††thanks: These authors have contributed equally to this work.††thanks:
E-mail<EMAIL_ADDRESS>E-mail<EMAIL_ADDRESS>
# Activity-induced Nonequilibrium Vaporization Leads to Reentrant Phase
Separation

Jie Su,∗ Mengkai Feng  Huijun Jiang  Zhonghuai Hou  Department of Chemical
Physics & Hefei National Laboratory for Physical Sciences at Microscales,
University of Science and Technology of China, Hefei, Anhui 230026, China

###### Abstract

Active Brownian particles (ABPs) with pure repulsion is an ideal model to
understand the effect of nonequilibrium on collective behaviors. It has long
been established that activity can create effective attractions leading to
motility-induced phase separation (MIPS), whose role is similar to that of
(inverse) temperature in the simplest equilibrium system with attractive
inter-particle interactions. Here, our theoretical analysis based on a kinetic
theory of MIPS shows that a new type of activity-induced nonequilibrium
vaporization is able to hinder the formation of dense phase when activity is
large enough. Such nonequilibrium vaporization along with the activity-induced
effective attraction thus lead to a MIPS reentrance. Numerical simulations
verify such nonequilibrium effect induced solely by activity on phase
behaviors of ABPs, and further demonstrate the dependence of MIPS on activity
and the strength of inter-particle interaction predicted by our theoretical
analysis. Our findings highlight the unique role played by the nonequilibrium
nature of activity on phase behaviors of active systems, which may inspire
deep insights into the essential difference between equilibrium and
nonequilibrium systems.

Active systems consisting of self-propelled units have been widely discovered
in nature on many scales, ranging from mesoscopic biological or manmade
swimmers such as _E. coli_ and Janus spheres to macroscopic objects like fish,
birds and horses[1, 2]. Due to the ability to take in and dissipate energy to
drive themselves far from equilibrium[3], active systems provide ideal model
systems to investigate the effect of nonequilibrium on collective behaviors
beyond their equilibrium counterparts[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15]. As one of the simplest active systems, active Brownian particles (ABPs)
with pure repulsion without any attraction can spontaneously undergo phase
separation between dense and dilute fluid phases[16, 17]. Such phase
separation resulting solely from the intrinsically nonequilibrium property,
i.e., activity, is so-called motility-induced phase separation (MIPS)[16].

Generally, for ABPs with pure repulsion, they tend to accumulate where they
move more slowly and will slow down at high density for steric reasons, which
then creates effective attractions leading to MIPS[17]. Quite interestingly,
it has been found that the phase diagram of ABPs[17, 18, 19, 20] is nearly the
same as that of the simplest equilibrium system with attractive inter-particle
interactions[21, 22, 23, 24], except that the role of (inverse) temperature is
replaced by activity. Is that all about the effect of activity on the phase
separation of ABPs? Will the nonequilibrium nature of activity bring unique
phase behaviors to the simplest ABPs beyond providing an alternative origin
for phase separation?

In this work, we report that activity can lead to a type of nonequilibrium
vaporization which lacks an analogue in equilibrium systems. Our theoretical
analysis based on a kinetic theory of MIPS in the simplest ABP system with
purely repulsive interactions shows that, such activity-induced nonequilibrium
vaporization brings a unique phase behavior to ABPs. That is, while activity-
induced effective attraction leads to MIPS, activity-induced nonequilibrium
vaporization hinders the formation of dense phase and thus results in a MIPS
reentrance. Besides, we find that the “softer” the repulsive interaction
potential is, the “stronger” the reentrance becomes. The MIPS reentrance is
then verified by numerical simulations. Furthermore, the simulated binodal
agrees well with the phase boundary derived by our theoretical analysis, and
the theoretically predicted dependence of MIPS on the strength of inter-
particle interactions is also demonstrated.

_Theoretical analysis_.– We start from a minimal active fluid theory
characterizing MIPS from a kinetics approach[18, 25, 19]. The theory describes
the steady state of phase separation with a dense clustering phase which is
set as close packed and a dilute phase as homogeneous and isotropic. Particles
transport between two phases through condensation and evaporation events with
rates $k_{\rm in}$ and $k_{\rm out}$, respectively. Absorption rate $k_{\rm
in}$ is assumed to be proportion to the dilute phase number density $\rho_{g}$
and the characteristic self-propulsion velocity $v_{0}$. For the vaporization,
the theory sets that active particles immediately escape from the dense phase
when the direction of the particle on the interface moves towards to dilute
phase. Therefore the escape rate $k_{\rm out}$ is proportion to the rotational
diffusion constants $D_{r}$. Then the steady state condition is achieved
through equating $k_{\rm in}$ and $k_{\rm out}$, leading to the dilute phase
density $\rho_{g}=\frac{\pi\kappa^{\prime}D_{r}}{\sigma v_{0}}$ where
$\kappa^{\prime}$ is a fitting parameter and $\sigma$ is the particle
diameter. For two dimensional ABP system with purely repulsive inter-particle
potential [18, 19] and Lennard-Jones potential [25], solutions of $\rho_{g}$
show very good agreement with simulated MIPS. However, it should be noted that
the particle current from cluster to dilute phase comprises not only
vaporization events due to rotational-diffusion-dependent active motion, but
also those due to active-motion-dependent translational diffusion. Good
agreement between solutions of $\rho_{g}$ and simulated MIPS indicates the
former one makes a major contribution to $k_{\rm out}$ for activity ranging
around the transition point of MIPS (i.e., small and moderate activity). Yet
for large activity, the contribution of translational diffusion process
becomes progressively more important as will be shown by the following
analysis.

To take account of the vaporization events of active-motion-dependent
translational diffusion, we need to firstly obtain the diffusivity of
particles in dense phase near the interface. In the gas phase, the
translational diffusion coefficient is close to the effective one for a free
active particle $D_{0}=D_{t}+\frac{v_{0}^{2}}{2D_{r}}$, where $D_{t}$ is
purely thermal diffusion constant [17]. In dense phase the local diffusion
coefficient $D[\rho]=D_{t}+\frac{v[\rho]^{2}}{2D_{r}}$ is much smaller than
$D_{0}$, where $[\rho]$ denotes an arbitrary functional of density field
$\rho(\vec{r})$ and $v[\rho]$ is a phenomenological motility parameter
satisfying $0<v[\rho]<v_{0}$ [17]. Around the dilute-dense phase interface, it
is convenient to introduce a parameter $\lambda$ to express the local
diffusion as $D[\rho]=D_{t}+\frac{\lambda v_{0}^{2}}{D_{r}}$, wherein
$\lambda\in(0,\frac{1}{2})$. We do notice that the divide of evaporation
event, translational and rotational diffusion, is a little ambiguous and there
might be double counting in this context. At least for a single event that a
particle leaves the dense phase, it is difficult to determine which type of
diffusion plays a major role. Nevertheless, we use a fitting parameter
$\kappa$ to compromise this defect, then the total evaporation current is
written as (see details in the supplemental information, SI)

$\displaystyle j_{\rm out}=$ $\displaystyle
D[\rho]\nabla\rho+\kappa\frac{D_{r}}{\sigma}$ $\displaystyle=$
$\displaystyle\left(D_{t}+\frac{\lambda
v_{0}^{2}}{D_{r}}\right)\frac{\rho_{d}-\rho_{g}}{\sigma}+\kappa\frac{D_{r}}{\sigma},$
(1)

where $\rho_{d}$ is the number density of the dense phase. Analogous to the
procedure in Ref.[25], by integrating the angles of ABPs over the direction of
self-propulsion toward the interface, the condensation current is written as
$j_{\rm in}={\rho_{g}v_{0}}/{\pi}$.

Based on the steady-state assumption, i.e., $j_{\rm in}=j_{\rm out}$, two
transition points can be immediately derived as

$\displaystyle v^{\pm}_{c}=$
$\displaystyle\frac{D_{r}\sigma}{2\lambda(\rho_{d}-\rho_{g})}\Bigg{[}\frac{\rho_{g}}{\pi}$
$\displaystyle\pm\sqrt{\frac{\rho_{g}^{2}}{\pi^{2}}-4\frac{\lambda(\rho_{d}-\rho_{g})}{D_{r}\sigma^{2}}\left[D_{t}(\rho_{d}-\rho_{g})+\kappa
D_{r}\right]}\Bigg{]}.$ (2)

We recall that, in the previous understanding of MIPS taking only rotational-
diffusion-dependent vaporization into account, there is only one transition
point $v_{c}$, since $j_{\rm out}$ is independent on $v_{0}$ and $j_{\rm in}$
increases linearly with $v_{0}$ [18]. Consequently, MIPS will be observed as
$v_{0}>v_{c}$, and activity is considered to mainly provide a nonequilibrium
origin for phase separation. Quite interestingly, Eq.(2) points out that,
increasing activity not only leads to MIPS when $v_{0}$ passes $v_{c}^{-}$ but
also destroys it when $v_{0}>v_{c}^{+}$. Such reentrant MIPS of the simplest
ABPs with purely repulsive inter-particle interactions indicates that, beyond
the activity-induced effective attraction when only rotational-diffusion-
dependent vaporization is considered, extra nonequilibrium vaporization events
due to active-motion-dependent translational diffusion further contributes a
contrary effect on phase separation by accelerating melting. Such unique phase
behavior induced by the nonequilibrium nature of activity has never been
observed in the equilibrium analogues with attractive inter-particle
interactions.

There are several further predictions can be made about the interesting phase
behavior. Firstly, the steady state assumption $j_{\rm in}=j_{\rm out}$ gives
a quantitative measurement of the dilute phase density,
$\rho_{g}=\left(\rho_{d}D[\rho]+\kappa
D_{r}\right)/\left(D[\rho]+v_{0}\sigma/\pi\right)$. Unlike the previous theory
where dilute phase density decreases with $v_{0}$ non-monotonically,
$\rho_{g}$ eventually increases to $\rho_{d}$ for very large activity. This
result strongly implies a homogeneous state when activity is sufficiently
large, which confirms the reentrance of MIPS from another point of view.

Secondly, the parameter $\lambda$ is physically dependent on the repulsive
inter-particle potential $U(r)=\epsilon\hat{U}(r)$, where $\epsilon$ is the
strength of the potential and $\hat{U}(r)$ is an arbitrary dimensionless
function of particle distance. Larger $\epsilon$ brings stronger repulsive
force which hinders particle motion in the dense phase, and consequently leads
to smaller diffusivity and smaller $\lambda$. Based on this picture, we
introduce a so-called “instant diffusion coefficient” [26], defined as
$D_{i}=D_{t}+\frac{v_{0}^{2}/(2D_{r})}{1+(2\gamma D_{r})^{-1}\sum_{j\neq
i}\nabla_{i}^{2}u_{ij}}$, to estimate the local diffusivity $D[\rho]$ and then
$\lambda$. Comparing with the formula of $D[\rho]$, we notice that $\lambda$
should be inversely proportional to $a+\epsilon$, where $a$ is introduced as
an interaction and activity independent parameter. According to Eq.(2),
$v_{c}^{+}$ prominently increases with interacting strength $\epsilon$, while
on the contrary $v_{c}^{-}$ slowly decreases. As a result, MIPS would also be
reentrant for finite interacting strength $\epsilon$, while increasing
$\epsilon$ would greatly broaden the parameter region for emergence of MIPS.
It is thus another testable prediction of our theory to investigate the
dependence of MIPS behavior on the interacting strength.

Figure 1: (a) Phase diagram in the ${\rm Pe}-\phi$ plane. The colored
background is obtained by the kinetics model with $\lambda=2.8\times 10^{-4}$
and $\kappa=0.9$. $f_{l}$ is the fraction of dense phase. The red and green
lines respectively represent the binodal and spinodal curves from simulations.
(b) Dependence of the ensemble-averaged waiting time $\langle t_{w}\rangle$
for the nucleation on ${\rm Pe}$ for active systems at $\phi=0.35$ inside the
gray dashed box in (a). Curves are the exponential fitting for the simulation
data. The inset is the dependence of $N_{\rm max}/N$ with $N_{\rm max}$ the
particle number of the largest cluster on $t$ for ten independent ensembles at
$\phi=0.35$ and ${\rm Pe}=900$. (c) Hysteresis at $\phi=0.35$ including the
gray dashed box in (a). Simulation data are reached from two initial
conditions: the final configurations of systems at slightly smaller ${\rm Pe}$
(red line and symbols) and the ones at slightly larger ${\rm Pe}$ (blue line
and symbols).

_Numerical simulation_.– In order to verify the unique phase behavior
predicted by the aforementioned theoretical analysis, we perform simulations
of a quasi two-dimensional system with size $L_{x}\times L_{y}$ and periodic
boundary conditions consisting of $N$ spherical ABPs with diameter $\sigma$
and friction coefficient $\gamma$. The pairwise inter-particle potential is
set as Weeks-Chandler-Andersen (WCA) potential
$U({r}_{ij})=4\epsilon[(\sigma/r_{ij})^{12}-(\sigma/r_{ij})^{6}+1/4]$ for
$r_{ij}<2^{1/6}\sigma$ and $U({r}_{ij})=0$ otherwise. We set the translational
diffusion coefficient $D_{t}=k_{B}T/\gamma$ so that the system satisfies the
fluctuation-dissipation relation with $k_{B}$ the Boltzmann constant and $T$
the temperature. The rotational diffusion coefficient $D_{r}$ is coupled with
the translational diffusivity as $D_{r}=3D_{t}/\sigma^{2}$. In this work, we
use two dimensionless variables as control parameters, i.e. the Péclet number
${\rm Pe}=v_{0}\sigma/D_{t}$ [18] to characterize the activity of ABPs and the
volume fraction $\phi=\pi\rho_{0}/(4\sigma^{2})$ to describe the density of
ABPs with $\rho_{0}=N/(L_{x}\times L_{y})$ the averaged number density of the
system (see details of the model in the SI).

The phase diagram in the ${\rm Pe}-\phi$ plane with $\epsilon=1$ is presented
in Fig. 1(a), wherein the color bar denotes the theoretical prediction of
particle number fraction in dense phase
$f_{l}=\frac{\rho_{d}}{\rho_{0}}\frac{\rho_{0}-\rho_{g}}{\rho_{d}-\rho_{g}}$
(since the number density of dense phase $\rho_{d}$ is not sensitive to the
Péclet number and total density $\rho_{0}$, we assume that $\rho_{d}$ is
constant and therefore, $f_{l}$ only depends on $\rho_{g}$ and $\rho_{0}$). It
can be found that the simulated binodal (red dotted line) confirms with the
phase boundary of the colored background predicted by our theoretical
analysis. Likewise, theoretically predicted reentrance of MIPS can be observed
as activity increases. For appropriate $\phi$ such as $\phi=0.35$, the active
system changes from a single phase to a coexisting phase and reenters to a
single phase as ${\rm Pe}$ increases (typical snapshots can be found in the
SI). Besides of the binodal, the spinodal (green dotted line) can also be
obtained by simulations too. It can be found that both the binodal and
spinodal curves shift to lower $\phi$ first, but reenter to higher $\phi$
again after an inflexion, forming a brand new metastable state located in a
much wider region at large ${\rm Pe}$.

Figure 2: Influence of the interaction strength. (a) Phase diagram in the
${\rm Pe}-\epsilon$ plane at $\phi=0.3$. The colored background is obtained by
the kinetics model. The blue symbols represent the single phase, the red ones
denote MIPS and the red-line boundaries is the binodal curves, which are all
obtained by simulations. The steady-state configurations at ${\rm Pe}=2500$
and $\epsilon=1.0$ (b), $1.5$ (c), $2.0$ (d) and $4.0$ (e) are respectively
marked as green stars in (a).

To investigate the nature of MIPS near the MIPS reentrance region, the growth
process of the largest cluster (Fig. 1(b)) and a hysteresis loop (Fig. 1(c))
are focused. The inset of Fig. 1(b) plots time series of $N_{\rm max}/N$ with
$N_{\rm max}$ the particle number of the largest cluster for ten independent
simulations with $\phi=0.35$ and ${\rm Pe}=900$ (time series for other
parameters can be found in the SI). It can be observed that $N_{\rm max}/N$
increases from about 0 quickly after a waiting time $t_{w}$, indicating a
first-order phase transition via nucleation [18]. Noticing that the waiting
time can be a good parameter to measure the nucleation barrier, the obtained
ensemble-averaged waiting time $\langle t_{w}\rangle$ as a function of ${\rm
Pe}$ is then presented in Fig. 1(b). As ${\rm Pe}$ increases across the upper
spinodal, $\langle t_{w}\rangle$ increases exponentially, demonstrating that
systems with larger ${\rm Pe}$ above the upper spinodal must go over a higher
nucleation barrier to attain MIPS. Besides, a very large hysteresis loop with
$\phi=0.35$ can be found around the upper MIPS transition point as shown in
Fig. 1(c), demonstrating again the nucleation behavior of a discontinuous
transition from a homogeneous initial state to MIPS[27]. In short, the
simulation results are consistent with our theoretical predictions, verifying
that activity can bring unique phase behavior to the simplest ABP system.

Fig. 2(a) verifies the theoretical prediction about the effect of interaction
strength $\epsilon$ on MIPS with fixed $\phi=0.3$, where blue and red symbols
indicate the single phase and the coexisting phase of MIPS, respectively.
Phase boundaries between these two phases are the binodal (red lines), and the
colored background is $f_{l}$ obtained by our theoretical analysis. Clearly,
the upper binodal increases remarkably as $\epsilon$ increases, while the
lower binodal stays nearly unchanged. Such observation agrees with the
theoretical prediction very well. To take a close look at the influence of
$\epsilon$, steady-state configurations of systems at ${\rm Pe}=2500$ and
$\epsilon=1.0$, $1.5$, $2.0$ and $4.0$ are shown in Fig. 2(b)-(e),
respectively. As $\epsilon$ increases, the dense phase emerges with a
relatively small cluster, and then grows to be a large cluster, indicating
more and more pronounced MIPS. In other words, “harder” repulsive interactions
between ABPs will lead to “weaker” reentrant MIPS while the “softer” ones will
result in “stronger” reentrant MIPS, which further demonstrates the unique
phase behavior induced by the nonequilibrium nature of activity.

In summary, reentrant MIPS induced solely by activity has been revealed. We
showed both theoretically and numerically that such reentrant MIPS results
from the competition between activity-induced effective attraction of ABPs
preferring particle accumulation and activity-induced nonequilibrium
vaporization hindering formation of large clusters. Our findings highlight the
unique role played by the nonequilibrium nature of activity on phase behaviors
of active systems, which may inspire deep insights into the essential
difference between equilibrium and nonequilibrium systems.

This work is supported by MOST(2018YFA0208702), Innovation Program for Quantum
Science and Technology (2021ZD0303306), NSFC (32090044, 21973085, 21833007,
21790350), Anhui Initiative in Quantum Information Technologies (AHY090200),
and the Fundamental Research Funds for the Central Universities
(WK2340000104).

## References

  * Bechinger et al. [2016] C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Rev. Mod. Phys. 88, 045006 (2016).
  * Vicsek and Zafeiris [2012] T. Vicsek and A. Zafeiris, Phys. Rep. 517, 71 (2012), ISSN 0370-1573.
  * Ramaswamy [2010] S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1, 323 (2010).
  * DiLuzio et al. [2005] W. R. DiLuzio, L. Turner, M. Mayer, P. Garstecki, D. B. Weibel, H. C. Berg, and G. M. Whitesides, Nature 435, 1271 (2005).
  * Riedel et al. [2005] I. H. Riedel, K. Kruse, and J. Howard, Science 309, 300 (2005).
  * Kümmel et al. [2013] F. Kümmel, B. ten Hagen, R. Wittkowski, I. Buttinoni, R. Eichhorn, G. Volpe, H. Löwen, and C. Bechinger, Phys. Rev. Lett. 110, 198302 (2013).
  * Gibbs et al. [2011] J. Gibbs, S. Kothari, D. Saintillan, and Y.-P. Zhao, Nano Lett. 11, 2543 (2011).
  * Mallory and Cacciuto [2019] S. A. Mallory and A. Cacciuto, J. Am. Chem. Soc. 141, 2500 (2019).
  * Gou et al. [2019] Y. Gou, H. Jiang, and Z. Hou, Soft Matter 15, 9104 (2019).
  * Du et al. [2019] Y. Du, H. Jiang, and Z. Hou, J. Chem. Phys. 151, 154904 (2019).
  * Yan et al. [2016] J. Yan, M. Han, J. Zhang, C. Xu, E. Luijten, and S. Granick, Nat. Mater. 15, 1095 (2016).
  * Sumino et al. [2012] Y. Sumino, K. H. Nagai, Y. Shitaka, D. Tanaka, K. Yoshikawa, H. Chaté, and K. Oiwa, Nature 483, 448 (2012).
  * Jiang et al. [2017] H. Jiang, H. Ding, M. Pu, and Z. Hou, Soft matter 13, 836 (2017).
  * Karani et al. [2019] H. Karani, G. E. Pradillo, and P. M. Vlahovska, Phys. Rev. Lett. 123, 208002 (2019).
  * Gou et al. [2020] Y.-l. Gou, H.-j. Jiang, and Z.-h. Hou, Chin. J. Chem. Phys. 33, 717 (2020).
  * Tailleur and Cates [2008] J. Tailleur and M. E. Cates, Phys. Rev. Lett. 100, 218103 (2008).
  * Cates and Tailleur [2015] M. E. Cates and J. Tailleur, Annu. Rev. Condens. Matter Phys. 6, 219 (2015).
  * Redner et al. [2013a] G. S. Redner, M. F. Hagan, and A. Baskaran, Phys. Rev. Lett. 110, 055701 (2013a).
  * Redner et al. [2016] G. S. Redner, C. G. Wagner, A. Baskaran, and M. F. Hagan, Phys. Rev. Lett. 117, 148002 (2016).
  * Takatori and Brady [2015] S. C. Takatori and J. F. Brady, Phys. Rev. E 91, 032117 (2015).
  * Gunton [1983] D. Gunton, Phase Transitions Crit. Phenom. 8, 267 (1983).
  * Onuki [2002] A. Onuki, _Phase transition dynamics_ (Cambridge University Press, 2002).
  * Laughlin and Hono [2014] D. E. Laughlin and K. Hono, _Physical metallurgy_ (Newnes, 2014).
  * Vuijk et al. [2019] H. D. Vuijk, J. M. Brader, and A. Sharma, Soft Matter 15, 1319 (2019).
  * Redner et al. [2013b] G. S. Redner, A. Baskaran, and M. F. Hagan, Phys. Rev. E 88, 012305 (2013b).
  * Feng and Hou [2017] M. Feng and Z. Hou, Soft Matter 13, 4464 (2017).
  * Speck et al. [2014] T. Speck, J. Bialké, A. M. Menzel, and H. Löwen, Phys. Rev. Lett. 112, 218304 (2014).


# The Mpemba effect demonstrated on a single trapped ion qubit

Shahaf Aharony Shapira†<EMAIL_ADDRESS>Yotam Shapira† Jovan
Markov  Gianluca Teza  Nitzan Akerman  Oren Raz  Roee Ozeri  Department of
Physics of Complex Systems
Weizmann Institute of Science, Rehovot 7610001, Israel
† These authors contributed equally to this work

###### Abstract

The Mpemba effect is a counter-intuitive phenomena in which a hot system
reaches a cold temperature faster than a colder system, under otherwise
identical conditions. Here we propose a quantum analog of the Mpemba effect,
on the simplest quantum system, a qubit. Specifically, we show it exhibits an
inverse effect, in which a cold qubit reaches a hot temperature faster than a
hot qubit. Furthermore, in our system a cold qubit can heat up exponentially
faster, manifesting the strong version of the effect. This occurs only for
sufficiently coherent systems, making this effect quantum mechanical, i.e. due
to interference effects. We experimentally demonstrate our findings on a
single ${}^{88}\text{Sr}^{+}$ trapped ion qubit.

ME

    Mpemba effect
GKSL

    Gorini-Kossakowski-Sudarshan-Lindblad
dof

    degree of freedom
ODE

    ordinary differential equation
QCs

    quantum computers
SM

    Supplemental Material

Physical systems undergoing relaxation can exhibit a wide range of rich and
non-trivial phenomena. A prominent example is the Mpemba effect (ME) [1, 2],
in which an initially hot system cools down faster than a colder, otherwise
identical, system. Some systems manifest a stronger version of this effect
[3], in which the hotter systems relaxes exponentially faster. The ME has been
experimentally demonstrated in various classical systems, e.g. water [2],
Clathrate hydrates [4], magnetic alloys [5], colloids diffusing in a potential
[6] and a few others [7, 8, 9]. An inverse-ME, in which an initially colder
system heats up faster than a warmer system, has been predicted [10, 11] and
recently measured [12]. Much theoretical insight was gained on this effect in
recent years, using various theoretical methods [13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23] and numerical results [24, 25, 26, 27].

Quantum versions of the ME have been recently proposed [28, 29, 30, 31, 32,
33], including its extensions in terms of relaxation of entanglement asymmetry
in quantum spin-systems [34]. The latter has been recently demonstrated using
trapped-ions [35].

Here we propose and experimentally demonstrate the existence of an inverse-ME
in the simplest quantum system - a single qubit. We consider a coherently
driven qubit that is coupled to a thermal Markovian bath, causing decoherence
of the qubit and its eventual relaxation to a non-equilibrium steady state.
Our only assumption on the qubit-bath coupling is that the qubit’s decoherence
rate is monotonically increasing with the bath’s temperature. This occurs,
e.g., for a black-body photon-emitting bath, such that the emission rate of
photons at resonance with the qubit’s transition energy increases with
temperature. Our analysis shows that a strong inverse-ME can occur, however,
only for a sufficiently coherent qubit, making this effect quantum mechanical,
i.e. due to interference.

We demonstrate the inverse-ME experimentally by implementing it on the Zeeman
qubit, defined on a single trapped ${}^{88}\text{Sr}^{+}$ ion. Figure 1 shows
our model and the corresponding implementation on the ion’s energy levels,
detailed further below.

Figure 1: Top left: The modeled quantum system exhibiting an inverse Mpemba-
effect. A thermal source of photons (fire) is coupled to coherently driven
qubit (blue), causing it to relax to a steady state. Bottom left: The
resulting coupling between the qubit’s levels, $\ket{\downarrow}$ and
$\ket{\uparrow}$, with a coherent drive (green) and decoherence terms causing
decay ($\gamma_{\text{decay}}$) and dephasing ($\gamma_{\text{dephase}}$).
Right: The qubit is mapped to the $5S_{\frac{1}{2}}$ levels of the Zeeman
ground state manifold of a trapped ${}^{88}\text{Sr}^{+}$ ion. Coherent
(green) and incoherent (purple) evolution is generated sequentially (pulse
schemes), by using various additional atomic transitions, detailed below.

The qubit’s dynamics is given by the Gorini-Kossakowski-Sudarshan-Lindblad
(GKSL) equation, $\partial_{t}\rho=\mathcal{L}\left[\rho\right]$, with
$\mathcal{L}$ a Lindblad super-operator, acting on $\rho\in\mathbb{C}^{2\times
2}$, the density matrix representing a statistical ensemble of a single qubit.
Specifically, the operation of the super-operator is given by ($\hbar=1$),

$\displaystyle\mathcal{L}\left[\rho\right]$
$\displaystyle=-\frac{i\Omega}{2}\commutator{\sigma_{x}}{\rho}+\gamma_{\text{decay}}L_{\outerproduct{\uparrow}{\downarrow}}\left[\rho\right]+\gamma_{\text{dephase}}L_{\outerproduct{\uparrow}{\uparrow}}\left[\rho\right]\,,$
(1)

with $\Omega$ the rate of the coherent driving of the qubit, set by the
$x$-Pauli matrix, $\sigma_{x}$. Open Markovian dynamics are generated by
$L_{A}\left[\rho\right]\equiv A\rho
A^{\dagger}-\frac{1}{2}\anticommutator{A^{\dagger}A}{\rho}$. We consider
decoherence due to decay (dephasing), generated by
$\outerproduct{\downarrow}{\uparrow}$ ($\outerproduct{\uparrow}{\uparrow}$),
with rates,

$\displaystyle\gamma_{\text{decay}}$
$\displaystyle=\alpha\gamma\left(T\right)\,,$
$\displaystyle\gamma_{\text{dephase}}$
$\displaystyle=\left(1-\alpha\right)\gamma\left(T\right)\,,$ (2)

where $0\leq\alpha\leq 1$, is the relative occurrence of decay. The overall
temperature-dependent decoherence rate due to coupling to the bath,
$\gamma\left(T\right)$, is assumed to be monotonically increasing with $T$,
such that we can characterize the bath by $T$ or $\gamma$ interchangeably.

Figure 2: Left: Steady state locus. We measure the qubit’s steady-state
position (points) on the y-z plane of the Bloch sphere (black line) at
different temperatures and compute the corresponding $\alpha$’s (see the main
text). Our data-sets (points) are fitted yielding $\alpha=0.21\pm 0.03$
(orange), $\alpha=0.51\pm 0.04$ (cyan) and $\alpha=0.94\pm 0.07$ (blue), used
hereafter. The steady state locus corresponding to the latter is presented in
color, showing values of $\gamma^{\prime}$ (log-scale). Right: Coefficient of
the slow-decaying eigenstate, $a_{-}$, as a function of
$\gamma_{\text{i}}^{\prime}$, for $\gamma_{\text{f}}^{\prime}=15$ (red star).
The coefficient shows a non-monotonic behaviour, implying the existence of a
ME. Furthermore, the curve is shown to vanish at
$\gamma_{\text{i}}^{\prime}=0.07\neq\gamma_{\text{f}}^{\prime}$, proving the
existence of a strong-ME. The highlighted circles correspond to the initial
states shown in Fig. 3 (with matching colors).

The dynamics is conveniently analyzed using the Bloch vector,
$\vec{r}=\left(x,y,z\right)$, with
$\rho=\nicefrac{{1}}{{2}}\left(1+\vec{r}\cdot\vec{\sigma}\right)$ (see the
Supplemental Material (SM) [36]). Since the system is driven, its fixed points
correspond to non-equilibrium steady states that do not obey detailed balance,
e.g. the qubit continuously scatters photons. The collection of steady-states,
$\vec{r}^{\text{ ss}}\left(\gamma\right)$, form a right-half of an ellipse in
the $y-z$ plane, with its center at $\left(0,0,-\nicefrac{{1}}{{2}}\right)$
and semiaxes
$\left(r_{y},r_{z}\right)=\left(\sqrt{\nicefrac{{\alpha}}{{2}}},\nicefrac{{1}}{{2}}\right)$,
shown in Fig. 2 (left). Each point on this curve, known as the steady state
locus, corresponds to a steady state at a given
$\gamma^{\prime}\equiv\gamma/\Omega$, with $\gamma^{\prime}\to
0\left(\infty\right)$ corresponding to the center (south-pole) of the Bloch
sphere.

Consider the relaxation path of an initial condition given by the steady state
solution of a cold temperature, $\vec{r}^{\text{
ss}}\left(\gamma_{\text{i}}\right)$, when coupled to a hot bath characterized
by $\gamma_{\text{f}}$. The solution of Eq. (1) is then given by,

$\vec{r}\left(t;\gamma_{\text{i}},\gamma_{\text{f}}\right)=\vec{r}^{\text{
ss}}\left(\gamma_{\text{f}}\right)+\sum_{n\in\left\\{+,-,x\right\\}}a_{n}\left(\gamma_{\text{i}},\gamma_{\text{f}}\right)\vec{v}_{n}\left(\gamma_{\text{f}}\right)e^{\lambda_{n}\left(\gamma_{\text{f}}\right)t},$
(3)

where $\vec{v}_{n}\left(\gamma_{\text{f}}\right)$ are the relaxation modes of
the system, $\lambda_{n}\left(\gamma_{\text{f}}\right)$ their rates, and
$a_{n}\left(\gamma_{\text{i}},\gamma_{\text{f}}\right)$ the corresponding
coefficients, determined by the initial and final temperatures [36].

We note that the $x$-coordinate has a stable fixed point at $x^{*}=0$, making
the $x$-direction trivially vanish throughout the system’s evolution.

The decay rates in the $y-z$ plane are given by the real part
$\real\left[\lambda_{\pm}\right]$, with

$\lambda_{\pm}=-\gamma_{\text{f}}^{\prime}\left(\alpha+\nicefrac{{1}}{{2}}\pm\sqrt{\left(\alpha-\nicefrac{{1}}{{2}}\right)^{2}-\nicefrac{{1}}{{\gamma_{\text{f}}^{{}^{\prime}2}}}}\right).$
(4)

The ME can exist only when $\real\left[\lambda_{\pm}\right]$ are distinct,
allowing for a slow and fast relaxation modes. This occurs for final
temperatures $\gamma_{\text{f}}^{\prime}>\gamma_{\text{b}}^{\prime}$, with the
bifurcation point
$\gamma_{\text{b}}^{\prime}\equiv\left|\alpha-\nicefrac{{1}}{{2}}\right|^{-1}$.

The relaxation at long times is determined by the slowest relaxation mode,
$\lambda_{-}$, and its coefficient, $a_{-}$, which clearly vanishes for
$\gamma_{\text{i}}=\gamma_{\text{f}}$. Fixing $\gamma_{\text{f}}^{\prime}$,
one might expect $a_{-}$ to be monotonic in the range,
$0\leq\gamma_{\text{i}}^{\prime}\leq\gamma_{\text{f}}^{\prime}$. However, for
an inverse-ME to take place, a cold system must reach the steady state faster
than a hotter one, i.e. $\left|a_{-}\right|$ is smaller for a cold system,
compared to a hotter system. It is therefore the non-monotonic behavior of
$a_{-}$ as a function of $\gamma_{\text{i}}$ which enables the existence of
the ME [3]. Indeed, the coefficient
$a_{-}\left(\gamma_{\text{i}},\gamma_{\text{f}}\right)$ (full expression in
the SM [36]) displays such a behavior, implying the existence of an inverse-
ME. An example with $\gamma_{\text{f}}^{\prime}=15$ is plotted in Fig. 2
(right).

A strong-ME occurs in the special case in which $a_{-}$ vanishes at an initial
temperature, $\gamma_{\text{i,SME}}\neq\gamma_{\text{f}}$. In that case, the
relaxation time is determined by the fast rate, $\lambda_{+}$, and as a
result, it is exponentially faster [3]. In other words, defining the distance
to steady state,
$d_{\text{ss}}^{\gamma_{\text{i}}}\left(t\right)\equiv\left|\vec{r}\left(t;\gamma_{\text{i}},\gamma_{\text{f}}\right)-\vec{r}^{\text{
ss}}\left(\gamma_{\text{f}}\right)\right|$, then
$d_{\text{ss}}^{\gamma_{\text{i}}\neq\gamma_{\text{i,SME}}}\left(t\right)/d_{\text{ss}}^{\gamma_{\text{i,SME}}}\left(t\right)$
is asymptotically exponentially increasing in time.

Here, $a_{-}$ vanishes at
$\gamma_{\text{i,SME}}^{\prime}=\gamma_{\text{f}}^{\prime}\left(\left(\alpha-\nicefrac{{1}}{{2}}\right)-\sqrt{\left(\alpha-\nicefrac{{1}}{{2}}\right)^{2}-\gamma_{f}^{{}^{\prime}-2}}\right)$.
For example, for $\gamma_{\text{f}}^{\prime}=15$, $a_{-}$ vanishes at
$\gamma_{\text{i,SME}}^{\prime}\approx 0.07$ as seen in Fig. 2 (right). The
strong-ME in this system is experimentally optimal to achieve the most
pronounced signal (see the SM [36]). Indeed, in our experimental
demonstrations we make use of values
$\gamma_{\text{i}}\approx\gamma_{\text{i,SME}}$. We mathematically prove the
strong-ME can only appear in a heating process, i.e. as an inverse-ME (see the
SM [36]).

Since $\gamma_{\text{i,SME}}^{\prime}>0$, the required $\alpha$ for a strong-
ME is bounded by
$\alpha>\nicefrac{{1}}{{2}}+\nicefrac{{1}}{{\gamma_{\text{f}}^{\prime}}}\geq\nicefrac{{1}}{{2}}$.
When $\alpha$ satisfies this condition, there exists a strong-ME for every
final temperature above the bifurcation point, at
$\gamma_{\text{i,SME}}^{\prime}$. Thus, an exponentially faster relaxation
occurs only in a coherent enough system, i.e. with a low excess dephasing on
top of that induced by the decay channel. Specifically, a classical bit, with
no coherence between its two states, cannot exhibit this strong effect.

This model describes, for example, a single trapped ${}^{88}\text{Sr}^{+}$ ion
qubit in a small-scale quantum computer [37]. Specifically we encode the
$\ket{\downarrow}$ ($\ket{\uparrow}$) qubit state on the
$5S_{\frac{1}{2},-\frac{1}{2}}$ ($5S_{\frac{1}{2},\frac{1}{2}}$) state in the
Zeeman ground state manifold, shown in Fig. 1 (right). The two states are
coherently coupled with a magnetic field (green), oscillating at the Zeeman
splitting frequency in the $5S_{\frac{1}{2}}$ manifold, generating the qubit’s
Hamiltonian, $H=\Omega\sigma_{x}$, with $\Omega$ the field’s Rabi frequency.

As shown in Fig. 1, we combine the coherent and open dynamics in discrete
steps, by interlacing small durations of coherent (green pulse) and open
Markovian evolution (purple pulse), i.e. by trotterization. Markovian open
dynamics are generated by coupling the qubit levels via fast decaying states
[38]. Control over $\alpha$ is gained by making use of sequential cascade of
pulses and transitions.

Specifically we use a narrow linewidth laser at 674 nm [39] (red) in order to
selectively couple the $\ket{\uparrow}$ state to the
$\ket{4D_{\frac{5}{2},\frac{5}{2}}}$ state in the $4D_{\frac{5}{2}}$
metastable manifold. An additional laser at 1033 nm (pink) couples the
$4D_{\frac{5}{2}}$ manifold to the short-lived $5P_{\frac{3}{2}}$. Due to
selection rules, only the $\ket{5P_{\frac{3}{2},\frac{3}{2}}}$ state is
populated, which quickly decays back to the $\ket{\uparrow}$ state (blue),
resulting in full dephasing, i.e. $\alpha=0$. By using an additional
$\pi$-pulse in the $4D_{\frac{5}{2}}$ manifold (grey), between the $674$ nm
and the $1033$ nm pulses, we map the $\ket{4D_{\frac{5}{2},\frac{5}{2}}}$
state to the $\ket{4D_{\frac{5}{2},-\frac{5}{2}}}$, which will ultimately
decay to the $\ket{\downarrow}$ state, yielding $\alpha\approx 1$.

We demonstrate this control experimentally by initializing the system to the
$\ket{\uparrow}$ state and letting it relax to a steady state under $n=100$
repetitions of interlaced dynamics, analogous to a decay time of
$7\gamma_{\text{f}}^{-1}$. After this evolution we perform state tomography to
determine the location of the steady state on the Bloch sphere. Figure 2
(left) shows the measured steady states for various values of
$\gamma^{\prime}$, forming the elliptically shaped steady state locus (blue
points), with a fitted value of $\alpha=0.94\pm 0.07$ (gradient line).

Intermediate values of $\alpha$ can be formed by replacing the $\pi$-pulse in
the $4D_{\frac{5}{2}}$ manifold (grey) with, e.g., a $\pi/2$-pulse or a
$\pi/5$-pulse, yielding a thinner steady state loci, fitted as $\alpha=0.51\pm
0.04$ and $\alpha=0.21\pm 0.03$, (cyan and orange) respectively.

A canonical experimental protocol for measuring the ME, comprises letting the
system relax to the steady state $\vec{r}^{\text{
ss}}\left(\gamma_{\text{i}}^{\prime}\right)$, then quenching it to a final
temperature, $\gamma_{\text{f}}^{\prime}$, while performing tomography of the
relaxation dynamics to $\vec{r}^{\text{
ss}}\left(\gamma_{\text{f}}^{\prime}\right)$. The measurements are then used
to obtain the Euclidean-distance on the Bloch sphere from the final steady
state, $d_{\text{ss}}\left(t\right)$. This protocol raises a technical
challenge, namely, the relaxation time to an initial cold system, with
$\gamma_{\text{i}}^{\prime}\ll 1$, requires a long evolution duration, which
may surpass the system’s natural coherence time, leading to an effectively
reduced and uncontrolled value of $\alpha$.

We mitigate this challenge by measuring the ME using two complementary
techniques: performing the experimental protocol with large trotter steps,
thus reducing the total duration of an experiment, or by effectively preparing
the qubit in the initial state, $\vec{r}^{\text{
ss}}\left(\gamma_{\text{i}}^{\prime}\right)$, thus circumventing the long
initial relaxation time.

The results obtained by the former technique, large trotter steps, are shown
in Fig. 3. Specifically we evolve the system to $t=4\gamma_{\text{f}}^{-1}$ in
14 trotter-steps (horizontal) and present $d_{\text{ss}}\left(t\right)$
(vertical) for various $\gamma_{\text{i}}^{\prime}$’s (color). These steps
form a less accurate approximation of the continuous model. Indeed, Fig. 3
(top) exhibits oscillations which are common to non-adiabatic digitized
evolution. We highlight an initially cold system,
$\gamma_{\text{i}}^{\text{C}}$ (blue) and an initially hot system,
$\gamma_{\text{i}}^{\text{H}}$ (cyan), which are analyzed in further detail.
Specifically the inset shows the distance between these two systems during
relaxation,
$d_{\text{ss}}^{\text{C}}\left(t\right)-d_{\text{ss}}^{H}\left(t\right)$,
which similarly exhibits oscillations.

Figure 3: The inverse-ME is demonstrated by relaxing qubits to a steady states
at various initial temperatures, $\gamma_{\text{i}}^{\prime}$ (color) and
tracking their relaxation as a function of time (horizontal) to a final steady
state at a fixed temperature,
$\gamma_{\text{f}}^{\prime}=15>\gamma_{\text{i}}^{\prime}$ and $\alpha=0.94$.
We consider the qubit’s Euclidean-distance to the final steady state,
$d_{\text{ss}}\left(t\right)$ (vertical). We highlight the distances of an
initially cold (thick blue) and hot (thick cyan) systems,
$d_{\text{ss}}^{\text{C}}$ and $d_{\text{ss}}^{\text{H}}$, respectively, and
analyze them further. The insets show
$d_{\text{ss}}^{\text{C}}-d_{\text{ss}}^{\text{H}}$. Error bars (shaded
regions) correspond to $\pm 2\sigma$ confidence intervals due to quantum shot-
noise. Top: $d_{\text{ss}}$ exhibits oscillations, not captured by the model
above, which occur due to the relatively large time steps used in the digital
evolution. Bottom: Post-processing the same data by polynomial smoothing
reproduces the inverse-ME. Specifically, $d_{\text{ss}}^{C}>d_{\text{ss}}^{H}$
at $t=0$, however their values cross at $t\approx 2\gamma_{\text{f}}^{-1}$,
after which $d_{\text{ss}}^{C}<d_{\text{ss}}^{H}$, seen as well in the inset
as a negative value, beyond the error bars.

We compensate for the oscillations by employing simple polynomial-smoothing of
the data, shown in Fig. 3 (bottom). This demonstrate an inverse-ME, as the
curves of the cold and hot systems cross, with the cold system reaching the
steady state before the hot system. Indeed, the inset shows that
$d_{\text{ss}}^{\text{C}}\left(0\right)-d_{\text{ss}}^{\text{H}}\left(0\right)>0$,
indicating the cold system is initially at a larger distance from steady
state. However, during the relaxation we observe a crossing time
$t_{\text{cross}}$ after which
$d_{\text{ss}}^{\text{C}}\left(t\right)-d_{\text{ss}}^{\text{H}}\left(t\right)<0$,
beyond $\pm 2\sigma$ error bars due to quantum shot noise.

Figure 4: The inverse-ME realized with initial state preperation. The qubit’s
distance to the final steady state, $d_{\text{ss}}\left(t\right)$, is shown
for qubits initialized at different temperatures (color) and quenched to
$\gamma_{\text{f}}^{\prime}=100$ and $\alpha=0.94$. We highlight
$d_{\text{ss}}^{\text{C}}$ (thick dark blue) and $d_{\text{ss}}^{\text{H}}$
(thick light blue), initially cold and hot systems, respectively, and present
their $\pm 2\sigma$ confidence intervals (shaded regions). As shown, the hot
qubit starts closer to the final steady state. However, at
$t_{\text{cross}}\approx 0.6\gamma_{f}^{-1}$ (vertical grey) the cold system
surpasses the hot system and relaxes first, manifesting the inverse Mpemba
effect. Inset:
$d_{\text{ss}}^{\text{C}}\left(t\right)-d_{\text{ss}}^{\text{H}}\left(t\right)$
highlighting the crossing of the two systems, beyond the confidence intervals.

Next, we directly prepare the qubit at an initial steady state. To do so we
write the initial steady state density matrix, $\rho_{\text{i}}^{\text{ss}}$,
as a linear combination of two pure-state density matrices. Then, the
evolution of $\rho_{\text{i}}^{\text{ss}}$ is equivalent to the same linear
combination of evolved pure states. Specifically, here the steady states are
all of the form,
$\rho_{\text{i}}^{\text{ss}}=\frac{1-p}{2}\outerproduct{+\theta}{+\theta}+\frac{1+p}{2}\outerproduct{-\theta}{-\theta}$,
with $\theta$ and $p$ representing the direction and distance of the steady
state from the Bloch sphere origin and
$\ket{\pm\theta}\equiv\exp\left[-\frac{i}{2}\left(\theta\pm\frac{\pi}{2}\right)\sigma_{x}\right]\ket{\uparrow}$
(see the SM [36]). An observable stemming from the evolution of
$\rho_{\text{i}}^{\text{ss}}$, is recovered by measuring the same observable
on the evolution of $\ket{\pm\theta}$, and using a weighted linear combination
of the measured observables, with weights $\left(1\mp p\right)/2$.

Figure 4 shows the dynamics of system initialized at steady states with
respect to various $\gamma_{\text{i}}^{\prime}$’s, and tracks their evolution
as a function of time (horizontal) towards a fixed
$\gamma_{\text{f}}^{\prime}=100>\gamma_{\text{i}}^{\prime}$. Similarly to Fig.
3, we show the distance to the final steady state, $d_{\text{ss}}$, and
highlight an initially cold (dark blue) and hot (light blue) systems. As
above,
$d_{\text{ss}}^{\text{C}}\left(0\right)>d_{\text{ss}}^{\text{H}}\left(0\right)$,
indicating the cold system is initially at a larger distance from steady
state, yet during the relaxation we observe a crossing time $t_{\text{cross}}$
after which
$d_{\text{ss}}^{\text{C}}\left(t\right)<d_{\text{ss}}^{\text{W}}\left(t\right)$,
beyond $\pm 2\sigma$ error bars due to quantum shot noise. This is also
reflected in the inset which shows
$d_{\text{ss}}^{\text{C}}-d_{\text{ss}}^{\text{H}}$ (vertical) initially
positive and at later times negative, beyond the error bars.

In conclusion, we have proposed and experimentally demonstrated the inverse ME
on as single qubit. Furthermore, we have proven that a strong, i.e.
exponentially faster relaxation, ME exists only for a sufficiently coherent
qubit. As our findings pertain the simplest quantum system, one expects to
find the ME in larger quantum systems, such as quantum computers, in which
maintaining a low temperature for long times is crucial.

###### Acknowledgements.

This work was supported by the Israel Science Foundation Quantum Science and
Technology (Grant 3457/21). O.R. is supported by the ISF (Grant 232/23). G. T.
is supported by the Center for Statistical Mechanics at the Weizmann Institute
of Science, the Simons Foundation (grant 662962), the grants HALT and
Hydrotronics of the EU Horizon 2020 program and the NSF-BSF (grant 2020765).

## References

  * Mpemba and Osborne [1969] E. B. Mpemba and D. G. Osborne, Cool?, Physics Education 4, 172 (1969).
  * Jeng [2006] M. Jeng, The Mpemba effect: When can hot water freeze faster than cold?, American Journal of Physics 74, 514 (2006).
  * Klich  _et al._ [2019] I. Klich, O. Raz, O. Hirschberg, and M. Vucelja, Mpemba index and anomalous relaxation, Phys. Rev. X 9, 021060 (2019).
  * Ahn  _et al._ [2016] Y.-H. Ahn, H. Kang, D.-Y. Koh, and H. Lee, Experimental verifications of Mpemba-like behaviors of clathrate hydrates, Korean J. Chem. Eng. 33, 1903 (2016).
  * Chaddah  _et al._ [2010] P. Chaddah, S. Dash, K. Kumar, and A. Banerjee, Overtaking while approaching equilibrium (2010), arXiv:1011.3598 [cond-mat.stat-mech] .
  * Kumar and Bechhoefer [2020] A. Kumar and J. Bechhoefer, Exponentially faster cooling in a colloidal system, Nature 584, 64 (2020).
  * Hu  _et al._ [2018] C. Hu, J. Li, S. Huang, H. Li, C. Luo, J. Chen, S. Jiang, and L. An, Conformation directed mpemba effect on polylactide crystallization, Crystal Growth & Design 18, 5757 (2018), https://doi.org/10.1021/acs.cgd.8b01250 .
  * Liu  _et al._ [2023] J. Liu, J. Li, B. Liu, I. W. Hamley, and S. Jiang, Mpemba effect in crystallization of polybutene-1, Soft Matter 19, 3337 (2023).
  * Chorazewski  _et al._ [2023] M. Chorazewski, M. Wasiak, A. V. Sychev, V. I. Korotkovskii, and E. B. Postnikov, The curious case of 1-ethylpyridinium triflate: Ionic liquid exhibiting the Mpemba effect, J. Solution Chem. (2023).
  * Lu and Raz [2017] Z. Lu and O. Raz, Nonequilibrium thermodynamics of the markovian mpemba effect and its inverse, Proceedings of the National Academy of Sciences 114, 5083 (2017).
  * Lasanta  _et al._ [2017] A. Lasanta, F. Vega Reyes, A. Prados, and A. Santos, When the hotter cools more quickly: Mpemba effect in granular fluids, Phys. Rev. Lett. 119, 148001 (2017).
  * Kumar  _et al._ [2022] A. Kumar, R. Chétrite, and J. Bechhoefer, Anomalous heating in a colloidal system, Proceedings of the National Academy of Sciences 119, e2118484119 (2022).
  * Walker and Vucelja [2021] M. R. Walker and M. Vucelja, Anomalous thermal relaxation of langevin particles in a piecewise-constant potential, Journal of Statistical Mechanics: Theory and Experiment 2021, 113105 (2021).
  * Holtzman and Raz [2022] R. Holtzman and O. Raz, Landau theory for the mpemba effect through phase transitions, Communications Physics 5, 280 (2022).
  * Degünther and Seifert [2022] J. Degünther and U. Seifert, Anomalous relaxation from a non-equilibrium steady state: An isothermal analog of the mpemba effect, Europhysics Letters 139, 41002 (2022).
  * Busiello  _et al._ [2021] D. M. Busiello, D. Gupta, and A. Maritan, Inducing and optimizing markovian mpemba effect with stochastic reset, New Journal of Physics 23, 103012 (2021).
  * Teza  _et al._ [2023a] G. Teza, R. Yaacoby, and O. Raz, Eigenvalue crossing as a phase transition in relaxation dynamics, Phys. Rev. Lett. 130, 207103 (2023a).
  * Schwarzendahl and Löwen [2022] F. J. Schwarzendahl and H. Löwen, Anomalous cooling and overcooling of active colloids, Physical review letters 129, 138002 (2022).
  * Zhang and Hou [2022] S. Zhang and J.-X. Hou, Theoretical model for the mpemba effect through the canonical first-order phase transition, Physical Review E 106, 034131 (2022).
  * Teza  _et al._ [2023b] G. Teza, R. Yaacoby, and O. Raz, Relaxation shortcuts through boundary coupling, Physical review letters 131, 017101 (2023b).
  * Biswas  _et al._ [2020] A. Biswas, V. Prasad, O. Raz, and R. Rajesh, Mpemba effect in driven granular maxwell gases, Physical Review E 102, 012906 (2020).
  * Santos and Prados [2020] A. Santos and A. Prados, Mpemba effect in molecular gases under nonlinear drag, Physics of Fluids 32 (2020).
  * Chétrite  _et al._ [2021] R. Chétrite, A. Kumar, and J. Bechhoefer, The metastable Mpemba effect corresponds to a non-monotonic temperature dependence of extractable work, Front. Phys. 9, 141 (2021).
  * Baity-Jesi  _et al._ [2019] M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvión, A. Gordillo-Guerrero, D. Iñiguez, A. Lasanta, A. Maiorano, E. Marinari, _et al._ , The mpemba effect in spin glasses is a persistent memory effect, Proceedings of the National Academy of Sciences 116, 15350 (2019).
  * Gal and Raz [2020] A. Gal and O. Raz, Precooling strategy allows exponentially faster heating, Physical review letters 124, 060602 (2020).
  * Gijón  _et al._ [2019] A. Gijón, A. Lasanta, and E. Hernández, Paths towards equilibrium in molecular systems: The case of water, Physical Review E 100, 032103 (2019).
  * Vadakkayil and Das [2021] N. Vadakkayil and S. K. Das, Should a hotter paramagnet transform quicker to a ferromagnet? monte carlo simulation results for ising model, Physical Chemistry Chemical Physics 23, 11186 (2021).
  * Carollo  _et al._ [2021] F. Carollo, A. Lasanta, and I. Lesanovsky, Exponentially accelerated approach to stationarity in markovian open quantum systems through the mpemba effect, Phys. Rev. Lett. 127, 060401 (2021).
  * Kochsiek  _et al._ [2022] S. Kochsiek, F. Carollo, and I. Lesanovsky, Accelerating the approach of dissipative quantum spin systems towards stationarity through global spin rotations, Phys. Rev. A 106, 012207 (2022).
  * Nava and Fabrizio [2019] A. Nava and M. Fabrizio, Lindblad dissipative dynamics in the presence of phase coexistence, Phys. Rev. B 100, 125102 (2019).
  * Chatterjee  _et al._ [2023] A. K. Chatterjee, S. Takada, and H. Hayakawa, Quantum mpemba effect in a quantum dot with reservoirs, Phys. Rev. Lett. 131, 080402 (2023).
  * Rylands  _et al._ [2023] C. Rylands, K. Klobas, F. Ares, P. Calabrese, S. Murciano, and B. Bertini, Microscopic origin of the quantum mpemba effect in integrable systems, arXiv preprint arXiv:2310.04419 (2023).
  * Ivander  _et al._ [2023] F. Ivander, N. Anto-Sztrikacs, and D. Segal, Hyperacceleration of quantum thermalization dynamics by bypassing long-lived coherences: An analytical treatment, Phys. Rev. E 108, 014130 (2023).
  * Murciano  _et al._ [2023] S. Murciano, F. Ares, I. Klich, and P. Calabrese, Entanglement asymmetry and quantum mpemba effect in the xy spin chain (2023), arXiv:2310.07513 [cond-mat.stat-mech] .
  * Joshi  _et al._ [2024] L. K. Joshi, J. Franke, A. Rath, F. Ares, S. Murciano, F. Kranzl, R. Blatt, P. Zoller, B. Vermersch, P. Calabrese, C. F. Roos, and M. K. Joshi, Observing the quantum mpemba effect in quantum simulations (2024), arXiv:2401.04270 [quant-ph] .
  * [36] Supplemental material, including additional derivation and technical information.
  * Manovitz  _et al._ [2022] T. Manovitz, Y. Shapira, L. Gazit, N. Akerman, and R. Ozeri, Trapped-ion quantum computer with robust entangling gates and quantum coherent feedback, PRX Quantum 3, 010347 (2022).
  * Ben Av  _et al._ [2020] E. Ben Av, Y. Shapira, N. Akerman, and R. Ozeri, Direct reconstruction of the quantum-master-equation dynamics of a trapped-ion qubit, Phys. Rev. A 101, 062305 (2020).
  * Peleg  _et al._ [2019] L. Peleg, N. Akerman, T. Manovitz, M. Alon, and R. Ozeri, Phase stability transfer across the optical domain using a commercial optical frequency comb system (2019), arXiv:1905.05065 [physics.atom-ph] .

Supplemental Material

##  I 1\. Analytic solution of the Lindblad equation

We provide additional information concerning the solution of the qubit’s
dynamics. The Lindbladian in the main text can be transformed, using the
Kronecker product, into a linear form:
$\partial_{t}\vec{\rho}=\hat{L}\vec{\rho}$, where $\vec{\rho}$ is a column
vector and $\hat{L}$ is a $4\times 4$ matrix. For our model, given in Eq. (1)
of the main text, we obtain

$\displaystyle\hat{L}\left(\Omega,\gamma,\alpha\right)$
$\displaystyle=\begin{pmatrix}-2\alpha\gamma&-\frac{i\Omega}{2}&\frac{i\Omega}{2}&0\\\
-\frac{i\Omega}{2}&-\gamma&0&\frac{i\Omega}{2}\\\
\frac{i\Omega}{2}&0&-\gamma&-\frac{i\Omega}{2}\\\
2\alpha\gamma&\frac{i\Omega}{2}&-\frac{i\Omega}{2}&0\end{pmatrix},$ (5)

such that the equation of motion reads,

$\displaystyle\left\\{\begin{array}[]{l}\dot{\rho}_{11}=-2\alpha\gamma\rho_{11}+\frac{i\Omega}{2}\left(\rho_{12}-\rho_{21}\right)\,,\\\
\dot{\rho}_{21}=\frac{i\Omega}{2}\left(\rho_{22}-\rho_{11}\right)-\gamma\rho_{21}\,,\\\
\dot{\rho}_{12}=\frac{i\Omega}{2}\left(\rho_{11}-\rho_{22}\right)-\gamma\rho_{12}\,,\\\
\dot{\rho}_{22}=2\alpha\gamma\rho_{11}+\frac{i\Omega}{2}\left(\rho_{21}-\rho_{12}\right)\,.\end{array}\right.$
(10)

Since $\rho=\rho^{\dagger}$ and $\text{Tr}\left[\rho\right]=1$, the state of a
single qubit can be equivalently written in the Bloch vector representation:

$\displaystyle\vec{r}$ $\displaystyle=\left(\rho_{12}+\rho_{21},\
i\left(\rho_{12}-\rho_{21}\right),\
\rho_{11}-\rho_{22}\right)\equiv\left(x,y,z\right)\,.$ (11)

Thus, the set of ordinary differential equations is

$\displaystyle\left\\{\begin{array}[]{l}\dot{x}=-\gamma x\,,\\\
\dot{y}=-\gamma y-\Omega z\,,\\\ \dot{z}=-2\alpha\gamma\left(1+z\right)+\Omega
y\,.\end{array}\right.$ (15)

The steady state solution of this system is given by:

$\displaystyle x^{*}$ $\displaystyle=0\,,$ $\displaystyle y^{*}$
$\displaystyle=\frac{\nicefrac{{\Omega}}{{\gamma}}}{1+\frac{\Omega^{2}}{2\alpha\gamma^{2}}}\,,$
$\displaystyle z^{*}$
$\displaystyle=-\frac{1}{1+\frac{\Omega^{2}}{2\alpha\gamma^{2}}}\,,$ (16)

where the first coordinate, $x$, is decoupled from the dynamics with a stable
fixed point (under the physical assumption $\gamma>0$). The remaining $2D$
first order differential equation for the $y-z$-coordinates is given by,

$\displaystyle\dot{\vec{r}}_{2}$
$\displaystyle=\begin{pmatrix}-\gamma&-\Omega\\\
\Omega&-2\alpha\gamma\end{pmatrix}\vec{r}_{2}+\begin{pmatrix}0\\\
-2\alpha\gamma\end{pmatrix}\,,$ $\displaystyle\vec{r}_{2}$
$\displaystyle\equiv\begin{pmatrix}y\\\ z\end{pmatrix}\,.$ (17)

Their steady state solution are written in terms of the dimensionless variable
$\gamma^{\prime}\equiv\nicefrac{{\gamma}}{{\Omega}}$:

$\displaystyle y^{*}$
$\displaystyle=\frac{1}{\gamma^{\prime}\left(1+\frac{1}{2\alpha\gamma^{{}^{\prime}2}}\right)}=-\frac{z^{*}}{\gamma^{\prime}}\,,$
$\displaystyle z^{*}$
$\displaystyle=-\frac{1}{1+\frac{1}{2\alpha\gamma^{{}^{\prime}2}}}=-\gamma^{\prime}y^{*}\,.$
(18)

These solutions, known as the steady state locus, form the right-half of an
ellipse described by

$\displaystyle\left(\sqrt{\frac{2}{\alpha}}\
y^{*}\right)^{2}+\left(2z^{*}+1\right)^{2}=1\,.$ (19)

The $\theta$ and $p$ representation, utilized in the main text for direct
state preparation, can be derived by setting,
$y^{\ast}=p\cos\left(\theta\right)$ and $z^{\ast}=p\sin\left(\theta\right)$,
yielding, $p=\frac{2\alpha\gamma^{\prime}\sqrt{1+\gamma^{\prime
2}}}{1+2\alpha\gamma^{\prime 2}}$ and
$\theta=-\arctan\left(\gamma^{\prime}\right)$.

The homogeneous part of Eq. (17) can be solved by diagonalization. The
corresponding eigenvalues are given in Eq. ($4$) of the main text and are
presented in Fig. 5. Since the rates are dictated by their real parts, there
exists a bifurcation point,
$\gamma_{b}^{\prime}\equiv\left|{\alpha-\nicefrac{{1}}{{2}}}\right|^{-1}$. The
strong ME exists only at final temperatures larger than the bifurcation point
$\left(\gamma_{\text{f}}>\gamma_{b}\right)$, i.e. the point at which the
degeneracy between eigenvalues breaks and there are distinct fast and slow
relaxation modes.

Figure 5: The real part of the system’s eigenvalues over final temperature,
for $\alpha=1$. The largest eigenvalue, corresponds to the steady state,
zeros. For $H\propto\sigma_{x}$, the equation for $x$ decouples and the set
can be solved analytically. In our case, $\lambda_{x}=-\gamma_{\text{f}}$. The
last couple eigenvalues, $\lambda_{\pm}$, correspond to the dynamics in the
$y-z$ plane and are therefore interesting. A ME can only occur after the
degeneracy is lifted.

The appropriate (normalized) eigenvectors are given by

$\displaystyle\vec{v}_{\pm}$
$\displaystyle=\frac{1}{2\alpha-1}\begin{pmatrix}\frac{1}{\gamma_{\text{f}}^{\prime}\left(\alpha-\nicefrac{{1}}{{2}}\pm\sqrt{\left(\alpha-\nicefrac{{1}}{{2}}\right)^{2}-\nicefrac{{1}}{{\gamma_{\text{f}}^{{}^{\prime}2}}}}\right)}\\\
1\end{pmatrix}\,.$ (20)

The coefficients of these modes are given by
$a_{n}\left(\gamma_{\text{i}},\gamma_{\text{f}}\right)\equiv\vec{u}_{n}(\gamma_{\text{f}})\cdot\vec{r}^{\text{
ss}}(\gamma_{\text{i}})$, where $\vec{u}_{n}$ are left eigenvectors, i.e row
vectors satisfying the equation,

$\vec{u}_{n}\begin{pmatrix}-\gamma&-\Omega\\\
\Omega&-2\alpha\gamma\end{pmatrix}=\lambda_{n}\vec{u}_{n}.$ (21)

Explicitly, we obtain

$\displaystyle a_{\pm}$
$\displaystyle=\frac{\alpha\cdot\left(\alpha-\nicefrac{{1}}{{2}}\right)}{\sqrt{\left(\alpha-\nicefrac{{1}}{{2}}\right)^{2}-\nicefrac{{1}}{{\gamma_{\text{f}}^{{}^{\prime}2}}}}\left(2\alpha+\nicefrac{{1}}{{\gamma_{\text{f}}^{{}^{\prime}2}}}\right)\left(2\alpha+\nicefrac{{1}}{{\gamma_{\text{i}}^{{}^{\prime}2}}}\right)}\left(\frac{\gamma_{\text{f}}^{{}^{\prime}}}{\gamma_{\text{i}}^{{}^{\prime}}}-1\right)\times$
$\displaystyle\left[4\alpha\left(\mp\alpha\pm\nicefrac{{1}}{{2}}+\sqrt{\left(\alpha-\nicefrac{{1}}{{2}}\right)^{2}-\nicefrac{{1}}{{\gamma_{\text{f}}^{{}^{\prime}2}}}}\right)-\frac{2}{\gamma_{\text{i}}^{{}^{\prime}}\gamma_{\text{f}}^{{}^{\prime}}}\left(\mp\alpha\mp\nicefrac{{1}}{{2}}+\sqrt{\left(\alpha-\nicefrac{{1}}{{2}}\right)^{2}-\nicefrac{{1}}{{\gamma_{\text{f}}^{{}^{\prime}2}}}}\right)\pm\frac{2}{\gamma_{\text{f}}^{{}^{\prime}2}}\right]\,.$
(22)

By inserting $\gamma_{\text{i,SME}}$ given in the main text, $a_{-}$ indeed
vanishes.

##  II 2\. The absence of a strong direct-ME

For a strong ME to exist, the coefficient of slow relaxation $a_{-}$ must
vanish. Therefore, in the case of a strong effect the dynamics is along the
fast relaxation eigenvector, $\vec{v}_{+}$, only. The fast relaxation
eigenvector of a given final temperature $\gamma_{\text{f}}^{\prime}$ must
therefore intersect the steady state locus in an additional point,
corresponding to the ideal initial temperature $\gamma_{\text{i,
SME}}^{\prime}$.

For $\alpha>\frac{1}{2}$, both entries of $v_{+}$ are positive (See (20)),
such that $v_{+}$ points to the up-right direction. On the steady state locus,
the temperature is monotonic in $z$ such that states located higher on the
locus necessarily correspond to lower temperatures. Moreover, since
$\gamma_{\text{f}}^{\prime}>\gamma_{\text{b}}^{\prime}$ for a ME to occur, and
the crossing of the steady state locus with the bifurcation-line of
$\alpha>\frac{1}{2}$ is located in the southern-hemisphere of the locus (See
Fig. 6), states located to the right on the locus also correspond to lower
temperatures. The implications are that all possible fast vectors point to
lower initial temperatures, relative to $\gamma_{\text{f}}^{\prime}$.
Accordingly, this system doesn’t exhibit a strong, direct, ME.

Figure 6: The steady state loci in the $y-z$ plane of the Bloch sphere, as
given by Eq. (16), for different values of $\alpha$, as a function of
temperature $\gamma^{\prime}\in(0,20]$. The ME exists only at temperatures
above $\gamma_{\text{b}}$, represented by circles at the crossing with the
bifurcation (purple dashed) line. We observe two bifurcation branches,
however, only the lower one, crosses loci of $\alpha\geq\nicefrac{{1}}{{2}}$,
is valid for a strong ME.

An explicit illustration of the orientation of $\vec{v}_{+}$, with respect to
the steady state locus of $\alpha=1$, is presented in Fig. 7.

Figure 7: The steady state locus for $\alpha=1$ (pink line), inside the Bloch
sphere (grey line), and three kinds of key points. The first is the
bifurcation point, $\gamma_{\text{b}}^{\prime}$ (purple dot), which is always
located in the southern-hemisphere of the locus. Since the strong ME exists
only at final temperatures above this point, the second kind is
$\gamma_{\text{f}}^{\prime}>\gamma_{\text{b}}^{\prime}$ (red dotted line). We
show the fast relaxation vector, $v_{+}$, for various possible final
temperatures (black lines). Each of these vectors intersects with the steady
state locus at a point corresponds to $\gamma_{\text{i, SME}}^{\prime}$, which
is the initial temperature for the strong effect. Blue dotted line: the
appropriate possible initial temperatures, for $\gamma_{f}>\gamma_{b}$. All
possible initial temperatures are smaller then the bifurcation point, i.e. no
strong direct effect. In other words, the absence of a vector pointing to
higher temperatures, e.g. purple line, refutes the existence of a strong
direct effect.

##  III 3\. The existence of a strong inverse-ME

In the main text we prove the existence of a strong inverse-ME by a vanishing
of the coefficient of the slow relaxation mode, $a_{-}$. As complementary
proof to the existence of the strong ME, one simply tracks the evolution of
states, near the final steady state. In the long-time limit, fast relaxation
modes have already decayed. Assuming $a_{-}$ crosses zero continuously - it
changes signs. Thus, states initialized before and after the zero-crossing
will approach the final steady state from opposite directions of $\vec{v}_{-}$
in the $y-z$ plane. An example is shown in Fig. 8.

An experimental observation of the ME involves comparing the dynamics of a
cold and hot initial states, $d_{\text{SS}}^{\text{C}}\left(t\right)$ and
$d_{\text{SS}}^{\text{H}}\left(t\right)$, respectively. Specifically, the
inverse ME is demonstrated by showing that the initially colder system reaches
the final steady state faster, although it starts further away. Namely, at the
beginning of the experiment
$d_{\text{SS}}^{\text{C}}\left(t=0\right)>d_{\text{SS}}^{\text{H}}\left(t=0\right)$,
whereas after some later time, $t_{\text{cross}}$, the colder system surpasses
the hotter such that the distances are reversed, i.e.
$d_{\text{SS}}^{\text{C}}\left(t>t_{\text{cross}}\right)<d_{\text{SS}}^{\text{H}}\left(t>t_{\text{cross}}\right)$
.

Measuring
$d_{\text{SS}}^{\text{C}}\left(t\right)<d_{\text{SS}}^{\text{H}}\left(t\right)$
is inherently challenging, as the two quantities are exponentially decaying.
The existence of a strong-ME is helpful due to two reasons: Early onset of the
crossing between the cold and hot distances from final steady state, and
maximal distance post-crossing. The former allows for short coherence-time of
the experimental apparatus and the latter requires less experimental
repetitions. Both parameters are optimized at
$\gamma_{\text{i}}\sim\gamma_{\text{i,SME}}$. Figure 9 shows this by fixing
$\gamma_{i,H}^{\prime}=0.77$ and plotting the crossing time (left), as well as
the maximal distance to various choices of $\gamma_{i,C}^{\prime}$ after the
two distances cross (right). Clearly, the minimal crossing time, as well as
maximal distance between the two decaying signals appear at
$\gamma_{\text{i}}^{\prime}=\gamma_{\text{i,SME}}^{\prime}$ (vertical gray).

Figure 8: Left: The dynamics of three initial states towards the same final
steady-state $\gamma_{\text{f}}^{\prime}=15$ (red dot). In order to achieve a
strong ME for that final temperature, the corresponding initial temperature is
$\gamma_{\text{SME,i}}^{\prime}\approx 0.07$. Since at
$\gamma_{\text{i}}^{\text{cold}\prime}\approx 0.07$ (blue line) the
coefficient of slow relaxation vanishes, it exhibits an exponentially faster
relaxation, compared, for example, to a colder
$\gamma_{\text{i}}^{\text{colder}\prime}=0.02$ (purple line) and hotter
$\gamma_{\text{i}}^{\text{hot}\prime}\approx 0.74$ (green line) initial
states. Right: The evolution of $\gamma_{\text{i}}^{\text{cold}}$ is along the
direction of $v_{+}$ only, and a zoom-in on the end of the process reveals how
lower and higher initial temperatures approach relaxation from opposite
directions, along $v_{-}$.

Figure 9: Left: The crossing time of states initialized at
$\gamma_{\text{i}}^{\prime}$ compared to $\gamma_{\text{i,h}}^{\prime}=0.77$,
while relaxing to $\gamma_{\text{f}}^{\prime}=15$. Right: The corresponding
maximal distance, post-crossing. The peak indicates the initial state yielding
the most pronounced effect, $\gamma_{\text{i,SME}}^{\prime}$ (grey line), for
which the effect is strong.

  *[ME]: Mpemba effect
  *[GKSL]: Gorini-Kossakowski-Sudarshan-Lindblad
  *[SM]: Supplemental Material


of this class will be referred to as _$\Sigma$ -bundles over $S^{1}$_. Let
$\operatorname{Diff}^{+}(\Sigma)$ be the group of orientation preserving
diffeomorphisms on $\Sigma$. Before formally defining the monodromy
representation, we give an intuitive understanding. The core idea is to treat
$S^{1}\cong{\left.\raisebox{1.99997pt}{$\mathbb{R}$}\middle/\raisebox{-1.99997pt}{$2\pi\mathbb{Z}$}\right.}$
so that $M$ can be ‘chopped’ along the fibre $\phi^{-1}(0)$ to form a bundle
over $[0,2\pi]$. As $[0,2\pi]$ is contractible, then this chopped bundle is
trivialisable. Any trivialisation essentially ‘untwists’ the fibre bundle. We
can then associate to $M$ the diffeomorphism $f\in\operatorname{Diff}^{+}(M)$
which twists and glues $\\{0\\}\times\Sigma$ to $\\{2\pi\\}\times\Sigma$ to
reform $M$. The resulting function $f$ depends on the choice of base point and
trivialisation. Associating to $M$ all possible diffeomorphisms $f$ is the
monodromy representation.

The monodromy representation of a $\Sigma$-bundle over $S^{1}$ can be
constructed through the use of an Ehresmann connection. Every fibre bundle
comes with the so-called vertical subbundle of $TM$ defined by
$\mathcal{V}M=\ker d\phi$. An Ehresmann connection is a smooth choice of
complement to $\mathcal{V}M$ in $TM$ at each point. The complement
$\mathcal{H}M$ is called the horizontal subbundle of $TM$. By definition
$TM=\mathcal{V}M\oplus\mathcal{H}M$. It will be assumed here that
$\mathcal{H}M|_{\partial M}\subset T\partial M$ so that the connection is
always a _complete connection_.

For $\Sigma$-bundles over $S^{1}$, there is a natural bijection between an
Ehresmann connection and vector fields on $M$ which are everywhere transverse
to the fibres of $M$ and tangent to $\partial M$. Consider
$S^{1}={\left.\raisebox{1.99997pt}{$\mathbb{R}$}\middle/\raisebox{-1.99997pt}{$2\pi\mathbb{Z}$}\right.}$
parameterised by $t$. If $\mathcal{H}M$ is given by a connection, then there
is a unique vector field $Y$, tangent to $\partial M$, defined by $d\phi(Y)=1$
and $Y_{x}\in\mathcal{H}_{x}M$ for all $x\in M$. Conversely, given a vector
field $Y$ such that $d\phi(Y)=1$ and $Y|_{\partial M}\in\Gamma(T\partial M)$,
we can define at each point $x\in M$ the horizontal subspace by
$\mathcal{H}_{x}M=\text{span}(Y_{x})$. Due to this correspondence, we will
henceforth define a connection on $M$ as a choice of vector field $Y$ such
that $d\phi(Y)=1$ and $Y|_{\partial M}\in\Gamma(T\partial M)$.

Every $\Sigma$-bundle over $S^{1}$ admits a connection.

######  Lemma E.1.

Every $\Sigma$-bundle over $S^{1}$ admits a connection.

###### Proof.

Let $\gamma:\mathbb{R}\to S^{1}$ be a covering map of $S^{1}$ and denote by
$\gamma^{*}M$ the pull-back bundle of $M$, defined as a subset of
$\mathbb{R}\times M$ by

$\gamma^{*}M=\\{(t,x)\in\mathbb{R}\times M\,|\,\gamma(t)\in\phi(x)\\}$

and with projections $\gamma^{*}\phi:\mathbb{R}\times M\to\mathbb{R}$ and
$\pi_{M}:\mathbb{R}\times M\to M$. Since the base space of $\gamma^{*}M$ is
contractible then $\gamma^{*}M$ is trivial. Let
$\tau:\mathbb{R}\times\Sigma\to\gamma^{*}M$ be a trivialisation such that
$(\gamma^{*}\phi)\circ\tau(t,x)=t$. Denote by $\partial_{t}$ the vector field
on $\mathbb{R}\times\Sigma$ with flow $\Phi_{s}(t,x)=(t+s,x)$. Then
$Y=(\pi_{M})_{*}\tau_{*}\partial_{t}$ is a well-defined vector field on $M$
satisfying $d\pi(Y)=1$, that is, $Y$ is a connection. ∎

In essence, a connection describes how to move from fibre to fibre in $M$. The
flow $\Phi:\mathbb{R}\times M\to M$ of a connection $Y$ _parallel transports_
points on any fibre $\Sigma_{t_{1}}:=\phi^{-1}(t_{1})$ to another
$\Sigma_{t_{2}}:=\phi^{-1}(t_{2})$ with $t_{2}>t_{1}$ via the map
$\Phi_{t_{1},t_{2}}:=\Phi(t_{2}-t_{1},\cdot):\Sigma_{t_{1}}\to\Sigma_{t_{2}}$.
In particular $\Phi_{t,t+2\pi}$ transports any fibre $\Sigma_{t}$ once around
the bundle and back to itself. As each fibre is diffeomorphic to $\Sigma$,
then local trivialisations give for each $t\in S^{1}$ a diffeomorphism
$h_{t}:\Sigma\to\Sigma_{t}$. Through these diffeomorphisms, it is possible to
consider
$\overline{\Phi}_{t_{1},t_{2}}:=h_{t_{2}}^{-1}\circ\Phi_{t_{1},t_{2}}\circ
h_{t_{1}}\in\operatorname{Diff}^{+}(\Sigma)$. Note that the diffeomorphisms
$h_{t}$ need not be smooth as a function of $t$; in fact, if the fibre bundle
is non-trivial, then necessarily they will not be.

######  Definition E.2.

The map $\overline{\Phi}_{0,2\pi}\in\operatorname{Diff}^{+}(\Sigma)$ is
denoted the _monodromy map with respect to $Y$_ of the fibre bundle
$(M,S^{1},\phi,\Sigma)$.

The monodromy map with respect to $Y$ has been defined with respect to a
specific choice of base point $t\in S^{1}$, namely $t=0$, through the
identification
$S^{1}={\left.\raisebox{1.99997pt}{$\mathbb{R}$}\middle/\raisebox{-1.99997pt}{$2\pi\mathbb{Z}$}\right.}$.
We now explore the consequence of changing the base from which the monodromy
map is defined. Let $t\in S^{1}$ be a base point. Then the monodromy map based
at $t$, $\overline{\Phi}_{t,t+2\pi}$, can be computed from the monodromy map
of $M$ using the semi-group property of flows,

$\overline{\Phi}_{t,t+2\pi}=\overline{\Phi}_{0,t}\circ\overline{\Phi}_{0,2\pi}\circ\overline{\Phi}_{0,t}^{-1}.$

It follows that changing the base from which the monodromy map is defined
amounts to conjugation by
$\overline{\Phi}_{0,t}\in\operatorname{Diff}^{+}(\Sigma)$. Hence, the
conjugacy class of $\overline{\Phi}_{0,2\pi}$ in
$\operatorname{Diff}^{+}(\Sigma)$ is an invariant of the bundle. In fact, as
we prove in what follows, it is a _complete_ invariant of $\Sigma$-bundles
over $S^{1}$ with a distinguished Ehresmann connection.

######  Definition E.3.

Let $\phi:M\to S^{1}$ and $\phi^{\prime}:M^{\prime}\to S^{1}$ be two
$\Sigma$-bundles with respective connections $Y,Y^{\prime}$. A diffeomorphism
$\Psi:M\to M^{\prime}$ is said to be an _isomorphism of bundles with
connection_ provided $\Psi$ is a bundle isomorphism and
$\Psi_{*}Y=Y^{\prime}$.

######  Theorem E.4 (Morita [50]).

The correspondence which sends any (orientable) $\Sigma$-bundle over $S^{1}$
with a given connection $Y$ to its monodromy map, induces a bijection

$\displaystyle\\{\text{isomorphism classes of $\Sigma$-bundles over $S^{1}$
with connection}\\}$ $\displaystyle\hskip 216.81pt\cong\\{\text{conjugacy
classes of $\operatorname{Diff}^{+}(\Sigma)$}\\}$

One direction of the theorem is clear; if there is an isomorphism $\Psi$ of
bundles with connection then the flow of the connections are $\Psi$-related by
definition. In consequence, the monodromy maps must be conjugate. The converse
is less clear. First, to any conjugacy class of diffeomorphisms, it must be
shown that there is a $\Sigma$-bundle over $S^{1}$ with a connection $Y$ whose
monodromy map is in the given conjugacy class. Second, if two fibre bundles
have conjugate monodromy maps, then they need to be isomorphic with
connection. The first requirement can be assured through a _mapping torus._

######  Definition E.5.

For any $f\in\operatorname{Diff}^{+}(\Sigma)$ the _mapping torus_ $\Sigma_{f}$
associated to $f$ is given by

$\Sigma_{f}:=\frac{\mathbb{R}\times\Sigma}{(t,f(x))\sim(t-2\pi,x)}.$

A mapping torus is naturally a $\Sigma$-bundle over $S^{1}$ where the
projection $\phi:\Sigma_{f}\to
S^{1}={\left.\raisebox{1.99997pt}{$\mathbb{R}$}\middle/\raisebox{-1.99997pt}{$2\pi\mathbb{Z}$}\right.}$
is defined by $[s,x]\mapsto[s]$ with $[s,x]$ denoting the equivalence class of
$(s,x)\in\mathbb{R}\times\Sigma$ under $(s,f(x))\sim(s-2\pi,x)$ and $[s]$ the
equivalence class under $s\sim s-2\pi$. Every mapping torus has a connection
with monodromy map $f$. Indeed, on the space $\mathbb{R}\times\Sigma$ there is
a flow $\Phi_{s}(t,x)=(t+s,x)$ with associated vector field $\partial_{t}$.
Using the quotient map $q:\mathbb{R}\times\Sigma\to\Sigma_{f}$, we obtain a
well-defined vector field $Y=q_{*}\partial_{t}$ on $\Sigma_{f}$. Moreover,
$d\phi(Y)=1$ and, if $\Sigma$ has boundary, then $Y$ is tangent to
$\partial\Sigma_{f}$ so that $Y$ is a complete connection on $\Sigma_{f}$.
Finally, observe that the monodromy map of $Y$ is exactly $f$, as desired.

It can be shown that every $\Sigma$-bundle over $S^{1}$ with monodromy map $f$
is isomorphic to the mapping torus $\Sigma_{f}$.

######  Lemma E.6.

Every $\Sigma$-bundle over $S^{1}$ with a connection $Y$ producing a monodromy
map $f$ is isomorphic to $\Sigma_{f}$.

###### Proof.

Define $\Psi:M\to\Sigma_{f}$ by
$\Psi(p)=\left[\phi(p),h_{0}^{-1}\circ\Phi_{0,\phi(p)}^{-1}(p)\right]$.
Observe that $\Psi$ is smooth, well-defined, and preserves fibres. Indeed, to
be well-defined we require $[\phi(p)+2\pi,f\circ
h_{0}^{-1}\circ\Phi_{0,\phi(p)+2\pi}^{-1}(p)]=[\phi(p),h_{0}^{-1}\circ\Phi_{0,\phi(p)}^{-1}(p)]$.
Now, $[\phi(p)]=[\phi(p)+2\pi]$ and

$h_{0}^{-1}\circ\Phi_{0,\phi(p)}^{-1}(p)=h_{0}^{-1}\circ\Phi_{0,2\pi}\circ\Phi_{0,\phi(p)+2\pi}^{-1}(p)=\bar{\Phi}_{0,2\pi}\circ
h_{0}^{-1}\circ\Phi^{-1}_{0,\phi(p)+2\pi}(p)=f\circ
h_{0}^{-1}\circ\Phi^{-1}_{0,\phi(p)+2\pi}(p).$

It follows that $\Psi$ is well-defined. ∎

We now show the second claim required for the proof of Theorem E.4, namely,
that two fibre bundles with conjugate monodromy maps are isomorphic with
connection. Observe that if $\Sigma_{f},\Sigma_{f^{\prime}}$ are two mapping
tori and $f,f^{\prime}$ are conjugate, that is there exists
$g\in\operatorname{Diff}^{+}(\Sigma)$ such that $g\circ f=f^{\prime}\circ g$,
then $\Psi([t,x])=[t,g(x)]$ is a bundle isomorphism. This completes the proof
of Theorem E.4. Hence, the monodromy map is a complete invariant of
$\Sigma$-bundles over $S^{1}$ with a connection.

In light of the fact that every $\Sigma$-bundle over $S^{1}$ admits a
connection (Lemma E.1), it is natural to view the monodromy map of a
$\Sigma$-bundle as an intrinsic invariant of the bundle itself, and not
associated to a specific connection. This intrinsic invariant is called the
_monodromy representation_ of $M$. Two different choices in connection give
rise to two different monodromy maps $f,f^{\prime}$. It follows that, in order
to make sure the monodromy representation is well-defined, we need to
understand how a change in choice of connection effects the monodromy map.

As $M$ is isomorphic to two different mapping tori
$\Sigma_{f},\Sigma_{f^{\prime}}$, finding a condition on the monodromy
representation so that it is well-defined amounts to determining a condition
on $f,f^{\prime}$ that guarantees that the two mapping tori are isomorphic as
bundles (but not necessarily as bundles with a connection).

######  Definition E.7.

Two functions $f,f^{\prime}\in\operatorname{Diff}^{+}(\Sigma)$ are _isotopic_
if there exists a smooth path $f_{t}\in\operatorname{Diff}^{+}(\Sigma)$ such
that $f_{0}=f,f_{1}=f^{\prime}$. Denote by $f\sim f^{\prime}$ the equivalence
relation by isotopy between $f,f^{\prime}\in\operatorname{Diff}^{+}(\Sigma)$.

The _mapping class group_ $\mathcal{M}(\Sigma)$ is the group of isotopy
classes in $\operatorname{Diff}^{+}(\Sigma)$. Equivalently, we have

$\mathcal{M}(\Sigma):={\left.\raisebox{1.99997pt}{$\operatorname{Diff}^{+}(\Sigma)$}\middle/\raisebox{-1.99997pt}{$\sim$}\right.}=\pi_{0}(\operatorname{Diff}^{+}(\Sigma))$

Two elements $[f],[f^{\prime}]\in\mathcal{M}(\Sigma)$ are considered
_conjugate_ if there exists $[g]\in\mathcal{M}(\Sigma)$ such that
$[f^{\prime}]=[g][f][g]^{-1}$. Equivalently, $[f^{\prime}]=[g\circ f\circ
g^{-1}]$.

######  Proposition E.8.

Two mapping tori $\Sigma_{f}$ and $\Sigma_{f^{\prime}}$ are bundle isomorphic
if and only if $f$ and $f^{\prime}$ are representatives of conjugate elements
$[f],[f^{\prime}]\in\mathcal{M}(\Sigma)$.

###### Proof.

$\implies$: Suppose that there exists a bundle isomorphism
$\Psi:\Sigma_{f}\to\Sigma_{f^{\prime}}$. For any base point $b\in S^{1}$ ,
denote by $\Psi_{b}$ the induced map from $\Psi$ of the fibre over $b$ in
$\Sigma_{f}$, denoted $\Sigma_{f,b}$, to the fibre over $b$ in
$\Sigma_{f^{\prime}}$, denoted $\Sigma_{f^{\prime},b}$. Hence
$\Psi_{b}:\Sigma_{f,b}\to\Sigma_{f^{\prime},b}$ and we can consider
$\Psi_{b}\in\operatorname{Diff}^{+}(\Sigma)$. We will show that
$[f]=[\Psi_{0}^{-1}\circ f^{\prime}\circ\Psi_{0}]$. In order to do this we
need to find an isotopy $f_{s}$ such that $f_{0}=f$ and
$f_{1}=\Psi_{0}^{-1}\circ f^{\prime}\circ\Psi_{0}$.

Let $Y_{f},Y_{f^{\prime}}$ be connections on respectively
$\Sigma_{f},\Sigma_{f^{\prime}}$ with monodromy $f,f^{\prime}$ and denote
their respective flows $\Phi^{f},\Phi^{f^{\prime}}$. Consider the map $f_{s}$
defined as the composition

$f_{s}:\Sigma_{f,0}\xrightarrow{\Psi_{0}}\Sigma_{f^{\prime},0}\xrightarrow{\Phi^{f^{\prime}}(s,\cdot)}\Sigma_{f^{\prime},s}\xrightarrow{\Psi_{s}^{-1}}\Sigma_{f,s}\xrightarrow{\Phi^{f}(1-s,\cdot)}\Sigma_{f,0}.$

Then $f_{0}=f$ and $f_{1}=\Phi_{0}^{-1}\circ f^{\prime}\circ\Phi_{0}$ as
desired.

$\impliedby$: We have already established that if $f$ is conjugate to
$f^{\prime}$ the mapping tori are isomorphic. It remains to show if
$[f]=[f^{\prime}]$ then $\Sigma_{f}$ is bundle isomorphic to
$\Sigma_{f^{\prime}}$. Indeed, let $f_{s}$ be the isotopy such that $f_{0}=f$
and $f_{1}=f^{\prime}$. Then the isomorphism is given by

$\Psi:[t,x]\mapsto[t,f_{1-t}\circ f^{-1}(x)]$

To check it is an isomorphism we need to make sure it is well-defined, that
is, $\Psi([2\pi,x])=\Psi([0,f(x)])$. We have $\Psi([2\pi,x])=[2\pi,f_{0}\circ
f^{-1}(x)]=[1,x]$ and $\Psi([0,f(x)])=[0,f_{1}\circ f^{-1}\circ
f(x)]=[0,f^{\prime}(x)]=[1,x]=\Psi(1,x)$ as desired. ∎

An immediate consequence of Proposition E.8 is that the monodromy
representation of $M$ is a complete invariant of bundles up to isomorphism.

######  Theorem E.9.

Let $\Sigma$ be a smooth, compact, orientable manifold (possibly with
boundary). Then the correspondence, which sends any $\Sigma$-bundle over
$S^{1}$ to its monodromy representation, induces a bijection

$\\{\text{isomorphism classes of {\hbox{\Sigma}}-bundles over
{\hbox{S^1}}}\\}\cong\\{\text{conjugacy classes of
{\hbox{\mathcal{M}(\Sigma)}}}\\}.$

###  E.2 Properties of the Mapping Class Group of Surfaces

Due to Theorem E.9, two $\Sigma$-bundles over $S^{1}$ are isomorphic if and
only if they have monodromy representations that are conjugate in
$\mathcal{M}(\Sigma)$. Consequently, by studying the conjugacy classes of
$\mathcal{M}(\Sigma)$ for a particular choice of $\Sigma$, all the possible
$\Sigma$-bundles over $S^{1}$ up to isomorphism can be determined. In
particular, if $\mathcal{M}(\Sigma)$ is trivial then all $\Sigma$-bundles are
isomorphic to the trivial bundle.

The mapping class groups for different surfaces are well understood. Thorough
accounts are given in [51, 52, 53]. However, most of the work has been
developed for the mapping class group of diffeomorphisms which leave the
boundary of the surface _point-wise_ invariant. We are interested in
diffeomorphisms which only leave the boundary _set-wise_ invariant. In this
section we review some important elements of the theory of mapping class
groups, converting where necessary proofs of well known results in the theory
for point-wise invariant boundary to the case of set-wise invariant boundary.

####  E.2.1 Curves and Dehn Twists

A _simple closed curve_ in $\Sigma$ is a smooth embedding $c:S^{1}\to\Sigma$
which does not intersect the boundary. The curve $c$ has an orientation and
the curve with the same image but opposite orientation will be denoted
$c^{-1}$. We will follow [51] by denoting a simple closed curve $c$ as
_essential_ if it does not bound a disk or it is isotopic to a boundary
component.222Note that this definition differs from that in [54].

We will write $a\simeq b$ if the two simple closed curves $a,b$ are isotopic.
The _index of intersection_ of two simple closed curves $a$ and $b$ is

$I(a,b)=\min\\{|a^{\prime}\cap b^{\prime}|\,:\,a^{\prime}\simeq
a,b^{\prime}\simeq b\\}.$

Through deformations of $a^{\prime}$ or $b^{\prime}$ it is always possible to
get a finite intersection so that $I(a,b)$ is well-defined. See [55] for more
details.

We will use the following propositions from [54] about simple closed curves.

######  Proposition E.10.

[54, Prop. 3.4] Suppose that $a_{1},\dots,a_{m}$ are essential simple closed
curves such that, if $i\neq j$, then $a_{i}\cap a_{j}=\emptyset$ and $a_{i}$
is neither isotopic to $a_{j}$ or $a_{j}^{-1}$. Then, for each $i$, there
exists a simple closed curve $b_{i}$ such that $a_{j}\cap b_{i}=\emptyset$ if
$i\neq j$, and $|a_{i}\cap b_{i}|=I(a_{i},b_{i})>0$.

######  Proposition E.11.

[54, Prop. 3.10] Let $a_{1}\dots,a_{m}$ and $b_{1},\dots,b_{m}$ be simple
closed curves none of which are null-homotopic in $\Sigma$. Suppose that for
$i\neq j$:

  1. 1.

$a_{i}\cap a_{j}=\emptyset$ and $b_{i}\cap b_{j}=\emptyset$.

  2. 2.

$a_{i}$ is not isotopic to $a_{j}^{\pm 1}$ and $b_{i}$ is not isotopic to
$b_{j}^{\pm 1}$.

  3. 3.

$a_{i}$ is isotopic to $b_{i}$.

Then there exists an isotopy of diffeomorphisms $h_{t}$ on $\Sigma$ such that
$h_{0}=Id$ and $h_{1}\circ a_{i}=b_{i}$ for all $i=1,\dots,p$.

The Dehn-Likorish Twist Theorem demonstrates that the mapping class group of
any closed surface (compact and without boundary) can be finitely presented by
so called Dehn twists. Even in the case of surfaces with boundary, Dehn twists
play a crucial role of understanding the mapping class group.

Let $A_{1,2}\subset\mathbb{C}$ be the annulus

$A_{1,2}=\\{z\in\mathbb{C}\,:\,1\leq|z|\leq 2\\}.$

Define the _standard Dehn twist_ on $A_{1,2}$ as the transformation

$\tau:z\mapsto e^{2\pi i(|z|)}z.$

Note that $\tau$ is the identity on each boundary component and ‘twists’ the
outer boundary one rotation anticlockwise with respect to the inner boundary.
Finally, $\tau$ is orientation preserving. In other words $\tau$ is in the
group of orientation preserving diffeomorphisms of $A_{1,2}$ which leave the
boundary point-wise invariant.

Let $a$ be a simple closed curve in a surface $\Sigma$. Let $U$ be a regular
neighbourhood of $a$. By regular neighbourhood, we mean a closed embedded
annulus $U$ of $\Sigma$ such that, if $i:U\to\Sigma$ is the inclusion, then
$i^{*}a$ is a simple closed curve in $U$. Choose an orientation preserving
diffeomorphism $\phi:A_{1,2}\to U$. We obtain a homeomorphism $\tau_{a}$ as
follows:

$\tau_{a}(x)=\begin{cases}\phi\circ\tau\circ\phi^{-1}(x)&\text{if $x\in U$}\\\
x&\text{if $x\in\Sigma\setminus U$ }.\end{cases}$

In other words, $\tau_{a}$ performs the standard Dehn twists $\tau$ on the
annulus $U$ and fixes every point outside of $U$. Denote the isotopy class of
$\tau_{a}$, namely $T_{a}:=[\tau_{a}]$, the _Dehn twist about $a$_.

We will state some key properties of Dehn twists. The first describes how a
Dehn twist changes index of intersection of two simple closed curves.

######  Proposition E.12.

[54, Prop. 3.3] Let $a,b:S^{1}\to\Sigma$ be two simple closed curves and $n$
any integer. Then

$I(\tau_{a}^{n}\circ b,b)=|n|I(a,b)^{2}.$

The second proposition is a modification of [54, Prop. 3.8] to the case of
set-wise invariant boundary.

######  Proposition E.13.

[54, Prop. 3.8] Suppose $a_{1},\dots,a_{m}:S^{1}\to\Sigma$ are essential
simple closed curves which are pairwise disjoint, and no curve $a_{i}$ is
isotopic to $a_{j}$ or $a_{j}^{-1}$, $i\neq j$. Consider the function

$h:\mathbb{Z}^{m}\to\mathcal{M}(\Sigma)$

defined by

$h(n_{1},\dots,n_{m})=T_{a_{1}}^{n_{1}}\dots T_{a_{m}}^{n_{m}}.$

Then $h$ is an injective homomorphism.

###### Proof.

First, because the curves $a_{1},\dots,a_{m}$ are pairwise disjoint, the Dehn
twists commute, and $h$ is a homomorphism. 333To see they commute note that if
they are disjoint we can find regular neighbourhoods $N_{i}$ of the respective
$a_{i}$ that are disjoint. Then the Dehn twists for each $a_{i}$ is the
identity out of the respective $N_{i}$, so the twisting of any two commutes.
To see that it is injective, suppose that $T_{a_{1}}^{n_{1}}\cdots
T_{a_{p}}^{n_{p}}=1$. Fix an index $i$. Proposition E.10 supplies a simple
closed curve $b$ in $\Sigma$ disjoint from $a_{j},j\neq i$ such that

$|a_{i}\cap b|=I(a_{i},b)>0.$

We calculate, using commutativity of the twists, $b\cap a_{j}=\emptyset$,
$i\neq j$, and Proposition E.12, that

$0=I(b,b)=I(\tau_{a_{1}}^{n_{1}}\circ\dots\circ\tau_{a_{m}}^{n_{m}}\circ
b,b)=I(\tau_{a_{i}}^{n_{i}}\circ b,b)=|n_{i}|I(a_{i},b)^{2}.$

Therefore $n_{i}=0$ and we have shown $h$ is injective. ∎

As an immediate consequence of Proposition E.13 we have the following.

######  Corollary E.14.

[54, Cor. 3.9] If $a:S^{1}\to\Sigma$ is an essential simple closed curve, then
the Dehn twist about $a$ has infinite order in $\mathcal{M}(\Sigma)$.

Note that, except for the cases of a disk, annulus, or pair of pants (a disk
with two open disks removed), every other compact surface has at least one
essential simple closed curve. For these compact surfaces in particular,
Corollary E.14 implies that $\mathcal{M}(\Sigma)$ contains a subgroup
isomorphic to $\mathbb{Z}$. As $\mathbb{Z}$ is abelian, the conjugacy classes
are singletons. That is, there are infinitely many conjugacy classes in
$\mathcal{M}(\Sigma)$. By Theorem E.9 there is thus infinitely many
isomorphism classes of $\Sigma$-bundles over $S^{1}$ whenever $\Sigma$ is a
compact surface that is not a disk, annulus, or pair of pants.

In the case that $\Sigma$ is a disk, annulus, or pair of pants, the mapping
class groups are finite. This follows from the following well established
result (see [51, 53, 56]).

######  Theorem E.15.

The following isomorphisms hold.

  1. 1.

If $\Sigma$ is a disk $D^{2}$ then $\mathcal{M}(D^{2})\cong 1$.

  2. 2.

If $\Sigma$ is an annulus $A_{1,2}$ then $\mathcal{M}(A)\cong 1$.

  3. 3.

If $\Sigma$ is a pair of pants $P$ then $\mathcal{M}(P)\cong\mathbb{Z}_{2}$.