# Multi-spin counter-diabatic driving in many-body quantum Otto refrigerators

Andreas Hartmann<EMAIL_ADDRESS>Institut für Theoretische
Physik, Universität Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria
Victor Mukherjee  Department of Physical Sciences, IISER Berhampur, Berhampur
760010, India  Glen Bigan Mbeng<EMAIL_ADDRESS>Institut für
Theoretische Physik, Universität Innsbruck, Technikerstraße 21a, A-6020
Innsbruck, Austria  Wolfgang Niedenzu  Wolfgang Lechner  Institut für
Theoretische Physik, Universität Innsbruck, Technikerstraße 21a, A-6020
Innsbruck, Austria

(December 21, 2020)

###### Abstract

Quantum refrigerators pump heat from a cold to a hot reservoir. In the few-
particle regime, counter-diabatic (CD) driving of, originally adiabatic, work-
exchange strokes is a promising candidate to overcome the bottleneck of
vanishing cooling power. Here, we present a finite-time many-body quantum
refrigerator that yields finite cooling power at high coefficient of
performance, that considerably outperforms its non-adiabatic counterpart. We
employ multi-spin CD driving and numerically investigate the scaling behavior
of the refrigeration performance with system size. We further prove that
optimal refrigeration via the exact CD protocol is a catalytic process.

##  1 Introduction

Heat engines and refrigerators are a cornerstone of modern physics and
indispensable in today’s society [1]. Unravelling their fundamental laws in
the few-particle regime has lead to the study of so-called quantum heat
engines and refrigerators [2, 3, 4, 5, 6, 7, 8, 9]. With the recent progress
in controling quantum systems [11, 12, 13], such quantum heat engines could
already be experimentally realized using various single-body quantum working
media (WM) [14, 15, 16, 17, 18, 19]. Whereas heat engines convert thermal
energy into work, their counterparts, namely refrigerators, cool down a cold
bath by pumping heat from the cold to the hot reservoir, thereby consuming
work [20, 21, 22, 23, 24, 7, 8, 25]. The maximum coefficient of performance
(CoP) of refrigerators is limited by the Carnot CoP [26]. For these infinitely
long (adiabatic) cycle times, the corresponding cooling power naturally
converges to zero. A natural question thus arises whether such quantum
refrigerators can be driven in finite time, yet produce a finite cooling
power.

So called shortcuts to adiabaticity (STA) [27, 28, 29, 30] are a promising
candidate to overcome this fundamental bottleneck, due to minimizing quantum
friction [31, 32, 33] during the work-exchange strokes. These STA methods [34,
35, 36, 37, 38, 39] including counter-diabatic (CD) driving [40, 39, 41, 42,
43, 44] where an additional CD Hamiltonian is applied to suppress any
transitions between the system’s eigenstates during the Hamiltonian’s
dynamics, have recently been applied in the field of quantum thermodynamics to
enhance the performance of quantum heat engines [45, 46, 47, 48, 49] and
refrigerators [50, 51] using _single-body_ quantum WM. For the latter, _exact_
local CD terms can be found analytically [36, 38].

In general, identifying the _exact_ CD term requires _a priori knowledge of
the system’s eigenstates_ at all times during the Hamiltonian’s dynamics and
which is numerically and experimentally impracticable for _many-body_ WM. With
this challenge in mind, Sels and Polkovnikov [42] have developed a variational
principle where _approximate_ multi-spin CD terms can be found. Based on this
method, a quantum heat engine using a many-body quantum WM and local 1-spin CD
Hamiltonian could be implemented [52]. However, the latter are poor
approximations and can only efficiently speed up an Otto cycle with weakly
coupled WM. The question whether a cycle with WM with complex all-to-all
connectivity – that provides versatile applications in state-of-the-art
experimental setups in the field of quantum computation and simulation, cf.
Refs. [53, 54, 55, 56] – can be sped-up is thus still open.

In this work, we present a _finite-time_ many-body quantum refrigerator (QR)
with finite cooling power using additional approximate _multi-spin_ CD terms.
The WM is described by an all-to-all connected Ising spin system and can
efficiently be driven using non-local, yet well approximated $p$-spin CD
terms. We numerically demonstrate a large enhancement in cooling power and CoP
along with improved scaling behaviour of the cooling power with the system
size for the sped-up QR compared to its original non-adiabatic and traditional
$1$-spin CD counterparts. For the QR with single-body quantum WM, we find an
analytical expression for the CoP. For the many-body WM, we provide an
analytical proof that _exact_ CD driving implies a zero work component of the
additional external control device and thus is a catalytic process.
Remarkably, the latter fully assists the piston to run the QR in this case,
mimicking the adiabatic quantum cycle, yet in finite time.

This work is structured as follows: In Sec. 2, we introduce the quantum Otto
refrigerator using a many-body spin system as its quantum WM and present
multi-spin local CD driving. In Sec. 3, we numerically analyze the performance
of the corresponding refrigerators and conclude our results in Sec. 4 while
giving an outlook on future research.

##  2 Methods

###  2.1 Quantum Otto refrigerator

Quantum Otto refrigerators cyclically pump heat from a cold to a hot reservoir
by consuming work. Its corresponding four-stroke quantum Otto cycle [8]
consists of two heat-exchange strokes – where the quantum WM is alternatingly
coupled to two heat baths – and two work-exchange strokes.

Figure 1: Entropy-working parameter diagram of quantum Otto refrigerator. The
quantum Otto cycle consists of two adiabatic (1 and 3) and two thermal strokes
(2 and 4). At the end of the cycle, the heat $Q_{\mathrm{c}}^{\mathrm{ad}}$ is
pumped from the cold bath at temperature $T_{\mathrm{c}}$ into the hot bath at
temperature $T_{\mathrm{h}}$ by consuming the work
$W^{1}_{\mathrm{ad}}+W^{3}_{\mathrm{ad}}>0$.

The four strokes are (cf. Fig. 1):

  1. 1.

_Adiabatic compression_ ($A\to B$): Initially, at cold temperature
$T_{\mathrm{c}}=1/\beta_{\mathrm{c}}$, Hamiltonian
$H_{0}(\lambda_{\mathrm{c}})$ with working parameter
$\lambda_{\mathrm{c}}\coloneqq\lambda(t=0)$ and in the thermal state
$\rho_{A}=e^{-H_{0}(\lambda_{\mathrm{c}})/T_{\mathrm{c}}}/\operatorname{Tr}[e^{-H_{0}(\lambda_{\mathrm{c}})/T_{\mathrm{c}}}]$,
the WM is changed adiabatically during the first isentropic stroke of duration
$\tau_{1}$ with final Hamiltonian $H_{0}(\lambda_{\mathrm{h}})$ and mixed
state $\rho_{B}$ with $\lambda_{\mathrm{h}}\coloneqq\lambda(t=\tau)$.

  2. 2.

_Hot isochore_ ($B\to C$): The WM is brought in contact with the hot thermal
bath at temperature $T_{\mathrm{h}}$ until it equilibrates to the thermal
state
$\rho_{C}=e^{-H_{0}(\lambda_{\mathrm{h}})/T_{\mathrm{h}}}/\operatorname{Tr}[e^{-H_{0}(\lambda_{\mathrm{h}})/T_{\mathrm{h}}}]$.
During this heat-exchange stroke of duration $\tau_{2}$, the heat
$Q_{\mathrm{h}}$ is imparted to the hot bath and the WM’s Hamiltonian
$H_{0}(\lambda_{\mathrm{h}})$ remains unchanged.

  3. 3.

_Adiabatic expansion_ ($C\to D$): The Hamiltonian
$H_{0}(\lambda_{\mathrm{h}})$ changes back to $H_{0}(\lambda_{\mathrm{c}})$
during the stroke duration $\tau_{3}$ until the WM attains the state
$\rho_{D}$.

  4. 4.

_Cold isochore_ ($D\to A$): The WM is brought in contact with the cold bath at
temperature $T_{\mathrm{c}}$ and working parameter $\lambda_{\mathrm{c}}$ and
cools down during the duration $\tau_{4}$ to its originally initial thermal
state $\rho_{A}$.

During one cycle, the WM extracts the heat
$Q_{\mathrm{c}}^{\mathrm{ad}}\coloneqq\langle
H_{0}(\lambda_{\mathrm{c}})\rangle_{\rho_{A}}-\langle
H_{0}(\lambda_{\mathrm{c}})\rangle_{\rho_{D}}>0$ from the cold thermal bath,
thereby consumes the work $W^{1+3}_{\mathrm{ad}}\coloneqq
W^{1}_{\mathrm{ad}}+W^{3}_{\mathrm{ad}}>0$ (here, $W^{1}_{\mathrm{ad}}=\langle
H_{0}(\lambda_{\mathrm{h}})\rangle_{\rho_{B}}-\langle
H_{0}(\lambda_{\mathrm{c}})\rangle_{\rho_{A}}$ and
$W^{3}_{\mathrm{ad}}=\langle
H_{0}(\lambda_{\mathrm{c}})\rangle_{\rho_{D}}-\langle
H_{0}(\lambda_{\mathrm{h}})\rangle_{\rho_{C}}$). A positive (negative) sign of
the work components corresponds to work performed on (extracted from) the WM.
Analogously, a positive (negative) sign of the heat corresponds to heat
extracted from (imparted to) the thermal bath.

The cooling power is defined as the pumped heat $Q_{\mathrm{c}}^{\mathrm{ad}}$
over the total cycle time $\tau_{\mathrm{cycle}}=\sum_{l=1}^{4}\tau_{l}$,
i.e.,

$\mathcal{J}_{\mathrm{ad}}=\dfrac{\mathrm{pumped\,heat}}{\mathrm{cycle\,duration}}=\dfrac{Q_{\mathrm{c}}^{\mathrm{ad}}}{\tau_{\mathrm{cycle}}}.$
(1)

The CoP of the Otto cycle is defined as the heat
$Q_{\mathrm{c}}^{\mathrm{ad}}$ pumped from the cold bath over the consumed
work $W_{\mathrm{ad}}^{1}+W_{\mathrm{ad}}^{3}$, i.e.,

$\epsilon_{\mathrm{ad}}=\dfrac{\mathrm{pumped\,heat}}{\mathrm{consumed\,work}}=\dfrac{Q_{\mathrm{c}}^{\mathrm{ad}}}{W_{\mathrm{ad}}^{1}+W_{\mathrm{ad}}^{3}}.$
(2)

In the adiabatic limit – where the isentropic strokes with durations
$\tau_{1}$ and $\tau_{3}$ are infinitely long – the cooling power of these
_adiabatic_ quantum refrigerators goes to zero, i.e.,
$\underset{\tau_{1},\tau_{3}\to\infty}{\lim}\mathcal{J}_{\mathrm{ad}}\to 0$.
To overcome this bottleneck, one can implement so called STA techniques for
the two work-exchange strokes.

###  2.2 Quantum working medium

As our quantum WM, we consider an all-to-all connected Ising spin chain with
Hamiltonian

$\displaystyle H_{0}(t)$
$\displaystyle=-[1-\vartheta(t)]\sum_{j=1}^{N}h_{j}\sigma_{j}^{x}$
$\displaystyle-\vartheta(t)\left[\sum_{j=1}^{N}b_{j}\sigma_{j}^{z}+\sum_{j=1}^{N}\sum_{k<j}J_{jk}\sigma_{j}^{z}\sigma_{k}^{z}\right]$
(3)

where $h_{j}$ and $b_{j}$, respectively, are the time-dependent transverse and
longitudinal magnetic field strengths at site $j$ and $J_{jk}$ the interaction
strength between spins at sites $j$ and $k$. $\vartheta(t)$ is a continuous
function that fulfills $\vartheta(t=0)=0$ and $\vartheta(t=\tau_{1})=1$. For
the second isentropic stroke the initial and final values of the function
$\vartheta(t)$ are interchanged.

Throughout this work, we parametrize the working parameters
$\lambda_{\mathrm{c}}$ and $\lambda_{\mathrm{h}}$ with the magnetic fields and
interaction strengths at each point of Fig. 1, i.e.
$\lambda_{\mathrm{c}}\coloneqq\\{h_{j,\mathrm{i}},b_{j,\mathrm{i}},J_{jk,\mathrm{i}}\\}$
and
$\lambda_{\mathrm{h}}\coloneqq\\{h_{j,\mathrm{f}},b_{j,\mathrm{f}},J_{jk,\mathrm{f}}\\}$.
Here, $h_{j,\mathrm{i}}$, $b_{j,\mathrm{i}}$ and $J_{jk,\mathrm{i}}$ are the
values of the magnetic fields and interaction strengths at points $A$ and $D$,
and $h_{j,\mathrm{f}}$, $b_{j,\mathrm{f}}$ and $J_{jk,\mathrm{f}}$ at points
$B$ and $C$, respectively. The explicit forms of the functions $h_{j}(t)$,
$b_{j}(t)$ and $J_{jk}(t)$ as well as the sweep function $\vartheta(t)$ are
given in Appendix A.

###  2.3 Multi-spin counter-diabatic driving

The underlying idea of CD driving [40, 39, 41, 42, 43, 44] is to efficiently
drive an adiabatic evolution of a Hamiltonian in _finite time_ by suppressing
transitions between its eigenstates. Thus, we always track the instantaneous
eigenstates during the whole sweep. Finding the _exact_ CD term requires a
priori knowledge of the system’s eigenstates for every time during the sweep
which is numerically and experimentally challenging.

In order to overcome this bottleneck, an analytical variational principle has
been developed recently [42, 43] to find _approximate_ CD terms.
In this work, we drive the WM during the isentropic strokes with the total
Hamiltonian

$H_{\mathrm{STA}}(t)=H_{0}(t)+H_{\mathrm{CD}}(t)$ (4)

where $H_{0}(t)$, Eq. (3), is the original non-adiabatic and
$H_{\mathrm{CD}}(t)$ the additional CD Hamiltonian that suppresses coherences
in the WM that cause quantum friction in finite-time sweeps [31, 32, 33].

The additional CD Hamiltonian reads

$H_{\mathrm{CD}}(t)=\dot{\vartheta}(t)\mathcal{A}_{\vartheta}(t)$ (5)

with $\mathcal{A}_{\vartheta}(t)$ the _exact_ adiabatic gauge potential [42,
57, 43] and $\dot{\vartheta}(t)$ the derivative of the sweep function of Eq.
(3). We rely on an _approximate_ adiabatic gauge potential
$\mathcal{A}^{*}_{\vartheta}$ that contains $p$-spin terms (with $p\leq N$)
and an odd number of $\sigma^{y}$ terms (e.g., $\sigma_{j}^{y}$ for $p=1$,
$\sigma_{j}^{y}$, $\sigma_{j}^{y}\sigma_{k}^{x}$ and
$\sigma_{j}^{y}\sigma_{k}^{z}$ for $p=2$, etc.). For $p=N$, we obtain the
solution of the _exact_ adiabatic gauge potential that entails all
combinations of $N$-spin terms (cf. Ref. [58] in the case of quantum
criticality). As an example, for $p=1$ we apply the ansatz
$\mathcal{A}^{*}_{\vartheta}=\sum_{j=1}^{N}\alpha_{j}\sigma_{j}^{y}$ and solve
for the optimal solution of each coefficient $\alpha_{j}$ by minimizing the
operator distance between the exact and approximate adiabatic gauge potential.
For more details, see Appendix B and Refs. [42, 57, 43].

We note, that we apply the CD Hamiltonian _only_ in the isentropic strokes as
these are normally much longer than the thermalization strokes. However,
techniques to speed up the latter have been also developed recently [59, 48,
60, 61, 62].

###  2.4 Quantum refrigerator under STA

The introduction of the additional _approximate_ CD term
$H^{*}_{\mathrm{CD}}(t)$ in the Hamiltonian’s dynamics during the, originally
non-adiabatic, work-exchange strokes requires a careful definition of work,
cooling power and CoP. In Ref. [52], the division

$W^{l}_{\mathrm{STA}}=W^{l}_{\mathrm{0}}+W^{l}_{\mathrm{CD}}$ (6)

with $W^{l}_{\mathrm{STA}}\equiv\Delta
E=\int_{0}^{\tau_{l}}\mathrm{Tr}\left[\rho(t)\dot{H}_{\mathrm{STA}}(t)\right]dt$
the total exchanged work for each of the two work-exchange strokes
$l\in\\{1,3\\}$ with duration $\tau_{l}$ has been introduced. The
corresponding work components thus read

$\displaystyle W_{0}^{l}$
$\displaystyle=\int_{0}^{\tau_{l}}\mathrm{Tr}\left[\rho(t)\dot{H}_{0}(t)\right]dt,$
(7) $\displaystyle W_{\mathrm{CD}}^{l}$
$\displaystyle=\int_{0}^{\tau_{l}}\mathrm{Tr}\left[\rho(t)\dot{H}_{\mathrm{CD}}(t)\right]dt$
(8)

where the work $W_{0}^{l}$ stems from the piston and $W_{\mathrm{CD}}^{l}$
from the external control device that implements $H^{*}_{\mathrm{CD}}(t)$.

As we will show in more detail in Appendix C, this work component
$W^{l}_{\mathrm{CD}}$ is _zero_ if the additional adiabatic gauge potential
and thus the CD Hamiltonian $H^{*}_{\mathrm{CD}}(t)$ is _exact_ , i.e.,
$\mathcal{A}^{*}_{\vartheta}(t)=\mathcal{A}_{\vartheta}(t)$ and thus
$H^{*}_{\mathrm{CD}}(t)=H_{\mathrm{CD}}(t)$. Hence, the work component
$W^{1+3}_{\mathrm{CD}}$ stemming from the external control device during one
cycle can be seen as an artifact of _inexact_ , e.g. $1$-spin CD driving (cf.
Ref. [52]). Experimentally it is advantageous to have a vanishing contribution
from the external control device as we want the external control device to
fully assist the piston instead of just exploiting it to run the quantum
refrigerator. In other words, the _exact_ CD drive is _catalytic_ in the sense
that it allows for speeding up the cycle without the external control device
being altered (charged or discharged) after a cycle. Therefore, as we will see
later, applying multi-spin CD terms can efficiently speed up the cycle without
excessively exploiting the external control device.

The cooling power under STA is given by

$\mathcal{J}_{\mathrm{STA}}\coloneqq\dfrac{\mathrm{pumped\,heat}}{\mathrm{cycle\,duration}}=\dfrac{Q_{\mathrm{c}}}{\tau_{\mathrm{cycle}}}$
(9)

and the CoP reads

$\epsilon_{\mathrm{STA}}\coloneqq\dfrac{\mathrm{pumped\,heat}}{\mathrm{consumed\,work}}=\dfrac{Q_{\mathrm{c}}}{W^{1}_{\mathrm{STA}}+W^{3}_{\mathrm{STA}}}$
(10)

where $W_{\mathrm{STA}}^{1+3}=W_{\mathrm{STA}}^{1}+W_{\mathrm{STA}}^{3}>0$ is
the total work performed on (consumed by) the WM per cycle. Note the
difference between the pumped heat $Q_{\mathrm{c}}^{\mathrm{ad}}$ for the
adiabatic [cf. Eqs. (1) and (2)] and $Q_{\mathrm{c}}$ for the sped-up cycle
[cf. Eqs. (9) and (10)].

For the case of a single-body quantum WM that is modelled by Eq. (3) with
$N=1$, i.e., this reduces to the Landau-Zener (LZ) model, the CoP evaluates to

$\epsilon_{\mathrm{LZ}}\coloneqq\dfrac{h_{\mathrm{x,i}}}{b_{\mathrm{z,f}}-h_{\mathrm{x,i}}}$
(11)

where $h_{\mathrm{x,i}}$ is the initial value of the transverse magnetic field
in the first isentropic stroke and $b_{\mathrm{z,f}}$ the final value of the
longitudinal magnetic field strength, respectively. This expression is
positive provided that $b_{\mathrm{z,f}}>h_{\mathrm{x,i}}$ and is limited by
the Carnot CoP
$\epsilon_{\mathrm{C}}=T_{\mathrm{c}}/(T_{\mathrm{h}}-T_{\mathrm{c}})$ [26]
(see Appendix D for more details).

##  3 Numerical Results

In this section, we present the numerical results of the proposed quantum Otto
refrigerator. To this end, we used the QuTip 4.2 [63] Python package to
simulate the quantum Otto cycle with (i) non-adiabatic [$H_{0}(t)$, Eq. (3)]
and (ii) STA Hamiltonian [$H^{*}_{\mathrm{STA}}(t)$, Eq. (4)] with $p$-spin CD
terms [$H^{*}_{\mathrm{CD}}(t)$, Eq. (5)]. We numerically solved the von
Neumann equation for the isentropic strokes and computed the heat
$Q_{\mathrm{c}}$ and $Q_{\mathrm{h}}$ for the two thermalization strokes as
the energy difference between points $B$ and $C$ as well as $D$ and $A$ (cf.
Fig. 1), respectively. Throughout this numerical performance analysis, the
temperatures for the cold and hot bath are set to $T_{\mathrm{c}}=0.2$ and
$T_{\mathrm{h}}=0.4$, respectively. The values for the working parameters at
points $A$ and $D$ as well as $B$ and $C$ in Fig. 1 read
$\lambda_{\mathrm{c}}=\\{h_{j,\mathrm{i}}=0.2,\,b_{j,\mathrm{i}}=0,J_{jk,\mathrm{i}}=0\\}$
and
$\lambda_{\mathrm{h}}=\\{h_{j,\mathrm{f}}=0,\,b_{j,\mathrm{f}}=0.5,J_{jk,\mathrm{f}}=0.1\\}$,
respectively, for both, non-adiabatic $H_{0}(t)$ and STA Hamiltonian
$H^{*}_{\mathrm{STA}}(t)$ with $p$-spin CD Hamiltonian
$H^{*}_{\mathrm{CD}}(t)$ for different $p$. At each point $A$, $B$, $C$ and
$D$, the additionally applied CD Hamiltonian is zero, i.e.,
$H^{*}_{\mathrm{CD}}(t=\sum_{j}\tau_{j})=0$ where $j\in\\{0,1,2,3\\}$ and
$\tau_{0}=0$. The durations $\tau_{2}$ and $\tau_{4}$ of each thermalization
stroke are set to $0.1$.

###  3.1 Scaling with system size

Our primary goal is to speed up the quantum refrigerator, i.e., we want to
pump as much heat $Q_{\mathrm{c}}$ as possible from the cold to the hot
reservoir in the shortest amount of time. Thus, we are particularly interested
in the cooling power $\mathcal{J}$ during one cycle and its scaling behaviour
for different system sizes $N$.

Figure 2: Cooling power $\boldsymbol{\mathcal{J}}$ over system size
$\boldsymbol{N}$. Cooling power $\mathcal{J}$ of a quantum Otto cycle for
isentropic stroke duration $\tau=\tau_{1}=\tau_{3}\approx 40$ for (i) non-
adiabatic Hamiltonian [$H_{0}(t)$, Eq. (3)] (red, bottom line) and (ii) STA
Hamiltonian [$H^{*}_{\mathrm{STA}}(t)$, Eq. (4)] for different $p$-spin CD
terms [$H^{*}_{\mathrm{CD}}(t)$, Eq. (5)] over different system sizes $N$.
Parameters: Cold and hot bath at temperatures $T_{\mathrm{c}}=0.2$ and
$T_{\mathrm{h}}=0.4$, respectively, and working parameters
$\lambda_{\mathrm{c}}$ with $h_{j,\mathrm{i}}=0.2$, $b_{j,\mathrm{i}}=0$ and
$J_{jk,\mathrm{i}}=0$ and $\lambda_{\mathrm{h}}$ with $h_{j,\mathrm{f}}=0$,
$b_{j,\mathrm{f}}=0.5$ and $J_{jk,\mathrm{f}}=0.1$, magnetic field and
interaction strengths, respectively. Duration of the thermalization strokes:
$\tau_{2}=\tau_{4}=0.1$. Black-dashed line denotes cooling power at adiabatic
limit where $\tau=\tau_{1}=\tau_{3}\to\infty$.

Figure 2 depicts the cooling power $\mathcal{J}$ for (i) non-adiabatic
[$H_{0}(t)$, Eq. (3)] and (ii) STA Hamiltonian [$H^{*}_{\mathrm{STA}}$, Eq.
(4)] for different $p$-spin CD terms [$H^{*}_{\mathrm{CD}}$, Eq. (5)] over
different system sizes $N$ of spins in the WM for an isentropic stroke
duration of $\tau=\tau_{1}=\tau_{3}\approx 40$. By applying $p$-spin CD terms,
we can efficiently enhance the cooling power of our sped-up refrigerator by
increasing the system size $N$ compared to the non-adiabatic counterpart. The
relative enhancement in cooling power decreases the higher $p$ becomes. The
major relative enhancement can be made by applying 1-spin or 2-spin CD terms.
Including multi-spin terms (e.g. $3$ and $4$-body in the cases studied) only
give a relatively slight improvement compared to 1-spin or 2-spin CD driving.
We note, that there is a trade-off between enhanced cooling power with
increasing $p$ and implementation complexity for an experiment. In the
adiabatic limit (black-dashed line), the cooling power naturally converges to
zero.

From a practical point of view, we deem it more favorable if the increased
cooling power $\mathcal{J}_{\mathrm{STA}}$ is due to the piston rather than
exploiting the external control device. Consequently, we are interested in the
work component $W^{1+3}_{\mathrm{CD}}$, Eq. (8). As shown in Appendix C the
latter is zero if the applied CD Hamiltonian is _exact_.

Figure 3: Work component $\boldsymbol{|W^{1+3}_{\mathrm{CD}}|}$ over system
size $\boldsymbol{N}$ for $\boldsymbol{p\leq 4}$. Absolute value of work
component $W^{1+3}_{\mathrm{CD}}$ of the STA Hamiltonian
[$H^{*}_{\mathrm{STA}}(t)$, Eq. (4)] for different $p$-spin CD terms
[$H^{*}_{\mathrm{CD}}(t)$, Eq. (5)] over different system sizes $N$.
Parameters: $\tau=\tau_{1}=\tau_{3}=1$. Other parameters as in Fig. 2.

Figure 3 depicts the absolute value of the work component
$W^{1+3}_{\mathrm{CD}}$ stemming from the external control device during one
cycle for different system sizes $N$ up to $p=4$. We see that
$W^{1+3}_{\mathrm{CD}}$ is zero as long as $p>N$. Namely, the CD Hamiltonian
must comprise all kinds of interactions up to order $N$, i.e., involving all
the spins in the chain. By contrast, for $N>p$, i.e., more spins $N$ in the WM
than order of interaction $p$ in $H^{*}_{\mathrm{CD}}(t)$, we see that the
absolute value of $W^{1+3}_{\mathrm{CD}}$ adopts a non-zero value which,
according to Appendix C, implies that the Hamiltonian $H^{*}_{\mathrm{CD}}(t)$
is _not_ exact anymore. We therefore conclude, that including a CD Hamiltonian
with all combinations up to $N$-body terms leads to an exact expression when
the WM contains $N$ spins which is consistent with Refs. [42, 43]. These
results encourage the goal to strive for an _exact_ CD drive. For the latter,
the external control device fully assists the piston that optimally
“compresses” and “expands” the quantum WM.

###  3.2 Dependence on cycle time

Figure 4: Pumped heat and coefficient of performance. (a) Pumped heat
$Q_{\mathrm{c}}$ per cycle and (b) CoP $\epsilon$ for (i) non-adiabatic
Hamiltonian [$H_{0}(t)$, Eq. (3)] and (ii) STA Hamiltonian
[$H^{*}_{\mathrm{STA}}(t)$, Eq. (4)] for different $p$-spin CD terms
[$H^{*}_{\mathrm{CD}}(t)$, Eq. (5)] over different isentropic stroke durations
$\tau=\tau_{1}=\tau_{3}$ and system size $N=6$. Yellow-shaded area
($\tau\gtrsim 28$) depicts the refrigerator regime where
$Q_{\mathrm{c}}^{\mathrm{na}}>0$ and the gray-shaded area ($\tau\geq 250$) the
adiabatic regime where $Q_{\mathrm{c}}^{\mathrm{na}}\approx
Q_{\mathrm{c}}^{\mathrm{ad}}$. Other parameters as in Fig. 2.

We are further interested in the pumped heat $Q_{\mathrm{c}}$ per cycle and
the corresponding CoP for different cycle durations $\tau$. Figure 4(a) shows
the heat $Q_{\mathrm{c}}$ extracted from the cold reservoir over one cycle for
a system size $N=6$. For the quantum Otto cycle with non-adiabatic Hamiltonian
$H_{0}(t)$, we see that heat $Q_{\mathrm{c}}^{\mathrm{na}}>0$ is pumped from
the cold reservoir to the hot bath only for isentropic stroke durations
$\tau=\tau_{1}=\tau_{3}\gtrsim 28$ (yellow-shaded area, refrigerator regime).
For shorter isentropic stroke durations, the obtained states
$\rho^{\prime}_{B}$ and $\rho^{\prime}_{D}$ at the end of each isentropic
stroke (cf. Fig. 1) are so far away from the ideal adiabatic states $\rho_{B}$
and $\rho_{D}$ that the quantum Otto cycle ceases to describe a refrigerator.
For large stroke durations, the refrigerator pumps the maximal possible
cooling heat (gray-shaded area, adiabatic regime, $\tau\geq 250$ where
$Q_{\mathrm{c}}^{\mathrm{na}}\approx Q_{\mathrm{c}}^{\mathrm{ad}}$).

By contrast, the sped-up cycle with Hamiltonian $H^{*}_{\mathrm{STA}}(t)$
including $1$\- up to $4$-spin CD terms $H^{*}_{\mathrm{CD}}(t)$ even pumps
heat from the cold reservoir in the quench limit $\tau_{1},\tau_{3}\to 0$
where the cycle duration is dominated by thermalization. For increasing $p$,
the cooling heat $Q_{\mathrm{c}}$ increases to its maximally possible value
$Q_{\mathrm{c}}^{\mathrm{ad}}$ for all durations studied. Note, however, that
$p=2$ appears to be efficient even for the intermediate regime where $p=1$
does not yield optimal results. This is in accordance with Fig. 2. Note that
for the adiabatic regime, the strength of the additional
$H^{*}_{\mathrm{CD}}(t)$ converges to zero (as $\dot{\vartheta}(t)\varpropto
1/\tau$, cf. Appendix A) and hence $Q_{\mathrm{c}}\to
Q_{\mathrm{c}}^{\mathrm{ad}}$.

Figure 4(b) depicts the corresponding CoP $\epsilon$, Eq. (10). For the sped-
up cycle with $H^{*}_{\mathrm{STA}}(t)$, Eq. (4), we see that the higher
$p$-spin terms we use for our CD Hamiltonian $H^{*}_{\mathrm{CD}}(t)$, Eq.
(5), the larger the CoP becomes. Note that the CoP appears to be more
sensitive to the value of $p$ than the pumped heat $Q_{\mathrm{c}}$. Namely,
converging to the optimal CoP $\epsilon^{\mathrm{ad}}$, Eq. (2), generally
requires a larger $p$ than for the optimal exctracted heat
$Q_{\mathrm{c}}^{\mathrm{ad}}$ which is due to a non-zero
$W^{1+3}_{\mathrm{CD}}$.

##  4 Discussion and Outlook

In this work, we have presented a _many-body_ quantum Otto refrigerator that
efficiently operates at _finite-time_. The many-body quantum WM is modelled by
an all-to-all connected Ising spin chain where the working parameter is
parametrized via magnetic fields and interactions. In order to speed up the
quantum Otto cycle, we apply an additional _approximate_ multi-spin CD
Hamiltonian. The latter outperforms its non-adiabatic and traditional $1$-spin
CD counterparts in pumped heat per cycle. The additional CD Hamiltonian
contains $p$-spin terms with an odd number of $\sigma^{y}$ terms and $p\leq
N$. For $p=N$, we obtain the exact CD Hamiltonian. We provide an analytical
proof that the work component stemming from the external control device is
_zero_ if the CD Hamiltonian is _exact_. In this case, the external control
device fully assists the piston to pump heat from a cold to a hot reservoir
and acts as a catalyst. Note that while an exact CD protocol implies zero work
contribution from the external control device over a cycle, the converse is
not true. It is therefore not possible to find an exact CD Hamiltonian via
minimizing this work component.

We numerically demonstrate an enhanced cooling power and coefficient of
perfomance for the sped-up quantum Otto refrigerator with higher $p$-spin CD
driving compared to its non-adiabatic and $1$-spin counterparts. Furthermore,
we show that increasing $p$ improves the scaling behaviour in cooling power
with system size. For the Otto cycle with single-body quantum WM, i.e.,
described via the Landau-Zener model, we find an analytical expression for the
CoP that only depends on the applied magnetic field strengths. The analytical
and numerical performance analysis reveals that quantum Otto refrigerators
using WM with even all-to-all connectivity can be scaled up efficiently by
applying multi-spin CD protocols while increasing the number of spins in the
WM.

We note that several cost identifiers for the additional CD Hamiltonian have
been introduced [64, 65, 66, 47, 46, 49, 67, 51]. We note, further, that these
_implementational_ costs are conceptually different to the _operational_ costs
described by the work $W^{1+3}_{\mathrm{CD}}$ stemming from the external
control device [52]. In fact, in the case of exact CD driving we go on the
optimal, i.e., adiabatic, path from points $A$ to $B$ and $C$ to $D$ in Fig.
1, respectively. Although the operational cost will then be zero, i.e.,
$W^{1+3}_{\mathrm{CD}}=0$, there will still be a cost of implementing the
additional CD Hamiltonian. In Appendix E we have computed the strongly setup-
dependent energetic cost of generating the additional magnetic fields of the
CD Hamiltonian. However, although these energetic costs can be quantified, the
actual implementation cost highly depends on the experimental setup and thus
cannot be answered unambigously.

The results presented in this work disclose some important operational
features. This work reveals a trade-off between experimental feasibility of
the additionally applied CD terms and vanishing work component stemming from
the external control device. Namely, for the latter we need to apply non-local
multi-spin CD terms (for example through additional laser fields) that are
hard to implement in current experiments. On the other hand, applying only
local $1$-spin CD terms may result in enhanced cooling power compared to its
non-adiabatic counterpart, yet under the price of extensive usage of the
external control device rather than the piston. This work further shows that
we can efficiently speed up the Otto cycle even for increasing system sizes by
applying sufficiently well approximated multi-spin CD Hamiltonians. This work
puts the results obtained in Ref. [52] into a wider context. In particular,
this work significantly advances the ideas of the latter by providing a full
assessment of experimental tuning parameters to consider and still being
applicable for strongly site-dependent magnetic fields and interactions in a
fully connected WM due to its increased complexity of external driving. While
we have numerical evidence that $1$-spin CD driving is just able to speed up
the cycle for weakly coupled spins in the WM, e.g. with nearest-neighbor
interaction, the higher-spin CD driving even generates cooling heat for
strongly coupled and site-dependent magnetic fields and interactions for
various all-to-all connected WM.

For future research, we intend to study the robustness of the applied CD
protocols with respect to external noise and possible cooperative effects
[68]. Furthermore, we aim at developing a many-body quantum refrigerator where
work and heat exchanges occur simultaneously.

## Acknowledgements

We thank Adolfo del Campo for valuable discussions. W. N. acknowledges support
from an ESQ fellowship of the Austrian Academy of Sciences (ÖAW). V. M.
acknowledges support from Science and Engineering Research Board (SERB)
through Start-up Research Grant (Project No.SRG/2019/000411) and from Seed
grant of IISER Berhampur. Work was supported by the Austrian Science Fund
(FWF) through a START grant under Project No. Y1067-N27 and the SFB BeyondC
Project No. F7108-N38, the Hauser-Raspe foundation, and the European Union’s
Horizon 2020 research and innovation program under grant agreement No. 817482.
This material is based upon work supported by the Defense Advanced Research
Projects Agency (DARPA) under Contract No. HR001120C0068. Any opinions,
findings and conclusions or recommendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of DARPA.

##  Appendix A Protocols for magnetic fields and interactions

For the many-body quantum WM in the text, the explicit time dependence of the
magnetic field and interaction strengths for the non-adiabatic [$H_{0}(t)$,
Eq. (3)] and STA Hamiltonian [$H^{*}_{\mathrm{STA}}(t)$, Eq. (4)] read

$\displaystyle h_{j}(t)$
$\displaystyle=h_{j,\mathrm{i}}+(h_{j,\mathrm{f}}-h_{j,\mathrm{i}})\vartheta(t),$
(A1a) $\displaystyle b_{j}(t)$
$\displaystyle=b_{j,\mathrm{i}}+(b_{j,\mathrm{f}}-b_{j,\mathrm{i}})\vartheta(t),$
(A1b) $\displaystyle J_{jk}(t)$
$\displaystyle=J_{jk,\mathrm{i}}+(J_{jk,\mathrm{f}}-J_{jk,\mathrm{i}})\vartheta(t)$
(A1c)

where

$\vartheta(t)\coloneqq\sin^{2}\left[\dfrac{\pi}{2}\sin^{2}\left(\dfrac{\pi
t}{2\tau_{l}}\right)\right]$ (A2)

is the sweep function and $h_{j,\mathrm{i}}=h_{j}(t=0)$,
$b_{j,\mathrm{i}}=b_{j}(t=0)$, $J_{jk,\mathrm{i}}=J_{jk}(t=0)$ and
$h_{j,\mathrm{f}}=h_{j}(t=\tau_{l})$, $b_{j,\mathrm{f}}=b_{j}(t=\tau_{l})$,
$J_{jk,\mathrm{f}}=J_{jk}(t=\tau_{l})$ the initial and final values for each
isentropic stroke $l\in\\{1,3\\}$ of duration $\tau_{l}$, respectively. Its
derivative with respect to time reads

$\dot{\vartheta}(t)=\dfrac{\pi^{2}}{4\tau_{l}}\sin\left(\dfrac{\pi}{\tau_{l}}t\right)\sin\left[\pi\sin^{2}\left(\dfrac{\pi}{2\tau_{l}}t\right)\right]$
(A3)

and is applied to the CD Hamiltonian, Eq. (5).

##  Appendix B Multi-spin CD driving

For the CD Hamiltonian $H_{\mathrm{CD}}(t)$, Eq. (5) in the text, we apply an
_approximate_ adiabatic gauge potential $\mathcal{A}^{*}_{\vartheta}$. Here,
we follow the variational method introduced in Ref. [42].

We make an ansatz $\mathcal{A}^{*}_{\vartheta}$ with $p$-spin Pauli matrices
and an odd number of $\sigma^{y}$ terms for the adiabatic gauge potential and
calculate the Hermitian operator
$G_{\vartheta}(\mathcal{A}^{*}_{\vartheta})=\partial_{\vartheta}H_{0}+i[\mathcal{A}^{*}_{\vartheta},H_{0}]$.
The goal is to minimize the operator distance
$D^{2}(\mathcal{A}^{*}_{\vartheta})=\mathrm{Tr}\\{[G_{\vartheta}(\mathcal{A}_{\vartheta})-G_{\vartheta}(\mathcal{A}^{*}_{\vartheta})]^{2}\\}$
between the exact, $\mathcal{A}_{\vartheta}$, and approximate,
$\mathcal{A}^{*}_{\vartheta}$ adiabatic gauge potential. This minimization is
equivalent to minimizing the action

$S(\mathcal{A}^{*}_{\vartheta})=\mathrm{Tr}[G^{2}_{\vartheta}(\mathcal{A}^{*}_{\vartheta})]$
(B1)

with respect to its parameters in front of every Pauli matrix term, i.e.,
$\delta\mathcal{S}(\mathcal{A}^{*}_{\vartheta})/\delta\mathcal{A}^{*}_{\vartheta}=0$.

As an example, for $p=1$, i.e., 1-spin CD driving, we apply the ansatz
$\mathcal{A}^{*}_{\vartheta}=\sum_{j=1}^{N}\alpha_{j}\sigma_{j}^{y}$ and
calculate the Hermitian operator $G_{\vartheta}(\mathcal{A}^{*}_{\vartheta})$
as well as the action $\mathcal{S}(\mathcal{A}^{*}_{\vartheta})$. By
minimizing the latter with respect to the coefficients $\alpha_{j}$, we find
the optimal solution for each spin. For $p=2$, we apply the ansatz
$\mathcal{A}^{*}_{\vartheta}=\sum_{j=1}^{N}\alpha_{j}\sigma_{j}^{y}+\sum_{k<j}\beta_{jk}(\sigma_{j}^{y}\sigma_{k}^{z}+\sigma_{j}^{z}\sigma_{k}^{y})+\gamma_{jk}(\sigma_{j}^{y}\sigma_{k}^{x}+\sigma_{j}^{x}\sigma_{k}^{y})$
and solve the corresponding action with respect to all coefficients
$\alpha_{j}$, $\beta_{jk}$ and $\gamma_{jk}$. With this variational method, we
can also include multi-spin terms, potentially up to $N$-body terms
$\sigma_{1}^{z}\cdots\sigma_{j}^{y}\cdots\sigma_{N}^{x}$. We solve the optimal
form of each coefficient numerically.

##  Appendix C Proof of zero work component $W_{\mathrm{CD}}$ for exact CD
driving

Here, we provide a detailed proof to the statement that the work contribution
[52]

$W_{\mathrm{CD}}^{l}=\int_{0}^{\tau_{l}}\mathrm{Tr}\left[\rho(t)\dot{H}_{\mathrm{CD}}(t)\right]dt$
(C1)

stemming from the external control device for each isentropic stroke
$l\in\\{1,3\\}$ becomes _zero_ if the additionally applied CD Hamiltonian
$H^{*}_{\mathrm{CD}}(t)$ is _exact_. The latter reads [35]

$H_{\mathrm{CD}}(t)=i\hbar\sum_{n}|\dot{n}\rangle\langle n|-\langle
n|\dot{n}\rangle|n\rangle\langle n|$ (C2)

where $|n\rangle$ denotes the instantaneous eigenstate at time $t$. The
density matrix $\rho(t)$ can be written as

$\rho(t)=\sum_{n}a_{n}|n\rangle\langle n|$ (C3)

where $a_{n}$ is a coefficient whose sum fulfils the normalization relation.
Here, we have used that under exact CD driving the density matrix remains
diagonal in the instantaneous eigenbasis [69].

We now calculate the integral over the energy function

$f(t)\coloneqq\mathrm{Tr}[\rho(t)\dot{H}_{\mathrm{CD}}(t)]$ (C4)

for one isentropic stroke of duration $\tau$ [where
$\dot{H}_{\mathrm{CD}}(t=0)=\dot{H}_{\mathrm{CD}}(t=\tau)=0$]. Employing the
completeness condition $\sum_{n}|n\rangle\langle n|=\mathds{1}$ of the
eigenbasis, we receive the following relation

$\displaystyle\sum_{n}\partial_{t}^{2}(|n\rangle\langle
n|)=\sum_{n}|\ddot{n}\rangle\langle
n|+2|\dot{n}\rangle\langle\dot{n}|+|n\rangle\langle\ddot{n}|=0$
$\displaystyle\Rightarrow\sum_{n}|\dot{n}\rangle\langle\dot{n}|=-\dfrac{1}{2}\sum_{n}\left(|n\rangle\langle\ddot{n}|+|\ddot{n}\rangle\langle
n|\right).$ (C5)

On the other hand, the normalization condition $\langle n|n\rangle=1$ for each
eigenstate separately leads to the following equations

$\displaystyle\partial_{t}(\langle n|n\rangle)=\langle\dot{n}|n\rangle+\langle
n|\dot{n}\rangle=0,$ (C6a) $\displaystyle\langle\dot{n}|n\rangle=-\langle
n|\dot{n}\rangle$ $\displaystyle\Rightarrow 2\langle n|\dot{n}\rangle=\langle
n|\dot{n}\rangle+\langle n|\dot{n}\rangle=\langle
n|\dot{n}\rangle-\langle\dot{n}|n\rangle.$ (C6b)

Consequently, the time-derivative of the Hamiltonian $H_{\mathrm{CD}}(t)$
reads

$\displaystyle\dot{H}_{\mathrm{CD}}(t)$
$\displaystyle=i\hbar\sum_{n}|\ddot{n}\rangle\langle
n|+|\dot{n}\rangle\langle\dot{n}|-\partial_{t}(\langle
n|\dot{n}\rangle|n\rangle\langle n|)$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\mathrm{(C5)}}}}{{=}}\,i\hbar\sum_{n}\dfrac{1}{2}(|\ddot{n}\rangle\langle
n|-|n\rangle\langle\ddot{n}|)-\partial_{t}(\langle
n|\dot{n}\rangle)|n\rangle\langle n|$ $\displaystyle-\langle
n|\dot{n}\rangle(|\dot{n}\rangle\langle n|+|n\rangle\langle\dot{n}|).$ (C7)

The function $f(t)=\mathrm{Tr}[\rho(t)\dot{H}_{\mathrm{CD}}(t)]$, Eq. (C4),
can consequently be rewritten (using $\mathrm{Tr}[\hat{H}]=\sum_{n}\langle
n|\hat{H}|n\rangle$) as

$\displaystyle f(t)$
$\displaystyle=i\hbar\sum_{n}a_{n}\left[\dfrac{1}{2}\left(\langle
n|\ddot{n}\rangle-\langle\ddot{n}|n\rangle\right)-\partial_{t}(\langle
n|\dot{n}\rangle)-\langle n|\dot{n}\rangle\left(\langle
n|\dot{n}\rangle+\langle\dot{n}|n\rangle\right)\right]$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\mathrm{(C6a)}}}}{{=}}\;\dfrac{i\hbar}{2}\sum_{n}a_{n}\left[\langle
n|\ddot{n}\rangle-\langle\ddot{n}|n\rangle-2\partial_{t}(\langle
n|\dot{n}\rangle)\right]$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\mathrm{(C6b)}}}}{{=}}\;\dfrac{i\hbar}{2}\sum_{n}a_{n}\left[\langle
n|\ddot{n}\rangle-\langle\ddot{n}|n\rangle-\partial_{t}(\langle
n|\dot{n}\rangle-\langle\dot{n}|n\rangle)\right]$
$\displaystyle=\dfrac{i\hbar}{2}\sum_{n}a_{n}\left[\langle
n|\ddot{n}\rangle-\langle\ddot{n}|n\rangle-\langle\dot{n}|\dot{n}\rangle-\langle
n|\ddot{n}\rangle+\langle\ddot{n}|n\rangle+\langle\dot{n}|\dot{n}\rangle\right]$
$\displaystyle=0$ (C8)

which proves the statement above.

##  Appendix D Single-body working medium

In analogy to Ref. [49], we derive the CoP of the quantum refrigerator with a
single-body WM.

The CoP, Eq. (10), in the text can be rewritten in terms of the energies
$E_{i}$ with $i\in\\{A,B,C,D\\}$ at each point (cf. Fig. 1) of the cycle. It
reads

$\epsilon_{\mathrm{LZ}}=\dfrac{E_{A}-E_{D}}{E_{B}-E_{A}+E_{D}-E_{C}}=\dfrac{1}{\dfrac{E_{B}-E_{C}}{E_{A}-E_{D}}-1}$
(D1)

where the energies are

$\displaystyle E_{A}$
$\displaystyle=-h_{\mathrm{x,i}}\tanh\left(\dfrac{h_{\mathrm{x,i}}}{T_{\mathrm{c}}}\right),$
$\displaystyle E_{B}$
$\displaystyle=-b_{\mathrm{z,f}}\tanh\left(\dfrac{h_{\mathrm{x,i}}}{T_{\mathrm{c}}}\right),$
$\displaystyle E_{C}$
$\displaystyle=-b_{\mathrm{z,f}}\tanh\left(\dfrac{b_{\mathrm{z,f}}}{T_{\mathrm{h}}}\right),$
$\displaystyle E_{D}$
$\displaystyle=-h_{\mathrm{x,i}}\tanh\left(\dfrac{b_{\mathrm{z,f}}}{T_{\mathrm{h}}}\right),$
(D2)

due to the uniform scaling of the energy levels in this two-level system after
each stroke. Inserting the latter, Eq. (D1) consequently reads

$\epsilon_{\mathrm{LZ}}=\dfrac{1}{\dfrac{b_{\mathrm{z,f}}}{h_{\mathrm{x,i}}}-1}$
(D3)

which is equivalent to Eq. (11) from the text.

The pumped heat

$Q_{\mathrm{c}}=E_{A}-E_{D}=h_{\mathrm{x,i}}\left[\tanh\left(\dfrac{b_{\mathrm{z,f}}}{T_{\mathrm{h}}}\right)-\tanh\left(\dfrac{h_{\mathrm{x,i}}}{T_{\mathrm{c}}}\right)\right]\geq
0$ (D4)

is positive for the Otto cycle to describe a refrigerator. From the latter and
the fact that $\tanh$ is a monotonously increasing function, it follows that

$\dfrac{b_{\mathrm{z,f}}}{h_{\mathrm{x,i}}}\geq\dfrac{T_{\mathrm{h}}}{T_{\mathrm{c}}}.$
(D5)

Inserting this into Eq. (D3), we finally obtain

$\epsilon_{\mathrm{LZ}}=\dfrac{1}{\dfrac{b_{\mathrm{z,f}}}{h_{\mathrm{x,i}}}-1}\leq\dfrac{1}{\dfrac{T_{\mathrm{h}}}{T_{\mathrm{c}}}-1}=\dfrac{T_{\mathrm{c}}}{T_{\mathrm{h}}-T_{\mathrm{c}}}=\epsilon_{\mathrm{C}}$
(D6)

with the Carnot CoP $\epsilon_{\mathrm{C}}$ as the upper bound.

##  Appendix E Implementation cost for multi-spin CD driving

In this section we will consider the cost of implementing the additional
Hamiltonian $H_{\mathrm{CD}}(t)$, Eq. (5) from the text. To this end, we will
compute the energetic cost

$\langle
H^{j}_{\mathrm{CD}}\rangle=\nu_{t,N}\int_{0}^{\tau}\mathrm{Tr}[H^{\dagger}_{\mathrm{CD}}(t)H_{\mathrm{CD}}(t)]dt$
(E1)

of generating the additional magnetic fields of the CD Hamiltonian as defined
in Ref. [66], for each isentropic stroke $j$ and squared Frobenius norm of the
CD Hamiltonian, i.e. $\|H^{j}_{\mathrm{CD}}(t)\|^{2}$. Here $\nu_{t,N}$ is an
experimental setup-dependent expression that can possibly be highly
complicated to be evaluated.

Figure 5 depicts this energetic cost for different $p$-spin CD driving for (a)
different sweep durations $1<\tau<10$ and system size $N=6$ and (b) different
system sizes $N$ and sweep duration $\tau\approx 40$ with same parameters as
in Fig. 2. For panel (a) we have studied the sweep times $1<\tau<10$ as the
strength of the additional magnetic fields are in the same order as the
original magnetic fields and interactions, respectively, which we consider to
be highly interesting for experimental realization.

Figure 5: Energetic costs. Energetic cost $\langle
H^{j}_{\mathrm{CD}}\rangle$, Eq. (E1), for implementation of the CD
Hamiltonian $H^{j}_{\mathrm{CD}}(t)$, Eq. (5) from the text, during each
isentropic stroke $j$ for (a) different sweep durations $1<\tau<10$ and system
size $N=6$ and (b) different system sizes $N$ and sweep duration $\tau\approx
40$. Parameters: $\nu_{t,N}=0.01$ and other parameters as in Fig. 2.

We see that the energetic costs decrease (increase) exponentially for
increasing sweep duration $\tau$ (system size $N$). Although the difference in
energetic cost between each $p$-spin CD Hamiltonian are rather small, in
particular in between $2$-spin and $4$-spin CD driving, the actual
implementation cost, i.e., implementing complicated non-local $N$-spin
Hamiltonian rather than only local $1$-spin terms, may differ considerably.

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# Real and Reciprocal Space Characterization of the 3-Dimensional Charge
Density Wave in Quasi-1-Dimensional CuTe

Fei Guo1,2 Michele Puppin1,2 Lukas Hellbrück1,2 Arnaud Magrez1 Eduardo B.
Guedes3 Igor Sokolović4 J. Hugo Dil1,2,3 1Institute of Physics, École
Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
2Lausanne Centre for Ultrafast Science (LACUS), École Polytechnique Fédérale
de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
3Photon Science Division, Paul Scherrer Institut, CH-5232 Villigen,
Switzerland
4Institute of Applied Physics, Technische Universität Wien, 1040 Vienna,
Austria

###### Abstract

Low-dimensional materials are susceptible to electronic instabilities such as
charge density waves (CDWs), originating from a divergence in the Lindhard
electron response function, combined with a finite electron-phonon coupling
strength. In this report, we present a detailed characterisation of the CDW in
the quasi-one-dimensional material CuTe, including (1) direct visualization of
lattice distortion seen with non-contact atomic force microscopy in real
space, (2) the out-of-plane momentum dependency of the CDW gap size of the
quasi-1-dimensional bands, by angle-resolved photoemission spectroscopy, (3)
coherent dynamics of a photoexcited phonon mode seen by time- and angle-
resolved photoemission spectroscopy, with frequency and wavevector
$\textbf{{q}}_{\text{CDW}}$ corresponding to the soft phonon modes predicted
by theory. Furthermore, we find that the CDW gap closes through a transient
band renormalisation. We thus confirm that, despite the quasi-1D
characteristics of CuTe, it hosts inherently 3-dimensional CDWs.

##  I Introduction

Low-dimensional materials are a vibrant field of research due to their rich
number of competing broken symmetry ground states [1, 2], which are promising
for novel electronic functionalities [3, 4] and the possibility of ultrafast
optical manipulation [5]. An ubiquitous effect is the susceptibility to charge
ordering instability, such as charge density waves (CDWs). In the past few
decades, there have been great efforts to understand the driving mechanisms of
several CDW-hosting systems [6], for example, quasi-2D transition metal
dichalcogenides (TMDCs) [7, 8], rare-earth tritellurides ($RTe_{3}$s) [9, 10],
as well as in quasi-1D systems [11, 12]. The formation of CDWs can be due to
an electronic instability driven by Fermi surface nesting [13, 14], combined
with electron-phonon coupling [15, 16]. Electron-electron couplings are
believed to be the driving force for CDW in different scenarios, for example
in excitonic insulators [17], or via the band Jahn-Teller effect [18]. In
reality, CDWs are likely to arise due to an interplay between multiple of the
above factors, which cannot be easily individually resolved. The
characterization of materials hosting CDWs requires a quantitative evaluation
of the direction and periodicity of charge density modulation represented by
the CDW wavevector $\textbf{{q}}_{\text{CDW}}$, along with the energy scale of
such modulation in the form of a transition temperature $T_{\text{CDW}}$, or
the magnitude of the energy gap, as well as a case-specific evaluation of the
driving many-body couplings.

In this report, we present an investigation of the quasi-1D CDW material
copper telluride (CuTe) [19]. We first directly visualize the spatial charge
ordering modulations via non-contact atomic force microscope (ncAFM), then, we
measure with angle-resolved photoemission spectroscopy (ARPES) the electronic
band dispersion of CuTe, including the dispersion orthogonal to the surface
plane $k_{z}$. Our data reveals a $k_{z}$-dependent gap, together with
evidence of inter-layer couplings in the ordering of the CDW. Finally we
investigate electron-phonon coupling in the system by femtosecond time-
resolved ARPES (trARPES), showing coherent oscillation of specific location of
the band structure. The oscillation correspond to a soft phonon mode linked to
the CDW, but are strongly coupled with inherently 3D states at the Fermi
surface. Further, we can follow the ultrafast closing of the CDW gap through a
renormalisation of the quasi 1D bands. In light of these results, we discuss
the dimensionality of the CDW in CuTe and show that under all typical
definitions of dimensionality the CDW in CuTe is 3-dimensional, despite the
fact that both the atomic and electronic structures show remarkable quasi-1D
features.

Figure 1: Schematic drawing of (a) undistorted and (b) Peierls distorted
lattices and electronic band dispersion; Fermi surface nesting of (c) 3D and
(d) 1D CDWs in the first definition in terms of $\textbf{{q}}_{\text{CDW}}$
[20], with Fermi surface (blue) and nesting wavevector
$\textbf{{q}}_{\text{CDW}}$ (red); lattice distortions of (e) 3D and (f) 1D
CDWs in the second definition in terms of inter-layer collectivity [12].

In the simplest model, the formation of a CDW in quasi-1D materials follows a
Peierls scenario, in which pairs of atoms dimerize. This modulation overcomes
Coulomb repulsion by opening up a gap at the Fermi energy and hence lowering
the total energy in the system. The formation of this superstructure
introduces an additional periodicity and ‘replica’ bands, as shown in Fig.1(a)
and (b). In this model, the Lindhard response function $\chi(\textbf{{q}})$ is
a key parameter:

$\chi(\textbf{{q}})=\sum_{k}\frac{f(\epsilon_{k})-f(\epsilon_{k+q})}{\epsilon_{k}-\epsilon_{k+q}}$
(1)

Here q is the electron wavevector, $\epsilon_{k}=\frac{\hbar^{2}k^{2}}{2m}$ is
the electron’s kinetic energy, and $f(\epsilon_{k})$ is the Fermi-Dirac
distribution. $\chi^{\prime}(\textbf{{q}})$ diverges whenever
$\epsilon_{k}=\epsilon_{k+q}$, i.e. two parts of the Fermi surface are
connected by a wavevector q, a condition referred to as Fermi surface nesting.
In the case of a strictly 1D system, the real part of $\chi(\textbf{{q}})$ can
be expressed as
$\chi^{\prime}(\textbf{{q}})=\frac{1}{q}\ln|\frac{q-2k_{F}}{q+2k_{F}}|$ and
has a weak divergence at $q=\pm 2k_{F}$ [21]. This susceptibility renormalizes
the phonon frequency $\omega_{q}$ to:

$\omega^{2}_{ren,q}=\omega^{2}_{q}+\frac{2g^{2}\omega_{q}}{\hbar}\chi(q,T)$
(2)

Where $g$ is the electron-phonon coupling constant [22]. With finite electron-
phonon coupling, the renormalized phonon frequency $\omega_{ren,q}$ shows a
sharp dip at divergence of $\chi^{\prime}(\textbf{{q}})$, and this indicates
an instability of the lattice structure and a tendency to charge density
reordering. However, the divergence of $\chi^{\prime}(\textbf{{q}})$ is not
protected with respect to carrier scattering and small geometrical deviations
from perfect nesting conditions [21]. Given this fact, one might expect that
CDWs could only survive in quasi-1D systems with nearly perfect nesting, but
there is plenty of evidence that CDWs exist in 2D [23, 24, 25] and even 3D
[26, 27] systems. Therefore, Fermi Surface nesting and electron-phonon
coupling are certainly not the only mechanism of CDW formation for higher-
dimensional materials.

The dimensionality of CDWs in low-dimensional materials is widely discussed in
the literature, but not uniquely defined. For instance, some authors define a
3-dimensional CDW as one where the CDW-associated lattice distortion is not
purely in-plane, and, as shown in Fig.1(c), there is a finite out-of-plane
component of the CDW wavevector $\textbf{{q}}_{\text{CDW}}$ [20]. Thus a
1-dimensional CDW in this picture would correspond to a 1D Fermi surface with
a nesting wavevector $\textbf{{q}}_{\text{CDW}}$ purely along one in-plane
direction (Fig.1(d)). Other authors define a 3-dimensional CDW where all
layers in the material have the same $\textbf{{q}}_{\text{CDW}}$ and modulate
coherently, but $\textbf{{q}}_{\text{CDW}}$ itself does not need an out-of-
plane component [12]. A 1-dimensional CDWs in this picture would be an
individual modulation on each atomic chain without any inter-layer
collectivity, despite having the same $\textbf{{q}}_{\text{CDW}}$. This
distinction is shown schematically in Fig.1(e) and (f). At this point, it is
worth mentioning that a purely 1D system cannot sustain stable long-range
orders due to quantum and thermal fluctuations, and inter-layer collectivity
is a prerequisite to the formation of charge and spin density waves [28, 29].

The subject of this article, CuTe, is a quasi-1D material, with the atomic
structure shown in Fig.2(a) [30], which has been reported to undergo a CDW
phase transition at $335\,$K. The electronic band structure of CuTe has been
investigated with ARPES [19], revealing a momentum-dependent gap as a large as
$190\,$meV which forms along quasi-1d Fermi surface sheets. The static
electronic band structure and the CDW gap were reported as a function of
temperature and doping, to visualize the closure of the gap. First-principle
calculations of the phonon spectrum were also presented showing soft phonon
modes at the observed nesting wavevector, which also corresponds to maxima of
the calculated electron scattering susceptibility. These results thus indicate
that electron-phonon coupling and Fermi surface nesting indeed contribute
dominantly to the formation of the CDW in CuTe. Recently, detailed electronic
structure and ultrafast nonequilibrium carrier dynamics near the Brillouin
zone center of CuTe has also been studied with low-photon-energy ARPES and
trARPES [31]. In this work, we expand on those results and elucidate the
dimensionality of the CDW in CuTe, by providing both real and reciprocal space
investigations.

##  II Methods

High quality single crystals of CuTe were synthesized using the flux
technique. A mixture of Cu and Te (atomic ratio 1:20) was sealed under vacuum
within a quartz ampoule. This ampoule was then placed in a vertical furnace
and heated to 600Â°C, where it remained for 12 hours before being gradually
cooled to 450Â°C at a cooling rate of 10Â°C per hour, and then further cooled
to 325Â°C at a slower rate of 1Â°C per hour. Following the crystal growth
process, the resulting crystals in the solidified Te flux were transferred in
a quartz tube furnace and maintained at 270Â°C under dynamic vacuum. After the
complete removal of the Te flux by sublimation, golden CuTe crystals were
obtained. The structure and stoichiometry were confirmed by single-crystal
diffraction and X-ray fluorescence spectroscopy.

Non-contact atomic force microscopy (ncAFM) measurements were performed using
a Omicron qPlus low-temperature head mounted in an ultrahigh vacuum (UHV)
chamber. Stiff qPlus sensors [32] ($k=1800$ N$\cdot$m-1, $Q$=5000–30000,
$f_{0}$ $\in$ [25-45] kHz) with a a sharp W tip [33] were used for imaging.
Cantilever deflection was supported with an in-vacuum cryogenic preamplifier
[34]. Prior to cleaving a CuTe sample and imaging the cleaved surface, sharp W
tips were additionally decorated with a sharp Cu pyramid at the apex by
deliberate tip restructuring on a Cu(110) surface. Damping of external
vibrations was assisted by suspending the whole chamber with 36 bungee cords
[35].

Single-crystal CuTe samples were prepared by attaching them to standardized
sample plates and attaching a stainless steel top post using conductive silver
epoxy. The silver epoxy was cured by baking this structure at $80^{\circ}$C
for 4 hours in ambient air. Samples were cleaved at room temperature in a
chamber with an UHV base pressure below $1\times 10^{-10}$ mbar, quickly
transferred $in-situ$ to a separate chamber with the base pressure below
$1\times 10^{-11}$ mbar where they were introduced to the measurement head and
subsequently imaged at $T=5\,$K. Such treatment of cleaved surfaces results in
a negligible amount of adsorbates, even on much more reactive surfaces [36].

For ARPES measurements samples were prepared in the same way as for the ncAFM
measurement. For the $k_{z}$ resolved ARPES measurements samples were cleaved
in-situ in ultrahigh vacuum below $5\times 10^{-10}$ mbar and at $T=24\,$K,
and for the trARPES measurements samples were cleaved in-situ in ultrahigh
vacuum at $T=70\,$K. High-resolution ARPES measurements as a function of
photon energy were taken at the SIS-ULTRA beamline of the Swiss Light Source,
Paul Sherrer Institut. All band structures were measured with circular $C+$
polarized light. trARPES measurement were taken at the LACUS facility at the
EPFL [37], which has a high-harmonic generation (HHG) source of extreme
ultraviolet (XUV) laser, along with an infra-red pump laser of wavelength 780
nm (photon energy 1.6 eV), which provides a time resolution better than 100
fs. trARPES data were collected with $p$-polarized HHG light with a photon
energy of 40.35 eV.

##  III Periodic Lattice Distortion: Direct Evidence of CDW in CuTe

The crystal structure of CuTe is shown in Fig.2(a), the 1d nature is reflected
by the appearance of chains of Cu atoms, each coordinated with 4 Te atoms.
Non-contact atomic force microscopy (ncAFM) was used to visualize the real
space CDW. A topographic image of a high-quality surface is shown in Fig.2(b).
The surface is seen to undergo a $5\times 1$ lattice modulation, as more
clearly observed by a fast Fourier transform in Fig.2(c) which shows a
unidirectional $5\times 1$ reconstruction. However, as shown by an ncAFM image
of a larger area in Fig.2(e), the CDW modulation is not simply $5\times 1$,
but proceed either as $2+1+2$ (red in Fig.2(e)) or $3+2$ (blue in Fig.2(e))
modulations, forming small domains coexisting on a patch of $20\times
20\,\text{nm}^{2}$.

Figure 2: (a) atomic structure of modulated CuTe, each Cu(black) is
coordinated with 4 Te(grey) atoms, 2 or 3 Te atoms group together in the CDW
state [30](b) small area detailed image: Te atoms are imaged as dark spheres
(attraction) while white spheres indicate absence of atoms (only long-range
forces). Te atoms are highlighted as blue spheres and the $5\times 1$ repeat
unit cell is indicated in red. (c) fast Fourier transform of ncAFM image, with
$5\times 1$ reconstruction in one direction and only $1\times 1$ peaks in the
perpendicular direction, (d) schematic drawing of the atomic structure of CuTe
and ncAFM contrast formation: Surface-terminating Te atoms act with chemical
attraction (negative $\delta f$) towards the Cu-terminated tip, while the Cu
atoms below the surface primarily contribute with long-range van der Waals
forces ($\delta f$ closer to zero), (e) ncAFM image on extended area, with 2
domains of CDWs with periodicities $2+1+2$(red) and $3+2$(blue) respectively;
schematic drawing of two different bilayer arrangements of the distorted
atomic chain: (f) 3+2 on 3+2, and (g) 3+2 on 2+3.

The modulation wavevector of the CDW is
$\textbf{{q}}_{\text{CDW}}=(0.4\textbf{a*},0\textbf{b*},\frac{1}{2}\textbf{c*})$
, in which a* is the reciprocal lattice vector in the chain direction, and c*
is the reciprocal lattice vector in the out-of plane direction [30]. The
magnitude of the c* component of $\textbf{{q}}_{\text{CDW}}$ is dictated by
the orthorhombic point group symmetry [38]. Therefore, in reality we should
expect an actual superstructure with a thickness of 2 layers. This could
potentially explain the structure of the $2+1+2$ domains seen in Fig.2(e). In
order for $\textbf{{q}}_{\text{CDW}}$ to overlap with the reciprocal lattice
vectors, it needs to be multiplied by $2$ in the c* direction and $2.5$ in the
a* direction. For a commensurate CDW, a superstructure with 2.5 lattice sites
is not possible, so the lattice reconstructs with either 2 or 3 sites,
alternating to finally form a periodicity of 5 lattice sites. Then, the
periodicity of 2 in the c* direction comes into play: the $3+2$ superstructure
has broken inversion symmetry, so there are 2 nonequivalent arrangements of 2
layers, $3+2$ on $3+2$, or $3+2$ on $2+3$. These are shown schematically in
Fig.2(f) and (g). With the $3+2$ on $2+3$ arrangement, the ncAFM tip can
detect different modulations on the 2 topmost layers, appearing like a $2+1+2$
modulation.

Given this 3-dimensional superstructure in real space, it is then instructive
to explore the electronic band dispersion in 3-dimensional reciprocal space.

##  IV Electronic Band Dispersion of CuTe

ARPES is the most direct way to visualize the electronic band structure, where
the onset of a CDW state is characterized by the opening of a band gap. The
Fermi surface of CuTe probed by ARPES is shown in Fig.3(a), which shows a
central feature extending in the $k_{x}$ direction, and 2 quasi-1D bands
extending in the $k_{y}$ direction [19]. In the CDW state, these bands display
a $k_{y}$-dependent gap, with a maximum of $190\,$meV at
$k_{y}=0.3\,\mathring{\text{A}}^{-1}$. Here we investigate the
$k_{z}$-dependence of the band structure, by tuning the photon energy.

We first focus on the cut $k_{y}=0$, the band structure of which is shown in
Fig.3(b), and vary the photon energy from $41\,$eV to $93\,$eV, with a step of
$2\,$eV. The resulting $k_{z}$ dispersion is shown in Fig.3(d). It is clear
that the bands at $k_{y}=0$ disperse strongly along the $k_{z}$ direction and
thus have a 3D nature, as also expected from calculations [30]. A similar
procedure was carried out for the cut at $k_{y}=0.4\,\mathring{\text{A}}^{-1}$
across the quasi-1D bands, the dispersion of which is shown in Fig.3(c). To
stay at a constant $k_{y}$ value, the tilt angle $\theta$ was adjusted at each
photon energy according to the relationship
$k_{y}=\frac{\sqrt{2mE_{k}}}{\hbar}\sin\theta$. The obtained dispersion is
shown in Fig.3(e). These bands barely show any dispersion as a function of
$k_{z}$, and have therefore indeed a clear quasi-1D character. The band
maximum is shown in Fig.3(f), and displays a periodic variation in $h\nu$. The
CDW gap size was extracted for each photon energy and plotted as a function of
the calculated $k_{z}$ in Fig.3(g), with an inner potential of $9.5\,$eV
extracted from the $h\nu$ periodicity of the 3D bands in Fig.3(d). The
difference in $k_{z}$ between the 2 gap size maxima in Fig.3(g) is
$0.59\,\mathring{\text{A}}^{-1}$. The observed periodicity of the CDW gap
along the $k_{z}$ direction further supports the 3D nature of the CDW.

Figure 3: (a) constant binding energy dispersion of CuTe taken at
$h\nu=65\,$eV, the $k_{y}$ values at which $h\nu$ scans were performed are
marked with red lines, (b) band map taken at $k_{y}=0\,\mathring{A}^{-1}$,
with $h\nu=65\,$eV, (c) band map taken at $k_{y}=0.4\,\mathring{A}^{-1}$, with
$h\nu=65\,$eV, (d) $h\nu$ dependency at $k_{y}=0\,\mathring{A}^{-1}$ showing a
clear dispersion, indicating 3D nature of the bands, (e) $h\nu$ dependency at
$k_{y}=0.4\,\mathring{A}^{-1}$, the bands are not dispersing in $k_{z}$,
indicating their 1D nature, (f) energy distribution as a function of photon
energy for $k_{y}=0.4\,\mathring{A}^{-1}$ and $k_{x}=0.46\,\mathring{A}^{-1}$
marked in (e), (g) CDW gap size variation with $k_{z}$

The above results indicate that CuTe hosts 3-dimensional CDWs in terms of the
modulation wavevector $\textbf{{q}}_{\text{CDW}}$, which has a finite
component in the z-direction. Also, the prerequisite of inter-layer
collectivity to the formation of CDWs, and the absence of Luttinger liquid-
like behaviour [39] suggests that all the layers modulate collectively at low
temperatures, hence the CDW is 3D inherently according to the second
definition. Overall, we find that at temperatures well below $T_{CDW}=335\,$K,
CuTe, albeit quasi-1D, hosts 3-dimensional CDWs with respect to both common
definitions.

##  V Coherent Dynamics of CuTe

Electron-phonon coupling is a critical ingredient in the CDW-forming
mechanism, here we investigate it by observing the coherent dynamics of the
band structure, initiated by a short optical perturbation. Through the
oscillation of the electronic bands, one can find the frequencies associated
to strongly coupled phonon modes that drives the Fermi surface nesting. For
this purpose, pump-probe time-resolved ARPES (trARPES) is a powerful tool, as
it can identify directly coupling to specific bands with momentum resolution
[40, 41].

To that end, we performed trARPES in our LACUS facility [37]. In this
experiment, $1.6\,$eV pump infrared laser pulses were used to excite electrons
to states above the Fermi level, then, after an adjustable time delay, an
extreme ultraviolet (XUV) laser of $40.35\,$eV photon energy was used to probe
the band structure by photoemission. Fig.4(a) demonstrates a time delay
difference spectrum measured by this technique: red denotes positive signal,
corresponding to an increased intensity at a time delay of 100 fs compared to
negative time delays (probe before the pump), and blue denotes an intensity
decrease. In Fig.4(a), the red area with kinetic energy $\geq 36.4\,$eV
indicates excited states above the Fermi level. Since the bands cross the
Fermi level, the excited states at high kinetic energies relax quickly
following an exponential decay with $\tau\approx 730\,$fs, as shown in
Fig.4(b).

Figure 4: (a) difference spectrum measured with trARPES at
$k_{x}=-0.34\,\mathring{\text{A}}^{-1}$ for a time delay of 100 fs: red
denotes extra intensity compared to negative time delay, and blue denotes
vanished intensity compared to negative time delay, (b) integrated intensity
in marked region in (a), as a function of time delay, (c) schematic drawing of
Fermi surface of CuTe, (d) bandmap with
$k_{x}=-0.34\,\mathring{\text{A}}^{-1}$ marked in (c), (e) time-delay
distribution of intensity integrated in the area marked in (d), along with
line fitting with a multi-exponential multiplied by a sinusoidal function, (f)
momentum distribution of intensity at constant energy and $k_{x}$ as a
function of time delay, (g) time-delay distribution of the negative-$k_{y}$
intensity peak position from (f). All of the above are taken at photon energy
$h\nu=40.35\,$eV.

We first focus on the bands at $k_{x}=-0.34\,\mathring{\text{A}}^{-1}$ in
reciprocal space, as marked on the schematic Fermi surface in Fig.4(c).
According to Fig.3(b), these bands show strong 3D characteristics. The
corresponding band map is shown in Fig.4(d). We follow the intensity evolution
in an area close to the Fermi level, marked in Fig.4(e). Clear oscillation of
intensity can be seen, and the time distribution of intensity can be fitted
with a multi-exponential decay with decay time of $2110\pm 120\,$fs and
oscillation frequency $f\approx 2.06\pm 0.1\,$THz, which corresponds to $69\pm
3\,\text{cm}^{-1}$. Another way to observe these oscillations is to track
directly the movement of bands as a function of pump-probe time delay.
Fig.4(f) shows the momentum distribution of intensity at a constant kinetic
energy directly below the Fermi level with varying time delays. The maximum of
the momentum intensity distribution at negative $k_{y}$ was extracted as a
function of time delay in Fig.4(g), on which similar oscillations were fitted
with a decay time of $1520\pm 120\,$fs and oscillation frequency of $f\approx
2.32\pm 0.2\,$THz, which corresponds to a phonon frequency of $77\pm
7\,\text{cm}^{-1}$. The above findings indicate that with the pump laser, we
have photo-excited a phonon mode of CuTe. Interestingly, ab-initio
calculations of the phonon spectrum [19] indicate softening for two phonon
modes at $\textbf{{q}}=(0.4,0,0)$ and $\textbf{{q}}=(0.4,0,0.5)$ respectively,
with softened phonon frequencies in the range of $60-85\,\text{cm}^{-1}$,
comparable with the observed phonon frequency. This is further evidence that
the predicted soft phonon mode q [19] is indeed present in CuTe.

Gap-filling dynamics has been studied extensively on CDW materials, and is
often accompanied by an oscillation of in-gap intensity or gap size [42, 43,
44, 45]. For this purpose, trARPES measurements of the 1D band were taken in a
diagonal $k_{x}-k_{y}$ direction, around the 1D band as shown by the diagonal
red line in Fig.4(c), and the corresponding band map is shown in Fig.5(a). For
each of the marked kinetic energies in Fig.5(a), momentum distribution of
intensity is visualized as a function of time, as representatively shown for
$E_{2}=35.7\,$eV and $E_{3}=35.3\,$eV in Fig.5(b). From these momentum
distributions we extracted the temporal evolution of the change of the
intensity peak position, as shown in Fig.5(c). To characterize the intensity
evolution, we calculated the peak area (= FWHM peak width$\times$peak
intensity) for each kinetic energy, as shown in Fig.5(d). At kinetic energy
$E_{1}=35.84\,$eV, which is within the CDW gap, we see a pronounced
renormalization in band position and a gap filling which decays on the order
of $700\,$fs, much shorter compared to the $2110\,$fs observed for the 3D
bands in Fig.4(e). However, even with the oscillation periodicity of the 3D
bands from Fig.4(e) superimposed, it is difficult to identify any clear
oscillations within the experimental uncertainty. At kinetic energy
$E_{2}=35.7\,$eV, comparable band renormalization is seen, along with an
intensity depletion which recovers with a slightly longer time scale. At the 2
lower kinetic energies, we see no pronounced band renormalization or intensity
change within the experimental signal to noise ratio.

Figure 5: (a) bandmap with diagonal $k_{//}$ marked in Fig.4(c), (b)
representative momentum distributions of intensity as a function of time
delay, at kinetic energies $E2$ and $E3$ marked in (a), (c) intensity peak
position change (as compared to negative time delays) as a function of time
delay, at all kinetic energies marked in (a), (d) peak area as a function of
time delay, at kinetic energies marked in (a) blue dashed lines stands for
time period extracted from 3D band coherent oscillation in Fig.4(e), (e)
Difference spectrum of quasi-1D band at time delay of $200\,$fs, as compared
to negative time delay, (f) schematic drawing of the quasi-1D band before
optically pumping (blue), and at $200\,$fs after optical pump arrives (red).
All of the above are taken at photon energy $h\nu=40.35\,$eV.

The phenomenon of band renormalization upon optical excitation is expected to
arise from transiently enhanced screening by high-energy electrons and holes.
It has been reported for gapless states of the nodal-line semimetal ZrSiSe
[46], and for CDW-state HoTe3 [45]. For the latter, enhanced screening
transiently reduces the tight binding parameter $t_{\perp}$, and nesting
conditions are improved as a result of this renormalization. On CuTe, in
contrast, this screening-induced renormalization has a consequence of
transient CDW melting. Fig.5(e) shows a difference spectrum taken at a time
delay of $200\,$fs, in which we see a renormalization of the band gradient. As
a result, the momentum at which the band crosses the Fermi level changes
accordingly. This situation is illustrated schematically in Fig.5(f). Since
the renormalized band (red) crosses the Fermi level with a momentum which
slightly differs from the nesting wavevector, the nesting condition is no
longer fulfilled and the CDW is transiently melted.

Overall, with our laser pump-probe trARPES experiment, we discover that, the
photoresponse of the system to femtosecond excitation comprises of a coherent
response to the softened phonon mode, which appear to be strongly coupled to
the 3D states, and an incoherent hot electron response lasting several
hundreds of femtoseconds. For the 1D states, only an incoherent dynamic was
observed, associated to an ultrafast gap filling and dispersion
renormalization towards worsened nesting conditions. Interestingly, the gap
filling relaxation time, 700 fs, is shorter than the hot electron relaxation
(around 2 ps) measured in the 3D states.

##  VI Conclusion

In this work, we combined different techniques and demonstrated for CuTe for
the first time to our knowledge, that the real space CDW superstructure and
the reciprocal space CDW gap size both have a periodicity in the out-of-plane
direction. These are evidences that the CDW wavevector
$\textbf{{q}}_{\text{CDW}}$ has a finite out-of-plane component. Together with
the prerequisite of inter-layer collectivity for CDW formation, we confirm
that the CDW in this quasi-1D material is 3D by all definitions. Our results
reveal that interlayer coupling effects play an important role in the CDW of
CuTe, and highlight the importance of combining multiple experimental probes.

## Acknowledgement

F.G. and J.H.D. acknowledge support from the Swiss National Science Foundation
(SNSF) Project No. 200021-200362

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