2021 longtable

[1]Peter Hannaford

[1]Optical Sciences Centre, Swinburne University of Technology, Hawthorn,
Victoria 3122, Australia

2]Institute of Theoretical Physics, Jagiellonian University, ulica Profesora
Stanisława Łojasiewicza 11, Kraków, PL-30-348, Poland

# Condensed Matter Physics in Big Discrete Time Crystals

<EMAIL_ADDRESS>ORCID Number: 0000-0001-9896-7284 Krzysztof Sacha
<EMAIL_ADDRESS>ORCID Number: 0000-0001-6463-0659 *  [

###### Abstract

We review the application of discrete time crystals created in a Bose-Einstein
condensate (BEC) of ultracold atoms bouncing resonantly on an oscillating atom
mirror to the investigation of condensed matter phenomena in the time
dimension. Such a bouncing BEC system can exhibit dramatic breaking of time-
translation symmetry, allowing the creation of discrete time crystals having
up to about 100 temporal lattice sites and suitable for hosting a broad range
of temporal condensed matter phenomena. We first consider single-particle
condensed matter phenomena in the time dimension which include Anderson
localization due to temporal disorder, topological time crystals, and quasi-
crystal structures in time. We then discuss many-body temporal condensed
matter phenomena including Mott insulator phases in time, many-body
localization in time, many-body topological time crystals and time crystals
having long-range exotic interactions. We also discuss the construction of two
(or three) dimensional time lattices, involving the bouncing of a BEC between
two (or three) orthogonal oscillating mirrors and between two oscillating
mirrors oriented at 45-degrees. The latter configuration supports a versatile
Möbius strip geometry which can host a variety of two-dimensional time
lattices including a honeycomb time lattice and a Lieb square time lattice.
Finally, we discuss the construction of a six-dimensional time-space lattice
based on periodically driven BECs trapped in a three-dimensional optical
lattice.

###### keywords:

Time crystals, Bose-Einstein condensate, ultra-cold atoms, condensed matter

##  1 Introduction

In 2012 Frank Wilczek speculated whether crystalline structures in space could
be extended to the fourth dimension – time Wilczek2012 ? To create such a
‘quantum time crystal’, he considered the possibility of spontaneous breaking
of continuous time-translation symmetry in the ground state of a many-body
system, in analogy with the formation of an ordinary crystal in space. While a
classical particle can perform periodic motion even if its energy is the
lowest possible and a classical time crystal can form Shapere2012 ; Das2018 ;
Shapere2019 , the quantum time crystal proposed by Wilczek turned out not to
be feasible Bruno2013b ; Watanabe2015 ; Watanabe2020 . However, his idea
inspired further proposals in time-independent Kozin2019  and time-dependent
quantum systems. In particular, it was shown that a _periodically driven_
many-body system can spontaneously break discrete time-translation symmetry
and self-reorganize its motion so that it evolves with a period longer than
the drive Sacha2015 . To create such a ‘discrete time crystal’, a range of
experimental platforms have been proposed, including a periodically driven
Bose-Einstein condensate Sacha2015 , driven spin-based systems Khemani16 ;
ElseFTC ; Yao2017 ; Russomanno2017 ; Pizzi2019a , and driven open dissipative
systems Gong2017 ; Sun2019 ; Cosme2019 . A number of experiments providing
evidence of discrete time crystals have been reported Zhang2017 ; Choi2017 ;
Pal2018 ; Rovny2018 ; Kreil2018 ; Smits2018 ; Autti2018 ; Kessler2020 ;
Taheri2020 ; Autti2021 ; Kyprianidis2021 ; Randall2021 ; Mi2021 . Several
review articles and a monograph on time crystals have been published
Sacha2017rev ; else2019discrete ; khemani2019brief ; Guo2020 ; SachaTC2020 ;
GuoBook2021 .

In this article we consider the case of a Bose-Einstein condensate (BEC) of
ultracold atoms bouncing resonantly on an oscillating atom mirror such that
the period of the bouncing atoms is equal to an integer multiple $s$ of the
period of the driving mirror Sacha2015 . Such a system has been shown to
exhibit dramatic breaking of discrete time-translation symmetry in which the
bouncing atoms can evolve with periods up to about 100 times longer than the
period of the drive Giergiel2018a ; Giergiel2020  (see also Surace2018 ;
Pizzi2019a ; Pizzi2021 ). This allows the creation of big discrete time
crystals possessing a large number of temporal lattice sites, which are
suitable for realizing a broad range of condensed matter phenomena in the time
dimension Sacha15a ; sacha16 ; delande17 ; Matus2021 ; Mierzejewski2017 ;
Giergiel2018b ; Giergiel2018 ; Giergiel2018c ; Kosior2018 ; Zlabys2021 ;
Kuros2021 ; Giergiel2021 . It has been shown for the case $s=2$ that such
discrete time crystals are robust against external perturbations Kuros2020
and quantum fluctuations Kuros2020 ; Wang2020  and live for extremely long
times Wang2020 ; Wang2021 . Condensed matter phenomena can also be
investigated in photonic time crystals Lustig2018 ; Sharabi2021  and phase
space crystals Guo2013 ; Guo2016 ; Guo2016a ; Liang2017 ; Guo2020 ; Guo2021 ;
GuoBook2021 .

A resonantly driven bouncing BEC system allows effective temporal lattice
potentials to be engineered almost at will, by choosing suitable Fourier
components in the periodic driving function of the atom mirror. This enables
the geometry of the time lattice to be readily varied and effective lattice
potentials to be constructed for a broad range of condensed matter phenomena
in the time dimension. Use of a resonantly driven many-body BEC also allows
one to precisely control the effective interparticle interaction and to
engineer exotic long-range interactions in a time lattice by modulating the
interparticle s-wave scattering length via a Feshbach resonance Giergiel2018 .

Temporal condensed matter phenomena that have been predicted to date include
Mott insulator phases in the time dimension Sacha15a ; Anderson localization
Sacha15a ; sacha16 ; delande17 ; Giergiel2018a ; Sharabi2021 ; Matus2021  and
many-body localization in the time dimension Mierzejewski2017 ; topologically
protected edge states in time Lustig2018 ; Giergiel2018b ; quasi-crystal
structures in time Giergiel2018 ; Giergiel2018c ; two-dimensional time
lattices supporting a Möbius strip geometry and flat-band physics Giergiel2021
; and time-space crystals exhibiting a six-dimensional quantum Hall effect
Zlabys2021 .

In Sec. 2, we describe the physical principles underlying the creation of big
discrete time crystals in a periodically driven bouncing BEC system and a
proposed experiment for realizing big discrete time crystals suitable for
investigating condensed matter physics in the time dimension. In Sec. 3, we
discuss single-particle condensed matter phenomena in the time dimension; in
Sec. 4 we describe many-body temporal condensed matter phenomena; and in Sec.
5 we describe temporal condensed matter phenomena in multi-dimensional time
lattices.

##  2 Big Discrete Time Crystals

Single-particle system

We first consider a single particle bouncing in the vertical direction $z$ on
a hard-wall potential mirror which is oscillating with frequency $\omega$ in
the presence of strong transverse confinement. Introducing gravitational units
of length, energy and time, $l_{0}=(\hbar^{2}/m^{2}g)^{1/3}$, $E_{0}=mgl_{0}$,
$t_{0}=\left(\hbar/mgl_{0}\right)^{1/3}$, and transforming to the frame moving
with the oscillating mirror, the Hamiltonian of the system can be expressed as
SachaTC2020

$H=\ H_{0}(z,p)+H_{1}(z)f(t),$ (1)

with the constraint $z\geq 0$, where $H_{0}=p^{2}/2+z$, $H_{1}=z$ and
$f\left(t+2\pi/\omega\right)=f\left(t\right)$ which describes how the mirror
oscillates in time. $p$ and _m_ are the particle’s momentum and mass and $g$
is the gravitational acceleration.

Here, we are interested in the case of resonant driving, i.e., when the
driving frequency $\omega$ of the mirror is an integer multiple $s$ of the
bouncing frequency $\mathrm{\Omega}$ of the particle. It is convenient to
switch from the Cartesian position-momentum variables, $z$ and $p$, to the
action-angle variables, $I$ and $\theta\ $ Lichtenberg1992 , where the
unperturbed Hamiltonian depends only on the new momentum (action $I$),
$H_{0}\left(I\right)$. Then, in the absence of the driving, the solutions of
the classical equations of motion are very simple because the action is a
constant of motion, $I\left(t\right)=\text{constant}$, and the conjugate
position variable (angle $\theta$) evolves linearly in time,
$\theta\left(t\right)=\Omega\left(I\right)t+\theta\left(0\right)$, where
$\Omega\left(I\right)=dH_{0}(I)/dI$.

If a periodically evolving particle is resonantly driven, i.e., if there is
another part of the Hamiltonian, $H_{1}\left(z\right)f\left(t\right)$, where
$f\left(t+2\pi/\omega\right)=f\left(t\right)$ and
$s=\omega/\Omega\left(I_{s}\right)$ is an integer number, then the classical
effective time-independent Hamiltonian describing the motion of the particle
in the vicinity of the resonant trajectory has the form Giergiel2018 ;
SachaTC2020

$H_{eff}=\frac{P^{2}}{2m_{eff}}+V_{eff}(\Theta),\quad
V_{eff}(\Theta)=\sum_{m}h_{ms}(I_{s})f_{-m}e^{ims\Theta},$ (2)

where $P=I-I_{s}$, $I_{s}$ is the value of the action that fulfills the $s:1$
resonance condition, and $m_{eff}^{-1}=d^{2}H_{0}(I_{s})/dI_{s}^{2}$ is the
inverse of the effective mass of the particle. The effective potential in Eq.
(2) depends on the components of the expansion of $H_{1}$ in the action-angle
variables, $H_{1}=\sum_{n}h_{n}\left(I\right)e^{in\Theta}$, and on the Fourier
components of $f\left(t\right)=\sum_{k}f_{k}e^{ik\omega t}$. The Hamiltonian
(2) is time-independent because it is obtained in the frame moving along the
resonant trajectory

$\Theta=\theta-\frac{\omega t}{s}.$ (3)

For pure harmonic driving, $f\left(t\right)=\lambda\cos\left(\omega t\right)$,
the effective Hamiltonian (2) can be expressed as Giergiel2018a

$H_{eff}=\frac{P^{2}}{2m_{eff}}+h_{s}(I_{s})\cos(s\Theta).$ (4)

Plots of the action $I$ (or $P=I-I_{s}$) versus angle $\Theta$ for a given
resonance number $s$ lead to $s$ resonance islands in phase space, which for a
hard-wall potential mirror are stable with minimal chaotic motion for a
driving strength $\lambda\lesssim 0.2$, independent of _s_ and the oscillation
frequency $\omega$ Giergiel2020 . Note that the amplitude of the mirror
oscillations in the laboratory frame is $\lambda/\omega^{2}$, not $\lambda$,
because in order to obtain (1) we have switched from the laboratory frame to
the frame oscillating with the mirror.

Switching to the quantum description (by quantizing the classical effective
Hamiltonian (4) or by performing a fully quantum effective description of the
resonant driving SachaTC2020 ; Giergiel2018 ; Berman1977 ), eigenenergies of
$H_{eff}$ form energy bands $E_{n}\left(\kappa\right)$, where $\kappa$ is a
temporal analog of the quasi-momentum, and the corresponding eigenstates have
the form of Bloch waves $e^{i\kappa\Theta}u_{n,\kappa}\left(\Theta\right)$,
with $u_{n,\kappa}\left(\Theta+2\pi/s\right)=u_{n,\kappa}\left(\Theta\right)$.
Such crystalline behavior, which we deal with in the moving frame, will be
observed in the time domain when we return to the laboratory frame. This is
due to the fact that the transformation between the laboratory and moving
frames is linear in time sacha16 ; SachaTC2020 ; Giergiel2018 , cf. Eq. (3).
This implies that if we fix the position in the laboratory frame close to the
resonant trajectory (i.e., $\theta=\text{constant}$ and $I\approx I_{s}$),
then the crystalline behavior that is observed in the moving frame versus
$\Theta$ will be observed as a function of time in the laboratory frame
because the Bloch waves take the form $e^{i\kappa\left(\theta-\omega
t/s\right)}u_{n,\kappa}\left(\theta-\omega t/s\right)$.

If we are interested in the first energy band of such a crystalline structure,
the description of the particle can be reduced to the tight-binding model
Dutta2015 . That is, there are $s$ Wannier-like states $w_{j}\left(z,t\right)$
corresponding to the first energy band which are localized wave-packets
evolving along the resonant trajectory with period $sT,\ $where
$T=2\pi/\omega$, i.e., $w_{j}\left(z,t+sT\right)=w_{j}\left(z,t\right)$ and
$w_{j}\left(z,t+T\right)=w_{j+1}\left(z,t\right)$ Sacha15a . The energy of the
particle in this subspace (more precisely the quasi-energy of the periodically
driven particle Shirley1965 ) takes the form

$H_{eff}\approx-\frac{J}{2}\sum_{j=1}^{s}\left(a_{j+1}^{*}a_{j}+c.c.\right),$
(5)

where we have assumed that the wavefunction of the particle
$\psi\left(z,t\right)=\sum_{j=1}^{s}a_{j}w_{j}\left(z,t\right)$. In Eq. (5),
$J$ is the tunneling (or hopping) amplitude of the particle between different
localized wave-packets $w_{j}$.

Many-body system

The second quantization formalism allows us to switch from a single particle
to many indistinguishable particles by introducing the bosonic field operator
$\hat{\psi}$, which can be expanded in terms of operators that annihilate a
boson in time-independent single-particle basis states Pethick2002 . In the
case of resonantly driven many-body systems, such an approach can be
reformulated in terms of a time-periodic basis if we consider a many-body
Floquet Hamiltonian and extend the Hilbert space of the system by the temporal
degree of freedom SachaTC2020 . In practice, this relies on expansion of the
field operator $\hat{\psi}$ in terms of operators $\hat{a}_{j}$ that
annihilate a particle in basis states $\phi_{j}\left(z,t\right)$ defined in
the extended Hilbert space. That is, the $\phi_{j}\left(z,t\right)$’s form a
complete basis for a particle moving in 1D space at any time $t$ and are
periodic with period $nT$, where $T=2\pi/\omega$ is the period of the external
(mirror) driving and $n$ can be any integer number not necessarily $n=1$ — in
such a basis the spectrum of the Floquet Hamiltonian is periodic with the
period $\hbar\omega/n$ SachaTC2020 . Clearly, we cannot afford a complete
expansion of the field operator $\hat{\psi}\left(z,t\right)$ and need to
introduce some truncation.

The description of a resonantly bouncing single particle, with the Hamiltonian
(1), can be reduced to the tight-binding model, Eq. (5), if we restrict the
Hilbert space to the resonant subspace spanned by $s$ localized wave-packets
$w_{j}\left(z,t\right)$ moving periodically along the $s:1$ resonant
trajectory. This subspace is related to the first energy band of the single-
particle effective Hamiltonian, Eq. (4). If the interaction energy per
particle is much smaller than the energy gap between the first and second
energy bands of the Hamiltonian (4), then, in the many-body case, we may
restrict our analysis to the first energy band, and the many-body Floquet
Hamiltonian takes the form of a Bose-Hubbard model in time SachaTC2020

$\displaystyle\hat{H}_{F}$ $\displaystyle=$
$\displaystyle\frac{1}{sT}\int\limits_{0}^{sT}dt\int
dx\;\hat{\psi}^{\dagger}\left[H(t)-i\frac{\partial}{\partial
t}+\frac{g_{0}}{2}\hat{\psi}^{\dagger}\hat{\psi}\right]\hat{\psi}$ (6)
$\displaystyle\approx$
$\displaystyle-\frac{J}{2}\sum_{j=1}^{s}\left(\hat{a}_{j+1}^{\dagger}\hat{a}_{j}+H.c.\right)+\frac{1}{2}\sum_{i,j=1}^{s}U_{ij}\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}\hat{a}_{j}\hat{a}_{i},$
(7)

where $g_{0}$ is the strength of the contact interaction between particles
which model the van der Waals interactions between ultracold atoms Pethick2002
. In Eq. (7) we have assumed bosonic atoms and restricted the Hilbert space to
the resonant subspace; that is,
$\hat{\psi}\left(z,t\right)\approx\sum_{j=1}^{s}w_{j}\left(z,t\right){\hat{a}}_{j}$,
where ${\hat{a}}_{j}$ are the standard bosonic annihilation operators and
$w_{j}$ are the Wannier-like localized wave-packets which evolve along the
resonant trajectory with period $sT$, i.e.,
$w_{j}\left(z,t+sT\right)=w_{j}\left(z,t\right)$, cf. Eq. (5). The tunneling
amplitude $J$ describes the hopping of atoms between neighboring wave-packets
in the time domain, i.e., between $w_{j}\left(z,t\right)$ and
$w_{j+1}\left(z,t\right)=w_{j}\left(z,t+T\right)$. The interaction
coefficients $U_{ij}$ depend on the overlap of the densities of the Wannier-
like states, i.e.,

$U_{ij}=\frac{2-\delta_{ij}}{sT}\int\limits_{0}^{sT}dt\int dz\;g_{0}\rvert
w_{i}(z,t)\rvert^{2}\;\rvert w_{j}(z,t)\rvert^{2},$ (8)

and in principle can describe effective long-range interactions in the Bose-
Hubbard model. However, in a typical situation, the effective long-range
interactions are negligible because they are orders of magnitude weaker than
the on-site interactions in Eq. (7) SachaTC2020 .

Within the mean-field approximation, we look for time-periodic $N$-body states
in the form of a product state
$\phi\left(z_{1},t\right)\phi\left(z_{2},t\right)\ldots.\phi\left(z_{N},t\right)$.
The resonant subspace is spanned by _s_ single-particle Floquet states or
equivalently by _s_ localized wave-packets $w_{j}\left(z,t\right)$. Thus, the
mean-field periodic solutions within the resonant subspace can be expanded in
the time-periodic basis
$\phi\left(z,t\right)=\sum_{j=1}^{s}a_{j}w_{j}\left(z,t\right)\text{.\ }$The
coefficients $a_{j}$, which satisfy $\sum_{j=1}^{s}{{{\rvert
a}_{j}\rvert}^{2}=1,}$ correspond to extremal values of the quasi-energy of
the system per particle

$E_{F}\approx-\frac{J}{2}\sum_{i=1}^{s}{(a_{i+1}^{*}}a_{i}+c.c.)+\frac{N}{2}\sum_{i,j=1}^{s}U_{ij}{{\rvert
a}_{i}\rvert}^{2}{{\rvert a}_{j}\rvert}^{2},$ (9)

which can be found by solving the corresponding Gross-Pitaevskii equation
Pethick2002 . The mean-field approach is very useful in the description of the
experiments on discrete time crystals in BECs and we will use it in the
following.

Discrete time crystal: $s=40$

As an example, we consider a BEC of ultracold atoms bouncing resonantly on an
oscillating atom mirror for the case of the $40:1$ resonance ($s=40$) with
driving amplitude $\lambda=0.2$ and driving frequency $\omega=4.9$ (both in
gravitational units) Giergiel2018a . When the attractive interactions are
sufficiently weak, i.e., $\rvert g_{1D}N\rvert\leq 1.6\times 10^{-3}$ (where
$g_{1D}=g_{0}{(\omega}_{\bot}/2\pi)<0$ and $\omega_{\bot}$ is the trapping
frequency of the transverse confinement) the lowest energy state of the Bose-
Hubbard Hamiltonian (7) is well approximated by the mean-field solution
$\psi\left(z_{1},\ldots..z_{N},t\right)=\prod_{i=1}^{N}{\phi\left(z_{i},t\right)}$,
where

$\phi\left(z,t\right)=\frac{1}{\sqrt{s}}\sum_{j=1}^{s}w_{j}\left(z,t\right).$
(10)

Although the Wannier states $w_{j}\left(z,t\right)$ are periodic with period
_sT_ , the wavefunction $\phi\left(z,t\right)$ is periodic with period _T_
because after each period _T_ the wave-packets exchange their role, i.e.,
$w_{j+1}\left(z,t\right)=w_{j}\left(z,t+T\right)\text{.\ }$Thus, the mean-
field time-periodic solution $\psi\left(z_{1},\ldots..z_{N},t\right)$
preserves the discrete time-translation symmetry of the many-body system.

However, when the attractive interactions are sufficiently strong, i.e.,
$\rvert g_{1D}N\rvert>1.6\times 10^{-3}$, the mean-field Gross-Pitaevski
solution (10) becomes dynamically unstable and _s_ new stable periodic
solutions are created which are a non-uniform superposition of the Wannier
wave-packets $w_{j}$ and therefore are evolving with a period $s=40$ times
longer than the driving period _T_ and consequently break the discrete time-
translation symmetry of the many-body Hamiltonian. For stronger interactions
${\rvert g}_{1D}N\rvert\gtrapprox 0.1$ the stable periodic solutions reduce
practically to the single localized wave-packets $\phi\left(z,t\right)\approx
w_{j}\left(z,t\right)$.

In order to recover the discrete time-translation symmetry of the many-body
system we can, in principle, prepare a superposition of the mean-field product
states

$\psi\left(z_{1},\ldots..z_{N}\right)\approx\frac{1}{\sqrt{s}}\sum_{j=1}^{s}{\prod_{i=1}^{N}{w_{j}\left(z_{i},t\right),}}$
(11)

which in the second quantization formalism reads

$|\psi\rangle\approx\frac{1}{\sqrt{s}}(|N,0,\ldots.,0\rangle+|0,N,0\ldots.,0\rangle+\ldots+|0,\
\ldots.,0,N\rangle.$ (12)

The state (11) is a good approximation to the many-body Floquet state
corresponding to the lowest energy state of the Hamiltonian (7) for
${|g}_{1D}N|\gtrapprox 0.1$. It is a superposition of $s=40$ Bose-Einstein
condensates and because $\langle
w_{j}|w_{j^{{}^{\prime}}}\rangle=\delta_{jj^{{}^{\prime}}}$ it is also a
Schrödinger cat-like state. It evolves with the driving period _T_ but after,
for example, measurement of the position of a particle, the Schrödinger cat
state collapses, the discrete time-translation symmetry is broken, and the
system starts evolving with a period $s=40$ times longer than the driving
period _T._

Figure 1: Discrete time crystal formation for _s_ = 40. In the absence of
interactions (blue curves), atoms tunnel from the initial state to neighboring
Wannier wave-packets which is indicated by the decrease of the overlap between
the time evolving wavefunction $\phi(z,t)$ and the initially chosen
$w_{1}(z,0)$ Wannier state (blue curve in (a)). When a sufficiently strong
attractive interaction is turned on ($g_{1D}N=-0.12)$, the system chooses a
periodic solution evolving with a period 40 times longer than the period
expected from the symmetry of the Hamiltonian, and the discrete time-
translation symmetry is broken (red curves). In (b)-(d), the densities of
interacting (noninteracting) atoms are presented with red (blue) curves at
different moments in time: (b) $t/T=1330$, (c) 2210 and (d) 5010. Parameters:
$\lambda=0.2$, $\omega=4.9$. Reprinted from Giergiel2018a .

In an experiment it will not be possible to prepare the Schrödinger cat state
(11). However, it should be straightforward to prepare a symmetry broken state

$\psi\left(z_{1},\ldots..z_{N},0\right)=\prod_{i=1}^{N}{\phi\left(z_{i},0\right)},$
(13)

with $\phi\left(z,0\right)\approx w_{j}\left(z,0\right)\ $if a BEC of
ultracold atoms is prepared in a harmonic trap located at the classical
turning point above the oscillating atom mirror and the trap is turned off at
the moment the mirror is in its downward position Giergiel2018a . Then the BEC
starts evolving along the classical $40:1$ resonant orbit. If the attractive
interactions are too weak, then atoms start tunneling to neighboring wave-
packets $w_{j-1}$ and $w_{j+1}$ and after time $t\approx 2.4/J\approx 2210T$
they totally leave the initial wave-packet $w_{j}$, as indicated by the
decrease of the overlap between the time-evolving wavefunction $\phi(z,t)$ and
the initially chosen Wannier wave-packet $w_{1}(z,0)$ in Fig. 1(a) (blue
curves) and by the probability density, which is the experimental observable,
in Fig. 1(b)-(d) (blue curves). However, when the attractive interactions are
sufficiently strong (e.g., $g_{0}N=-0.12$), the initial wavefunction
$\phi\left(z,0\right)\approx w_{1}\left(z,0\right)$ reproduces the symmetry
broken state and no tunnelling to neighboring wave-packets is observed, even
for a time evolution as long as 5000 _T_ (Fig. 1, red curves), indicating the
stability of the discrete time crystal.

Gaussian-shape potential mirror

In the above analysis we have assumed bouncing of particles from a hard-wall
potential mirror, for which the optimal driving strength to produce stable
resonance islands is $\lambda\approx 0.2$ for $s\gg 1$ Giergiel2020 . In an
experiment it is convenient to use a repulsive light-sheet as the atom mirror,
which is well approximated by a Gaussian shape potential. For such a mirror,
the same time crystal phenomena as for a hard-wall potential mirror can be
realized, but the optimal driving strength needed to have the same effect is
typically an order of magnitude larger, which makes the driving more
accessible experimentally Giergiel2020 . In the case of a light-sheet atom
mirror, the driving can be conveniently implemented by modulating the light
intensity.

Robustness of discrete time crystals

The robustness of a discrete time crystal created in a BEC bouncing resonantly
on a periodically driven mirror has been investigated for the cases $s=2$ and
4 Kuros2020 . The simulated effects of imperfections in the driving and in the
initial state are summarized in the form of a phase diagram of the
displacement $\epsilon$ of the initial state from the classical turning point
for perfect resonant bouncing ($z=h_{0}$) versus the interaction strength
$g_{0}N$ required to create a discrete time crystal, with the average atom
momentum and the width of the initial wave-packet chosen randomly Kuros2020 .
For $s=2$, the range of displacements around the classical turning point is
about $\Delta\epsilon/h_{0}\approx 0.2$ for $g_{0}N\approx-0.02$,
$\lambda=0.12.$ For $s\gg 1$, the range of displacements was estimated to be
$\Delta\epsilon/h_{0}\approx 8\sqrt{\lambda}/s\pi$ Kuros2020 , which is about
$0.03$ for $s=40$, $\lambda\approx 0.2$.

Many-body effects

It has been observed that a BEC initially prepared in the ground state of an
optical lattice potential or ultra-cold atoms in free space can be gradually
depleted due to single-particle heating and two-particle processes when a time
periodic perturbation is turned on Choudhury2014 ; Reitter2017 ; Li2019 .
However, in the present article we consider a different situation to that
where ultra-cold atoms are not initially prepared in an eigenstate of the
time-independent Hamiltonian where the mirror does not move but the atoms are
loaded to a resonant Hilbert subspace of the periodically driven system. If,
in the course of time evolution, the atoms do not leave this subspace, our
predictions of the time crystal phenomena are valid.

So far, we have mainly used the mean-field approximation. The effects of
quantum many-body phenomena and possible heating on the formation of a
discrete time crystal in a BEC bouncing resonantly on a periodically driven
mirror have been studied for _s_ = 2 using the Bogoliubov approach Kuros2020
and also using a full multi-mode quantum treatment based on the truncated-
Wigner approximation (TWA) Wang2020 . The Bogoliubov and TWA calculations
agree broadly with the mean-field (single-mode) calculations for times out to
at least 2000 driving periods, except for interaction strengths $g_{0}N$ very
close to the critical value for discrete time crystal formation where the TWA
calculations predict a quantum depletion up to about 260 atoms out of a total
of 600 which results from atoms escaping to essentially just a second mode
(for $s=2$). The TWA calculations also predict that the mean energy per
particle does not increase significantly out to at least 2000 driving periods,
indicating that the system reaches a steady state with no net energy pumped
from the drive, and thus there is no evidence of thermalization on this time
scale, see Fig. 2. Additional many-body calculations for $s=2$ based on a two-
mode model derived from standard quantum field theory indicate that the
discrete time crystals survive for times out to at least 250,000 driving
periods Wang2021 .

Figure 2: Mean energy of the atoms (i.e., the sum of the kinetic energy, the
potential energy in the gravitational field and the interaction energy) which
are released from a harmonic trap and start bouncing on the oscillating mirror
along the $2:1$ ($s=2$) resonant trajectory. The presented results are
obtained within the TWA for $N=600$ Wang2020  and they correspond to the
interaction strength $g_{0}N$ close to the critical value ($g_{0}N=-0.012$
Wang2021 ) for the formation of a DTC and deep in the DTC regime
($g_{0}N=-0.1$). In all the cases, there is no signature of heating. Reprinted
from Wang2020 .

Size of discrete time crystals

For a BEC bouncing resonantly on a periodically driven hard-wall potential
mirror, the drop height required to tune to a given resonance _s_ can be
expressed as Giergiel2020

$h_{0}=(\pi/2)^{4/3}(\alpha/\lambda)^{1/3}s^{4/3},$ (14)

where all quantities are in gravitational units and
$\alpha=2\pi^{2}s^{2}\lambda/\omega^{6}$ is a universal parameter which we
choose to be $\alpha=0.456$ so that the energy gap between the first and
second energy bands of the quantum version of the effective Hamiltonian (4) is
$\Delta E/J\approx 10$ for $\lambda=0.2$ Giergiel2020 . This $\Delta E/J$
value is sufficently large to ensure validity of the single-band Hamiltonian
(7) while sufficiently small to allow all atoms to tunnel to neighbouring
wave-packets during a reasonably small number of bounces ($N_{b}=54$) in the
absence of interactions. The driving frequency $\omega$ required for a given
resonance $s$ and drop height _h_ 0 is

$\omega=\pi s/\sqrt{2h_{0}}.$ (15)

Table 1: Dependence of parameters for the case of a 39K atom on the size of a discrete time crystal for a hard-wall potential mirror and $\lambda=0.2$, $\Delta E/J=10$, $N_{b}=54$. Gravitational units for 39K are $l_{0}=0.647\mu$m, $t_{0}=0.256$ms. Adapted from Giergiel2020 . $s$ | $\lambda l_{0}/\omega^{2}$ | $h_{0}l_{0}$ | $\omega/(2\pi t_{0})$ | $t_{tunnel}t_{0}$
---|---|---|---|---
| [nm] | [$\mu$m] | [kHz] | [s]
10 | 13.6 | 34 | 1.92 | 0.28
20 | 8.56 | 85 | 2.42 | 0.45
30 | 6.53 | 145 | 2.77 | 0.59
40 | 5.38 | 213 | 3.05 | 0.71
50 | 4.64 | 281 | 3.28 | 0.83
60 | 4.11 | 365 | 3.49 | 0.93
70 | 3.71 | 449 | 3.67 | 1.03
80 | 3.40 | 536 | 3.84 | 1.13
90 | 3.14 | 627 | 3.99 | 1.22
100 | 2.93 | 722 | 4.13 | 1.31
|  |  |  |

Table 1 summarizes calculated parameters for different size discrete time
crystals in the range $s=10-100$ for the case of a 39K atom and a hard-wall
potential mirror with $\lambda=0.2$, $\Delta E/J=10$, $N_{b}=54$ Giergiel2020
. In a time crystal experiment, we require a relatively large drop height
($h_{0}l_{0}$) to allow high spatial resolution probing of the atom density
over a range of distances between the classical turning point and the
oscillating mirror, a relatively large mirror oscillation amplitude ($\lambda
l_{0}/\omega^{2}$ in the laboratory frame), and a relatively short tunnelling
time ($t_{tunnel}t_{0}$) to allow the experiment to be performed in times
shorter than the lifetime of the bouncing BEC. For $s=10$, the drop height is
only 34 $\mu$m which limits the spatial resolution for probing over a range of
distances $z$. For $s=100$, the amplitude of the drive is only about 2.9 nm
for the case of a hard-wall potential mirror, but the amplitude will be about
an order of magnitude larger for a realistic soft Gaussian potential mirror.

Experimental system

A suitable atomic system for realizing a discrete time crystal in a BEC
bouncing resonantly on an oscillating mirror is bosonic potassium-39.
Potassium-39 has a broad Feshbach resonance centred at 402 G DErrico2007
which allows one to magnetically tune the inter-atomic s-wave scattering
length, and hence the interaction strength, in the region near $a_{s}=0$ with
very high precision, i.e., $\rvert da_{s}/dB\rvert_{a_{s}=0}\approx\;\rvert
a_{bg}/\Delta\rvert=0.56a_{0}/$G (where $a_{bg}$ is the background scattering
length, $\Delta$ is the width of the Feshbach resonance, and _a_ 0 is the Bohr
radius). Potassium-39 also has resonance lines at a convenient wavelength for
diode laser sources: 767 nm (D2) and 770 nm (D1).

In a proposed experiment currently being set up in Melbourne, a 39K BEC is
prepared in a crossed optical dipole trap Salomon2014  in the vertical plane,
and positioned at height $z=h_{0}$ corresponding to the selected $s:1$
resonance above an oscillating light-sheet atom mirror. The BEC is then
released from the optical dipole trap to fall on to the oscillating light-
sheet mirror under strong transverse confinement. The light-sheet atom mirror
is driven with frequency $\omega$ corresponding to the drop height $h_{0}$,
Eq. (15), and the amplitude $\lambda$ of the drive is adjusted to produce
stable resonance islands located around periodic orbits with period $sT$
Giergiel2018a ; Giergiel2020 .

The atom density is recorded at fixed positions between the classical turning
point and the oscillating atom mirror and at different moments in time for
different particle interaction strengths $g_{1D}N$ by varying the
interparticle scattering length $a_{s}$ via the broad Feshbach resonance at
402 G DErrico2007 . The time for atoms to tunnel from the initial wave-packet
to neighboring wave-packets (which is the time scale of the system dynamics in
a time lattice) in the absence of interactions is about 0.7 s for $s=40$
Giergiel2020 . When the particle interaction strength $\rvert g_{1D}N\rvert$
is raised above a critical value to spontaneously break the time-translation
symmetry, a stable localized wave-packet evolves without tunneling to other
wave-packets, i.e., a discrete time crystal is formed (Table 1), see Fig. 1
(red curves).

##  3 Single-Particle Condensed Matter Phenomena in the Time Dimension

Spontaneous breaking of continuous space translation symmetry into discrete
space translation symmetry results in the formation of ordinary space
crystals. However, crystalline structures in space can also be created
externally without any spontaneous process. For example, by imposing periodic
behavior of the refractive index in space in dielectric materials, one obtains
photonic crystals in which the propagation of electromagnetic waves possesses
similar properties to the transport of an electron in a space crystal
Joannopoulos_Book . Another example is ultracold atoms in optical lattice
potentials which are routinely created in the laboratory by means of
electromagnetic standing waves Lewenstein2007Adv . There, ultracold atoms
behave like solid state systems but the crystalline structures in space do not
emerge spontaneously.

Crystalline structures can also be created by means of a proper periodic
driving of the system. It has been shown by Guo and coworkers that a
resonantly driven harmonic oscillator reveals a crystalline structure but in
phase space Guo2013 ; Guo2016 ; Guo2016a ; Liang2017 ; Guo2020 ; Guo2021 ;
GuoBook2021 . Consider a periodically kicked harmonic oscillator where an
additional perturbation is turned on periodically during very short periods of
time, $H=(p^{2}+x^{2})/2+KT\cos x\sum_{n}\delta(t-nT)$, where the driving
period $T$ is $s$ times shorter than the harmonic oscillator period ($s$ is
integer) and $K$ is the strength of the kicking. Applying the rotating wave
approximation, one obtains an effective time-independent Hamiltonian that
describes different crystalline structures in the 2D phase space for different
$s:1$ resonant drivings. The $x$ and $p$ operators do not commute, and to plot
the crystal structures one has to use, for example, the coherent state
representation Liang2017 . For $s=4$, a square lattice is observed in phase
space; for $s=3$ and $s=6$, a hexagonal lattice structure is observed; while
for $s=5$ and $s\geq 7$, a quasi-crystal structure is observed. The non-
commutative geometry in phase space can lead to topological properties of the
system. For $s=4$, the effective Hamiltonian has a simple form,
$H_{eff}=K(\cos x+\cos p)/2$, and its energy spectrum depends on the value of
the effective Planck constant, $\hbar_{eff}$, which can be controlled
experimentally Liang2017 . If $\hbar_{eff}$ is a rational number, then there
exists an abelian group related to translations in phase space which is a
symmetry group of the system. Energy levels of $H_{eff}$ form topologically
nontrivial bands which, similar to the case of the quantum Hall effect, reveal
a characteristic Hofstadter butterfly structure as a function of different
rational values of $\hbar_{eff}$ Liang2017 .

In the present article we focus on periodically driven systems which reveal
crystalline structures but in the time domain. In the case of space crystals
one is interested in a regular distribution of particles in space which can be
observed at a fixed moment of time, i.e., at the moment of experimental
detection. Switching from space to time crystals, we have to exchange the
roles of space and time sacha16 . We fix the position in space, i.e., we
locate a detector at a certain space point, and ask if the probability of
clicking of the detector is periodic in time. Obviously, the periodic
evolution of many different systems fulfills this criterion but we are
interested in systems whose periodic behavior in time is described by
condensed matter models Sacha2017rev ; SachaTC2020 .

In Sec. 2 we introduced the general approach for the realization of condensed
matter in the time domain and now we are ready to present different examples.

Figure 3: Comparison of Anderson localization in a 1D space crystal with
periodic boundary conditions (left) with Anderson localization in the time
domain (right). A particle moving in the space crystal with disorder can
Anderson localize, i.e., if we go around the ring, we observe an exponential
localization of a particle around a certain space point. Switching from
Anderson localization in space to Anderson localization in time, we have to
exchange the role of space and time. We fix the position in space and ask if
the probability of clicking of a detector is exponentially localized around a
certain moment of time — such behavior is repeated periodically due to the
periodic boundary conditions in time. This circulating ring system is
equivalent to an atom bouncing resonantly on an oscillating mirror. Reprinted
from Sacha2017rev .

Anderson localization in time

The standard Anderson localization phenomenon is exponential localization of
eigenstates of a particle in configuration space due to the presence of a
disordered potential in space Anderson1958 . Anderson localization can also be
observed in momentum space (often called dynamical localization) and it is
related to the quantum suppression of classical diffusion of a particle in
classically chaotic systems Fishman:LocDynAnders:PRL82 ;
MuellerDelande:Houches:2009 . It turns out that yet another version of
Anderson localization is possible: Anderson localization in time due to the
presence of disorder in time Sacha15a ; sacha16 ; Giergiel2017 ; delande17 ;
Giergiel2018a ; Matus2021 .

Suppose that a single particle, described by the Hamiltonian (1), is driven in
time like $f(t)=\lambda\cos(\omega t)+\sum_{k}f_{k}e^{ik\omega t/s}$, where
the $f_{k}$’s are random complex numbers with $\lvert f_{k}\rvert\ll\lambda$.
Then, the effective Hamiltonian (2) reveals a similar crystalline structure as
in the Hamiltonian (4) but with a weak disordered contribution Sacha15a ;
sacha16 . In the tight-binding approximation, Eq. (5), there are now
additional terms, $\sum_{j=1}^{s}\epsilon_{j}\lvert a_{j}\rvert^{2}$, where
the $\epsilon_{j}$’s are real random numbers, and we arrive at the Anderson
model and Anderson localization can be observed. That is, if we locate a
detector close to the resonant trajectory, the probability of clicking of the
detector is exponentially localized around a certain moment of time Sacha15a .
In Fig. 3 we illustrate Anderson localization in the time domain and compare
it with the standard Anderson localization in a space crystal. Actually,
Anderson localization in time does not require an underlying crystalline
structure in time. It is sufficient to drive a particle in a disordered way,
$f(t)=\sum_{k}f_{k}e^{ik\omega t/s}$, where the $f_{k}$’s are the random
numbers, and the particle can reveal localization in the time domain sacha16 ;
Giergiel2017 . It is even possible to observe a localized-delocalized Anderson
transition in the time domain if a particle moving in 3D space is properly
driven delande17 . Recently, signatures of Anderson localization in time have
also been analyzed in photonic time crystals, i.e., in dielectric materials
where the refractive index is modulated periodically in time Sharabi2021 .

Figure 4: A particle driven resonantly in time so that the effective
Hamiltonian $H_{eff}$, Eq. (2), reproduces the topological Su-Schrieffer-
Heeger model with an edge Su1979 ; Asboth2016short . There are two kinds of
eigenstates of $H_{eff}$: bulk states and topologically protected edge states.
In the laboratory frame, the bulk states are delocalized along the entire
resonant orbit while the edge states are localized close to the edge which
appears periodically in time. The case of a $42:1$ resonance is presented in
the plot. Reprinted from SachaTC2020 .

Single-particle topological time crystals

As a second example, we present topological time crystals Giergiel2018b .
Topological insulators are solid state systems which are insulators in the
bulk but possess topologically protected conducting edge or surface states
which are responsible for quantization of the Hall conductance in the quantum
Hall effect Hasan2010 . If a particle is resonantly driven with the help of
two harmonics, i.e., $f(t)=\lambda\cos(s\omega t)+\lambda^{\prime}\cos(s\omega
t/2)$, then the effective potential in Eq. (2) reveals a crystalline structure
with a two-point basis,
$V_{eff}(\Theta)=V_{0}\cos(s\Theta)+V_{0}^{\prime}\cos(s\Theta/2)$. In the
quantum description of the particle, the first two energy bands of $H_{eff}$
can be described by the Su-Schriefer-Heeger (SSH) model which is a tight-
binding model with staggered hopping amplitudes Su1979 ; Asboth2016short .
Depending on the ratio of the hopping amplitudes, the system can be
characterized by a zero or nonzero winding number. In the latter case, the
system is topologically nontrivial and when we introduce a narrow barrier in
$V_{eff}(\Theta)$, we deal with a topological system with an edge and
eigenstates localized close to it. The barrier can be realized by means of an
additional modulation of the system in time, i.e., $f(t)=\lambda\cos(s\omega
t)+\lambda^{\prime}\cos(s\omega t/2)+\sum_{k}f_{k}e^{ik\omega t}$, where the
$f_{k}$’s are chosen so that $V_{eff}(\Theta)$ acquires an additional
contribution in the form of a narrow barrier Giergiel2018b . In the presence
of the edge, apart from bulk states which are delocalized along the entire
lattice, there are also two eigenstates (with eigenenergies located in the gap
of the spectrum) which are localized close to the edge. These edge states
appear localized in time when we observe the particle in the laboratory frame
Giergiel2018b , see Fig. 4.

Lusting et al. have considered a topological photonic time crystal with edges,
i.e., the photonic crystal was finite because it was turned on at some moment
of time and later turned off. They observed an exponential increase of
amplitudes of electromagnetic waves propagating in a dielectric material if
the wavenumber was located in the gap Lustig2018 .

Figure 5: Left panel: quasi-crystal structure in time. The motion of the
mirror is chosen so that the effective potential $V_{eff}(\Theta)$ in Eq. (2)
is a finite fragment of the Fibonacci sequence of small ($S$) and big ($B$)
potential wells. Right panel: Fourier components of the mirror oscillations,
$f(t)/\omega^{2}=\sum_{k}[f_{k}^{c}\cos(k\omega t)+f_{k}^{s}\sin(k\omega t)]$
— the inset shows $f(t)/\omega^{2}$ over one period $T=2\pi/\omega$. Reprinted
from Giergiel2018 .

Time quasi-crystal structures

We have seen that different crystalline structures can be realized in the time
domain by means of resonant driving of a periodically moving particle.
However, one can realize nearly any shape of the effective potential in Eq.
(2) if the Fourier components of $f(t)$ are properly chosen Giergiel2018 .
Indeed, any potential can be expanded in the series
$V_{eff}(\Theta)=\sum_{m}V_{m}e^{im\Theta}$ and choosing
$f_{-m}=V_{m}/h_{m}(I_{1})$ in the $1:1$ resonant driving of a particle allows
one to realize such an effective potential in Eq. (2). Figure 5 shows an
example where $V_{eff}(\Theta)$ possesses a quasi-crystal structure Janot1994
, i.e., it is a series of small ($S$) and big ($B$) potential wells ordered
according to the Fibonacci sequence $BSBBSBSB\dots$ and transport properties
of the particle in a time quasi-crystal can be investigated Giergiel2018 .

##  4 Many-Body Condensed Matter Phenomena in the Time Dimension

We have seen that a resonantly driven single particle can behave in the time
domain like an electron moving in a space crystal. One may raise the question
if many-body condensed matter phenomena can be observed in the time dimension
too? Having the Bose-Hubbard Hamiltonian (7) that describes the resonant
driving of a many-body system, the investigation of many-body solid state
phenomena is quite straightforward which we show in this section.

Figure 6: (a) Schematic plot of the Mott insulator phase in a 1D space crystal
with periodic boundary conditions. (b) Similar Mott insulator phase but in the
time domain. In order to switch from space to time crystals, one has to
exchange the role of space and time. That is, the position in space is fixed
and we analyze how the probability of detection of particles at this fixed
point changes in time. In the Mott insulator phase, a well defined number of
atoms are appearing at this fixed point like on a conveyor belt or in a
machine gun. Reprinted from Sacha15a .

Mott insulator phases in time

If in a resonantly driven $N$-atom system, Eq. (7), the effective repulsive
on-site interactions dominate over the long-range repulsion ($U_{ii}\gg
U_{ij\neq i}$) and they are sufficiently strong compared with the hopping rate
($U_{ii}N\gg J$), the system reveals a Mott-insulator phase Sacha15a . That
is, the gap between the ground and excited states of the Bose-Hubbard
Hamiltonian (7) is opened, fluctuations of the number of atoms in each lattice
site are suppressed and the system is not compressible. This behavior is
observed in the time domain. When we look at the system in the laboratory
frame at a fixed position close to the resonant trajectory, we will see that
the atoms arrive periodically at the observation point in well-defined
portions of $N/s$ and there is no coherence between them Sacha15a , see Fig.
6. This is in contrast to the superfluid phase where the repulsive
interactions are weak and in a finite lattice ($s<+\infty$) the ground state
of the Bose-Hubbard model (7) is a Bose-Einstein condensate. The superfluid-
Mott insulator transition can be realized in the time lattice described here
by changing the hopping rate $J$ which can be done by changing the amplitude
of the periodic driving of the system Sacha15a .

Many-body localization in time

If a resonantly driven many-body system experiences temporal disorder, the
many-body localization phenomenon in a time crystalline structure can be
investigated Mierzejewski2017 . Indeed, when the driving of the system
possesses a fluctuating contribution, there are additional terms in the
effective Bose-Hubbard model (7), i.e.,
$\sum_{j=1}^{s}\epsilon_{j}\hat{a}_{j}^{\dagger}\hat{a}_{j}$, where the
$\epsilon_{j}$’s are random numbers (see Anderson localization in time
described in the previous section), and the system can reveal many-body
localization if the disorder is sufficiently strong. Many-body localization is
characterized by a lack of dc transport, extremely slow dynamics of various
correlation functions and a logarithmic growth of the entanglement entropy,
and it has been extensively investigated in many-body systems in the presence
of time-independent disorder pal10 ; schreiber15 . Here, we see that many-body
localization can also be induced by temporal disorder Mierzejewski2017 ;
SachaTC2020 .

Figure 7: Left panel: effective long-range interaction coefficients as a
function of the distance $i-j$ between temporal lattice sites in the Bose-
Hubbard Hamiltonian (7) in units of the tunneling amplitude $J$. The character
of the interactions changes between repulsive ($U_{ij}>0$) and attractive
($U_{ij}<0$) in an oscillatory way vs. $i-j$. Right panel: periodic changes of
the original contact interaction strength $g_{0}(t)$ between ultracold atoms
that results in the effective interactions presented in the left panel, cf.
Eq. (8). Reprinted from Giergiel2018 .

Long-range exotic interactions in time

In a typical situation, the effective long-range interactions in the Bose-
Hubbard model (7) are negligible compared to the on-site interactions.
However, if we modulate periodically in time the contact interaction strength
$g_{0}(t)$ in Eq. (8), we can make the long-range interactions significant and
even engineer how they change with the distance $\rvert i-j\rvert$ between
lattice sites Giergiel2018 . In ultracold atoms experiments $g_{0}$ can be
modulated in time by modulating the external magnetic field around a Feshbach
resonance Chin2010 . If $g_{0}(t)$ changes in time with the period $sT$, the
interaction coefficients can be written as
$U_{ij}=\int_{0}^{sT}dt\;u_{ij}(t)g_{0}(t)$ where
$u_{ij}(t)=(2-\delta_{ij})\int dx\rvert w_{i}\rvert^{2}\rvert
w_{j}\rvert^{2}/sT$. The latter can be considered as a matrix with $(i,j)$ and
$t$ treated as indices of rows and columns, respectively. The left singular
vectors of the matrix $u_{ij}(t)$ determine which sets of the interaction
coefficients $U_{ij}$ can be realized in a system while the right singular
vectors tell us how we should modulate $g_{0}(t)$ in order to achieve them.
Such an approach allows one to engineer very exotic long-range interactions
which do not exist in nature — for example, interactions that change the
repulsive and attractive character in an oscillatory way with an increase of
$\rvert i-j\rvert$, see Fig. 7 Giergiel2018 .

In the case of many-body phase space crystals investigated by Guo and
coworkers Guo2016 ; Liang2017 ; Guo2020 , it is also possible to realize
effective long-range interactions between ultracold atoms but they are long-
range in phase space.

Many-body topological time crystals

If $g_{0}$ can be modulated not only in time but also in space, there are
additional possibilities to engineer different effective long-range
interactions. For example, if $g_{0}(z,t)=\sum_{m=0}^{2}\alpha_{m}(t)z^{m}$,
the coefficients $\alpha_{m}(t)$ can be chosen so that the Bose-Hubbard
Hamiltonian (7) reduces to Giergiel2018b

$\hat{H}_{F}\approx-\frac{J}{2}\sum_{j=1}^{s}\left(\hat{a}_{j+1}^{\dagger}\hat{a}_{j}+H.c.\right)+\frac{U}{2}\sum_{j=1}^{s}\hat{n}_{j}(\hat{n}_{j}-1)+V\sum_{j=1}^{s}\hat{n}_{j+1}\hat{n}_{j},$
(16)

where $\hat{n}_{j}=\hat{a}_{j}^{\dagger}\hat{a}_{j}$. This system can reveal
the bosonic analog of the topological Haldane insulator in a spin-1 chain
Haldane83 ; Haldane83a ; Torre2006 ; Rossini2012 . That is, between the Mott
insulator phase, which is observed if the repulsive on-site interactions are
dominant, and the density-wave phase, observed for dominant repulsive nearest-
neighbor interactions, there is the topological Haldane insulator phase
related to a highly nonlocal string order parameter Kennedy92 ; Torre2006 .
All of these phases can be realized in ultracold atoms bouncing resonantly on
an oscillating atom mirror in the presence of an external Feshbach magnetic
field that changes in time and space in a proper way Giergiel2018b .

##  5 Higher-dimensional time lattices

Although time is a single degree of freedom, crystalline structures in time
with the properties of 2D (or 3D) condensed matter systems can be created by a
BEC bouncing resonantly between two (or three) oscillating atom mirrors
SachaTC2020 ; Giergiel2018 ; Zlabys2021 ; Kuros2021 ; Giergiel2021 .

Figure 8: Trajectories of atoms bouncing between two oscillating mirrors,
located at $x=0$ and $y=0$, on a two-dimensional $5\times 5$ time lattice.
Reprinted from Giergiel2018 .

We consider the case of atoms bouncing resonantly between two orthogonal
mirrors, located at $x=0$ and $y=0$, oscillating at the same frequency
$\omega$ and inclined at 45o to the gravitational force (Fig. 8), but this can
be extended to three orthogonal oscillating mirrors. In this case the motion
of a single particle separates into two _independent_ one-dimensional motions
along the directions orthogonal to the mirrors. Averaging the 2D equivalent of
the single-particle Hamiltonian over time leads to the 2D classical secular
effective Hamiltonian, cf. Eq. (4), SachaTC2020

$H_{\text{eff}}=\ \frac{P_{+}^{2}+P_{-}^{2}}{2{\rvert m}_{\text{eff}}\rvert}+\
V_{0}[\cos\left(s\Theta_{x}\right)+\cos{\left(s\Theta_{y}\right)]}.$ (17)

Switching to $N$-boson particles and restricting to the resonant Hilbert
subspace, one obtains a 2D Bose-Hubbard model similar to the 1D model, Eq.
(7). Restricting to the first energy band in the quantized version of Eq. (17)
leads to the tight-binding approximation. In the Hilbert subspace
corresponding to the first energy band we can define a basis of the Wannier
states $W_{\mathbf{i}}\left(\Theta_{x}{,\Theta}_{y}\right)$ localized in
minima of the potential  $\ignorespaces V_{0}$ in Eq. (17). In the laboratory
frame, these 2D Wannier states
$W_{\mathbf{i}=i_{x}{,i}_{y}}\left(x,y,t\right)$ are products of the 1D
localized wave-packets $w_{i_{x}}\left(x,t\right)$ and
$w_{i_{y}}\left(y,t\right)$ moving along the classical orbits.

When the bouncing atoms fulfill the $s:1$ resonance criterion with each
mirror, a square $s\times s$ time lattice is created (see Fig. 8 for the case
of a $5\times 5$ time lattice). In the Bose-Hubbard model of a 2D time
lattice, the on-site interactions $U_{\mathbf{ii}}$ typically dominate over
the long-range inter-site interactions $U_{\mathbf{ij\neq i}}$ and for
sufficiently strong repulsive interactions, the ground state of the system
corresponds to a Mott insulating phase, which for unit filling of the lattice
is a Fock state in which a single bosonic atom occupies each localized wave-
packet.

Use of a 2D time lattice allows a range of 2D condensed matter phenomena to be
investigated with two oscillating mirrors. For example, it is possible to
realize spontaneous formation of time quasi-crystals when the ratio of the
resonance numbers along the $x$ and $y$ directions, $s_{x}/s_{y}$,
approximates the golden number $(1+\sqrt{5})/2\approx 1.618$ Giergiel2018c .
Then, when the discrete time translation symmetry (defined by the oscillation
period of the mirrors) is spontaneously broken, the sequence of atoms
reflected from one mirror and the other mirror forms a Fibonacci quasi-crystal
in time SachaTC2020 ; Giergiel2018c .

In a 2D time lattice, one can also realize gradual breaking of the discrete
time-translation symmetry when the motion along one of the two orthoganol
directions breaks the symmetry and with a range of system parameters the
motion along the other direction also reveals symmetry breaking Kuros2021 .

_Möbius strip geometry_

We now consider the case when the two mirrors are oriented at 45o to each
other to form a wedge with one mirror located in the vertical direction
$x-y=0$ and the other located at $x=0$ Giergiel2021 . When a particle strikes
the vertical mirror, its momenta are exchanged $p_{x}\leftrightarrow p_{y}$,
whereas when it strikes the mirror located at $x=0$, the momentum $p_{y}$
remains the same while the momentum $p_{x}\rightarrow-p_{x}$. In the following
we suppose the mirror located at  $\ignorespaces x=0$ oscillates with the
driving function $f_{x}(t)=-(\lambda_{1}/\omega^{2})\cos(\omega
t)-(\lambda_{2}/4\omega^{2})\cos(2\omega t)$ while the vertical mirror
oscillates with $f_{x-y}\left(t\right)=-(\lambda_{3}/4\omega^{2})\cos(2\omega
t+\phi)$, where $\lambda_{1,2,3}$ are the oscillation amplitudes and $\phi$ is
a constant phase.

Transforming to the action-angle variables $I_{\pm}=I_{y}\pm I_{x}$ and
$\theta_{\pm}=(\theta_{y}\pm\theta_{x})/2$ and switching to the frame
oscillating with the mirrors, $\Theta_{+}=\theta_{+}-\omega t/s$ and
$\Theta_{-}=\theta_{-}$, leads to the classical secular effective Hamiltonian
Giergiel2021

$\displaystyle H_{eff}$ $\displaystyle=$
$\displaystyle-\frac{P_{+}^{2}+P_{-}^{2}}{2{|m}_{eff}|}-\frac{\lambda_{2}}{2\omega^{2}}\cos(2s\Theta_{+})\cos(2s\Theta_{-})-\frac{{2\lambda}_{1}}{\omega^{2}}\cos(s\Theta_{+})\cos(s\Theta_{-})$
(19) $\displaystyle+\frac{\lambda_{3}}{4\omega^{2}}\cos(2s\Theta_{+}+\phi),$

where $P_{\pm}=I_{\pm}-I_{\pm}^{0}$,
$|m_{eff}|=\left(3I_{\pm}^{0}\right)^{4/3}/\left(2\pi^{2}\right)^{1/3}$ and
$I_{\pm}^{0}$ is the resonant value of the action $I_{\pm}$. The Hamiltonian
(19) describes a particle with negative effective mass $-{|m}_{eff}|$ in the
presence of an _inseparable_ lattice potential which is moving on a Möbius
strip because at $\Theta_{+}=\pi$, $\Theta_{\pm}$ flips to $\pi-\Theta_{\pm}$.

Figure 9: (a) Honeycomb and (b) Lieb time lattices presented in the frame
moving along a resonant trajectory. Atoms loaded to lattices fulfill the
boundary conditions of the Möbius strip geometry, i.e., points
$\\{\Theta_{+}=\pi,\Theta_{-}\\}$ are identified with
$\\{\Theta_{+}=0,\pi-\Theta_{-}\\}$. Dark blue regions correspond to the
lowest energies of a particle with negative effective mass. Reprinted from
Giergiel2021 .

Use of different parameters of the mirrors’ oscillations allows one to realize
different crystalline structures of the effective potential in (19). For
example, for $\lambda_{3}/\lambda_{1}=4$, $\lambda_{2}=0$ and $\phi=0$, a
honeycomb time lattice can be realized (Fig. 9(a)), while for
$\lambda_{3}/\lambda_{1}=4$, $\lambda_{3}/\lambda_{2}=1.62$ and $\phi=\pi/4$,
a Lieb ‘decorated’ square time lattice with a flat band – in which the
dynamics of the atoms is governed solely by interactions – is created (Fig.
9(b)) Giergiel2021 .

Six-dimensional time-space crystals

We live in a 3D space and spatial crystals of dimension up to three can be
observed and investigated. In the previous sections we have seen that time
appears as a degree of freedom where a crystalline behavior can be created
too. Combining space and time lattices, we should be able to create crystals
of dimension higher than three. Surprisingly, such time-space crystals can be
even six-dimensional as we will show in the following Zlabys2021 .

Suppose that ultracold atoms are prepared in a 1D optical lattice potential
created by means of an optical standing wave Pethick2002 . Suppose also that
the optical lattice is periodically shaken. If the atoms do not interact, the
description of the system can be reduced to the following single-particle
Hamiltonian

$H=\frac{p_{x}^{2}}{2}+V_{0}\sin^{2}(x-\lambda\cos\omega t),$ (20)

where $V_{0}$ is the amplitude of the optical potential while $\lambda$ and
$\omega$ denote the amplitude and frequency of the shaking of the potential.
We assume that the depth of the optical lattice potential is sufficiently
large that the atoms can perform periodic motion in the lattice wells which is
resonant with the shaking. If the $s:1$ resonance condition is fulfilled, $s$
localized wave-packets can form in each well which evolve along the resonant
trajectories with the period $s2\pi/\omega$. Restricting to the resonant
Hilbert subspace we obtain an effective tight-binding Hamiltonian Zlabys2021

$H_{eff}\approx-\frac{1}{2}\sum_{i,\alpha,j,\beta}J_{i,\alpha}^{j,\beta}\;a^{*}_{j,\beta}\;a_{i,\alpha},$
(21)

which describes a 2D time-space lattice, where $(i,j)$ and $(\alpha,\beta)$
denote indices of spatial and temporal lattice sites, respectively. To obtain
Eq. (21) we have assumed that the atom is described by the wavefunction
$\psi(x,t)=\sum_{i,\alpha}a_{i,\alpha}w_{i,\alpha}(x,t)$, where
$w_{i,\alpha}(x,t)$ is a localized wave-packet evolving periodically along the
resonant trajectory in the $i$-th well of the optical lattice potential. In
each well there are $s$ such wave-packets, i.e., $\alpha=1,\dots,s$. The
parameters $J_{i,\alpha}^{j,\beta}$ describe hopping of the atom between the
spatial sites $i$ and $j$ and between the temporal sites $\alpha$ and $\beta$.

The 1D optical lattice problem can be easily generalized to a 3D optical
lattice potential which is periodically shaken along the three independent
spatial directions

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2}+V_{0}[\sin^{2}(x-\lambda\cos\omega
t)+\sin^{2}(y-\lambda\cos\omega t)$ (23)
$\displaystyle+\sin^{2}(z-\lambda\cos\omega t)].$

If the $s:1$ resonance condition is fulfilled for the motion of the atom along
each spatial direction, then we can create 3D localized wave-packets which are
products of the wavefunctions of the 1D wave-packets, i.e.,

$W_{\vec{i},\vec{\alpha}}(\vec{r},t)=w_{i_{x},\alpha_{x}}(x,t)\;w_{i_{y},\alpha_{y}}(y,t)\;w_{i_{z},\alpha_{z}}(z,t).$
(24)

Within each well of the 3D optical lattice potential there is a $s\times
s\times s$ time lattice whose temporal sites are labeled by three indices
$\alpha_{x}$, $\alpha_{y}$ and $\alpha_{z}$. The wells of the optical lattice
potential are labeled by three spatial indices $i_{x}$, $i_{y}$ and $i_{z}$.
Thus, all together the wave-packets $W_{\vec{i},\vec{\alpha}}(\vec{r},t)$ are
labeled by six indices corresponding to a 6D time-space lattice, and 6D
condensed matter phenomena can be investigated. In Ref. Zlabys2021  it was
shown how to realize an artificial gauge potential by tilting the 6D time-
space lattice along the temporal directions and employing so-called photon-
assisted tunneling. This allowed the realization of the 6D quantum Hall
effect.

It is worth mentioning that we consider here deep optical lattice potentials
which can host many energy bands, and highly excited bands (with energies
close to the barrier between the potential wells) are resonantly coupled by
the shaking. This is in contrast to a more typical situation when an optical
lattice potential is shallow and resonant coupling of the ground and first
excited energy bands are investigated Sowinski2012 ; Lacki2013 ; Li2016 .

##  6 Conclusions

Big discrete time crystals created by a Bose-Einstein condensate of ultracold
atoms bouncing resonantly on an oscillating atom mirror provide a highly
flexible platform for investigating a broad range of condensed matter
phenomena in the time dimension. By choosing suitable Fourier components in
the periodic drive of the atom mirror, such a system allows us to construct
effective temporal lattice potentials of almost any shape and to readily vary
the geometry of the time lattice. By modulating the interparticle scattering
length via a magnetically tuneable Feshbach resonance, such a system also
allows us to precisely control the effective interparticle interaction and to
engineer exotic long range interactions in the time lattice. Use of a bouncing
BEC system also allows us to construct higher-dimensional time lattices,
involving the bouncing of a BEC between two (or three) orthogonal oscillating
mirrors and also between two oscillating mirrors oriented at 45-degrees. The
latter configuration supports a versatile Möbius strip geometry that can host
a variety of time lattices such as a honeycomb time lattice and a Lieb square
time lattice. By combining 3D time lattices with 3D space lattices, it is
possible to create 6D time-space lattices.

Condensed matter phenomena in the time dimension that have been predicted to
date include Anderson and many-body localization due to temporal disorder,
topological time crystals, quasi-crystal structures in time, and Mott
insulator phases in time. The application of big discrete time crystals to
condensed matter physics in the time dimension has opened up a rich new area
of research ready to be exploited experimentally.

##  7 Statements and Declarations

Ethical approval and consent to participants

The authors declare they have upheld the integrity of the scientific record.

Consent for publication

The authors give their consent for publication of this article.

Availability of data and materials

The data generated during the current study are available from the
contributing author upon reasonable request.

Competing interests

The authors have no competing interests to declare that are relevant to the
content of this article.

Funding

This work was supported by the Australian Research Council Discovery Grant No.
DP100100815 and the National Science Centre, Poland via Project No.
218/31/B/ST2/00319.

Authors’ contributions

PH and KS equally contributed to all aspects of the manuscript. Both authors
read and approved the final manuscript.

Acknowledgements

The authors thank K. Giergiel, E. Anisimovas, B. Dalton, D. Delande, Chu-Hui
Fan, W. Golletz, C. Gunawardana, A. Kosior, A. Kuroś, M. Lewenstein, M.
Mierzejewski, A. Sidorov, A. Singh, A. Syrwid, S. Tojo, T. Tran, J. Wang, A.
Zaheer, J. Zakrzewski and G. Žlabys for fruitful discussions.

Authors’ information

Peter Hannaford

Optical Sciences Centre

Swinburne University of Technology

Hawthorn

Victoria 3122, Australia

<EMAIL_ADDRESS>
ORCID Number: 0000-0001-9896-7284

Krzysztof Sacha

Institute of Theoretical Physics

Jagiellonian University

ulica Profesora Stanislawa Lojasiewicza 11

PL-30-348 Krakow, Poland

<EMAIL_ADDRESS>
ORCID Number: 0000-0001-6463-0659

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