverse magnetization
$\langle\sigma^{x}_{n}\rangle-\langle\sigma^{x}_{n}\rangle_{GS}$ during the
ramp as a function of time, where $\langle\sigma^{x}_{n}\rangle_{GS}$ is its
value in the adiabatic ground state and $\hat{\xi}=\sqrt{\tau_{Q}}$ is the KZ
length. Its oscillation amplitude suddenly increases in the middle of the
ferromagnetic phase. The inset shows the zigzag ladder. Bottom: NN
ferromagnetic interaction energy as a function of time.

In Ref. 1, two models were studied. The simplest quantum Ising chain with only
nearest-neighbor (NN) couplings is integrable and a paradigmatic example of
quantum phase transitions. The Ising chain with both NN and next to NN (NNN)
couplings is nonintegrable:

$H=-\sum_{n=1}^{L}\left(g\sigma^{x}_{n}+\sigma^{z}_{n}\sigma^{z}_{n+1}+J_{2}\sigma^{z}_{n}\sigma^{z}_{n+2}\right).$
(1)

Here, $\sigma_{n}^{x,y,z}$ are the Pauli operators on site $n$, $g$ is a
transverse field, and $J_{2}$ is the NNN ferromagnetic coupling. It can be
implemented with only NN couplings on a zigzag ladder in Fig. 1. Both models
belong to the same universality class, so their behavior in the immediate
vicinity of the critical point is expected to be essentially the same. Indeed,
we have demonstrated [1] that both models exhibit similar oscillatory behavior
with KZM scalings following the quench.

Dramatic differences in the behavior between the two Ising chains arise away
from the transition point. In the integrable quantum Ising chain with only NN
couplings, post-critical coherent oscillations gradually decay. By contrast
(and to our surprise), in the nonintegrable Ising chain with both NN and NNN
couplings, the post-KZM oscillation decays, but before it disappears, it
becomes enhanced and gives rise to persistent oscillation that does not decay
on the timescales explored by our numerical simulations. These reincarnated
oscillations have features that are no longer subject to KZM scalings (see
Fig. 1). In particular, the frequencies of such oscillations are no longer
controlled by the size of the Ising chain gap associated with a single
quasiparticle.

The aim of this paper is to investigate properties and to characterize the
nature of these oscillations.

Integrable NN chain; $\mathbf{J_{2}=0}$. We linearly ramp the transverse field
[50, 52, 58, 102, 86, 87] as

$\displaystyle
g(t)=g_{c}\left[1-\epsilon(t)\right]=g_{c}-g_{c}(t-t_{c})/\tau_{Q},$ (2)

from a strong field at $t=-\infty$, across the critical point at
$g(t_{c})=g_{c}=1$, to $g(t_{s})=0$ when the transverse field vanishes and the
ramp stops. The Jordan-Wigner transformation maps the model to a set of
independent Landau-Zener systems that can be solved analytically. In
particular, the final density of excited quasiparticles/kinks scales like [50,
52, 103]

$\rho=\frac{1}{2\pi\sqrt{2\tau_{Q}}}\propto\hat{\xi}^{-1}.$ (3)

Here, $\hat{\xi}$ is the KZ length. The KZM operates within $\pm\hat{t}$ of
$t_{c}$, where $\hat{t}\propto\hat{\xi}^{1}$. Prior to the kink count, the
final state is a quantum superposition of different numbers [58, 73] and
correlated locations of kinks [84, 86, 87] that manifests in coherent
oscillations of the transverse magnetization $\sigma^{x}$ [1].

Two new results [103] make observation of the coherent oscillations more
feasible for the present quantum simulation platforms. The first is an
analytical formula for the transverse magnetization after $t_{c}+\hat{t}$:

$\delta^{x}=\langle\sigma^{x}\rangle-\langle\sigma^{x}\rangle_{\rm GS}\approx
2\rho+\rho^{2}d\frac{57\sqrt{6\pi}}{80}\cos\phi.$ (4)

Here, $d=(1+f^{2})^{-3/4}$ is a dephasing factor, where

$f=\frac{3}{4\pi}\left[\gamma_{E}-2\frac{t-t_{c}}{\tau_{Q}}+\ln\frac{4(t-t_{c})^{2}}{\tau_{Q}}\right]$
(5)

with Euler’s constant $\gamma_{E}$, the phase of the oscillations is

$\phi=\frac{\pi}{4}+2\frac{(t-t_{c})^{2}}{\tau_{Q}}+\frac{3}{2}\arctan f,$ (6)

and $\langle\sigma^{x}\rangle_{\rm GS}$ is the magnetization in the adiabatic
ground state. In (4) both terms are accurate to their leading order in $\rho$.
The second result is the NN ferromagnetic correlator:

$\displaystyle\delta^{zz}$ $\displaystyle=$
$\displaystyle\langle\sigma^{z}_{n}\sigma^{z}_{n+1}\rangle-\langle\sigma^{z}_{n}\sigma^{z}_{n+1}\rangle_{\rm
GS}$ (7) $\displaystyle\approx$ $\displaystyle-2\rho-g\leavevmode\nobreak\
\rho^{2}d\frac{57\sqrt{6\pi}}{80}\cos\phi,$

which may be more directly accessible in some quantum simulation platforms.
Together with (4), it satisfies $-\delta^{zz}-g\delta^{x}\approx 2(1-g)\rho.$
The excitation energy density on the left-hand side is given by the density of
quasiparticles $\rho$ times the quasiparticle gap opening with the distance
from the critical point as $2(1-g)$, in consistency with the extended QKZM
[85].

The oscillations (4) are damped by dephasing of non-interacting
quasiparticles. Breaking the integrability with $J_{2}=1$, one might expect
them to become even more scrambled, but it turns out to be the other way
around. Not only do they become more persistent, but even their amplitude is
enhanced (see Fig. 1).

Figure 2: BCS versus MPS.  The first derivative of the gap function $\Delta$
with respect to the field $g$ in the ground state from the BCS theory and
matrix product states (MPS). Estimates of $g_{c}$ from BCS and MPS are
$g_{c}=2.48135$ and $g_{c}=2.47725$, respectively.

Nonintegrable chain with $\mathbf{J_{2}=1}$; Kink pairs. With a fermionic
operator that annihilates a kink on a bond between sites $n$ and $n+1$ [87],

$f_{n}=\left(\prod_{l\leq
n}\sigma^{x}_{l}\right)\frac{\sigma^{z}_{n}-\sigma^{z}_{n+1}}{2i},$ (8)

one can rewrite the Hamiltonian (1) with $J_{2}=1$ as

$\displaystyle H$ $\displaystyle=$
$\displaystyle-2L+6\sum_{n}f_{n}^{\dagger}f_{n}-4\sum_{n}f_{n}^{\dagger}f_{n}f_{n+1}^{\dagger}f_{n+1}$
(9) $\displaystyle-g\sum_{n}\left(f_{n}^{\dagger}f_{n+1}+f_{n+1}f_{n}+{\rm
h.c.}\right).$

The first term is the energy of the ferromagnetic state and the second is that
of individual kinks in this state. The transverse field in the second line can
create/annihilate a pair of kinks on NN bonds by flipping the spin between
them. A spin flip can also make a kink hop between NN bonds. The integrability
is broken by the quartic attraction between NN kinks.

In the BCS theory, the ground state is assumed to be a fermionic Gaussian
state $\ket{0}$. After a Fourier transform

$f_{n}=e^{i\pi/4}\sum_{k}\frac{e^{ik(n+1/2)}}{\sqrt{L}}f_{k},$ (10)

and a Bogoliubov transformation,

$f_{k}=u_{k}\gamma_{k}+v_{-k}^{*}\gamma_{-k}^{\dagger},$ (11)

the energy is minimized by a vacuum, $\gamma_{k}\ket{0}=0,$ where
$(u_{k},v_{k})$ solve the Bogoliubov-de Gennes equations:

$\omega_{k}\left(\begin{array}[]{c}u_{k}\\\
v_{k}\end{array}\right)=\left(\begin{array}[]{cc}H_{k}&D_{k}\\\
D^{*}_{k}&-H_{k}\end{array}\right)\left(\begin{array}[]{c}u_{k}\\\
v_{k}\end{array}\right).$ (12)

Here, $H_{k}=2(3-4\rho)-2(g-4t_{f})\cos k$, $D_{k}=-2(g+4\Delta)\sin k$, and
the quasiparticle dispersion $\omega_{k}=\sqrt{H_{k}^{2}+D_{k}^{2}}$. Here,
$\rho=\bra{0}f_{n}^{\dagger}f_{n}\ket{0}$ is density of kinks,
$\Delta=\bra{0}f_{n+1}f_{n}\ket{0}$ is their anomalous density, and
$t_{f}=\bra{0}f_{n+1}^{\dagger}f_{n}\ket{0}$ is a correction to the kink
hopping rate. The BCS solution compares well with the matrix product states
(MPS) [104] (see Fig. 2 and [103]). The critical $g_{c}=2.48135$ by BCS lies
within $0.2\%$ of $g_{c}=2.47725$ by MPS VUMPS algorithm.

The accuracy of the BCS theory makes it a reliable approximation for the
ground state. For $g\ll g_{c}$ it yields:

$\rho=\frac{g^{2}}{32}+{\cal O}(g^{4}),\leavevmode\nobreak\
\leavevmode\nobreak\ \Delta=\frac{g}{8}+{\cal O}(g^{3}),\leavevmode\nobreak\
\leavevmode\nobreak\ t_{f}={\cal O}(g^{3}),$ (13)

and the Bogoliubov coefficients:

$\displaystyle u_{k}$ $\displaystyle\approx$
$\displaystyle\left(1-\frac{g^{2}}{64}\right)+\frac{g^{2}}{64}\cos 2k+{\cal
O}(g^{3}),\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle v_{k}$
$\displaystyle\approx$ $\displaystyle-\frac{g}{4}\sin k-\frac{g^{2}}{24}\sin
2k+{\cal O}(g^{3}).$ (14)

With (11), we obtain the kink operator

$\displaystyle f_{n}$ $\displaystyle\approx$
$\displaystyle\left(1-\frac{g^{2}}{64}\right)\gamma_{n}+\frac{g^{2}}{128}\left(\gamma_{n+2}+\gamma_{n-2}\right)$
(15)
$\displaystyle+\frac{g}{8}\left(\gamma_{n+1}^{\dagger}-\gamma_{n-1}^{\dagger}\right)+\frac{g^{2}}{48}\left(\gamma_{n+2}^{\dagger}-\gamma_{n-2}^{\dagger}\right).$

Here, the position representation $\gamma_{n}$ is the transform (10) of
$\gamma_{k}$. It is a kink $f_{n}$ dressed with quantum fluctuations. Beyond
BCS, the exact Hamiltonian (9) becomes

$H=E_{0}+\sum_{k}\omega_{k}\gamma_{k}^{\dagger}\gamma_{k}-4\sum_{n}:f_{n}^{\dagger}f_{n}f_{n+1}^{\dagger}f_{n+1}:.$
(16)

Here, $E_{0}=\bra{0}H\ket{0}$, the normal ordering is with respect to
$\gamma$, and the quasiparticle dispersion is

$\displaystyle\omega_{k}$ $\displaystyle=$
$\displaystyle\omega_{\gamma}-2t_{\gamma}\cos k-2t^{\prime}_{\gamma}\cos
2k+{\cal O}(g^{3}),$ (17)

where $\omega_{\gamma}=6+g^{2}/8$, $t_{\gamma}=g$ and
$t^{\prime}_{\gamma}=3g^{2}/16$. With (15) the Hamiltonian (16) becomes

$\displaystyle H$ $\displaystyle=$ $\displaystyle
E_{0}+\omega_{\gamma}\sum_{n}\gamma_{n}^{\dagger}\gamma_{n}-\left(4-\frac{g^{2}}{8}\right)\sum_{n}\gamma_{n}^{\dagger}\gamma_{n+1}^{\dagger}\gamma_{n+1}\gamma_{n}$
(18) $\displaystyle-
t_{\gamma}\sum_{n}\left(\gamma_{n}^{\dagger}\gamma_{n+1}+{\rm
h.c.}\right)+\frac{g}{2}\sum_{n}V_{n}^{(1)}$
$\displaystyle-t^{\prime}_{\gamma}\sum_{n}\left(\gamma_{n}^{\dagger}\gamma_{n+2}+{\rm
h.c.}\right)+\frac{g^{2}}{16}\sum_{n}V^{(2)}_{n}$
$\displaystyle+g^{2}\tilde{V}^{(2)}+{\cal O}(g^{3}),$

where

$\displaystyle V_{n}^{(1)}$ $\displaystyle=$
$\displaystyle\left(\gamma_{n+2}^{\dagger}\gamma_{n+1}^{\dagger}+\gamma_{n-1}^{\dagger}\gamma_{n-2}^{\dagger}\right)\gamma_{n}^{\dagger}\gamma_{n}+{\rm
h.c.},$ (19) $\displaystyle V_{n}^{(2)}$ $\displaystyle=$
$\displaystyle\left(\gamma^{\dagger}_{n+1}\gamma^{\dagger}_{n}+\gamma^{\dagger}_{n+2}\gamma^{\dagger}_{n+1}\right)\gamma_{n}\gamma_{n-1}+{\rm
h.c.},$ (20)

and $\tilde{V}^{(2)}$ collects remaining terms that do not contribute to (21)
[103]. For $g\ll g_{c}$, the first line of (18) dominates, and pairs of NN
quasiparticles form tight bound states. These are single reversed spins
dressed in quantum fluctuations.

Figure 3: Oscillation frequency.  The plot compares (22) with the gap between
the ground and first excited state obtained with exact diagonalization (ED)
extrapolated to infinite periodic chain size ($L\to\infty$), and transverse
oscillation frequency after the periodic drive (24) with small amplitude
obtained by a fit to MPS simulations. The ED and MPS error bars are on the
order of the marker stroke width.

Accordingly, we introduce NN pair operators $b_{n}=\gamma_{n+1}\gamma_{n}$.
They are approximately bosonic at low pair density where an effective
Hamiltonian can be obtained as [103]

$\displaystyle H_{b}$ $\displaystyle=$
$\displaystyle\omega_{b}\sum_{n}b_{n}^{\dagger}b_{n}-t_{b}\sum_{n}\left(b^{\dagger}_{n}b_{n+1}+{\rm
h.c.}\right)$ (21)
$\displaystyle-t^{\prime}_{b}\sum_{n}\left(b^{\dagger}_{n}b_{n+2}+{\rm
h.c.}\right)+{\cal O}\left(g^{3}\right).$

Here, $\omega_{b}=8-g^{2}/4$, $t_{b}=g^{2}/8$, and $t^{\prime}_{b}=g^{2}/8$. A
pair has minimal energy for zero pair quasimomentum:

$\displaystyle\omega=\omega_{b}-2t_{b}-2t^{\prime}_{b}=8-\frac{3}{4}g^{2},$
(22)

which agrees with the gap from exact diagonalization in Fig. 3.

Within a dynamic BCS theory, the KZ ramp results in a fermionic Gaussian state
where pairs of $\gamma$-quasiparticles are not bound but free to move along
the chain [87]. Bound pairs would not affect the long-range order, while the
actual state has finite ferromagnetic domains of typical size $\hat{\xi}$. The
free quasiparticles occupy quasimomenta up to $\propto\hat{\xi}^{-1}$, and
their non-trivial dispersion makes dephasing inevitable.

However, as $g$ is slowly ramped further down, the individual
$\gamma$-quasiparticles become unstable towards the lower energy branch of
pair excitations. This happens when the pair gap in (22) becomes lower than
twice the $\gamma$-gap in (17), $\omega<2\omega_{0}$, near $g_{0}\approx 1.12$
(see Fig. 4). The pair dispersion is weaker than the dispersion of
$\gamma$-quasiparticles ($\propto g^{2}$ in (21) versus $\propto g$ in (17),
respectively) but not weak enough to explain the absence of dephasing that we
observe for times much longer than $\propto g^{-2}$. It is made irrelevant by
a Bose-enhanced transfer of $\gamma$-quasiparticles into pairs with zero
quasimomentum. The appearance of pairs manifests in Fig. 1 by the
amplification of oscillations near $g_{0}$, as can be explained with a simple
example: for $\ket{\psi}=\alpha\ket{0}+\beta
b_{n-1}^{\dagger}\ket{0}e^{-i\omega t}$ we obtain

$\displaystyle\bra{\psi}\sigma^{x}_{n}\ket{\psi}\approx$
$\displaystyle\bra{\psi}\gamma_{n-1}^{\dagger}\gamma_{n}+\gamma_{n}^{\dagger}\gamma_{n-1}-\gamma_{n-1}\gamma_{n}-\gamma_{n}^{\dagger}\gamma_{n-1}^{\dagger}\ket{\psi}=$
$\displaystyle\alpha^{*}\beta e^{-i\omega t}+\alpha\beta^{*}e^{+i\omega t}.$
(23)

As the pair is a reversed $n$-th spin (dressed in quantum fluctuations), it
manifests in oscillations of the spin-reversing $\sigma^{x}_{n}$.

Figure 4: Oscillations after KZ ramp.  Top: the energy gap for creating a pair
of fermionic quasiparticles, $2\omega_{0}$, versus that for their bound pair,
$\omega$. They crossover near $g_{0}=1.12$. Bottom: oscillations after a
smooth KZ ramp to $g=0.5$ with $\tau_{Q}=16$, using a smooth ramp
$g(t)=g_{c}[2-(1-0.5/(2g_{c}))(1+\sin(t/\tau_{Q}))],t\in[-\tau_{Q}\times\pi/2,\tau_{Q}\times\pi/2]$.
The dashed line indicates the end of the ramp, followed by a free evolution at
$g=0.5$. The oscillations during the free evolution have a Q factor of
${\approx}4012$ for $\tau_{Q}=16$, which increases to $Q\approx 8000$ for
$\tau_{Q}=32$.  Figure 5: Periodic driving.  Top: coherent transverse field
oscillations at $g=0.25$ after driving $\delta g(t)=0.005\sin(8t)$ close to
resonance for time $2\pi$. The plot compares (28) with MPS simulations. The
maximum percentage difference between (28) and MPS simulations is less than
$0.8\%$. Bottom: the same at $g=0.5$ after driving $\delta g(t)=0.05\sin(8t)$
for time $\pi$. The Q factors of the top and bottom plots are $Q\approx
461504$ and $Q\approx 8307$, respectively.

Periodic drive.  The KZ ramp is not the most efficient way to excite coherent
oscillations. Starting from a ground state at $g$, the pair condensate can
also be readily excited by a time-dependent perturbation to the Hamiltonian

$\delta H=\delta g(t)\sum_{n}\sigma^{x}_{n}.$ (24)

Assuming a small density of pairs, where their hard-core nature can be ignored
and there is a gap between paired and unpaired quasiparticles ($g\ll g_{0}$),
we can approximate the transverse magnetization as

$\sum_{n}\left(\sigma^{x}_{n}-\langle\sigma^{x}_{n}\rangle_{GS}\right)\approx\sum_{n}\left(b_{n}+b_{n}^{\dagger}\right)=\sqrt{L}\left(\tilde{b}_{0}+\tilde{b}_{0}^{\dagger}\right),$
(25)

where $\tilde{b}_{k}=e^{i\pi/4}b_{k}$, and the pair Hamiltonian (21) becomes

$\displaystyle H^{\prime}_{b}$ $\displaystyle=$
$\displaystyle\sum_{k}\omega_{b,k}\tilde{b}_{k}^{\dagger}\tilde{b}_{k}+\delta
H.$ (26)

Here, $\omega_{b,k}=\omega_{b}-2t_{b}\cos k-2t^{\prime}_{b}\cos 2k$. Since
only $k=0$ is driven by $\delta H$ in (25), it reduces to a driven oscillator:

$\displaystyle H^{\prime}_{\rm b,0}$ $\displaystyle=$
$\displaystyle\omega\tilde{b}_{0}^{\dagger}\tilde{b}_{0}+\delta
g(t)\sqrt{L}\left(\tilde{b}_{0}+\tilde{b}_{0}^{\dagger}\right).$ (27)

The driving generates a coherent state of $\tilde{b}_{0}$ bosons. Starting
from the ground state, the transverse magnetization evolves as

$\left(\frac{d^{2}}{dt^{2}}+\omega^{2}\right)\left(\langle\sigma^{x}_{n}\rangle-\langle\sigma^{x}_{n}\rangle_{GS}\right)=-2\omega\leavevmode\nobreak\
\delta g(t).$ (28)

The pairs manifest themselves also in the correlators:

$\displaystyle-\sum_{n}\langle\sigma^{z}_{n}\sigma^{z}_{n+1}\rangle\approx-\sum_{n}\langle\sigma^{z}_{n}\sigma^{z}_{n+2}\rangle\approx$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ -L+4\sum_{n}\langle
b_{n}^{\dagger}b_{n}\rangle+\frac{g}{2}\sqrt{L}\left\langle\tilde{b}_{0}+\tilde{b}_{0}^{\dagger}\right\rangle.$
(29)

A weak driving for a finite time results in persistent oscillations with a
well-defined frequency. Fig. 5 shows their examples for $g=0.25,0.5$. Fig. 3
demonstrates the frequency to be consistent with the gap estimate from exact
diagonalization and the perturbative formula (22).

Conclusion.  In the NN and NNN quantum Ising chain (or the NN zigzag ladder),
the fermionic Bogoliubov quasiparticles below the critical transverse magnetic
field give way to their tightly bound Cooper pairs as the lowest branch of
excitations. They can be readily excited by the KZ ramp or periodic driving.
Despite nonintegrability, the excitation manifests by persistent coherent
oscillations contributing an example to a variety of phenomena like quantum
many-body scars [105, 106, 107, 108] or quantum time crystals [109].

Acknowledgements. This research was funded by the National Science Centre
(NCN), Poland, under project 2021/03/Y/ST2/00184 within the QuantERA II
Programme that has received funding from the European Union’s Horizon 2020
research and innovation programme under Grant Agreement No 101017733 (FB), NCN
under projects 2019/35/B/ST3/01028 (JD) and 2020/38/E/ST3/00150 (MMR) , and
Department of Energy under the Los Alamos National Laboratory LDRD Program
(WHZ). The research was also supported by a grant from the Priority Research
Area DigiWorld under the Strategic Programme Excellence Initiative at
Jagiellonian University (JD, MMR).

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The supplementary material can be divided into two parts. The first one, from
App. A to App. C, outlines the complete solution of the integrable NN quantum
Ising chain that naturally leads to the new results presented in the main
text. The standard parts of the solution are based on four papers: two older
items [52, 59] and two more recent ones [86, 87, 1]. They are included here to
make the supplementary material self-contained. App. B derives oscillations
between $t_{c}+\hat{t}$ and the end of the ramp and App. C estimates their
dephasing rate and the quality factor. The second part, from D to G, deals
with the nonintegrable chain.

Both the first part and the BCS theory in App. D use a fermionic Bogoliubov
theory. The two theories are not the same and should not be confused even
though we use the same symbols like $u_{k},v_{k}$. The former is in the
representation of Jordan-Wigner fermions, while the latter is in that of
kinks.

##  Appendix A  Quantum Ising chain

The Hamiltonian for transverse field quantum Ising chain reads

$H\leavevmode\nobreak\ =\leavevmode\nobreak\
-\sum_{n=1}^{L}\left(\sigma^{z}_{n}\sigma^{z}_{n+1}+g\sigma^{x}_{n}\right)\leavevmode\nobreak\
,$ (S1)

where we consider a system of $L$ spins one-half with periodic boundary
conditions, $\vec{\sigma}_{L+1}\leavevmode\nobreak\ =\leavevmode\nobreak\
\vec{\sigma}_{1}$. In the limit of $L\to\infty$, there are quantum critical
points at $g_{c}=\pm 1$, respectively, that separate the paramagnetic phase
for $|g|>1$ from the ferromagnetic phase for $|g|$<1\. For simplicity of
presentation, we additionally assume that $L$ is even. The Jordan-Wigner
transformation,

$\displaystyle\sigma^{x}_{n}\leavevmode\nobreak\ =\leavevmode\nobreak\
1-2c^{\dagger}_{n}c_{n}\leavevmode\nobreak\ ,$ (S2)
$\displaystyle\sigma^{z}_{n}\leavevmode\nobreak\ =\leavevmode\nobreak\
-\left(c_{n}+c_{n}^{\dagger}\right)\prod_{m<n}(1-2c^{\dagger}_{m}c_{m})\leavevmode\nobreak\
,$ (S3)

introduces fermionic annihilation ($c_{n}$) and creation ($c^{\dagger}_{n}$)
operators. It maps the Hamiltonian in Eq. (S1) to

$H\leavevmode\nobreak\ =\leavevmode\nobreak\ P^{+}\leavevmode\nobreak\
H^{+}\leavevmode\nobreak\ P^{+}\leavevmode\nobreak\ +\leavevmode\nobreak\
P^{-}\leavevmode\nobreak\ H^{-}\leavevmode\nobreak\ P^{-}\leavevmode\nobreak\
.$ (S4)

The projectors on subspaces with even ($+$) and odd ($-$) numbers of
$c$-quasiparticles read

$P^{\pm}=\frac{1}{2}\left[1\pm\prod_{n=1}^{L}\sigma^{x}_{n}\right]=\frac{1}{2}\left[1\leavevmode\nobreak\
\pm\leavevmode\nobreak\
\prod_{n=1}^{L}\left(1-2c_{n}^{\dagger}c_{n}\right)\right].$ (S5)

The reduced Hamiltonians in each parity subspace,

$H^{\pm}\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{n=1}^{L}\left[g\left(c_{n}^{\dagger}c_{n}-\frac{1}{2}\right)-c_{n}^{\dagger}c_{n+1}+c_{n}c_{n+1}\right]+{\rm
h.c.},$ (S6)

differ in boundary conditions. Namely, in $H^{-}$ we assume periodic boundary
conditions, $c_{L+1}=c_{1}$, and in $H^{+}$ we have antiperiodic boundary
conditions, $c_{L+1}=-c_{1}$.

The parity of the number of $c$-quasiparticles commutes with the Hamiltonian.
As the ground state for $g\gg 1$ has even parity, we limit ourselves to that
relevant subspace. The next step in the diagonalization of $H^{+}$ is a
Fourier transform,

$c_{n}\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{e^{-i\pi/4}}{\sqrt{L}}\sum_{k}c_{k}e^{ikn}\leavevmode\nobreak\ ,$ (S7)

with half-integer pseudo-momenta consistent with the antiperiodic boundary
conditions,

$k\leavevmode\nobreak\ =\leavevmode\nobreak\
\pm\frac{1}{2}\frac{2\pi}{L},\pm\frac{3}{2}\frac{2\pi}{L},\dots,\pm\frac{L-1}{2}\frac{2\pi}{L}\leavevmode\nobreak\
.$ (S8)

After this transformation, the Hamiltonian takes the form

$\displaystyle H^{+}\leavevmode\nobreak\ =$ $\displaystyle\sum_{k}$
$\displaystyle\left[(g-\cos
k)\left(c_{k}^{\dagger}c_{k}-c_{-k}c_{-k}^{\dagger}\right)+\right.$ (S9)
$\displaystyle\left.\leavevmode\nobreak\ \leavevmode\nobreak\ \sin
k\left(c^{\dagger}_{k}c^{\dagger}_{-k}+c_{-k}c_{k}\right)\right]\leavevmode\nobreak\
.$

Its diagonalization is completed by a Bogoliubov transformation,

$c_{k}\leavevmode\nobreak\ =\leavevmode\nobreak\
u_{k}\gamma_{k}+v_{-k}^{*}\gamma^{\dagger}_{-k}\leavevmode\nobreak\ ,$ (S10)

where the Bogoliubov modes $(u_{k},v_{k})$ follow as eigenstates of the
Bogoliubov-de Gennes equations,

$\displaystyle\epsilon\leavevmode\nobreak\ u_{k}$ $\displaystyle=$
$\displaystyle+2(g-\cos k)u_{k}+2\sin k\leavevmode\nobreak\ v_{k},$
$\displaystyle\epsilon\leavevmode\nobreak\ v_{k}$ $\displaystyle=$
$\displaystyle-2(g-\cos k)v_{k}+2\sin k\leavevmode\nobreak\ u_{k}.$ (S11)

There are two eigenstates for each value of $k$, with eigenfrequencies
$\epsilon=\pm\epsilon_{k}$,

$\epsilon_{k}\leavevmode\nobreak\ =\leavevmode\nobreak\ 2\sqrt{(g-\cos
k)^{2}+\sin^{2}k}\leavevmode\nobreak\ .$ (S12)

The eigenstates with positive frequency,
$(u^{+}_{k},v^{+}_{k})=(\cos(\theta_{k}/2),\sin(\theta_{k}/2))$, define a
fermionic quasiparticle operator $\gamma_{k}\leavevmode\nobreak\
=\leavevmode\nobreak\ u_{k}^{+*}c_{k}+v^{+}_{-k}c_{-k}^{\dagger}$, where
angles $\theta_{k}$ satisfy
$(\cos\theta_{k},\sin\theta_{k})=\frac{2}{\epsilon_{k}}(g-\cos k,\sin k)$. The
negative frequency ones, with $(u^{-}_{k},v^{-}_{k})=(v^{+}_{k},-u^{+}_{k})$,
formally define
$\gamma_{k}^{-}=u_{k}^{-*}c_{k}+v_{-k}^{-}c_{-k}^{\dagger}=-\gamma_{-k}^{\dagger}$.
After the Bogoliubov transformation, the Hamiltonian reads

$H^{+}\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{k}\epsilon_{k}\leavevmode\nobreak\
\left(\gamma_{k}^{\dagger}\gamma_{k}\leavevmode\nobreak\ -\frac{1}{2}\right).$
(S13)

Note that, due to the projector $P^{+}$ in Eq. (S4), only states with even
numbers of $c$-quasiparticles belong to the spectrum of $H$.

The quasiparticle dispersion in Eq. (S12) implies a linear dispersion for
small $k$ at the critical $g=1$, $\epsilon_{k}\approx 2|k|$, and the dynamical
exponent $z$ is equal to $1$. Moreover, for $k=0$, we have
$\epsilon_{0}\propto|g-1|^{1}$ and $z\nu=1$. Finally, the correlation length
exponent $\nu$ is equal to $1$.

##  Appendix B  Linear quench and the Landau-Zener problem

The Hamiltonian follows a linear ramp in the transverse field,

$g(t\leq 0)\leavevmode\nobreak\ =\leavevmode\nobreak\ -\frac{t}{\tau_{Q}},$
(S14)

with the quench rate $\tau_{Q}$. For convenience, here we fix the time when
the ramp reaches $g(t_{s})=0$ at $t_{s}=0$ (we use $t_{s}$ in the main text
for clarity). As such, time $t$ runs from $-\infty$ to $0$ when the ramp stops
at transverse field $g=0$, crossing the critical point at $g_{c}=1$ when
$t_{c}=-\tau_{Q}$. The system starts in the ground state at $g\to\infty$,
where $(u_{k},v_{k})=(1,0)$, and is the vacuum state annihilated by all
corresponding Bogoliubov operators, $\gamma_{k}|0\rangle=0$.

In addressing the dynamical problem, it is convenient to employ the Heisenberg
picture. The state of the system stays as the vacuum of Bogoliubov operators
$\gamma_{k}$, while the Bogoliubov modes evolve according to the Heisenberg
equation of motion $i\frac{d}{dt}c_{k}=[c_{k},H^{+}]$. Following the time-
dependent Bogoliubov transformation,

$c_{k}=u_{k}(t)\gamma_{k}+v_{-k}^{*}(t)\gamma^{\dagger}_{-k},$ (S15)

this gives time-dependent Bogoliubov-de Gennes equations,

$\displaystyle i\frac{d}{dt}u_{k}$ $\displaystyle=$
$\displaystyle+2\left(g(t)-\cos k\right)u_{k}+2\sin k\leavevmode\nobreak\
v_{k}\leavevmode\nobreak\ ,$ $\displaystyle i\frac{d}{dt}v_{k}$
$\displaystyle=$ $\displaystyle-2\left(g(t)-\cos k\right)v_{k}+2\sin
k\leavevmode\nobreak\ u_{k}\leavevmode\nobreak\ ,$ (S16)

and the initial condition is $(u_{k}(-\infty),v_{k}(-\infty))=(1,0)$.

Introducing a new time variable,

$\tau\leavevmode\nobreak\ =\leavevmode\nobreak\ 4\tau_{Q}\sin
k\left(\frac{t}{\tau_{Q}}+\cos k\right),$ (S17)

that runs from $-\infty$ to $\tau^{\rm final}_{k}=2\tau_{Q}\sin(2k)$ for
$t=0$, allows one to map Eq. (S16) to the Landau-Zener (LZ) problem [52, 59],

$\displaystyle i\frac{d}{d\tau}u_{k}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\tau\Delta_{k}\leavevmode\nobreak\
u_{k}+\frac{1}{2}v_{k},$ $\displaystyle i\frac{d}{d\tau}v_{k}$
$\displaystyle=$ $\displaystyle+\frac{1}{2}\tau\Delta_{k}\leavevmode\nobreak\
v_{k}+\frac{1}{2}u_{k}.$ (S18)

Here, $\Delta_{k}=(4\tau_{Q}\sin^{2}k)^{-1}$ sets an efficient rate of the
transition for given $k$.

Only modes with small $k$ that have small energy gaps at their anti-crossing
point can get excited when the ramp is slow. For such modes, $\tau^{\rm
final}_{k}$ is much longer than the time when the anti-crossing is completed
and we are allowed to use the LZ formula,

$p_{k}\leavevmode\nobreak\ \approx\leavevmode\nobreak\
e^{-\frac{\pi}{2\Delta_{k}}}\leavevmode\nobreak\ \approx\leavevmode\nobreak\
e^{-2\pi\tau_{Q}k^{2}}\leavevmode\nobreak\ ,$ (S19)

where approximations become accurate for $\tau_{Q}\gg 1$. Eq. (S19) gives the
probability that a pair of quasiparticles with quasimomenta $+k$ and $-k$ got
excited. The mean density of kinks at $g=0$ is simply given by
$\rho=\sum_{k}p_{k}/L$ [52]. Taking the limit $L\to\infty$,

$\rho=\lim_{L\to\infty}\frac{1}{L}\sum_{k}p_{k}=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\leavevmode\nobreak\
p_{k}\approx\frac{1}{2\pi\sqrt{2\tau_{Q}}}.$ (S20)

The density scales as an inverse of $\hat{\xi}\propto\tau_{Q}^{1/2}$, in full
consistency with KZM prediction for $\nu=z=1$. For convenience, we can use the
density of kinks to supplement the numerical prefactor

$\hat{\xi}\equiv\frac{1}{\rho}=2\pi\sqrt{2\tau_{Q}},$ (S21)

making its inverse equal to the mean density of kinks at the end of the ramp
at $g=0$.

###  B.1  Ramp to zero transverse field

To characterize the oscillations, we require more than just the excitation
spectrum in Eq. (S19). A general solution to Eqs. (S18) has the form [110,
59],

$\displaystyle v_{k}(\tau)$ $\displaystyle=$ $\displaystyle-
aD_{-s-1}(-iz)-bD_{-s-1}(iz),$ $\displaystyle u_{k}(\tau)$ $\displaystyle=$
$\displaystyle\left(-\Delta_{k}\tau+2i\frac{\partial}{\partial\tau}\right)v_{k}(\tau),$
(S22)

where $D_{m}(x)$ is a Weber function, $s^{-1}=4i\Delta_{k}$, and
$iz=\sqrt{\Delta_{k}}\tau e^{i\pi/4}$. Constants $a,b$ are fixed by initial
conditions. From the asymptotic behavior of the Weber function when
$\tau\to-\infty$, one gets $a=0$, and

$|b|^{2}=\frac{e^{-\pi/8\Delta_{k}}}{4\Delta_{k}}\leavevmode\nobreak\ .$ (S23)

On the other hand, at the end of the ramp when $t=0$ and
$\tau=2\tau_{Q}\sin(2k)$ the argument of the Weber function reads
$iz=2\sqrt{\tau_{Q}}e^{i\pi/4}{\rm sign}(k)\cos(k)$. Its absolute value is
large for slow transitions (except near $k=\pm\frac{\pi}{2}$), and one can
again use the asymptotic behavior of the Weber function [59].

When $(t-t_{c})\gg\hat{t}$ the evolution becomes adiabatic and the time-
dependent Bogoliubov modes can be accurately decomposed as

$\left(\begin{array}[]{c}u_{k}\\\
v_{k}\end{array}\right)=\sqrt{1-p_{k}}\left(\begin{array}[]{c}u^{+}_{k}\\\
v^{+}_{k}\end{array}\right)+\sqrt{p_{k}}\left(\begin{array}[]{c}v^{+}_{k}\\\
-u^{+}_{k}\end{array}\right)e^{i\varphi_{k}(t)},$ (S24)

were $(u^{+}_{k},v^{+}_{k})$ is a positive-frequency stationary Bogoliubov
mode at $g(t)$. With the asymptotic behavior of the Weber function the phase
at $t=0$ reads [59]:

$\displaystyle\varphi_{k}(0)$ $\displaystyle=$
$\displaystyle\frac{\pi}{4}+2\tau_{Q}-(2-\ln
4)k^{2}\tau_{Q}+k^{2}\tau_{Q}\ln\tau_{Q}+$ (S25) $\displaystyle-$
$\displaystyle\arg\left[\Gamma\left(1+ik^{2}\tau_{Q}\right)\right]\leavevmode\nobreak\
.$

Above, $\varphi_{k}$ is a dynamical phase acquired by a pair of excited
quasiparticles $(k,-k)$, with $\Gamma(x)$ being the gamma function. We follow
[86], and approximate ${\rm
arg}[\Gamma\left(1+i\tau_{Q}k^{2}\right)]\approx-\gamma_{E}\tau_{Q}k^{2}$ to
make the phase more tractable. Here $\gamma_{E}$ is the Euler gamma constant.
The approximation is valid for small enough $\tau_{Q}k^{2}$, which is
consistent with the fact that excited quasiparticles have at most
$\tau_{Q}k^{2}\approx 1/2\pi$, see Eq. (S19). This makes $\varphi_{k}$
conveniently quadratic in $k$,

$\displaystyle\varphi_{k}(0)=\frac{\pi}{4}+2\tau_{Q}+\left(\ln\tau_{Q}+\ln
4-2+\gamma_{E}\right)k^{2}\tau_{Q}$ (S26)

This formula completes the full characterization of the Bogoliubov modes when
the linear ramp ends at $g=0$.

###  B.2  Ramp after $\mathbf{+\hat{t}}$

In order to obtain phase $\varphi_{k}(t)$ at earlier time $t<0$, we have to
subtract from (S26) the dynamical phase acquired between $t$ and the final
time when $g=0$. The dynamical phase depends on the spectrum of quasiparticles
that for small $k$ can be approximated as

$\epsilon_{k}(g)\approx 2(1-g)+\frac{g}{(1-g)}k^{2}.$ (S27)

The dynamical phase at time $t$ can be obtained as

$\displaystyle\varphi_{k}(t)$ $\displaystyle=$
$\displaystyle\varphi_{k}(0)-2\int_{t}^{0}dt^{\prime}\epsilon_{k}[g(t^{\prime})]$
$\displaystyle=$
$\displaystyle\varphi_{k}(0)-2\int_{t}^{0}dt^{\prime}\left[2[1-g(t^{\prime})]+\frac{g(t^{\prime})}{[1-g(t^{\prime})]}k^{2}\right]$
$\displaystyle=$
$\displaystyle\frac{\pi}{4}+\frac{2t_{+}^{2}}{\tau_{Q}}+\left(\gamma_{E}-\frac{2t_{+}}{\tau_{Q}}+\ln\frac{4t_{+}^{2}}{\tau_{Q}}\right)k^{2}\tau_{Q},$

where $t_{+}=t-t_{c}$. This formula is valid for small $k$ within the support
of $p_{k}$.

Within the same support we can approximate $u_{k}^{+}\approx k/2/(1-g)+{\cal
O}(k^{3})$ and $v_{k}^{+}\approx 1+{\cal O}(k^{2})$. With this approximation
relevant products of Bogoliubov modes that follow from (S24) become

$\displaystyle|v_{k}|^{2}$ $\displaystyle\approx$
$\displaystyle|v_{k}^{+}|^{2}-p_{k}-\sqrt{p_{k}(1-p_{k})}\frac{k\cos\varphi_{k}(t)}{(1-g)}$
$\displaystyle u_{k}v_{k}^{*}$ $\displaystyle\approx$ $\displaystyle
u_{k}^{+}v_{k}^{+}-p_{k}\frac{k}{(1-g)}+\sqrt{p_{k}(1-p_{k})}e^{i\varphi_{k}(t)}.$

The last terms with $\varphi_{k}(t)$ — that contribute to oscillations — and
the middle terms with $p_{k}$ are approximated by their leading terms in
powers of $k$. They will become the leading terms in $\rho$.

In order to make the following integrals analytically tractable we approximate
[86]:

$\displaystyle\sqrt{p_{k}(1-p_{k})}\approx
e^{-a\pi\tau_{Q}k^{2}}A\sqrt{2\pi}\left(\tau_{Q}k^{2}\right)^{1/2}.$ (S29)

Here, $A$ and $a$ are variational parameters that can be optimally chosen as
$A\approx 19/20$ and $a\approx 4/3$. The transverse field is

$\langle\sigma^{x}_{n}\rangle=1-2\int_{0}^{\pi}\frac{dk}{\pi}\left|v_{k}\right|^{2}.$
(S30)

The integration yields (4). The NN ferromagnetic correlator is

$\langle\sigma^{z}_{n}\sigma^{z}_{n+1}\rangle=2\int_{0}^{\pi}\frac{dk}{\pi}\left|v_{k}\right|^{2}\cos
k+2{\rm Re}\int_{0}^{\pi}\frac{dk}{\pi}u_{k}v_{k}^{*}\sin k.$ (S31)

After approximating $\cos k\approx 1$ and $\sin k\approx k$ to leading order
in $k$ and then performing the integral, we obtain (7) that is accurate to
leading order in $\rho$. The other correlator is

$\langle\sigma^{y}_{n}\sigma^{y}_{n+1}\rangle=2\int_{0}^{\pi}\frac{dk}{\pi}\left|v_{k}\right|^{2}\cos
k-2{\rm Re}\int_{0}^{\pi}\frac{dk}{\pi}u_{k}v_{k}^{*}\sin k.$ (S32)

With the same approximations, we obtain

$\delta^{yy}=-2\rho-(2-g)\rho^{2}d\frac{57\sqrt{6\pi}}{80}\cos\phi.$ (S33)

It does not vanish at $g=0$.

##  Appendix C Oscillations and dephasing after $\mathbf{+\hat{t}}$

The expansion in Eq. (S24) becomes accurate at times later than $\hat{t}$
after the phase transition. The phase $\varphi_{k}$ increases as

$\varphi_{k}(t)=\int^{t}dt^{\prime}\leavevmode\nobreak\
2\epsilon_{k}[g(t^{\prime})].$ (S34)

After $\hat{t}$, the quasiparticle spectrum for KZM excitations that are
localized near $k=0$ can be considered flat and equal to the gap
$\epsilon_{0}(g)$ that opens with the increasing distance from the critical
point. Therefore, the transverse field oscillates with frequency given by
twice the instantaneous gap. With (S27)

$T=\frac{2\pi}{2\epsilon_{0}(g)}=\frac{\pi}{2(1-g)}$ (S35)

is the period of the oscillations.

Beyond the approximation of flat dispersion, there is some dephasing. The
dephasing time can be estimated with the help of the approximate dispersion
relation in Eq. (S27). The difference between $\epsilon_{k}$ for
$\hat{k}\approx 1/\sqrt{\tau_{Q}}$ and $k=0$ is
$\delta\epsilon\approx\frac{g}{|1-g|\tau_{Q}}$. Therefore, the phase gets
scrambled on a timescale

$\tau_{D}=\frac{\pi}{\delta\epsilon}\approx\frac{\pi|1-g|\tau_{Q}}{g}.$ (S36)

When combined with the period (S35) it yields a quality factor:

$Q=\frac{\tau_{D}}{T}\approx 2\tau_{Q}\frac{(1-g)^{2}}{g}.$ (S37)

It diverges at $g=0$ when the dispersion is flat and is the smallest soon
after $\hat{t}$ when $1-g\approx 1/\sqrt{\tau_{Q}}$ and $Q\approx 2$. For a
later $g$ it improves with increasing $\tau_{Q}$ that makes the excitations
more narrow in $k$ reducing the dispersion.

Figure S1: BCS versus MPS.  Comparison between expectation values (from top to
bottom) $\Delta$, $\rho$, and $t_{f}$ obtained within the BCS theory and with
infinite matrix product states (MPS). The dotted lines are the perturbative
formulas (13).

##  Appendix D  BCS theory

By Wick’s theorem, the energy per site is

$\displaystyle L^{-1}\bra{0}H\ket{0}$ $\displaystyle=$
$\displaystyle-(J_{1}+J_{2})+2\left(J_{1}+2J_{2}\right)\rho$ (S38)
$\displaystyle-g\left(t_{f}+t_{f}^{*}+\Delta+\Delta^{*}\right)$
$\displaystyle-4J_{2}\left(\rho^{2}+\Delta^{*}\Delta-t_{f}^{*}t_{f}\right).$

It is energetically favorable we assume $\Delta$ and $t_{f}$ real. Self-
consistency requires the expectation values to satisfy:

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\int_{-\pi}^{\pi}\frac{dk}{2\pi}v_{k}^{2},$ (S39) $\displaystyle
t_{f}$ $\displaystyle=$
$\displaystyle\int_{-\pi}^{\pi}\frac{dk}{2\pi}v_{k}^{2}\cos k,$ (S40)
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle-\int_{-\pi}^{\pi}\frac{dk}{2\pi}u_{k}v_{k}\sin k.$ (S41)

Here we used that $u_{k},v_{k}$ are real and have opposite parities with
respect to $k$. A self-consistent solution of the Bogoliubov-de Gennes
equations is shown in Fig. S1 where it is compared with results obtained with
the matrix product states (MPS). Derivatives of the averages with respect to
$g$ are shown in Fig. 2 in the main text and here in Fig. S2. The derivatives
allow to locate the critical point at $g_{c}=2.48135$ (BCS) and
$g_{c}=2.47725$ (MPS). These estimates agree within $0.2\%$.

Figure S2: BCS versus MPS.  First derivatives with respect to the field $g$ of
$\rho$, and $t_{f}$ in the ground state from the BCS theory and matrix product
states (MPS) in Fig. S1. A derivative of $\Delta$ is shown in Fig. 2 in the
main text. Estimates of $g_{c}$ from BCS and MPS are $g_{c}=2.48135$ and
$g_{c}=2.47725$, respectively. The dotted lines are the derivatives of the
perturbative formulas (13).

##  Appendix E  Effective pair Hamiltonian

The effective pair Hamiltonian (21) follows from the kink Hamiltonian in (18),
(19), and (20). It operates in a Hilbert space spanned by Fock states

$b_{n_{1}}^{\dagger}\dots b_{n_{M}}^{\dagger}|0\rangle,$ (S42)

where $n_{i}+1<n_{i+1}$. From the second and third term in the first line of
(18) we obtain a contribution to the pair energy $\omega_{b}$ in (21)

$\omega_{b}^{(1)}=2\left(6+\frac{g^{2}}{8}\right)-\left(4-\frac{g^{2}}{8}\right)=8+\frac{3g^{2}}{8}.$
(S43)

In terms of pair operators, the third line in (18) reads

$\displaystyle t^{\prime}_{\gamma}\sum_{n}\left(b_{n}^{\dagger}b_{n+1}+{\rm
h.c.}\right)+$
$\displaystyle-\frac{g^{2}}{16}\sum_{n}\left(b_{n}^{\dagger}b_{n-1}+b^{\dagger}_{n+1}b_{n-1}+{\rm
h.c.}\right).$ (S44)

Its contribution to the NN and NNN hopping terms in (21) are

$\displaystyle t_{b}^{(1)}$ $\displaystyle=$
$\displaystyle-t^{\prime}_{\gamma}+\frac{g^{2}}{16}=-\frac{g^{2}}{8},$ (S45)
$\displaystyle t^{\prime(1)}_{b}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{16}.$ (S46)

The second line of (18), which is linear in $g$, breaks/creates/annihilates
pairs. Its contributions are second-order perturbative corrections to the pair
energy and the NNN hopping. The $t_{\gamma}$-term contributes:

$\displaystyle\omega^{(2)}_{b}$ $\displaystyle=$
$\displaystyle-2\frac{t_{\gamma}^{2}}{4}=-\frac{g^{2}}{2},$ (S47)
$\displaystyle t^{(2)}_{b}$ $\displaystyle=$
$\displaystyle\frac{t_{\gamma}^{2}}{4}=\frac{g^{2}}{4},$ (S48)

and the $V^{(1)}$-terms contribute:

$\displaystyle\omega^{(3)}_{b}$ $\displaystyle=$
$\displaystyle-2\frac{(g/2)^{2}}{4}=-\frac{g^{2}}{8},$ (S49) $\displaystyle
t^{\prime(3)}_{b}$ $\displaystyle=$
$\displaystyle\frac{(g/2)^{2}}{4}=\frac{g^{2}}{16}.$ (S50)

Adding up all contributions, we obtain
$\omega_{b}=\omega^{(1)}_{b}+\omega^{(2)}_{b}+\omega^{(3)}_{b}=8-g^{2}/4$,
$t_{b}=t_{b}^{(1)}+t_{b}^{(2)}=g^{2}/8$, and
$t^{\prime}_{b}=t^{\prime(1)}_{b}+t^{\prime(3)}_{b}=g^{2}/8$.

The second order term:

$\displaystyle g^{2}\tilde{V}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{8}\sum_{n}\gamma_{n+1}^{\dagger}\gamma_{n-1}^{\dagger}\gamma_{n-1}\gamma_{n+1}+$
(S51)
$\displaystyle-\frac{g^{2}}{16}\sum_{n}\left(\gamma_{n}\gamma_{n+1}\gamma_{n+2}\gamma_{n+3}+{\rm
h.c.}\right)+$
$\displaystyle-\frac{g^{2}}{16}\sum_{n}\left(\gamma_{n-3}^{\dagger}\gamma_{n}^{\dagger}\gamma_{n}\gamma_{n-1}+{\rm
h.c.}\right)+$
$\displaystyle-\frac{g^{2}}{16}\sum_{n}\left(\gamma_{n+3}^{\dagger}\gamma_{n}^{\dagger}\gamma_{n}\gamma_{n+1}+{\rm
h.c.}\right)+$
$\displaystyle-\frac{g^{2}}{6}\sum_{n}\left(\gamma_{n}^{\dagger}\gamma_{n-1}\gamma_{n}\gamma_{n+1}+{\rm
h.c.}\right)+$
$\displaystyle-\frac{g^{2}}{12}\sum_{n}\left(\gamma_{n}^{\dagger}\gamma_{n}\gamma_{n-1}\gamma_{n-3}+{\rm
h.c.}\right)+$
$\displaystyle-\frac{g^{2}}{12}\sum_{n}\left(\gamma_{n}^{\dagger}\gamma_{n}\gamma_{n+3}\gamma_{n+1}+{\rm
h.c.}\right).$

does not contribute here. Its leading contribution ${\cal O}(g^{4})$ could be
obtained as a second-order perturbation against quasiparticle energy ($6$ \-
for terms that change the quasiparticle number) or the pair binding energy
($4$ \- for terms that break pairs).

Figure S3: Crash test at $\mathbf{g=0}$.  Top row: The periodic driving
$\delta g(t)=A\sin(8t)$ at $g=0$ with amplitude $A=0.1$ for time $2\pi$
results in a large amplitude of transverse oscillations. The amplitude is
lower than predicted by (28) due to the hard-core nature of bosons $b_{n}$
that is not quite negligible when their density $0.1$. Almost all
quasiparticles/kinks are bound into pairs; densities of longer trains are at
$10^{-4}$ or lower. Bottom row: Upon closer inspection, the oscillations for
$A=0.1$, and even more $0.5$, have an admixture of frequency $4$ in addition
to the main $\omega=8$. It is a manifestation of $4$-kink trains with density
$0.00008$ and $0.001$ respectively.

##  Appendix F Crash test at $\mathbf{g=0}$

Here, we test the theory for strong periodic driving. It is the simplest at
$g=0$ when a Bogoliubov quasiparticle is just a kink,

$\gamma_{n}=k_{n},$ (S52)

and the Hamiltonian simplifies to

$\displaystyle H$ $\displaystyle=$
$\displaystyle-L+6\sum_{n}\gamma_{n}^{\dagger}\gamma_{n}-4\sum_{n}\gamma_{n}^{\dagger}\gamma_{n+1}^{\dagger}\gamma_{n+1}\gamma_{n}.$
(S53)

The basic theory ignores that the pairs created by
$b_{n}^{\dagger}=\gamma^{\dagger}_{n}\gamma^{\dagger}_{n+1}$ are not exactly
bosons. It also ignores trains of kinks/quasiparticles longer than the pair.
The trains also make bound states. Their energy at $g=0$ is
$6n-4(n-1)=2n+4=6,8,10,12,...$ for $n=1,2,3,4,...$. At nonzero $g$ the terms
(19) can transform a train of $n>2$ quasiparticles into one with $n-2$ — that
has lower energy — making the longer trains unstable. This mechanism does not
exist at $g=0$.

In Fig. S3, we drive $\delta g(t)=A\sin(8t)$ with $A=0.1,0.5$ for time $2\pi$.
$A=0.1$ is enough to excite large transverse field oscillations that do not
quite agree with (28). In addition to the lower amplitude of the main
frequency $\omega=8$, there is an admixture of frequency $4$. Both effects
become stronger for the stronger driving with $A=0.5$, which increases the
density of excited kinks. The frequency $4$ originates from a superposition
between a train of 4 kinks (energy $12$) and a pair of kinks (energy $8$),
e.g.

$\alpha\ket{...\uparrow\uparrow\downarrow\uparrow\uparrow\uparrow\uparrow...}e^{-8it}+\beta\ket{...\uparrow\uparrow\downarrow\uparrow\downarrow\uparrow\uparrow...}e^{-12it},$
(S54)

where the corresponding $...$ are the same in both states. The states differ
by one reversed spin; hence, $\sigma^{x}$ has a non-zero matrix element
between them, and its expectation value oscillates with frequency $12-8=4$.

Figure S4: Amplitude of oscillations.  The amplitude of oscillations after a
smooth KZ ramp ending at $g=0.5$ with a protocol
$g(t)=g_{c}[2-(1-0.5/(2g_{c}))(1+\sin(t/\tau_{Q}))]$,
$t\in[-\tau_{Q}\times\pi/2,\tau_{Q}\times\pi/2]$).

##  Appendix G  Amplitude of oscillations after KZ ramp

After a KZ ramp the number of kinks $\rho\propto\tau^{-1/2}$ as expected for
this universality class [1]. The frequency of the persistent oscillations is
consistent with Fig. 3. Their amplitude depends on the ramp time with a power
law $\tau^{-0.7}$, see Fig. S4.

A simple estimate for the exponent is as follows. After crossing the phase
transition the density of excited $\gamma$-quasiparticles is
$\rho\propto\tau^{-1/2}$. Near the crossover at $g_{0}$, when the bound pairs
begin to have lower energy, the number of pairs can be roughly estimated as
$\rho_{p}\propto\rho^{2}\propto\tau^{-1}$. The amplitude is proportional to
the square root of $\rho_{p}$, compare the simple example in (23), and should
scale as $\tau^{-0.5}$. The $0.5$ is close to the $0.7$ but appreciably
different. Some more complex physics is missing in the simple argument.