# Probing coherent quantum thermodynamics using a trapped ion

O. Onishchenko∗ Institut für Physik, Universität Mainz, Staudingerweg 7, 55128
Mainz, Germany  G. Guarnieri∗ Dahlem Center for Complex Quantum Systems, Freie
Universität Berlin, 14195 Berlin, Germany  P. Rosillo-Rodes  Institute for
Cross-Disciplinary Physics and Complex Systems, Campus Universitat de les
Illes Balears, E-07122, Palma, Spain  D. Pijn  J. Hilder  U. G. Poschinger
Institut für Physik, Universität Mainz, Staudingerweg 7, 55128 Mainz, Germany
M. Perarnau-Llobet  Department of Applied Physics, University of Geneva, 1211
Geneva, Switzerland  J. Eisert  Dahlem Center for Complex Quantum Systems,
Freie Universität Berlin, 14195 Berlin, Germany  F. Schmidt-Kaler  Institut
für Physik, Universität Mainz, Staudingerweg 7, 55128 Mainz, Germany

###### Abstract

Quantum thermodynamics is aimed at grasping thermodynamic laws as they apply
to thermal machines operating in the deep quantum regime, a regime in which
coherences and entanglement are expected to matter. Despite substantial
progress, however, it has remained difficult to develop thermal machines in
which such quantum effects are observed to be of pivotal importance. In this
work, we report an experimental measurement of the genuine quantum correction
to the classical work fluctuation-dissipation relation (FDR). We employ a
single trapped ion qubit, realizing thermalization and coherent drive via
laser pulses, to implement a quantum coherent work protocol. The results from
a sequence of two-time work measurements display agreement with the recently
proven quantum work FDR, violating the classical FDR by more than $10.9$
standard deviations. We furthermore determine that our results are
incompatible with any SPAM error-induced correction to the FDR by more than
$10\sigma$. Finally, we show that the quantum correction vanishes in the high-
temperature limit, again in agreement with theoretical predictions.

**footnotetext: These authors contributed equally to this work.

One of the pillars on which modern physics rests is classical phenomenological
thermodynamics. Born out of the scrutiny of the functioning of heat engines in
the 19th century, it is one of the most profound physical theories available,
offering a wealth of applications within physics and in everyday life. Its
strength originates from the fact that while it applies to the description of
complex systems, consisting of a large number of microscopic constituents, it
largely abstracts from details about these individual physical constituents:
It offers a description in terms of a small number of quantities, capturing
the relationship between heat, work, temperature, and energy. In recent years,
it has become increasingly clear that the rules of thermodynamics must be
sharpened for thermal machines operating in the quantum regime, a regime in
which coherence, entanglement and quantum fluctuations are expected to play a
role Goold  _et al._ (2016); Millen and Xuereb (2016); Kosloff (2013);
Vinjanampathy and Anders (2016); Gogolin and Eisert (2016). While such a
development was unforeseeable for the founders of thermodynamics, the question
of how thermodynamic notions should be altered has become highly relevant in
light of experimental progress in quantum engineering. In fact, the past few
decades have seen huge leaps towards experimental realizations of meso- and
nano-scale devices Dubi and Di Ventra (2011); Blickle and Bechinger (2012);
Martínez  _et al._ (2016); Roßnagel  _et al._ (2016); Josefsson  _et al._
(2018); Peterson  _et al._ (2019a), culminating in the emerging quantum
technologies Acin  _et al._ (2018). The tantalising prospect of harnessing
quantum features and exploiting them in order to outperform classical
counterparts has boosted research across many fields, ranging from quantum
biology Huelga and Plenio (2013) to quantum computation Arute  _et al._
(2019); Hangleiter and Eisert (2022); Boixo  _et al._ (2018).

Figure 1: Experimental implementation scheme for detecting quantum work
fluctuations using a trapped-ion qubit. a) Sequence of operations for
obtaining a single measured value of work. The flame represents thermalization
to a Gibbs state in the computational basis. $\ket{0}$ represents qubit
initialization. A two-point measurement is realized around a work step, given
by a qubit rotation $\hat{R}(\theta)$. The inset shows an non-demolition
energy measurement, which is emulated via conditional re-initialization. b)
The relevant energy levels of a ${}^{40}\text{Ca}^{+}$ ion. The qubit is
encoded in the Zeeman sub-levels of the $4^{2}\text{S}_{1/2}$ ground state.
The Raman detuning $\Delta_{\text{R}}$ is typically 250 GHz. c) Discrete
protocol visualized on the Bloch sphere.

A rich body of theoretical work has provided guidelines of what to expect in
this exciting development in quantum thermodynamics Goold  _et al._ (2016);
Millen and Xuereb (2016); Kosloff (2013); Vinjanampathy and Anders (2016);
Gogolin and Eisert (2016); Brandão  _et al._ (2015). However, to see evidence
of _genuine quantum effects_ in experimentally realized microscopic thermal
machines seems to be harder to come across. Thermal machines operating _with_
single quantum systems and featuring signatures of quantum coherence have been
devised Myers  _et al._ (2022); Maslennikov  _et al._ (2019); von Lindenfels
_et al._ (2019); Peterson  _et al._ (2019b); Bouton  _et al._ (2021); Wenniger
_et al._ (2022) and compared to their classical counterpart Klatzow  _et al._
(2019); but the task of actually verifying and statistically significantly
observing genuine quantum signatures that have no classical analogue in
thermodynamics remains elusive. This is in particular relevant when assessing
the important question how quantum coherence affects protocols of actual work
extraction.

It has become clear, again from a theoretical perspective, that fluctuations
as inspired by the notions of quantum _stochastic thermodynamics_ Esposito
_et al._ (2009); Campisi  _et al._ (2011); Bäumer  _et al._ (2018) may be the
tool that offers to discriminate quantum from classical prescriptions and
actually provides the answer. Within this general framework, all thermodynamic
quantities (such as work) become stochastic variables at the microscopic
level, that can be defined at the level of single trajectories or dynamical
realizations Esposito  _et al._ (2009); Campisi  _et al._ (2011); Miller  _et
al._ (2021) and are therefore distributed according to probability
distributions from which fluctuations can be computed. Quantum effects most
prominently manifest themselves as such fluctuations Allahverdyan (2014);
Talkner and Hänggi (2016). In particular, these have a prominent impact on the
so-called _work fluctuation-dissipation relation_ (FDR) Hermans (1991);
Jarzynski (1997); Speck and Seifert (2004)

$\displaystyle\frac{\beta}{2}\mathrm{Var}(W)=\langle W\rangle-\Delta F,$ (1)

which is valid in linear response, i.e., when the system remains close to
thermal equilibrium at all times Speck and Seifert (2004); Miller  _et al._
(2019); Scandi  _et al._ (2020). Eq. (1) relates the first two cumulants of
the work distribution, i.e., $\langle W\rangle$ and $\mathrm{Var}(W)$, for a
slowly driven system in contact with a thermal bath at inverse temperature
$\beta>0$, with $\Delta F$ being the change in free energy between the two
endpoints of the process. Eq. (1) has moreover been confirmed in various
experimental platforms in mesoscopic systems Jun  _et al._ (2014); Barker  _et
al._ (2022).

However, it has recently been shown Miller  _et al._ (2019); Scandi  _et al._
(2020) that Eq. (1) is violated in the presence of _quantum friction_ , i.e.,
the generation of coherence in the instantaneous energy eigenbasis. The
observation of these quantum violations requires two basic ingredients: (i)
coherent driving of the system of interest, in the sense that the Hamiltonian
at different times does not commute, $[H(t),H(t^{\prime})]\neq 0$ for some
$t^{\prime}\neq t$, so that coherence is dynamically generated, and (ii)
coupling of the system to a thermal bath that drives it into a Gibbs state
with respect to the instantaneous Hamiltonian $H(t)$. Trapped ion qubits offer
an excellent degree of control in terms of coherent and dissipative operations
and thus represent an ideal platform for studying microscopic thermodynamics
An  _et al._ (2014); Maslennikov  _et al._ (2019); Pijn  _et al._ (2022); Yan
_et al._ (2022).

In this work, we report on the experimental observation of such a _genuine
quantum correction_ using a single trapped-ion qubit. Our results clearly show
a positive quantum correction to the work FDR Eq. (1) due to quantum coherent
driving, which is shown to vanish in the high-temperature limit, where thermal
fluctuations dominate. Importantly, we demonstrate that such correction
measurements are incompatible by more than $10.9\sigma$ with values obtained
from any incoherent (i.e., classical) protocol at finite driving speed and by
more than $12.1\sigma$ with values stemming from _state preparation and
measurement_ (SPAM) errors, thus statistically certifying the genuine quantum
nature of our findings.

Experimental implementation. We encode a qubit in the spin of the valence
electron of a ${}^{40}\text{Ca}^{+}$ ion Poschinger  _et al._ (2009) confined
in a segmented Paul trap Schulz  _et al._ (2008). Fig. 1 b) shows the relevant
energy levels and transitions. A static magnetic field gives rise to an energy
splitting of $\omega_{q}\approx 2\pi\times 10.5~{}\text{MHz}$ between the
Zeeman sub-levels of the $4^{2}\text{S}_{1/2}$ ground state, which are taken
to be the logical basis vectors $\ket{0}:=\ket{m_{j}=+1/2}$ and
$\ket{1}:=\ket{m_{j}=-1/2}$ Ruster  _et al._ (2016). Qubit initialization to
state vector $\ket{0}$ is realized via optical pumping on the
$4^{2}\text{S}_{1/2}\leftrightarrow 4^{2}\text{P}_{1/2}$ transition using a
circularly polarized driving field. A state preparation fidelity of better
than 99% is obtained by an additional pumping stage, consisting of repeated
selective depletion of one of the qubit levels via a $\pi$-pulse driving the
narrow $4^{2}\text{S}_{1/2}\leftrightarrow 3^{2}\text{D}_{5/2}$ electric
quadrupole transition at about 729 nm, followed by depletion of the meta-
stable state by driving the $3^{2}\text{D}_{5/2}\leftrightarrow
4^{2}\text{P}_{3/2}$ transition at about 854 nm.

A (thermal) Gibbs state in the logical basis $\\{\ket{0},\ket{1}\\}$ is
prepared by controlled incomplete optical pumping. We perform the pumping
scheme on the quadrupole transition as mentioned above, selectively depleting
$\ket{0}$. The control over the transfer probability from state vector
$\ket{0}$ via the pulse area enables us to prepare any Gibbs state of the form

$\displaystyle\hat{\rho}=\frac{1}{\mathcal{Z}}\left(|0\rangle\langle
0|+e^{-\beta\hbar\omega_{q}}|1\rangle\langle 1|\right),$ (2)

where the partition function $\mathcal{Z}$ ensures normalization. The
effective inverse temperature $\beta>0$ is controlled by the initial excited
state probability $p$ via von Lindenfels  _et al._ (2019)

$\text{Tr}(\hat{\sigma}_{z}\hat{\rho})=1-2p=\mathrm{tanh}\left(\frac{\beta\hbar\omega_{q}}{2}\right).$
(3)

We calibrate $\beta$ by repeatedly performing incomplete optical pumping and
projective measurement in the logical basis.

Coherent qubit rotations are performed via stimulated Raman transitions driven
by a pair of off-resonant beams, far red-detuned from the
$4^{2}\text{S}_{1/2}\leftrightarrow 4^{2}\text{P}_{1/2}$ transition, see Fig.
1b. The large detuning ensures that the qubit rotations is performed without
significant parasitic dissipation. The drive Hamiltonian
$\hat{H}_{d}=\hbar\Omega\hat{\sigma}_{x}/2$ generates the evolution
$\hat{R}(\theta)=\exp(-i(\theta/2)\hat{\sigma}_{x})$, where the “pulse area”
$\theta$ is accurately controlled with acousto-optic modulators. Starting from
a logical basis state, the state after applying $\hat{U}(\theta)$ is an
eigenstate of the Hamiltonian

$\hat{H}_{\text{coh}}(\theta)=\frac{\hbar\omega_{q}}{2}\left(\sin(\theta)\hat{\sigma}_{y}-\cos(\theta)\hat{\sigma}_{z}\right).$
(4)

The operation $\hat{R}(\theta)$ creates coherence in the logical basis, such
that the Gibbs state (2) can be prepared in an arbitrary basis. In the
following, we set $\hbar\omega_{0}:=1$ without loss of generality. Qubit
readout is performed by selective population transfer from $\ket{0}$ to the
meta-stable state $3^{2}\text{D}_{5/2}$ Poschinger  _et al._ (2009). After
that, the detection of state-dependent fluorescence using 397-nm light reveals
the result $\ket{1}$ (“bright”) or $\ket{0}$ (“dark”). This qubit readout is
destructive, as the post-measurement state ends up being completely
depolarized. Realizing a projection-valued measurement therefore requires re-
initializing the qubit after a measurement by optical pumping, followed by a
$\pi$-pulse conditional on the previous measurement result, see Fig. 1a.

_Quantum fluctuation-dissipation relation protocol._ We realize a discrete
protocol of alternating driving and thermalization steps as proposed in Ref.
Scandi  _et al._ (2020), which has the additional benefit of not requiring any
dynamical detail about the relaxation process. Throughout the process, the
angle $\theta$ in (4) is varied from $\theta=0$ to $\theta=\pi/2$ in discrete
steps $j=0,\dots,N-1$, changing the effective Hamiltonian from
$\hat{H}^{i}_{\text{coh}}=-{\hbar\omega_{q}}\hat{\sigma_{z}}/2$ to
$\hat{H}^{f}_{\text{coh}}=\hbar\omega_{q}\hat{\sigma}_{y}/2$. For each step,
the following operations are carried out: (i) The qubit is prepared in a Gibbs
state at inverse temperature $\beta$, followed by a readout in the
computational basis, which yields the outcome $e_{j}\in\\{0,1\\}$, (ii) the
qubit is prepared in the state vector $\ket{0}$, followed by a $\pi$-pulse
only if $e_{j}=1$, (iii) a coherent rotation $\hat{R}(\Delta\theta)$ by angle
$\Delta\theta=\pi/2N$ is implemented, and finally (iv) a readout, which gives
$e_{j}^{\prime}\in\\{0,1\\}$. We consider the difference of these measurements
as work after step $j$: $w_{j}=e_{j}^{\prime}-e_{j}$; which follows the
standard _two-point-measurement_ (TPM) scheme Talkner  _et al._ (2007);
Campisi  _et al._ (2009); Wang  _et al._ (2022), based on projective
measurements of the qubit energy at the beginning and at the end of each
driving step. The full protocol is schematically illustrated in Fig. 1.

The rotation angle $\Delta\theta$ is calibrated experimentally by preparing
the qubit in the state vector $\ket{0}$, then making a pulse of length
$t_{\text{Raman}}$ with the Raman beam pair, and finally performing a
measurement. This experiment is repeated a few thousand times for different
$t_{\text{Raman}}$, while keeping the intensity, and thus the two-photon Rabi
frequency $\Omega$ constant. The probability of reading out “bright”
(corresponding to 1) is given by $P_{\text{bright}}=\sin^{2}(\Omega
t_{\text{Raman}})/2)$ Foot (2005), and that can be related to the probability
of detecting “bright” after a rotation on the Bloch sphere by angle $\theta$
from the pole, $P_{\text{bright}}=\sin^{2}(\theta/2)$. Even though
$P_{\text{bright}}$ is not a linear function $t_{\text{Raman}}$, we have
chosen closely-spaced values of $t_{\text{Raman}}$ so that $P_{\text{bright}}$
is linearized. We can then determine the required $t_{\text{Raman}}$ by linear
regression.

For each step, the probabilities to measure positive or negative work within
the TPM are given by $P(w_{j}=+1)=(1-p)\;\sin^{2}(\Delta\theta/2)$,
$P(w_{j}=-1)=p\;\sin^{2}(\Delta\theta/2)$, and
$P(w_{j}=0)=1-P(w_{j}=+1)-P(w_{j}=-1)$, with the initial thermal excited state
population $p$ from Eq. (3). Since each thermalization step resets any
information that has been available in the previous state, the protocol is
effectively Markovian and the step work probabilities $w_{j}$ are
statistically identical and independent. This justifies starting each step in
$\ket{0}$ for the sake of experimental simplicity. The total work accumulated
throughout the process is the sum of individual $w_{j}$:
$W=\sum_{j=1}^{N}w_{j}$. Due to thermal and eventual quantum fluctuations, $W$
is a stochastic quantity.

From the first and second moments, i.e., its mean $\langle W\rangle$ and its
variance ${\rm Var}(W)$, we quantify violations of the work FDR Eq. (1) by
introducing the quantity

$\mathcal{Q}=\frac{\beta}{2}{\rm Var}(W)-\left[\langle W\rangle-\Delta
F\right].$ (5)

Note that for the processes considered, the free energy change $\Delta F$ in
Eq. (1) is invariant under a basis change given by the effective Hamiltonian
Eq. (4), and therefore $\Delta F=0$ and the total work $W$ equals the
dissipated work. We also recall that the regime of validity of the work FDR
(1) is the linear-response or slow driving regime; for the protocols
considered here, this means focusing on protocols with $N\gg 1$ and keeping
the magnitudes of interest at order $\mathcal{O}\left(N^{-1}\right)$ Scandi
_et al._ (2020). Explicit analytical expressions for the quantities appearing
in Eq. (5) are provided in the Supplementary Material.

#### Results and discussion.

We first characterize the quantum correction $\mathcal{Q}$ to the FDR by
collecting work samples for different values of the step number $N$ at a fixed
inverse temperature $\beta=3.413\pm 0.025$ (in units of $\hbar\omega_{0}$),
corresponding to a excited state population $p=0.032\pm 0.001$. For each value
of $N$, we repeat the protocol 8000 times and compute the sample mean and
sample variances from the total work values $W$ pertaining to each work
sample. This allows us to reveal the quantum correction Eq. (5) for different
values of $N$, see Fig. 2.

Figure 2: Quantum correction $N\mathcal{Q}/\|\Delta H\|$ as a function of the
inverse of the process velocity $v^{-1}=N/\|\Delta H\|$. The slow driving
regime $v\ll 1$ corresponds to the large number of subdivisions $N\gg 1$. The
results of the experimental measurements (black dots) are shown together with
the theoretical curve (solid red line) calculated for a discrete fully
coherent protocol. The plot also displays the simulations of thousands of
incoherent processes (blue markers) obtained by both varying all possible
qubit’s energy gap $\delta\omega=\omega_{f}-\omega_{0}=2\|\Delta H\|$ and the
number of subdivisions $N$ and such that their ratio, i.e., the speed
$v=\|\Delta H\|/N$ is within the range plotted. Finally, the values of
$N\mathcal{Q}_{\text{SPAM}}/\|\Delta H\|$ are also displayed (black empty
circle marker). This plot clearly shows that the experimental points have a
finite separation from the region (in light-blue color) of incoherent
processes and from the SPAM error region (in light-purple color), thus
quantitatively proving that the measured quantum correction
$N\mathcal{Q}/\|\Delta H\|$ is a genuine quantum signature.

To quantitatively certify that this value of $\mathcal{Q}$ is a genuine
quantum effect and not one caused by thermal fluctuations, finite $N$, or
experimental imperfections, we have performed an in-depth two-step analysis.
First of all, it is important to notice that a deviation from the standard FDR
Eq. (1) is theoretically demonstrated to vanish for incoherent processes
(i.e., ones where only the eigenvalues of the Hamiltonian change over time,
but not the eigenbasis) only within the slow-driving regime. This means that,
for finite values of the driving speed, one can observe a positive deviation
from the classical FDR even for incoherent processes, which would not be a
quantum signature. Such incoherent-stemming FDR deviations, however, would
decay as $N\mathcal{Q}\sim 1/N$, in contrast to those of quantum origin that
would instead reach an asymptotic positive value $N\mathcal{Q}\sim 1$, see
Fig. 2. To rule this possibility out on a quantitative basis, and therefore
prove that our observed values of $\mathcal{Q}$ are truly due to the coherent
rotations of the qubit’s Hamiltonian, we first introduce the notion of a
_speed_ as $v={\|\Delta H\|}/{N}$, where $\|\Delta H\|$ denotes the operator
norm of the change in the system’s Hamiltonian and where the slow driving
regime is recovered when $N\gg\hbar\omega_{0}(=1)$. We then proceed by
simulating hundreds of thousands of incoherent protocols, i.e., such that the
Hamiltonian is driven in time from
$\hat{H}_{\text{incoh}}^{i}={-\hbar\omega_{0}}\hat{\sigma}_{z}/2$ to
$\hat{H}_{\text{incoh}}^{f}={-\hbar\omega_{f}}\hat{\sigma}_{z}/2$, thus
changing in $N$ discrete steps the qubit’s energy gap
$\hbar\omega_{j}=\hbar\omega_{0}+(\omega_{f}-\omega_{0})j/N$, while keeping
the energy eigenbasis fixed to $\hat{\sigma}_{z}$. It is immediate to see
that, for this class of protocols, $\|\Delta H\|=(\omega_{f}-\omega_{0})/2$,
while for the coherent protocol performed in the experiment, one has $\|\Delta
H\|=1/\sqrt{2}$. In order to quantitatively compare our experimental
observations with the values of FDR corrections compatible with incoherent
processes for any possible driving speed, we plot the rescaled quantity
$N\mathcal{Q}/\|\Delta H\|\equiv\mathcal{Q}/v$. The results of these
simulations are displayed with blue markers in Fig. 2, together with the
experimentally measured values of $N\mathcal{Q}/\|\Delta H\|$ (black dots) and
the corresponding fully coherent driving protocol (red solid curve). This
analysis clearly shows that our experimental points lie beyond the region of
values for $N\mathcal{Q}/\|\Delta H\|$ attainable by any incoherent process
(light blue shaded region) and therefore provide a striking evidence that
those measured corrections are only compatible with a genuinely quantum
coherent process.

Point | $\sigma^{\text{stat}}$ | $\delta^{\text{inc}}$ | $\delta^{\text{SPAM}}$
---|---|---|---
$(2.828,0.438)$ | $\sigma_{1}=0.021$ | $11\sigma_{1}$ | $19.8\sigma_{1}$
$(4.243,0.530)$ | $\sigma_{2}=0.027$ | $14.8\sigma_{2}$ | $18.9\sigma_{2}$
$(5.657,0.537)$ | $\sigma_{3}=0.028$ | $15.5\sigma_{3}$ | $17.5\sigma_{3}$
$(7.071,0.568)$ | $\sigma_{4}=0.032$ | $15.2\sigma_{3}$ | $15.5\sigma_{3}$
$(8.845,0.619)$ | $\sigma_{5}=0.033$ | $16.8\sigma_{3}$ | $15.7\sigma_{3}$
$(9.899,0.581)$ | $\sigma_{6}=0.036$ | $14.4\sigma_{3}$ | $12.1\sigma_{3}$
Table 1:  Verification of the quantum fluctuation dissipation relation. The
statistical margin $\sigma^{\text{stat}}$, the distance to the classical
prediction from the incoherent process $\delta^{\text{inc}}$ and to that due
to SPAM $\delta^{\text{SPAM}}$ is given.

Furthermore, in contrast to the ideal case where energy measurements in the
TPM scheme are error-free, experimental measurement-readout errors may occur.
In our setup, the second measurement of each TPM has a small but non-zero
conditional probability $p_{d|1}$ of incorrectly reading out the qubit as
“dark” when it was in the ‘bright” state $\ket{1}$, and vice-versa. As we show
in detail in the Supplementary Material, the introduction of these measurement
readout errors leads to a spurious non-zero correction
$N\mathcal{Q}_{\text{SPAM}}/\|\Delta H\|$. As trapped-ion qubits feature low
SPAM, we can show that the quantum correction is above $12.1\sigma$ away from
the spurious broadening of the FDR due to SPAM for all the experimentally
measured points. We, moreover, show that, at a striking difference from both
the quantum coherent $N\mathcal{Q}/\|\Delta H\|$ (which reaches an asymptotic
constant value for increasing $N$) and the fully incoherent correction (which
decreases to zero as $1/N$), $N\mathcal{Q}_{\text{SPAM}}/\|\Delta H\|$
linearly increases with the number of steps $N$. This has the important
consequence that, in presence of such readout errors, any observation of the
genuinely quantum correction Eq. (5) would be completely hindered in the
quasi-static limit $N\rightarrow\infty$. It is precisely the outstanding
control offered by ultra-cold trapped ion platforms that allowed us to keep
track and have very small readout error probabilities, thus allowing to
witness a quantum correction to the classical FDR that was statistically shown
to be incompatible with both incoherent (i.e., classical) processes and also
with any value due to the above-mentioned SPAM error. The black empty circles
and the underlying purple region in Fig. 2 show the region of values of
$N\mathcal{Q}/\|\Delta H\|$ that would be indistinguishable from a worst-case
scenario $N\mathcal{Q}_{\text{SPAM}}/\|\Delta H\|$ computed by performing
energy measurements without any in-between qubit rotations. In particular, our
result shows that the experimental points have a statistical distance above
$12.1\sigma$ both to the region of values of $N\mathcal{Q}$ compatible with
incoherent processes and with SPAM errors (see Table 1 for all the specific
values).

Figure 3: Measured quantum correction $N\mathcal{Q}/\|\Delta H\|$ as a
function of the inverse temperature for $N=5$. Error bars are due to the
counting statistics with 8000 repetitions. The theory prediction Eq. (Results
and discussion.) is plotted from high (red) to low (blue) temperatures,
without free parameters. The maximum SPAM-induced fluctuation readout is
denoted by the dashed line.

We complete our analysis by measuring $N\mathcal{Q}$ as a function of the
temperature for a fixed number of subdivisions $N=5$, see Fig. 3. First of
all, it can be clearly seen that the quantum correction $N\mathcal{Q}$
correctly reproduces the behaviour

$\displaystyle N\mathcal{Q}/\|\Delta H\|$
$\displaystyle=(\pi^{2}\sqrt{2}/4)\left[\beta/2-\tanh\left(\beta/2\right)\right]+\mathcal{O}(1/N)$
$\displaystyle\simeq(\pi^{2}\sqrt{2}/4)\left[0+\mathcal{O}(\beta^{2})\right]+\mathcal{O}(1/N)$
(6)

where the second line shows a quadratic decrease to zero in the high-
temperature limit, where thermal fluctuations dominate (see Supplementary
Material for details). Instead, at low temperatures, the excess fluctuations
$\mathcal{Q}$ emerge from quantum coherence, which also leads to a non-
Gaussian work distribution Scandi  _et al._ (2020).

#### Conclusions and outlook.

In this work, we have exploited the controllability of a trapped-ion qubit
platform to perform a measurement of a quantum thermodynamic signature, namely
the quantum correction to the work fluctuation-dissipation relation. This has
been realized by performing a sequence of $N$ alternating coherent drives,
thermalization steps and energy non-demolition measurements on a single qubit.
Our result has revealed a quantum correction to the classical work FDR Eq. (1)
which was proven to be statistically incompatible by more than $10.9$ standard
deviations with any incoherent protocol, and incompatible by more than $12.1$
standard deviations with any SPAM-induced error. This conclusion thus
certifies the genuine quantum nature of our measurements. Moreover, we have
shown that the spurious correction to the FDR induced by small, but non-zero,
error probabilities in the measurement readout, which we called
$N\mathcal{Q}_{\text{SPAM}}/\|\Delta H\|$, has the general property of
linearly growing with the number of subdivisions $N$, a scaling in stark
contrast both with the incoherent $N\mathcal{Q}_{\text{inco}}/\|\Delta H\|$,
which decreases as $~{}1/N$, and with the quantum correction
$N\mathcal{Q}/\|\Delta H\|$, which approaches a constant positive asymptote.

We believe that our research represents a significant result in the direction
of experimental observation and certification of genuine quantum effects in
small-scale platforms, thus pushing forward the still painfully scarce body of
literature reporting first experimentally observed quantum effects in quantum
thermodynamics, relating to squeezed resources Maslennikov  _et al._ (2019),
quantum effects in spin quantum heat engines Peterson  _et al._ (2019b), steps
towards quantum heat engines driven by atomic collisions Bouton  _et al._
(2021), coherence between solid-state qubits and light fields Wenniger  _et
al._ (2022), signatures of internal coherence Klatzow  _et al._ (2019), or
verifications of quantum fluctuation relations Micadei  _et al._ (2021).
Furthermore, our findings tie in with recent insights into _coherence as a
resource_ Streltsov  _et al._ (2017), which in our case resulted in measurable
quantum signatures in the work statistics. Finally, this invites further
exciting endeavours: for example, observing quantum effects in quantum field
thermal machines in the realm of _quantum many-body physics_ Gluza  _et al._
(2021). It also seems conceivable that further quantum corrections can be
measured in work extraction experiments, witnessing temporal coherence
reflecting highly _non-Markovian quantum dynamics_ Wolf  _et al._ (2008);
Breuer  _et al._ (2016); Rivas  _et al._ (2014), a feature that could be
exploited in order to further further curb down the SPAM measurement-readout
errors, allowing to access slower protocols at higher $N$. One may also bring
these experimental results into contact with a body of theoretical work on
_single-shot work extraction_ Brandão  _et al._ (2015); Gallego  _et al._
(2016); Alhambra  _et al._ (2016). It is the hope that the present work will
stimulate further efforts of experimentally exploring the deep quantum regime
in quantum thermodynamics.

#### Acknowledgements.

We warmly thank J. Anders for helpful comments, and H. Miller and M. Scandi
for insightful discussions. This work has been funded by the DFG (FOR 2724,
for which this is an inter-node collaboration reaching an important milestone,
and CRC 183) and the FQXI. G. G. acknowledges fundings from European Union’s
Horizon 2020 research and innovation programme under the Marie Sklodowska-
Curie Grant Agreement INTREPID, No. 101026667. MPL acknowledges funding from
Swiss National Science Foundation through an Ambizione Grant No.
PZ00P2-186067.

Supplementary Material

This Supplementary Material contains additional details about the data
analysis and the theoretical calculations. In particular, Section A provides
further insight and derivation of Eq. (6) of the main text, while Section B
contains a detailed discussion, organized in subsections, of all relevant
errors used in the generation of Figs. 2 and 3 of the main text.

##  I Section A: Analytic expressions for the cumulants of the work
distribution

In this Section, we provide explicit expressions for the first and second
cumulants of the work distribution distribution, i.e., its mean $\langle
W\rangle$ and its variance ${\rm Var}(W)$, calculated for the proposed
protocol. In order to do so, we need to make use of the expressions for the
step work probabilities to measure positive or negative work within the TPM
protocol, i.e.,

$\displaystyle P(w_{j}=+1)=(1-p)\;p_{f},$ $\displaystyle
P(w_{j}=-1)=p\;p_{f},$ $\displaystyle P(w_{j}=0)=1-P(w_{j}=+1)-P(w_{j}=-1).$
(S1)

Here, $p$ denotes the thermal excited state population from Eq. (3) of the
main text and $p_{f}$ denotes the flip probability induced by the qubit
rotation.

In the case of the coherent protocol describing the driving
$\hat{H}^{i}_{\text{coh}}=-{\hbar\omega_{q}}\hat{\sigma_{z}}/2$ to
$\hat{H}^{f}_{\text{coh}}=\hbar\omega_{q}\hat{\sigma}_{y}/2$, the ideal flip
probability for the case of $N$ subdivisions is given by

$p_{f}^{(\text{ideal})}=\sin^{2}\left(\frac{\pi}{4N}\right).$ (S2)

Using this probability, one can compute the first two cumulants of the total
work accumulated, which read

$\displaystyle\langle
W\rangle=N\left(1-2p\right)\sin^{2}\left({\frac{\pi}{4N}}\right),$
$\displaystyle\text{Var}(W)=N\sin^{2}\left({\frac{\pi}{4N}}\right)\left(1-\sin^{2}\left({\frac{\pi}{4N}}\right)(1-2p)^{2}\right).$
(S3)

Expanding these expressions at leading order in the number of steps
corresponds to the slow-driving regime,

$\displaystyle\langle
W\rangle=\tanh\left(\frac{\beta}{2}\right)\frac{\pi^{2}}{4N}+\mathcal{O}(1/N^{2}),$
$\displaystyle\text{Var}(W)={\frac{\pi^{2}}{4N}}+\mathcal{O}(1/N^{2}).$ (S4)

This implies that the quantum correction $\mathcal{Q}$, in the protocol we
experimentally implement, has the following analytical expression

$\displaystyle\mathcal{Q}=\frac{\beta}{2}\text{Var}(W)-\langle W\rangle$
$\displaystyle=N\sin^{2}\left({\frac{\pi}{4N}}\right)\left[\frac{\beta}{2}\left(1-\sin^{2}\left({\frac{\pi}{4N}}\right)(1-2p)^{2}\right)-(1-2p)\right]$
(S5)
$\displaystyle=\frac{\pi^{2}}{4N}\left[\frac{\beta}{2}-\tanh\left(\frac{\beta}{2}\right)\right]+\mathcal{O}(1/N^{2}),$
(S6)

where we use the relation $1-2p=\tanh{\beta/2}$ (we remind that we set
$\hbar\omega_{0}=1$) . First of all, this expression clearly shows that
$\mathcal{Q}\geq 0$ for all $N$, since

$\displaystyle\frac{\beta}{2}\geq\tanh\left(\frac{\beta}{2}\right)$ (S7)

for all $\beta>0$. Furthermore, it highlights that the quantum correction
vanishes in the high-temperature limit $\beta\to 0$ as $\tanh(x)\approx
x+\mathcal{O}(x^{2})$, thus leading to the standard fluctuation-dissipation
relation (see Eq. (1) of the main text).

##  II Section B: Error estimation

We discuss how statistical and systematic errors for the quantity
$N\mathcal{Q}/\|\Delta H\|$ shown in Fig. 2 and 3 of the main text (with $N$
being the number of subdivisions and $\|\Delta H\|$ being the norm of the
change in the qubit’s Hamiltonian) are estimated.

###  II.1 Subsection B.1: Parameter extraction

Each measured value of the quantity $N\mathcal{Q}/\|\Delta H\|$ is estimated
from $M=~{}8000$ independent runs, each consisting of $N$ _two-point energy
measurements_ (TPM). Each TPM is comprised of two single-qubit readouts in the
computational basis. The inverse effective temperature $\beta$ is determined
by the probability $p$ of the qubit to be in $\ket{0}$ after the first
measurement, according to Eq. (3) of the main text. With a binomial
statistical error $\sigma^{\text{(b)}}=\sqrt{p(1-p)/(N\;M)}$, the statistical
error of $\beta$ is given by

$\displaystyle\sigma_{\beta}=\frac{d\beta}{dp}|_{p}\sigma^{\text{(b)}}.$ (S8)

The probability of the result of the second measurement for each TPM being
different from the first result is given by the flip probability $p_{f}$ and
can therefore be estimated from the overall frequency of flipped results. It
is also subject to the same binomial statistical errors
$\sigma^{\text{(b)}}_{f}=\sqrt{p_{f}(1-p_{f})/(N\;M)}$.

###  II.2 Subsection B.2: Statistical errors of $\mathcal{Q}$

The measured values of the quantum correction to the FDR $\mathcal{Q}$ are
computed from: $\beta$, the average work $\langle W\rangle$ and its variance
$\text{Var}(W)$ (see Eq. (5) of the main text). The resulting statistical
error of $\mathcal{Q}$ is computed via bootstrapping, where for each parameter
set $(N,\beta,\theta=\pi/(4N))$, 200 artificial data sets are generated, each
consisting of $NM$ statistically independent TPMs. This is done by randomly
initializing the qubit in the “dark” state $\ket{0}$ with a probability $p$
and then flipping it with a probability $p_{f}$. For each of the 200 data
sets, the mean work, work variance and $\beta$ are determined, from which a
bootstrapping value of $\mathcal{Q}$ is computed. The bootstrapping
statistical error $\sigma_{\mathcal{Q}}^{(BS)}$ of a given $\mathcal{Q}$ value
is the sample standard deviation of the the 200 $\mathcal{Q}$ values obtained
for each data set. Finally, $\sigma_{\mathcal{Q}}^{(BS)}$ includes the
bootstrapping errors $\sigma_{\beta}^{(BS)}$ on the inverse temperature
(through Eq. (S5)), which are shown as horizontal error bars in Fig. (3) of
the main text.

###  II.3 Subsection B.3: State preparation and measurement (SPAM) errors

Both state preparation and measurement readout of the qubit are error-prone.
After the first measurement of each TPM, the qubit is reinitialized via
optical pumping on the quadrupole transition and an optional $\pi$-pulse. The
state preparation error for both these processes is very small $\lesssim$
0.1%, and therefore it can be neglected. For each TPM, the state is re-
prepared based on the result of the first measurement, and the true value of
$p$ is given by the dark event probability of the first measurement slot. The
determination of $\beta$ can be considered to be free of state preparation and
measurement errors.

However, measurement errors have to be taken into account for the second
measurement of each TPM, based on the conditional error probabilities
$p_{b|0}~{}(p_{d|1})$ of incorrectly reading out qubit as “bright” (“dark”)
when it has actually been in $\ket{0}(\ket{1})$. These conditional error
probabilities are determined from separate calibration measurements, which
merely consist of state preparation both in $\ket{0}$ and $\ket{1}$ and
readout. Using at least 10000 shots for each calibration measurement, we
determine readout error rates of about $0.4\pm 0.1$ %.

The measurement error leads to modified probabilities to measure nonzero work
events within each TPM protocol. In particular, Eq. (I) becomes

$\displaystyle P^{\prime}(w_{j}=+1)$ $\displaystyle=$
$\displaystyle(1-p_{d|1})P(w_{j}=+1)+p_{b|0}P(w=0),$ $\displaystyle
P^{\prime}(w_{j}=-1)$ $\displaystyle=$
$\displaystyle(1-p_{b|0})P(w_{j}=-1)+p_{d|1}P(w_{j}=0),$ (S9)

where $P(w_{j}=\pm 1,0)$ are the probabilities defined in Eq. (I). Eq. (II.3)
gives rise to spurious values of the correction to the classical FDR we we
call $\mathcal{Q}_{\text{SPAM}}$. Specifically, in order to single out the
effect of such SPAM errors, we compute $\mathcal{Q}_{\text{SPAM}}$ when
coherent drive on the qubit’s Hamiltonian is performed (i.e., $\theta=0$) and
only TPM readouts are carried out according to Eq. (II.3). This quantity
provides the worst-case estimation of a spurious FDR correction, which reads

$\mathcal{Q}_{\text{SPAM}}=N\left[\frac{\beta}{2}\left[-(p\,p_{d|1}-(1-p)p_{b|0})^{2}-pp_{b|0}+p\,p_{d|1}+p_{b|0}\right]+\left[p\,p_{d|1}-(1-p)p_{b|0}\right]\right].$
(S10)

Eq. (S10) clearly shows that this quantity linearly grows with the number of
subdivisions $N$, thus becoming the predominant contribution in the slow
driving regime $N\gg 1$.

###  II.4 Subsection B.4: Parameter drifts

The parameters $\beta$ and $\theta$ are determined by laser intensities, which
are subject to drifts over the course of the data acquisition. For each value
of $\mathcal{Q}$, the parameters are estimated from $MN$ runs. In order to
analyze the impact of such potential drifts, these runs are partitioned into
$(MN)/K$ bins of size $K$ each, and the probabilities $p$ (from which $\beta$
is computed) and $p_{f}$ (from which $\theta$ is computed) are calculated for
each bin. Then, the standard deviation among the various $p$ and $p_{f}$ is
computed and compared to the standard deviation expected from a binomial
distribution, as explained in Subsection B.1. For all data sets, no
significant excess spread is observed, i.e., $\delta p,\delta p_{f}<$ 0.01
throughout each estimation of $\mathcal{Q}$. We conclude that drift errors are
therefore negligible.

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