# Non-equilibrium many-body dynamics in supersymmetric quenching

Christopher Campbell<EMAIL_ADDRESS>Thomás Fogarty  Quantum
Systems Unit, Okinawa Institute of Science and Technology Graduate University,
Onna, Okinawa 904-0495, Japan  Thomas Busch  Quantum Systems Unit, Okinawa
Institute of Science and Technology Graduate University, Onna, Okinawa
904-0495, Japan

###### Abstract

We study the dynamics induced by quenching an ultracold quantum many-body
system between two supersymmetric Hamiltonians. Such a quench can be created
by carefully changing the external trapping potential and leads to a situation
where the eigenspectra before and after the quench are nearly identical. We
show that the dynamics originating from this can be conveniently described
using knowledge about the initial state only and apply this insight to the
specific example of a fermionic gas that is initially trapped in an infinite
box potential. Quenching to different, higher order supersymmetric partners
potentials we observe the appearance of many-body revivals in the survival
probability and show that some of these are robust at finite temperatures.
This is in contrast to the well known Talbot effect, which is the standard
example for quenching into a system with a quadratic spectrum.

##  I Introduction

The ongoing advances in the control and measurement of ultracold quantum gases
has established these systems as highly promising laboratories to study many-
body quantum dynamics Mistakidis  _et al._ (2022). Of particular interest in
this area is the evolution of a system after a quench of its internal or
external parameters and the question of how the large number of degrees of
freedom leads to equilibration and thermalization over time Eisert  _et al._
(2015). To study these, a system is usually considered to be prepared in a
pure state, typically the ground state of a potential, and a sudden change in
the Hamiltonian parameters or the addition of an external perturbation is used
to excite it. Many ways to do this have been proposed and studied in recent
years, ranging from from giving a momentum kick to a quantum gas Kinoshita
_et al._ (2006), changing the trapping potential del Campo and Muga (2006);
Collura  _et al._ (2013); Nagao  _et al._ (2019), adjusting the interparticle
interaction strengths Mistakidis  _et al._ (2014); García-March  _et al._
(2016), or introducing an impurity Knap  _et al._ (2012a); Campbell  _et al._
(2014).

Once the system is excited, the subsequent non-equilibrium dynamics is often
rather complex. However, over the past decade a number of quantities and tools
have been developed that allow one to capture typical behaviour and identify
underlying mechanisms, such as the Loschmidt echo Keller and Fogarty (2016);
Fogarty  _et al._ (2020), the work probability distribution Fusco  _et al._
(2014); Campbell (2016) and the amount of information scrambling Chenu  _et
al._ (2018, 2019); Mikkelsen  _et al._ (2022). Most quenches that rely on a
change of the Hamiltonian parameters induce the non-equilibrium dynamics via a
change in the eigenspectrum of the system. In this case states that were
previously stationary become superpositions of eigenstates, which leads to
dynamics that are strongly affected by the relation between the initial and
the final spectrum. Interesting examples here are quenches in shape invariant
potentials Deffner  _et al._ (2014), such as harmonic traps Minguzzi and
Gangardt (2005); Rohringer  _et al._ (2015); Collura  _et al._ (2018), and box
potentials Friesch  _et al._ (2000); del Campo and Boshier (2012); Vicari
(2019) where the initial and final spectrum are related through a scaling
factor. These systems can exhibit perfect revivals of the initial state for
both single particle and many-body systems, similar to the well-known Talbot
effect in infinite box potentials which has also been thoroughly studied for
both bosonic and fermionic cold atoms Grochowski  _et al._ (2020); Łebek  _et
al._ (2021). It is also worth noting that the recurrences of many-body states
are related to the concepts of quantum scars, which signal weak ergodicity
breaking after a global quench Surace  _et al._ (2020); Dooley and Kells
(2020); Turner  _et al._ (2018).

In this work we study a specific type of quench that leaves the eigenspectrum
unperturbed, but removes a fixed number of states from below. This situation
appears when quenching between two different Hamiltonians that are related
through a supersymmetric algebra, which results in the eigenspectra between
the two potentials being near identical Cooper  _et al._ (1995); Sukumar
(1985a). The supersymmetric relationship also provides a number of tools that
can be used to transform a wavefunction between the two partner potentials,
namely the supersymmetric operators and the superpotential. We will show that
these allow us to recast the non-equilibrium dynamics of the post quench
system purely in terms of the the initial Hamiltonian and its supersymmetric
operators. While this has many advantages from a practical perspective, it
also allows us to highlight and understand the unique dynamics that emerge in
supersymmetric quenches. This is therefore another example where
supersymmetric setups can be used in quantum systems for controlled dynamics,
similar to approaches developed for the ground state preparation of ultracold
atoms arrays Luo  _et al._ (2021) or mode filtering in optical setups Heinrich
_et al._ (2014).

In the following we will first briefly review the concepts of supersymmetric
quantum mechanics relevant to our work and introduce the main quantities we
will use to describe a quench between two supersymmetric Hamiltonians. We then
study the ensuing dynamics at zero and at finite temperatures and in
particular the appearance and stability of full revivals in the survival
probability. These results are also compared to the related Talbot effect.

Figure 1: Infinite box potential, $V^{(1)}$, and its first supersymmetric
partner, $V^{(2)}=\frac{\pi^{2}}{2L^{2}}\left(2\sec^{2}(\frac{\pi
x}{L})-1\right)$. The spectrum is indicated by the dashed lines and the
densities of the eigenfunctions, $|\psi_{n}^{(1)}|^{2}$ and
$|\psi_{m}^{(2)}|^{2}$, are given by the red curves.

###  I.1 Supersymmetric Hamiltonians

A supersymmetric algebra relates two Hamiltonians that can be written in a
factorized manner as $H^{(1)}=A_{1}^{\dagger}A_{1}$ and
$H^{(2)}=A_{1}A_{1}^{\dagger}$, where Infeld and Hull (1951); Cooper  _et al._
(1995),

$\displaystyle A_{1}=$
$\displaystyle\frac{\hbar}{\sqrt{2m}}\frac{d}{dx}+\mathcal{W}(x),$ (1)
$\displaystyle A_{1}^{\dagger}=$
$\displaystyle-\frac{\hbar}{\sqrt{2m}}\frac{d}{dx}+\mathcal{W}(x).$ (2)

The function $\mathcal{W}(x)$ is known as the superpotential and writing the
Hamiltonians in the explicit form

$\displaystyle H^{(1)}$
$\displaystyle=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}-\frac{\hbar}{\sqrt{2m}}\mathcal{W}^{\prime}(x)+\mathcal{W}(x)^{2},$
(3) $\displaystyle H^{(2)}$
$\displaystyle=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+\frac{\hbar}{\sqrt{2m}}\mathcal{W}^{\prime}(x)+\mathcal{W}(x)^{2},$
(4)

allows one to find two supersymmetric potentials as

$\displaystyle V^{(1)}(x)$
$\displaystyle=\mathcal{W}(x)^{2}-\frac{\hbar}{\sqrt{2m}}\mathcal{W}^{\prime}(x),$
(5) $\displaystyle V^{(2)}(x)$
$\displaystyle=\mathcal{W}(x)^{2}+\frac{\hbar}{\sqrt{2m}}\mathcal{W}^{\prime}(x).$
(6)

The superpotential itself can be determined in a number of ways Cooper  _et
al._ (1995), with a straightforward one being given by the logarithmic
derivative of the ground state wavefunction of a given $H^{(1)}$ as

$\mathcal{W}(x)=-\frac{\hbar}{\sqrt{2m}}\partial_{x}\ln\left(\psi^{(1)}_{0}\right).$
(7)

An important property of supersymmetric Hamiltonians is that they possess near
identical eigenspectra, with the one for the second Hamiltonian shifted
upwards by one quantum number, i.e. $E_{m}^{(2)}=E_{m+1}^{(1)}$ (see Fig. 1
for the example of an infinite well Gutierrez  _et al._ (2018)).

From the factorized form of the Hamiltonians one can immediately see that an
intertwining relationship between them exists

$\displaystyle A_{1}H^{(1)}$ $\displaystyle=H^{(2)}A_{1},$ (8) $\displaystyle
H^{(1)}A_{1}^{\dagger}$ $\displaystyle=A_{1}^{\dagger}H^{(2)},$ (9)

which can be used to connect the respective eigenfunctions as Cooper  _et al._
(1989)

$\psi^{(2)}_{m}=\frac{A_{1}}{\sqrt{\Delta E_{m}}}\psi^{(1)}_{m+1},$ (10)

where $\Delta E_{m}=E^{(2)}_{m}-E^{(1)}_{0}$. The operators $A$ and
$A^{\dagger}$ therefore map even wavefunctions into odd ones.

The intertwining property between the two Hamiltonians also extends to the
time propagation operator $U(t)=e^{-\frac{i}{\hbar}Ht}$ Lahrz  _et al._
(2017). This can be easily seen by expanding the exponential in a power series
as

$e^{-\frac{i}{\hbar}H^{(2)t}}=\sum^{\infty}_{k=0}\frac{1}{k!}\left(-\frac{i}{\hbar}H^{(2)}t\right)^{k},$
(11)

and noting that all terms in the sum are of the form
$(A_{1}A_{1}^{\dagger})^{k}$. Applying an annihilation operator from the right
then leads to $A_{1}(H^{(1)})^{k}$ and therefore to

$A_{1}U^{(1)}(t)=U^{(2)}(t)A_{1}.$ (12)

Iterating on the supersymmetric relation between $H^{(1)}$ and $H^{(2)}$, one
can create a hierarchy of supersymmetric partner potentials by finding a
factorization of $H^{(2)}=A_{2}^{\dagger}A_{2}$ Fernández C. and Fernández-
García (2004). The resulting partner Hamiltonian,
$H^{(3)}=A_{2}A_{2}^{\dagger}$, has a spectrum that is degenerate with the
ones of $H^{(1)}$ and $H^{(2)}$, however it is missing the ground state
energies of each. The transformations between higher order Hamiltonians
require their own set of operators, however the connection between the
eigenfunctions of the, say, first and third supersymmetric Hamiltonian is
simply given by

$\psi^{(1)}_{m+2}=\frac{1}{\sqrt{E^{(3)}_{m}-E^{(1)}_{0}}}\frac{1}{\sqrt{E^{(3)}_{m}-E^{(2)}_{0}}}A^{\dagger}_{1}A^{\dagger}_{2}\psi^{(3)}_{m}\;.$
(13)

###  I.2 Survival Probability and Dynamical Overlap

In the following we will consider an effectively one-dimensional, spin-
polarized Fermi gas of $N$ neutral atoms at low temperatures Liao  _et al._
(2010). The gas is assumed to be confined in a potential that can be changed
time-dependently with a high degree of flexibility, which can be
experimentally realised in the recently emerging setups that use SLMs
Henderson  _et al._ (2009); Barredo  _et al._ (2018). The many-body states
$\mathcal{H}^{(\alpha)}|\Psi_{n}^{(\alpha)}\rangle=\mathcal{E}_{n}^{(\alpha)}|\Psi_{n}^{(\alpha)}\rangle$
are determined by the Hamiltonian

$\mathcal{H}^{(\alpha)}=\sum^{N}_{j=1}\left[-\frac{\hbar^{2}}{2m}\nabla^{2}_{j}+V^{(\alpha)}(x_{j})\right],$
(14)

where $\alpha$ represents the order of the supersymmetric hierarchy and
$\alpha=1$ is the original potential we quench from. The initial state at
inverse temperature $\beta=(k_{B}T)^{-1}$ is given by
$\rho^{(1)}=\sum_{n}\frac{1}{Z_{0}}e^{\beta(\mathcal{E}_{n}^{(1)}-N\mu)}|\Psi_{n}^{(1)}\rangle\langle\Psi_{n}^{(1)}|$,
where $\mu$ is the chemical potential of the system, $k_{B}$ is the Boltzmann
constant and $Z_{0}$ is the grand canonical partition function.

The non-equilibrium dynamics induced by a quench is generally rather complex
and a good quantity to characterise it is the survival probability, which
describes the squared overlap of the initial state with the time-evolved one.
For a quench from $\mathcal{H}^{(1)}$ to $\mathcal{H}^{(2)}$ it is given by
$F(t)=|\mathcal{O}(t)|^{2}=\left|\Tr\left[e^{\frac{i}{\hbar}\mathcal{H}^{(2)}t}e^{-\frac{i}{\hbar}\mathcal{H}^{(1)}t}\rho^{(1)}\right]\right|^{2}$.
The calculation of the dynamical overlap $\mathcal{O}(t)$ can be eased
considerably by rewriting the many-body states $|\Psi_{n}^{(\alpha)}\rangle$
as a Slater determinant of single particle states
$H^{(\alpha)}|\psi^{(\alpha)}_{k}\rangle=E^{(\alpha)}_{k}|\psi^{(\alpha)}_{k}\rangle$,
which gives Abanin and Levitov (2005); d’Ambrumenil and Muzykantskii (2005);
Knap  _et al._ (2012b); Schmidt  _et al._ (2018)

$\mathcal{O}(t)=\det\left[1-\hat{n}+\hat{n}e^{\frac{i}{\hbar}H^{(2)}t}e^{-\frac{i}{\hbar}H^{(1)}t}\right],$
(15)

where $\hat{n}=(e^{\beta(H^{(1)}-\mu)}+1)^{-1}$ is the Fermi–Dirac
distribution and
$H^{(\alpha)}=-\frac{\hbar^{2}}{2m}\nabla^{2}+V^{(\alpha)}(x)$ is the
corresponding single particle Hamiltonian.

At zero temperature the initial many-body ground state is a perfect Fermi sea
described by

$\Psi_{0}^{(1)}\left(x_{1},x_{2},...,x_{N}\right)=\frac{1}{\sqrt{N!}}\det_{k,j=1}^{N}[\psi^{(1)}_{k}(x_{j})],$
(16)

which allows one to write the survival probability in a simpler way as del
Campo (2011); Goold  _et al._ (2011); Knap  _et al._ (2012b)

$F_{T=0}(t)=\left|\bra{\Psi^{(1)}_{0}}e^{\frac{i}{\hbar}\mathcal{H}^{(2)}t}e^{-\frac{i}{\hbar}\mathcal{H}^{(1)}t}\ket{\Psi^{(1)}_{0}}\right|^{2}.$
(17)

This can be simplified further by introducing the dynamical overlap
probabilities of the single particle states as

$\displaystyle\mathcal{O}_{kl}(t)=$
$\displaystyle\bra{\psi^{(1)}_{k}}e^{\frac{i}{\hbar}H^{(2)}t}e^{-\frac{i}{\hbar}H^{(1)}t}\ket{\psi^{(1)}_{l}}$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{\infty}\bra{\psi^{(1)}_{k}}\ket{\psi^{(2)}_{m}}\bra{\psi^{(2)}_{m}}\ket{\psi^{(1)}_{l}}e^{-\frac{i}{\hbar}\left(E_{l}^{(1)}-E_{m}^{(2)}\right)t},$
(18)

leading to

$F_{T=0}(t)=\big{|}\det\left[\mathcal{O}_{kl}(t)\right]\big{|}^{2}\;,$ (19)

which is equivalent to Eq. (15) at $T=0$.

It is interesting to note that the time-dependence of the survival probability
can be fully predicted from the knowledge of the initial single particle
Hamiltonian since, by using the transformation between the two sets of
eigenstates of the two supersymmetric Hamiltonians given in Eq. (10), the
single particle overlaps can be expressed purely in terms of the eigenstates
of the initial Hamiltonian as

$\displaystyle\mathcal{O}_{kl}=\sum_{m=1}^{\infty}\bra{\psi^{(1)}_{k}}A\ket{\psi^{(1)}_{m+1}}$
$\displaystyle\bra{\psi^{(1)}_{m+1}}A^{\dagger}\ket{\psi^{(1)}_{l}}$
$\displaystyle\times\frac{e^{-\frac{i}{\hbar}\left(E_{l}^{(1)}-E_{m+1}^{(1)}\right)t}}{\Delta
E_{m}}\;.$ (20)

The survival probability indicates how distinguishable the non-equilibrium
state after the quench $|\Psi(t)\rangle$ is from the state at $t=0$. The
initial decay of the survival probability is proportional to the strength of
the quench, i.e. how different $H^{(2)}$ is from $H^{(1)}$, and is related to
the amount of irreversible excitations created García-March  _et al._ (2016).
Complete revivals of the survival probability, $F(t)=1$, indicate the
recurrence of the initial state and are usually the result of finite size
effects stemming from the trapping potential García-March  _et al._ (2016);
Grochowski  _et al._ (2020); Łebek  _et al._ (2021).

##  II Quench at Zero Temperature

To explore quenching between supersymmetric potentials we start by assuming
that the $N$ particle system is initially confined in an infinite square well
of width $L$

$V^{(1)}(x)=\begin{cases}\infty,&\text{if}\;|x|>\frac{L}{2}\;,\\\
0,&\text{if}\;|x|<\frac{L}{2}\;.\end{cases}$ (21)

For this system the single particle eigenstates are well known and given by

$\psi^{(1)}_{n}(x)=\begin{cases}\sqrt{\frac{2}{L}}\text{sin}(\frac{nx\pi}{L}),&\;n=\text{even;},\\\
\sqrt{\frac{2}{L}}\text{cos}\left(\frac{nx\pi}{L}\right),&\;n=\text{odd},\end{cases}$
(22)

with the corresponding eigenenergies being
$E^{(1)}_{n}=n^{2}\pi^{2}\hbar^{2}/(2mL^{2})$. The infinite box potential is a
common system for which supersymmetric partners have been studied extensively
Gutierrez  _et al._ (2018), and, indeed, the superpotential for $V^{(1)}(x)$
and its partner potential $V^{(2)}(x)$ can be calculated as
$\mathcal{W}^{(2)}(x)=\frac{\pi}{\sqrt{2}L}\tan\left(\frac{\pi x}{L}\right)$
where the superscript $(2)$ references the target potential. The exact
eigenfunctions of the supersymmetric Hamiltonian $H^{(2)}$ are known and the
expression for the general superpotential for higher partners is given by
Gutierrez  _et al._ (2018); Sukumar (1985b)

$\mathcal{W}^{(\alpha)}=(\alpha-1)\frac{\pi}{\sqrt{2}L}\text{tan}\left(\frac{x\pi}{L}\right)\;.$
(23)

In the following we focus on the dynamics of a quench from an initial box
potential $V^{(1)}(x)$ directly to a higher order supersymmetric potential
$V^{(2)}(x)$, $V^{(3)}(x)$ or $V^{(4)}(x)$. The initial state at $T=0$ is the
groundstate $|\Psi_{0}^{(1)}\rangle$ of the Hamiltonian $H^{(1)}$ and, for a
gas of $N=30$ fermions, the survival probability is shown in Figs. 2(a-c). One
can see that in all cases the survival probability decays quickly immediately
after the quench, but regular complete recurrences of the initial state
($F(t)=1$) appear as well. In fact, these occur at integer multiples of
$t_{r}/4$, where the standard, state-independent revival time for a box is
given by Aronstein and Stroud (1997); Robinett (2000)

$t_{r}=\frac{4mL^{2}}{\pi\hbar}=\frac{2\pi\hbar}{E^{(1)}_{1}},$ (24)

and where $E^{(1)}_{1}$ is the ground state energy of the initial Hamiltonian.

Figure 2: Survival probabilities for different quenches of a system of $N=30$
fermions at $T=0$ starting with an infinite box of width $L=4$. In panels
(a-c) the final potentials are the supersymmetric partner potentials
$V^{(2)}$, $V^{(3)}$ and $V^{(4)}$, respectively. In panel (d) the quench is
from a box of length $L^{\prime}=3.9$ to one of length $L=4$. Note that the
parameters for the quench in (d) were chosen to compare to the dynamics
observed in (a).

The occurrence of the revivals at multiples of $t_{r}/4$ can be easily
understood by realising that at these times the phase evolution of all single
particle eigenstates is independent of $n$ Styer (2001); Bluhm  _et al._
(1996); Aronstein and Stroud (1997); Robinett (2000, 2004); Aronstein and
Stroud (1997). This can be shown by separating the evolving single particle
eigenstates into their even and odd sub-spaces, which have the associated
energies

$E_{n}^{e}=E_{1}(2n+1)^{2}\quad\text{and}\quad E_{n}^{o}=4E_{1}(n+1)^{2}\;,$
(25)

for $n=0,1,2,3\dots$. The time evolution is then given by

$\displaystyle e^{-\frac{i}{\hbar}E^{e}_{n}t}$
$\displaystyle=e^{-i\phi^{e}_{n}(t)}=e^{-\frac{2\pi i(2n+1)^{2}}{t_{r}}t}\;,$
(26) $\displaystyle e^{-\frac{i}{\hbar}E^{o}_{n}t}$
$\displaystyle=e^{-i\phi^{o}_{n}(t)}=e^{-\frac{8\pi i(n+1)^{2}}{t_{r}}t}\;.$
(27)

and one can see that for $t=t_{r}/4$ the single particle phases phases become
independent of $n$

$\displaystyle\phi_{n}^{e}(t_{r}/4)$ $\displaystyle=\frac{\pi}{2}\pmod{2\pi}$
(28) $\displaystyle\phi_{n}^{o}(t_{r}/4)$ $\displaystyle=0\pmod{2\pi},$ (29)

and in particular are purely real or imaginary. This means that the
orthonormality that was present at $t=0$ between the states is recovered at
integer multiples of $t_{r}/4$ and the overlap matrix $\mathcal{O}_{kl}(t)$
contains only diagonal elements. For a quench $V^{(1)}\rightarrow V^{(2)}$ (or
every quench where the difference in the order of the supersymmetric partner
potentials is odd) the diagonal elements describe a transition from a state
with an even quantum number to one with an odd quantum number (in the original
basis) and the phases of the overlap matrix elements in Eq. (I.2) can be
calculated to be

$\Delta\phi=\phi^{\text{e,o}}\left(\frac{t_{r}}{4}\right)-\phi^{\text{o,e}}\left(\frac{t_{r}}{4}\right)=\pm\frac{\pi}{2}\pmod{2\pi}$
(30)

for all possible combinations of overlaps.

For the quench $V^{(1)}\rightarrow V^{(3)}$ (or every quench where the
difference in the order of the supersymmetric partner potentials is even), the
corresponding phase difference is the difference between like parity states,
which is simply zero or a multiple of $2\pi$. Since in both cases the
determinant is just the absolute value of the product of the diagonal
elements, the revivals in the survival probability in Eq. (19) occur.

From these insights above one can classify the revivals of the survival
probability into two different groups Robinett (2000). For times where the
phases of the overlaps do not align at a multiple of $2\pi$ the wavefunction
density overlaps with the density at $t=0$, however the imaginary and real
parts of the wavefunction do not align. Such revivals are called quasi-
revivals. Conversely, at integer multiples of $t_{r}$ the overlap phases align
at multiples of $2\pi$ and are called true revivals.

In between the full revivals the dynamics in each quench is different and one
can see that for higher order quenches the survival probability is approaching
zero in these regions. It is worth noting that the initial decay of the
survival probability is directly related to the average work done on the
system Fusco  _et al._ (2014), which can therefore be used to quantify the
strength of the quench. For the quenches between supersymmetric Hamiltonians
we can calculate the quantity explicitly to be (taking advantage of the fact
that we only need to know properties of the initial Hamiltonian)

$\displaystyle\langle W^{(\alpha)}\rangle$
$\displaystyle=-i\frac{d}{dt}\mathcal{O}(t)\big{|}_{t=0}$
$\displaystyle=\sum_{m}\left(\mathcal{E}^{(\alpha)}_{m}-\mathcal{E}^{(1)}_{0}\right)\left|\langle\Psi_{m}^{(\alpha)}|\Psi_{0}^{(1)}\rangle\right|^{2}$
$\displaystyle=E^{(1)}_{1}N(N+1)(\alpha^{2}-\alpha).$ (31)

From this expression one can immediately see that quenches to higher order
potentials result in larger amounts of work. Furthermore, the irreversible
work can quantify the amount of non-equilibrium excitations induced by the
quench and can be calculated to be

$\langle W_{\text{irr}}^{(\alpha)}\rangle=\langle
W^{(\alpha)}\rangle-\Delta\mathcal{E}^{(\alpha)}_{0}=E_{1}N^{2}(\alpha-1)^{2}\,,$
(32)

where

$\displaystyle\Delta\mathcal{E}_{0}^{(\alpha)}$
$\displaystyle=\sum_{l=1}^{N}E_{l+\alpha-1}^{(1)}-\sum_{l=1}^{N}E_{l}^{(1)}$
$\displaystyle=(\alpha-1)(N^{2}+\alpha N)E^{(1)}_{1}$ (33)

is the energy difference between the ground states of the initial and the
final potential.

Figure 3: Work probability distribution corresponding to the quenches shown in
Fig. 2. The transition between ground states of both potentials is marked with
a black star, with other transitions colour coded for single particle
transitions (blue), two particle transitions (red) and three particle
transitions (green).

Additional insight into the quench dynamics can be obtained by examining the
work probability distribution (WPD). This quantity is given by the Fourier
transform of the dynamical overlap

$\displaystyle P(W)$ $\displaystyle=\int dt\,e^{-iWt}\mathcal{O}(t)$ (34)
$\displaystyle=\sum_{m}\left|\langle\Psi_{m}^{(\alpha)}|\Psi_{0}^{(1)}\rangle\right|^{2}\delta\left(W-\left(\mathcal{E}^{(\alpha)}_{m}-\mathcal{E}^{(1)}_{0}\right)\right)\,,$
(35)

and allows one to understand how the excitations are spread in the Hilbert
space of the final Hamiltonian Campisi  _et al._ (2011). We show this quantity
in Fig. 3(a-c) for the supersymmetric quenches corresponding to the quench
dynamics shown in Fig. 2(a-c). The weakest quench is to $V^{(2)}$, where the
survival probability is small, but still finite, in between revivals. This
indicates that the initial state is not completely destroyed by the quench
and, correspondingly, the WPD shows a significant peak at
$\Delta\mathcal{E}_{0}^{(2)}$ (black star in Fig. 3(a)). This is the lowest
possible excitation, which corresponds to the transition probability between
the respective groundstates $|\Psi_{0}^{(1)}\rangle$ and
$|\Psi_{0}^{(2)}\rangle$. One can also distinguish the higher energy
excitations through the number of particles that are excited from the initial
Fermi sea: a single particle excitation (blue diamonds) corresponds to the
lowest $N-1$ states being occupied and one particle occupying a state above
$N$; a two particle excitation (red diamonds) leads to the lowest $N-2$ states
being occupied and two particles occupying states above $N-1$, and higher
order excitations are defined similarly.

One can see from Fig. 3(a) that single particle excitations possess the
largest probability and exist in a single band. For excitations consisting of
more particles there is a larger set of different two particle combinations,
however the majority of these can be seen to possess a lower probability,
similarly for the three particle excitations. For the stronger quenches to
$V^{(3)}$ and $V^{(4)}$, the WPD broadens and the probability to occupy the
groundstate $|\langle\Psi_{0}^{(\alpha)}|\Psi_{0}^{(1)}\rangle|^{2}$ decreases
(black star in panels 3(b) and (c)), which corresponds to the survival
probability in Fig. 2 essentially vanishing between revivals. In fact, the WPD
for these quenches shows that two particle excitations dominate the dynamics
and single particle excitations become less important in the quench to
$V^{(4)}$. Nevertheless, for each supersymmetric quench an underlying
structure can be observed in the WPD which is related to the appearance of the
perfect revivals. While for weaker quenches the majority of the excitations
are happening just above the Fermi surface, for stronger quenches this is no
longer true.

Finally, it is interesting to compare the quench dynamics observed through
$F(t)$ to the well-known Talbot effect. In this an initial state is released
into a box potential and in the resulting dynamics repeated images of the
initial state are formed at regular and well defined times Bluhm  _et al._
(1996); Aronstein and Stroud (1997); Robinett (2000); Stifter  _et al._
(1997); Großmann  _et al._ (1997). In Fig. 2(d) we show the survival
probability for a Fermi gas after a sudden expansion of the infinite well
potential. The initial state is the ground state before the expansion and one
can see that also here the full revivals occur at integer multiples of
$t=t_{r}/4$ owing to the quadratic energy spectrum of the final potential.
Note that, for a fair comparison, we have chosen the change in the box-width
such that the survival probability in between the revivals is visually close
to the one from the supersymmetric quench to $V^{(2)}$. Nevertheless, the WPD
shown in Fig. 3(d) can be seen to possess quite a different structure,
indicating that the supersymmetric quench is intrinsically different to the
Talbot effect. This will become even more clear when looking at quenches at
finite temperatures.

##  III Quench at Finite Temperature

Figure 4: Survival probability $F(t)$ as a function of temperature and time
for a quench $V^{(1)}\rightarrow V^{(2)}$.

While the revivals in the survival probability at zero temperature can be
connected to the existence of a Fermi-sea (all occupation probabilities in Eq.
(15) are either zero or one) which restricts the dynamics of all lower lying
particles Busch  _et al._ (1998); DeMarco  _et al._ (2001), the behaviour at
finite temperatures can be more complicated: with increasing temperature the
Fermi-edge becomes less sharp and the occupation probabilities for states
below the Fermi edge decreases below one. This expands the number of possible
transitions after the quench and therefore leads to a stronger decay of the
initial state Knap  _et al._ (2012b); Mitchison  _et al._ (2020).

In Fig. 4 we show the survival probability for the quench $V^{(1)}\rightarrow
V^{(2)}$ as a function of time and temperature in units of the Fermi
temperature $T_{F}=N^{2}E_{1}/k_{B}$. One can immediately see that the
revivals at integer multiples of $t_{r}$ are not affected by temperature, but
all other ones present at zero temperature decay as the temperature increases.
For longer times this is also shown in Fig. 5(a) and a similar behaviour can
be observed for a quench $V^{(1)}\rightarrow V^{(4)}$ in Fig. 5(c). However,
for the quench $V^{(1)}\rightarrow V^{(3)}$ all the revivals are unaffected by
temperature. This phenomenon is also observed for quenches between potentials
in a hierarchy from $V^{(\alpha)}\rightarrow V^{(\alpha+2n)}$.

Figure 5: Survival probability for finite temperatures of $T/T_{F}=0.05$
(blue) and $T/T_{F}=0.1$ (red) for the same quenches as presented in Fig. 2.

To calculate the WPD for finite temperatures we include the thermal ensemble
of the initial state
$p^{(1)}_{n}=\frac{1}{Z_{0}}e^{\beta(\mathcal{E}^{(1)}_{n}-N\mu)}$, which in
analogy with Eq. (35) gives

$P(W)=\sum_{m,n}p^{(1)}_{n}\left|\langle\Psi_{m}^{(\alpha)}|\Psi_{n}^{(1)}\rangle\right|^{2}\delta\left(W-\left(\mathcal{E}^{(\alpha)}_{m}-\mathcal{E}^{(1)}_{n}\right)\right)\,.$
(36)

In Fig. 6 one can see that, when compared to the zero temperature case, the
distributions for the quenches to $V^{(2)}$ and $V^{(4)}$ are much denser and
less structured, while for the quench to the potential $V^{(3)}$ the
distribution remains comparatively sparse and ordered.

Figure 6: Survival probability and work probability plotted for finite
temperatures of $T/T_{F}=0.05$ and $T/T_{F}=0.1$ for the same quenches as
presented in Fig. 2. The black star is the same as in Fig. 3 (for $T=0$) and
included for easier comparison.

At finite temperatures the difference between the supersymmetric quenches and
the Talbot effect in a quenched expanding box becomes especially clear. One
can see from Fig. 5(d) that, despite the similarity in the eigenspectra of
these two class of quenches, all revivals completely disappear for the box
expansion. This difference is not obvious from the WPD (see Fig. 6(d)) which
shows similar broadening of the probability distribution and an increase in
the number of excitations. Instead, the presence and absence of constructive
interference of the single particle states at the revival times can be best
seen by looking at the elements of the determinant in Eq. (15) at integer
multiples of $t_{r}/4$. For full revivals of the initial state all off-
diagonal elements should vanish and the remaining diagonal elements should sit
on the unit circle. While this is true at zero temperature as shown in Fig. 7
(blue dots show the diagonal elements, whereas the cross indicates the off-
diagonal ones), at finite temperatures (red dots) this is no longer the case
for the supersymmetric quenches $V_{1}\rightarrow V_{2}$ and $V_{1}\rightarrow
V_{4}$ at $t_{r}/4$ and $t_{r}/2$, as well as the quench of the box size. This
is a consequence of the finite temperature Fermi–Dirac distribution in Eq.
(15) which means that the diagonal elements of the determinant are no longer
just $\pm 1$ or $\pm i$, and therefore the survival probability is smaller
than one.

However, for the supersymmetric quench $V_{1}\rightarrow V_{3}$ at any
multiple of $t_{r}/4$ and for all supersymmetric quenches at $t_{r}$, the
elements of the determinant are lying on the unit circle with a well defined
phase of $2\pi$. This corresponds to the revivals of the exact single particle
wavefunctions at these times, and these real revivals are independent of
temperature. Therefore for the supersymmetric quenches only the quasi-revivals
are affected by finite temperatures. This also sheds light on why all revivals
vanish for the Talbot quench, despite the energy spectrum having a quadratic
form before and after the quench. From Fig. 7(d) one can see that at $T=0$ the
diagonal elements are distributed around the unit circle (blue dots) at any
integer multiple of $t_{r}/4$, indicating that all revivals are just quasi-
revivals. This is consistent with the observation that at finite temperatures
(red dots) the elements of the determinant no longer have a magnitude of one,
and therefore the revivals vanish. This highlights the resilience of the
supersymmetric systems to thermal effects.

Figure 7: Diagonal elements of the overlaps for each quench at $T/T_{F}=0$
(blue dots) and finite temperature of $T/T_{F}=0.5$ (red dots). (a-c) show
supersymmetric quenches where target potentials are partner potentials
$V^{(2)}$, $V^{(3)}$ and $V^{(4)}$ respectively and (d) shows the expanding
box. The off-diagonal elements are zero at these times, which is indicted by
the black crosses in the center of each panel.

##  IV Conclusion

In this work we have investigated the many-body out-of-equilibrium dynamics
for a gas of spin-polarised fermions following a sudden quench in the trapping
potential. In particular we have focused on quenches between an infinite box
potential to a hierarchy of supersymmetric potentials which are connected
through a supersymmetric algebra. What is unique about this scenario is the
similarity of the eigenspectra between these different potentials, which
results in interesting and accessible non-equilibrium dynamics. Indeed we have
shown that quenches between these supersymmetric potentials at zero
temperature produce periodic revivals in the survival probability and compared
this to the revivals present in Talbot oscillations in the infinite well.
Quenches to higher order potentials lead to a stronger decay of the survival
probability in between these revivals, which can be understood by quantifying
the post-quench excitations through the average work done. We have shown that,
due to the existence of the intertwining properties between supersymmetric
systems, the average and the irreversible work at zero temperature can be
exactly calculated. For finite temperatures we have observed that the quasi-
revivals decay in the many-body dynamics, whereas the true revivals remain and
are temperature independent. These different revivals can be identified by
analyzing the elements of the overlap matrix at zero and finite temperatures,
showing that the presence of perfect revivals at finite temperature is
strongly dependent on the order of the potential quench.

Comparing these supersymmetric quenches to the well known Talbot oscillations,
where a quench corresponds to changing the width of an infinite box, it is
clear that supersymmetric quenches of such potentials are fundamentally
different, possess interesting many-body dynamics, and are highly analytically
accessible. Supersymmetric quenches are therefore an interesting addition to
better understand non-equilibrium dynamics of quantum many-body systems.
Finally, while our work has focused on quenches from the infinite box
potential, a similar analysis can be readily extended to quenches between
other well known superpotentials Cooper  _et al._ (1995). However, as these
possess a different structure of the energy spectrum, many-body revivals might
not be present and one would have to look for other distinctive measures. This
poses interesting questions about the role of supersymmetry in different
correlation functions and related concepts in information scrambling that we
leave for future work.

###### Acknowledgements.

This work was supported by the Okinawa Institute of Science and Technology
Graduate University and utilized the computing resources of the Scientific
Computing and Data Analysis section of the Research Support Division at OIST.
TF acknowledges support under JSPS KAKENHI-21K13856.

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