# Two-point measurement energy statistics from particle scattering

Samuel L. Jacob<EMAIL_ADDRESS>Department of Physics, Trinity College Dublin,
Dublin 2, Ireland  Gabriel T. Landi  Department of Physics and Astronomy,
University of Rochester, Rochester, New York 14627, USA
<EMAIL_ADDRESS>Massimiliano Esposito
<EMAIL_ADDRESS>Complex Systems and Statistical Mechanics,
Physics and Materials Science Research Unit, University of Luxembourg, L-1511
Luxembourg, G.D. Luxembourg  Felipe Barra<EMAIL_ADDRESS>Departamento
de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile,
837.0415 Santiago, Chile

(August 27, 2024)

###### Abstract

We show that the energy statistics resulting from a two-point measurement of
an isolated quantum system subject to a time-dependent driving protocol can be
probed by subjecting the same system to a collision with a suitably prepared
incoming particle. This means that the particle acts both as an external drive
and as an energy measurement device for the quantum system and that energy
fluctuations can be defined within a fully autonomous setting.

_Introduction. —_ Fluctuations are ubiquitous at the micro-scale [1, 2], but
properly defining quantum fluctuations is not always an easy task. This is
particularly true out-of-equilibrium when considering fluctuations of
quantities which characterize flows across a system, for instance work and
particle currents. The most widely adopted method resorts to a two-point
measurement (TPM) which consists in projectively measuring the observable of
interest at two different times [3, 4, 5]. The resulting probability
distribution for changes in outcomes can then be used to recover, for
instance, the celebrated work and current fluctuation relations [6, 7, 8, 4,
3, 5, 9].

However, these projective measurements are challenging from an experimental
point of view. To our knowledge, a direct implementation in a quantum system
has only been realized in Ref. [10]. This has led to a flurry of research to
find alternative ways of measuring energy statistics. One famous approach is
to perform Ramsey interferometry on the system [11, 12] and has been
successfully implemented in nuclear magnetic resonance experiments [13].
Alternatively, one can use generalized quantum measurements, in which case the
energy statistics can be inferred from a single measurement on an auxiliary
system [14, 15, 16]. A different approach is based on dynamic Bayesian
networks, used to study energy exchanges between two quantum systems that are
initially correlated and locally thermal [17, 18]. Yet another approach
exploits entanglement between a system and an ancilla, in such a way that the
first of the two measurements is translated into a measurement of the
entangled ancilla instead [19, 20].

In this paper, we show that the energy statistics of a quantum system can be
inferred from its collision with a particle prepared in a suitable initial
state. Contrary to TPM and the alternatives mentioned above, our scattering
setup is fully autonomous so that the colliding particle acts effectively both
as an external driving and as an energy measurement device. Previous studies
using a scattering approach have focused on identifying the conditions on the
initial state of the incoming colliding particle that are needed to interpret
the average energy exchanges in the system as heat or work [21, 22, 23]. In
this study, we identify the conditions for the state of the particle after the
collision to encode the energy fluctuations that would be obtained from a two-
point projective measurement of the system energy (often interpreted as
fluctuating work when the system is initially in a canonical state). We find
that this happens when the incoming colliding particle is semiclassical and
narrow in energy compared to the system’s energies, and we provide the
relation between the time-dependent driving energy and the spatial potential
energy of the scattering process. We also show that when its energy
uncertainty is large, the change in kinetic energy due to the collision
encodes the untouched work, i.e. the average energy change that would result
from a non-autonomous driving without performing any projective measurement
[9].

Scattering theory is used from high-energy to condensed matter physics to link
quantum theory with experiments [24, 25, 26, 27]. It is also widely used to
study open quantum systems [28, 29, 30, 31, 32, 33, 34, 34], quantum transport
[35, 36, 37], quantum thermodynamics [38, 39, 21, 22, 23] and quantum
metrology [40, 41, 42]. Our results should therefore contribute to a better
understanding of quantum fluctuations in a broad class of realistic systems.
As we shall see, since scattering maps are energy-preserving operations, our
results can be used to provide realistic ways to implement otherwise highly
idealized quantum resource theory operations [43, 44, 45].

_Two-point measurement scheme. —_ We consider a closed quantum system $S$, of
dimension $N$, undergoing a time-dependent dynamics according to the
Hamiltonian $H_{S}(t)=H_{S}+V(t)\nu$, where $H_{S}$ is the bare Hamiltonian,
$V(t)$ is a non-vanishing function only in the interval $t\in(0,\tau)$ and
$\nu$ is a time-independent operator. The bare energies of the system are
defined by $H_{S}\ket{j}=e_{j}\ket{j}$, where $\\{\ket{j}\\}_{j=1}^{N}$ is the
basis of eigenvectors associated to its energy spectrum
$\\{e_{j}\\}_{j=1}^{N}$. We denote the energy gap (or Bohr transition
frequency) between eigenstates $\ket{j}$ and $\ket{j^{\prime}}$ by
$\Delta_{j^{\prime}j}\equiv e_{j^{\prime}}-e_{j}$.

In the interaction picture, the time-dependent protocol is implemented by a
unitary operator $U_{I}(\tau)$ which obeys the Schrödinger equation in the
interaction picture

$\displaystyle\frac{d}{dt}U_{I}(t)=-\frac{i}{\hbar}V_{I}(t)U_{I}(t)\;,$ (1)

with $U_{I}(0)=\mathbb{I}_{S}$ and $V_{I}(t)=V(t)e^{iH_{S}t/\hbar}\nu
e^{-iH_{S}t/\hbar}$. According to the two-point measurement scheme, we perform
two projective measurements in the eigenbasis of $H_{S}$, one before and one
after the interaction takes place [3, 4, 5, 9]. The energy distribution
associated to this scheme is then

$\displaystyle
P^{\mathrm{TPM}}(W)=\sum_{j^{\prime},j}\delta(W+\Delta_{j^{\prime}j})|\braket{j^{\prime}}{U_{I}(\tau)}{j}|^{2}\braket{j}{\rho_{S}}{j}\;,$
(2)

where $\rho_{S}\equiv\rho_{S}(0)$ is the state of the system before the
projective measurements are performed. Although the last expression is valid
for arbitrary energy spectrum, we consider without loss of generality that the
spectrum is non-degenerate, i.e. we assume that $\Delta_{j^{\prime}j}=0$
implies $j^{\prime}=j$. This is warranted since no energy statistics can be
inferred within an energy-degenerate subspace, although the spectrum can be
still have degenerate energy gaps. The average energy is the first moment of
the distribution

$\displaystyle\langle W\rangle^{\mathrm{TPM}}$ $\displaystyle=\int
dWP^{\mathrm{TPM}}(W)W$
$\displaystyle=\mathrm{Tr}_{S}[H_{S}\rho_{S}]-\mathrm{Tr}_{S}[H_{S}\rho^{\mathrm{TPM}}_{S}(\tau)]\;.$
(3)

The second line expresses the average energy in a basis-independent fashion,
where

$\displaystyle\rho^{\mathrm{TPM}}_{S}(\tau)=\sum_{j^{\prime},j}\Pi_{j^{\prime}}U_{I}(\tau)\Pi_{j}\rho_{S}\Pi_{j}U^{\dagger}_{I}(\tau)\Pi_{j^{\prime}}\;$
(4)

is the state of the system after the non-selective, two-point measurement
scheme is performed and $\Pi_{j}\equiv\ket{j}\bra{j}$.

If the time-dependent protocol is implemented without measurements, then the
probability distribution in Eq. (2) cannot be defined. However, we can still
define the average energy change in the system before and after the protocol

$\displaystyle\langle
W\rangle^{0}=\mathrm{Tr}_{S}[H_{S}\rho_{S}]-\mathrm{Tr}_{S}[H_{S}U_{I}(\tau)\rho_{S}U^{\dagger}_{I}(\tau)]\;,$
(5)

which is the so-called untouched work [9]. Note that Eqs. (5) and (Two-point
measurement energy statistics from particle scattering) coincide if the
initial state is diagonal in the energy eigenbasis [9].

_Scattering setup. —_ We consider now a time-independent scattering process
involving the same quantum system $S$ with bare Hamiltonian $H_{S}$. In
addition, we also include the motional degrees of freedom $X$ of a particle of
mass $m$ with kinetic energy operator $(\hat{p}^{2}/2m)\ket{p}=E_{p}\ket{p}$.
Here $\\{\ket{p}\\}$ are improper (non-normalizable) eigenstates whose
position representation are plane waves
$\braket{x}{p}=\exp(\mathrm{i}px/\hbar)/\sqrt{2\pi\hbar}$ and
$E_{p}=p^{2}/2m\geq 0$ is the kinetic energy. The total Hamiltonian reads

$\displaystyle
H=H_{0}+\mathcal{V}(x)=H_{S}\otimes\mathbb{I}_{X}+\mathbb{I}_{S}\otimes\hat{p}^{2}/2m+\mathcal{V}(x)\;,$
(6)

where $\mathbb{I}_{S}$ and $\mathbb{I}_{X}$ are the identity operators on the
Hilbert space of $S$ and $X$. The interaction operator is given by
$\mathcal{V}(x)=\nu\otimes\lambda(x)$, where $\lambda(x)$ is non-vanishing
only inside the interval $x\in(0,a)$. In this scattering process, we picture
the system $S$ as being at rest in the region $x\in(0,a)$, thus playing the
role of a scatterer for the incoming particle.

We take the full system to be initially in a factorized state
$\rho_{S}\otimes\rho_{X}$, with $\rho_{X}$ describing the initial state of the
motion of the particle before the collision. According to scattering theory,
the final state of the full system after a single collision

$\displaystyle\rho^{\prime}=S\big{(}\rho_{S}\otimes\rho_{X}\big{)}S^{\dagger}\;,$
(7)

where $S$ is the unitary scattering operator. Crucially, it is possible to
show that $S$ is also energy-preserving $[S,H_{0}]=0$ [26, 25, 21]. From Eq.
(7), we can obtain the final state of motion of the particle

$\displaystyle\rho^{\prime}_{X}=\mathrm{Tr}_{S}[S\big{(}\rho_{S}\otimes\rho_{X}\big{)}S^{\dagger}]\;.$
(8)

In the same way, we can obtain the final state of the system
$\rho^{\prime}_{S}$ by performing the partial trace over $X$.

We are now interested in obtaining the final kinetic energy distribution from
Eq. (8). It can be formally expressed in terms of the final momentum
distribution $\rho^{\prime}_{X}(p)\equiv\braket{p}{\rho^{\prime}_{X}}{p}$ as
follows

$\displaystyle\rho^{\prime}_{X}(E_{p})=\frac{m}{p(E_{p})}\sum_{\alpha=\pm}\rho^{\prime}_{X}(\alpha
p(E_{p}))\;,$ (9)

where $p(E_{p})=\sqrt{2mE_{p}}$ and the sum is over positive and negative
momenta of the final distribution (see discussion below).

We obtain an explicit expression for $\rho^{\prime}_{X}(p)$ from Eq. (8) by
expressing the scattering operator in the eigenbasis of $H_{0}$, denoted by
$\ket{p,j}\equiv\ket{p}\otimes\ket{j}$

$\displaystyle\braket{p^{\prime},j^{\prime}}{S}{p,j}$
$\displaystyle=\frac{\sqrt{|pp^{\prime}|}}{m}\delta(E_{p}-E_{p^{\prime}}-\Delta_{j^{\prime}j})s_{j^{\prime}j}^{(\alpha\beta)}(E)\;.$
(10)

This expression follows from energy conservation $[S,H_{0}]=0$ [26, 25, 21].
In Eq. (10) $s_{j^{\prime}j}^{(\alpha\beta)}(E)$ is the scattering matrix at
total energy $E=E_{p}+e_{j}$ and $\alpha={\rm sign}(p^{\prime})$ and
$\beta={\rm sign}(p)$ accounts for the final and initial direction of the
momenta, which can be positive or negative. The pairs $(++),(+-),(-+),(--)$
correspond to transmission from the left, reflection from the left, reflection
from the right and transmission from the right probability amplitudes, which
can be obtained from the solutions of the stationary Schrödinger equation [46,
47]. Using Eq. (10) it can be shown that the unitarity of $S$ implies the
unitarity of the scattering matrix
$\sum_{j^{\prime},\alpha}s^{(\alpha\beta)}_{j^{\prime}j}(E)[s^{(\alpha\beta^{\prime})}_{j^{\prime}k}(E)]^{*}=\delta_{jk}\delta_{\beta\beta^{\prime}}$
[21]. We add that, strictly speaking, Eq. (10) implies that
$s^{(\alpha\beta)}_{j^{\prime}j}(E)$ is only defined if
$E\geq\max{\\{e_{j^{\prime}},e_{j}\\}}$, in which case we say that the channel
$\ket{j}\rightarrow\ket{j^{\prime}}$ is open [21]. However, we can extend the
scattering matrix to closed channels by defining
$s^{(\alpha\beta)}_{j^{\prime}j}(E)=\delta_{j^{\prime}j}\delta_{\alpha\beta}$
if $E<\max{\\{e_{j^{\prime}},e_{j}\\}}$ so that unitarity holds for both
closed and open channels.

In this study, we consider that the initial state of motion $\rho_{X}$ is
travelling to the right so that
$\rho_{X}(p,p^{\prime})\equiv\braket{p}{\rho_{X}}{p^{\prime}}$ has support
only in $p,p^{\prime}\geq 0$ (a similar analysis holds for states travelling
to the left). In practice, this happens if $p_{0}\gg\sigma_{p}$ where
$p_{0}=\mathrm{Tr}_{X}[\hat{p}\rho_{X}]$ is the average momentum of the
initial state and
$\sigma_{p}^{2}=\mathrm{Tr}_{X}[(\hat{p}^{2}-p^{2}_{0})\rho_{X}]$ is its
variance. We then obtain an explicit form for Eq. (9)

$\displaystyle\rho^{\prime}_{X}(E_{p})$
$\displaystyle=\sum_{j^{\prime},j,k}\braket{j}{\rho_{S}}{k}\rho_{X}(E_{p}+\Delta_{j^{\prime}j},E_{p}+\Delta_{j^{\prime}k})$
$\displaystyle\times\sum_{\alpha=\pm}s_{j^{\prime}j}^{(\alpha+)}(E_{p}+e_{j^{\prime}})[s_{j^{\prime}k}^{(\alpha+)}(E_{p}+e_{j^{\prime}})]^{*}\;.$
(11)

In the last expression, the initial state of the particle is defined in terms
of the kinetic energy

$\displaystyle\rho_{X}(E_{p},E_{p^{\prime}})=\frac{m}{\sqrt{p(E_{p})p(E_{p^{\prime}})}}\rho_{X}(p(E_{p}),p(E_{p^{\prime}}))\;.$
(12)

Note that the diagonal elements $\rho_{X}(E_{p})\equiv\rho_{X}(E_{p},E_{p})$
yield the initial kinetic energy distribution. The change in average kinetic
energy due to the collision can now be retrieved from

$\displaystyle\Delta E_{X}$
$\displaystyle=\int_{0}^{\infty}dE_{p}\leavevmode\nobreak\
E_{p}\leavevmode\nobreak\ [\rho^{\prime}_{X}(E_{p})-\rho_{X}(E_{p})]$
$\displaystyle=\mathrm{Tr}_{S}[H_{S}\rho_{S}]-\sum_{j^{\prime},j,k}e_{j^{\prime}}\braket{j}{\rho_{S}}{k}\int_{e_{j^{\prime}}}^{\infty}dE\leavevmode\nobreak\
\rho_{X}(E-e_{j},E-e_{k})$
$\displaystyle\times\sum_{\alpha=\pm}s_{j^{\prime}j}^{(\alpha+)}(E)[s_{j^{\prime}k}^{(\alpha+)}(E)]^{*}\;.$
(13)

The last expression follows from substituting Eq. (Two-point measurement
energy statistics from particle scattering) in the first line of Eq. (Two-
point measurement energy statistics from particle scattering) after some
algebraic manipulation. Throughout the rest of the paper, we show how to
connect Eq. (Two-point measurement energy statistics from particle scattering)
and (Two-point measurement energy statistics from particle scattering) of
scattering theory to Eqs. (2) and (Two-point measurement energy statistics
from particle scattering) of the two-point measurement scheme. We then show
how to recover the notion of untouched work and finally connect our results
with quantum resource theories.

_Connection to the two-point measurement scheme. —_ Under certain conditions,
to be discussed below, Eq. (Two-point measurement energy statistics from
particle scattering) can be written as

$\displaystyle\rho^{\prime}_{X}(E_{p})=\int dWP(E_{p},W)\rho_{X}(E_{p}-W)\;,$
(14)

where

$\displaystyle
P(E_{p},W)\equiv\sum_{j^{\prime},j}\delta(W+\Delta_{j^{\prime}j})P_{j^{\prime}j}(E_{p}+e_{j^{\prime}})\braket{j}{\rho_{S}}{j}\;,$
(15)

and
$P_{j^{\prime}j}(E)\equiv\sum_{\alpha}|s^{(\alpha+)}_{j^{\prime}j}(E)|^{2}$ is
the probability for a transition $\ket{j}\rightarrow\ket{j^{\prime}}$ at total
energy $E=E_{p}+e_{j^{\prime}}$ due to the collision. The result in Eq. (14)
follows from Eq. (Two-point measurement energy statistics from particle
scattering) when the state of the particle is narrow in energy with respect to
the system 111Alternatively, the result also follows if the system is
initially diagonal
$\braket{j}{\rho_{S}}{k}=\delta_{jk}\braket{j}{\rho_{S}}{j}$; however, in this
paper we generally make no such assumption, meaning

$\displaystyle\frac{|\Delta_{j^{\prime}j}-\Delta_{k^{\prime}k}|}{v_{0}}\gg\sigma_{p}\;,$
(16)

for all non-degenerate energy gaps
$\Delta_{j^{\prime}j}\neq\Delta_{k^{\prime}k}$, with $v_{0}=p_{0}/m$ being the
particle’s average velocity. As studied in Refs. [21, 22], when condition (16)
is obeyed, the initial state appearing in Eq. (Two-point measurement energy
statistics from particle scattering) can be written as
$\rho_{X}(E_{p}+\Delta_{j^{\prime}j},E_{p}+\Delta_{j^{\prime}k})\simeq\delta_{e_{j},e_{k}}\rho_{X}(E_{p}+\Delta_{j^{\prime}j})$.
Thus, initial system coherences between non-degenerate states do not
contribute to Eq. (Two-point measurement energy statistics from particle
scattering) and Eq. (14) follows directly.

The function $P(E_{p},W)$ defined in Eq. (15) differs from
$P^{\mathrm{TPM}}(W)$ in Eq. (2) in the transition probabilities. In the
former, the transition probability $P_{j^{\prime}j}(E_{p}+e_{j^{\prime}})$ is
independent of time but dependent on the energy of the system $S$, while in
the latter $\braket{j^{\prime}}{U_{I}(\tau)}{j}|^{2}$ is dependent on time but
independent of the system’s energy. However, as shown in Ref. [23], it is
possible to simplify the scattering matrix when the kinetic energy — or
equivalently, the mass — is the largest parameter involved, so that it depends
only on the kinetic energy

$\displaystyle
s_{j^{\prime}j}^{(\alpha+)}(E_{p}+e_{j^{\prime}})\simeq\delta_{\alpha,+}\braket{j^{\prime}}{s(E_{p})}{j}\;,$
(17)

where $s(E_{p})$ acts only on $S$. This implies that $E_{p}\gg\lambda(x)$ and
$pa_{\mathrm{min}}\gg\hbar$, where $a_{\mathrm{min}}$ is the minimum length
scale over which the potential varies significantly. This ensures that the
particle travels undisturbed through a given potential $\lambda(x)$ which can
otherwise be arbitrary. Second, it implies that $E_{p}\gg\Delta_{S}$, so the
kinetic energy is much larger than the maximum energy gaps of the system
$\Delta_{S}\equiv\max(\Delta_{j^{\prime}j})\leavevmode\nobreak\ \forall
j^{\prime},j$.

The unitary operator $s(E_{p})$, acting only on $S$, can be found by solving
the Schrödinger equation

$\displaystyle\frac{\partial U_{I}(E_{p},t)}{\partial
t}=-\frac{i}{\hbar}V_{I}(E_{p},t)U_{I}(E_{p},t)\;,$ (18)

with $U(E_{p},-\infty)=\mathbb{I}_{S}$, $U(E_{p},+\infty)=s(E_{p})$, and
$V_{I}(E_{p},t)\equiv\lambda(p(E_{p})t/m)e^{iH_{S}t/\hbar}\nu
e^{-iH_{S}t/\hbar}$. In other words, in the semiclassical limit a particle
with kinetic energy $E_{p}$ induces an effective time-dependent interaction on
the system $S$ equivalent to Eq. (1), provided we identify
$V(t)\equiv\lambda(p(E_{p})t/m)$. Such a time $t$ is related to the position
$x$ at a fixed kinetic energy $E_{p}$ through the relation $t=mx/p(E_{p})$,
which can be used to write Eq. (18) as an equivalent position-dependent
equation. Note that since in our study $\lambda(x)$ is only non-zero inside an
interval of length $a$, the previous relation automatically defines an
interaction time $ma/p(E_{p})$ at a fixed kinetic energy $E_{p}$.

Finally, the condition $p_{0}\gg\sigma_{p}$, already fulfilled for the states
of motion considered, guarantees that $s(E_{p})$ can be well approximated at
$E_{p_{0}}$ [23]. This means that $U_{I}(\tau)=s(E_{p_{0}})$, provided we
identify $\tau\equiv a/v_{0}$ as the time of the collision. Therefore, in this
regime Eq. (15) implies $P(E_{p},W)\simeq P^{\mathrm{TPM}}(W)$. Substituting
in Eq. (14) we obtain our main result

$\displaystyle\rho^{\prime}_{X}(E_{p})=\int
dWP^{\mathrm{TPM}}(W)\rho_{X}(E_{p}-W)\;.$ (19)

The last expression shows that the final kinetic energy is the convolution of
the two-point measurement distribution with the initial kinetic energy
distribution. Moreover, applying the semiclassical regime to the integral in
Eq. (Two-point measurement energy statistics from particle scattering) and
taking initially diagonal states or narrow states obeying (16), we find
$\Delta E_{X}=\langle W\rangle^{\mathrm{TPM}}$.

_Connection to untouched work. —_ We now inquire what happens when the
coherences are initially present and cannot be ignored, so that condition (16)
no longer holds. In particular, we look at states of motion which are broad in
energy and narrow in position

$\displaystyle\sigma_{p}\geq\frac{\hbar}{2\sigma_{x}}\gg\frac{|\Delta_{j^{\prime}j}-\Delta_{k^{\prime}k}|}{v_{0}}\;,$
(20)

where $\sigma_{x}=\mathrm{Tr}_{X}[(\hat{x}^{2}-x^{2}_{0})\rho_{X}]$ and
$x_{0}=\mathrm{Tr}_{X}[\hat{x}\rho_{X}]$ is the average position. As studied
in Refs. [21], the semiclassical regime together with conditions (20) and
$\tau\ll\hbar/\Delta_{S}$ imply that
$\rho_{X}(E_{p}+\Delta_{j^{\prime}j},E_{p}+\Delta_{j^{\prime}k})\simeq\rho_{X}(E_{p})$.
In this case, we find from Eq. (Two-point measurement energy statistics from
particle scattering) that $\rho^{\prime}_{X}(E_{p})\simeq\rho_{X}(E_{p})$, so
that the final kinetic energy distribution is approximately unaffected by the
collision with the system, carrying no information about its energy
statistics. However, the second line of Eq. (Two-point measurement energy
statistics from particle scattering) now yields $\Delta E_{X}=\langle
W\rangle^{0}$ and we recover the untouched work 222Note that we cannot use the
first line of Eq. (13) to compute $\Delta E_{X}$ with the zero-order
approximation $\rho_{X}^{\prime}(E_{p})\simeq\rho_{X}(E_{p})$ for very broad
states.. We can thus interpolate between the notion of touched $\langle
W\rangle^{\mathrm{TPM}}$ and untouched $\langle W\rangle^{\mathrm{0}}$ by
varying the energy width of the particle scattering with the system.

_Numerics. —_ We illustrate our results with a $1/2$-spin system with
Hamiltonian $H_{S}=(\Delta_{S}/2)\sigma_{z}$ and system interaction
$\nu=J\sigma_{x}$, where $\Delta_{S}$ is the energy gap and $J\in\mathbb{R}$.
We consider two different initial states for the system: a thermal state
$\rho_{S}^{\beta}=\exp(-\beta H_{S})/\mathrm{Tr}_{S}[\exp(-\beta H_{S})]$ and
a pure state with thermal populations
$\rho_{S}^{\mathrm{coh}}=\rho_{S}^{\beta}+C$, where $C$ is only non-zero in
the off-diagonal elements. The state of motion of the particle is modelled by
a Gaussian wave packet $\rho_{X}(p,p^{\prime})=\phi(p)\phi^{*}(p^{\prime})$
where
$\phi(p)=(2\pi\sigma_{p})^{-1/4}\exp[-(p-p_{0})^{2}/(4\sigma^{2}_{p})]\exp(-ipx_{0}/\hbar)$
and the potential in space is given by a barrier $\lambda(x)=V_{0}$ for
$x\in(0,a)$ and zero otherwise. The exact scattering matrix
$s_{j^{\prime}j}^{(\alpha\beta)}(E)$ is computed by solving the coupled and
non-linear equations of multi-channel scattering theory [24]. We compare our
results in the semiclassical regime with TPM by solving Eq. (1) with a time-
dependent potential $V(t)=V_{0}$ for $t\in(0,\tau)$ and zero otherwise, where
$\tau=a/v_{0}$.

In Fig. 1, we show the final kinetic energy distribution in Eq. (Two-point
measurement energy statistics from particle scattering) and the initial
distribution for different values of normalized momentum uncertainty
$\bar{\sigma}_{p}\equiv\sigma_{p}v_{0}/\Delta_{S}$ in the semiclassical
regime. For narrow wave packets (first panel), we see that the final
distribution yields well-defined and non-overlapping peaks corresponding to
the different energy gaps of the system, independently of initial coherences.
Accordingly, as we show in the third panel the average change in kinetic
energy is equivalent to $\langle W\rangle^{\mathrm{TPM}}$. As we increase
$\sigma_{p}$, the wave packet becomes broad in energy and the final kinetic
energy distribution presents overlapping peaks whose amplitude depends on
initial coherences (second panel). In the limit of very broad wave packets,
the kinetic energy change approaches the untouched work $\langle
W\rangle^{\mathrm{0}}$.

Figure 1:  First and second panels: Initial and final kinetic energy
distribution for different values of normalized momentum uncertainty
$\bar{\sigma}_{p}\equiv\sigma_{p}v_{0}/\Delta_{S}$ (computed from Eqs. (12)
and (Two-point measurement energy statistics from particle scattering),
respectively). Third panel: Average change in kinetic energy as a function of
$\bar{\sigma}_{p}$, along with the average energy from the TPM and untouched
work (computed from Eqs. (Two-point measurement energy statistics from
particle scattering), (Two-point measurement energy statistics from particle
scattering) and (5), respectively). Parameters: $\Delta_{S}=\beta=V_{0}=J=1$,
$a=1$, $v_{0}=1$ and $m=1000$.

_Connection to quantum resource theories. —_ We now connect our results to
quantum resource theories [43, 44, 45, 50]. In Ref. [43], the system $S$
interacts with another quantum system $A$ whose Hamiltonian is
$H_{A}=\gamma\hat{p}$, where $\gamma$ has units of velocity. The unitary $U$
describes the evolution of $S$ and $A$ after they interacted

$\displaystyle\rho^{\prime}_{SA}=U(\rho_{S}\otimes\rho_{A})U^{\dagger}\;.$
(21)

It is assumed to satisfy energy conservation $[U,H_{S}+H_{A}]=0$ and
$[U,\hat{x}]=0$, implying that $U$ can be written as

$\displaystyle
U=e^{-iH_{S}\otimes\hat{x}/(\hbar\gamma)}(V_{S}\otimes\mathbb{I}_{A})e^{iH_{S}\otimes\hat{x}/(\hbar\gamma)}\;,$
(22)

where $V_{S}$ is an arbitrary unitary operator that acts on the system. A
connection with the scattering setup is made by computing the final momentum
distribution of $A$ given by
$\rho^{\prime}_{A}(p)=\braket{p}{\rho^{\prime}_{A}}{p}$, which is obtained by
tracing over $S$ in Eq. (21) and using Eq. (22), yielding

$\displaystyle\rho^{\prime}_{A}(p)$
$\displaystyle=\sum_{j^{\prime},j,k}\braket{j}{\rho_{S}}{k}\rho_{A}(p+\Delta_{j^{\prime}j}/\gamma,p+\Delta_{j^{\prime}k}/\gamma)$
$\displaystyle\times\braket{j^{\prime}}{V_{S}}{j}\braket{j^{\prime}}{V_{S}}{k}^{*}\;,$
(23)

where $\rho_{A}(p,p^{\prime})=(2\pi\hbar)^{-1}\int dx^{\prime}\int
dxe^{-ipx/\hbar}e^{ip^{\prime}x^{\prime}/\hbar}\braket{x}{\rho_{A}}{x^{\prime}}$
is the Fourier transform of system $A$ in position space
$\braket{x}{\rho_{A}}{x^{\prime}}$. Performing a change of variables from
momentum to energy $E_{A}\equiv\gamma p$ in Eq. (Two-point measurement energy
statistics from particle scattering), we see that the resulting final energy
distribution $\rho_{A}(E_{A})\equiv\gamma^{-1}\rho_{A}(E_{A}/\gamma)$ has the
same form as Eq. (Two-point measurement energy statistics from particle
scattering) in the semiclassical regime (see condition (17) and accompanying
discussion). It follows that Eq. (Two-point measurement energy statistics from
particle scattering) can be interpreted as the final kinetic energy
distribution of a very massive particle $A$ after it collides with system $S$.
Indeed, such an interpretation is corroborated by the fact that the
Hamiltonian $H_{A}=\gamma\hat{p}$ has been used before to model very massive
particles whose position in space can be used as a clock (see discussion after
Eq. (18) and Refs. [51, 52, 53]). As we have shown in this study, the energy
statistics of $S$ can then be directly retrieved from Eq. (Two-point
measurement energy statistics from particle scattering) when the system is
initially diagonal or when the particle is narrow in energy, thus bypassing
the need for a direct implementation of TPM.

_Discussion and conclusion. —_ Inferring the energy statistics through
semiclassical scattering with narrow energy states is an example of a “which-
path” experiment [54, 55, 56, 57]. The colliding particle plays the dual role
of inducing an energy jump in the system and carrying away “which-jump”
information in its kinetic energy distribution. However, acquiring “which-
jump” information automatically precludes the acquisition of information about
coherences initially present in the system — in this sense, information about
“which-jump” and information about superpositions are complementary [54, 55].
On the opposite extreme, an incoming particle which is very broad in energy
drives the system without carrying away information about it in its energy
distribution. Accordingly, the average kinetic energy change corresponds to
the untouched work, which depends on initial coherences in the system. We
illustrated the transition between average work in the TPM scheme and the
untouched work by changing the energy width of the incoming particle.

In summary, we have shown that retrieving the energy statistics for a driven
time-dependent isolated quantum system is possible by devising an appropriate
scattering setup. In contrast to previous approaches, we provide a method to
probe the energy statistics directly which can be experimentally realized in
controlled scattering experiments e.g. in cavity quantum electrodynamics [58,
59] or ultracold atoms setups [40, 42, 39]. In this way, scattering emerges as
a versatile tool to study open quantum systems and quantum thermodynamics if
one can control the state of the incoming particle.

## Acknowledgments

SLJ acknowledges the financial support from a Marie Skłodowska-Curie
Fellowship (Grant No. 101103884). GTL acknowledges the financial support of
the São Paulo Funding Agency FAPESP (Grant No. 2019/14072-0.). F. B. thanks
Fondecyt project 1191441 and ANID – Millennium Science Initiative Program-
NCN19-170.

## References

  * Jarzynski [2011] C. Jarzynski, Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale, Annu. Rev. Condens. Matter Phys. 2, 329 (2011).
  * Seifert [2012] U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  * Talkner  _et al._ [2007] P. Talkner, E. Lutz, and P. Hänggi, Fluctuation theorems: Work is not an observable, Phys. Rev. E 75, 050102 (2007).
  * Esposito  _et al._ [2009] M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys. 81, 1665 (2009).
  * Campisi  _et al._ [2011] M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and applications, Rev. Mod. Phys. 83, 771 (2011).
  * Crooks [1999] G. E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60, 2721 (1999).
  * Evans and Searles [2002] D. J. Evans and D. J. Searles, The fluctuation theorem, Advances in Physics 51, 1529 (2002).
  * Maes [2003] C. Maes, On the origin and the use of fluctuation relations for the entropy, Séminaire Poincaré 2, 29 (2003).
  * Talkner and Hänggi [2016] P. Talkner and P. Hänggi, Aspects of quantum work, Phys. Rev. E 93, 022131 (2016).
  * Masuyama  _et al._ [2018] Y. Masuyama, K. Funo, Y. Murashita, A. Noguchi, S. Kono, Y. Tabuchi, R. Yamazaki, M. Ueda, and Y. Nakamura, Information-to-work conversion by Maxwell’s demon in a superconducting circuit quantum electrodynamical system, Nat. Commun. 9, 10.1038/s41467-018-03686-y (2018).
  * Mazzola  _et al._ [2013] L. Mazzola, G. De Chiara, and M. Paternostro, Measuring the characteristic function of the work distribution, Phys. Rev. Lett. 110, 230602 (2013).
  * Dorner  _et al._ [2013] R. Dorner, S. R. Clark, L. Heaney, R. Fazio, J. Goold, and V. Vedral, Extracting quantum work statistics and fluctuation theorems by single-qubit interferometry, Phys. Rev. Lett. 110, 230601 (2013).
  * Batalhão  _et al._ [2014] T. B. Batalhão, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, Experimental reconstruction of work distribution and study of fluctuation relations in a closed quantum system, Phys. Rev. Lett. 113, 140601 (2014).
  * Roncaglia  _et al._ [2014] A. J. Roncaglia, F. Cerisola, and J. P. Paz, Work measurement as a generalized quantum measurement, Phys. Rev. Lett. 113, 250601 (2014).
  * De Chiara  _et al._ [2015] G. De Chiara, A. J. Roncaglia, and J. P. Paz, Measuring work and heat in ultracold quantum gases, New J. Phys. 17, 035004 (2015).
  * Cerisola  _et al._ [2017] F. Cerisola, Y. Margalit, S. Machluf, A. J. Roncaglia, J. P. Paz, and R. Folman, Using a quantum work meter to test non-equilibrium fluctuation theorems, Nat. Commun. 8, 10.1038/s41467-017-01308-7 (2017).
  * Micadei  _et al._ [2020] K. Micadei, G. T. Landi, and E. Lutz, Quantum fluctuation theorems beyond two-point measurements, Phys. Rev. Lett. 124, 090602 (2020).
  * Micadei  _et al._ [2021] K. Micadei, J. P. Peterson, A. M. Souza, R. S. Sarthour, I. S. Oliveira, G. T. Landi, R. M. Serra, and E. Lutz, Experimental validation of fully quantum fluctuation theorems using dynamic bayesian networks, Phys. Rev. Lett. 127, 180603 (2021).
  * Aguilar  _et al._ [2022] G. H. Aguilar, T. L. Silva, T. E. Guimarães, R. S. Piera, L. C. Céleri, and G. T. Landi, Two-point measurement of entropy production from the outcomes of a single experiment with correlated photon pairs, Phys. Rev. A 106, L020201 (2022).
  * Solfanelli  _et al._ [2021] A. Solfanelli, A. Santini, and M. Campisi, Experimental verification of fluctuation relations with a quantum computer, PRX Quantum 2, 030353 (2021).
  * Jacob  _et al._ [2021] S. L. Jacob, M. Esposito, J. M. Parrondo, and F. Barra, Thermalization induced by quantum scattering, PRX Quantum 2, 020312 (2021).
  * Tabanera  _et al._ [2022] J. Tabanera, I. Luque, S. L. Jacob, M. Esposito, F. Barra, and J. M. R. Parrondo, Quantum collisional thermostats, New J. Phys. 24, 023018 (2022).
  * Jacob  _et al._ [2022] S. L. Jacob, M. Esposito, J. M. R. Parrondo, and F. Barra, Quantum scattering as a work source, Quantum 6, 750 (2022).
  * Razavy [2003] M. Razavy,  _Quantum Theory of Tunneling_ (World Scientific Publishing, Singapore, 2003).
  * Belkić [2004] D. Belkić,  _Principles of quantum scattering theory_ (Institute of Physics Publishing, Bristol, 2004).
  * Taylor [2006] J. R. Taylor,  _Scattering Theory: The Quantum Theory of Nonrelativistic Collisions_ (Dover Publications, New York, 2006).
  * Furrer  _et al._ [2009] A. Furrer, J. Mesot, and T. Strässle,  _Neutron Scattering in Condensed Matter Physics_ (World Scientific, Singapore, 2009).
  * Alicki [1981] R. Alicki, On the scattering theory for quantum dynamical semigroups, Ann. Inst. H. Poincaré Sect. A 35, 97 (1981).
  * Alicki and Frigerio [1983] R. Alicki and A. Frigerio, Scattering theory for quantum dynamical semigroups. - II, Ann. Inst. H. Poincaré Sect. A 38, 187 (1983).
  * Dümcke [1985] R. Dümcke, The low density limit for an N-level system interacting with a free Bose or Fermi gas, Commun. Math. Phys. 97, 331 (1985).
  * Hornberger  _et al._ [2003] K. Hornberger, S. Uttenthaler, B. Brezger, L. Hackermüller, M. Arndt, and A. Zeilinger, Collisional decoherence observed in matter wave interferometry, Phys. Rev. Lett. 90, 160401 (2003).
  * Hornberger [2006] K. Hornberger, Master equation for a quantum particle in a gas, Phys. Rev. Lett. 97, 060601 (2006).
  * Hornberger [2007] K. Hornberger, Monitoring approach to open quantum dynamics using scattering theory, EPL 77, 50007 (2007).
  * Hornberger and Vacchini [2008] K. Hornberger and B. Vacchini, Monitoring derivation of the quantum linear boltzmann equation, Phys. Rev. A 77, 022112 (2008).
  * Landauer [1957] R. Landauer, Spatial variation of currents and fields due to localized scatterers in metallic conduction, IBM Journal of Research and Development 1, 223 (1957).
  * Büttiker [1992] M. Büttiker, Scattering theory of current and intensity noise correlations in conductors and wave guides, Physical Review B 46, 12485 (1992).
  * Nazarov and Blanter [2009] Y. V. Nazarov and Y. M. Blanter,  _Quantum Transport_ (Cambridge University Press, Cambridge, 2009).
  * Bruch  _et al._ [2018] A. Bruch, C. Lewenkopf, and F. von Oppen, Landauer-büttiker approach to strongly coupled quantum thermodynamics: Inside-outside duality of entropy evolution, Phys. Rev. Lett. 120, 107701 (2018).
  * Bouton  _et al._ [2021] Q. Bouton, J. Nettersheim, S. Burgardt, D. Adam, E. Lutz, and A. Widera, A quantum heat engine driven by atomic collisions, Nat. Commun. 12, 10.1038/s41467-021-22222-z (2021).
  * Hohmann  _et al._ [2016] M. Hohmann, F. Kindermann, T. Lausch, D. Mayer, F. Schmidt, and A. Widera, Single-atom thermometer for ultracold gases, Phys. Rev. A 93, 043607 (2016).
  * Schmidt  _et al._ [2019] F. Schmidt, D. Mayer, Q. Bouton, D. Adam, T. Lausch, J. Nettersheim, E. Tiemann, and A. Widera, Tailored single-atom collisions at ultralow energies, Phys. Rev. Lett. 122, 013401 (2019).
  * Bouton  _et al._ [2020] Q. Bouton, J. Nettersheim, D. Adam, F. Schmidt, D. Mayer, T. Lausch, E. Tiemann, and A. Widera, Single-atom quantum probes for ultracold gases boosted by nonequilibrium spin dynamics, Phys. Rev. X 10, 011018 (2020).
  * Alhambra  _et al._ [2016] A. M. Alhambra, L. Masanes, J. Oppenheim, and C. Perry, Fluctuating work: From quantum thermodynamical identities to a second law equality, Phys. Rev. X 6, 041017 (2016).
  * Lostaglio [2019] M. Lostaglio, An introductory review of the resource theory approach to thermodynamics, Rep. Prog. Phys. 82, 114001 (2019).
  * Chitambar and Gour [2019] E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019).
  * Landau and Lifshitz [1977] L. D. Landau and E. M. Lifshitz,  _Quantum Mechanics : Non-relativistic Theory_ (Pergamon Press, Oxford, 1977).
  * Sakurai [1993] J. J. Sakurai,  _Modern Quantum Mechanics (Revised Edition)_ (Addison-Wesley, 1993).
  * Note [1] Alternatively, the result also follows if the system is initially diagonal $\mathinner{\langle{j|\rho_{S}|k}\rangle}=\delta_{jk}\mathinner{\langle{j|\rho_{S}|j}\rangle}$; however, in this paper we generally make no such assumption.
  * Note [2] Note that we cannot use the first line of Eq. (13) to compute $\Delta E_{X}$ with the zero-order approximation $\rho_{X}^{\prime}(E_{p})\simeq\rho_{X}(E_{p})$ for very broad states.
  * Landi and Paternostro [2021] G. T. Landi and M. Paternostro, Irreversible entropy production: From classical to quantum, Rev. Mod. Phys. 93, 035008 (2021).
  * Aharonov and Kaufherr [1984] Y. Aharonov and T. Kaufherr, Quantum frames of reference, Phys. Rev. D 30, 368 (1984).
  * Aharonov  _et al._ [1998] Y. Aharonov, J. Oppenheim, S. Popescu, B. Reznik, and W. G. Unruh, Measurement of time of arrival in quantum mechanics, Phys. Rev. A 57, 4130 (1998).
  * Woods  _et al._ [2018] M. P. Woods, R. Silva, and J. Oppenheim, Autonomous quantum machines and finite-sized clocks, Annales Henri Poincaré 20, 125 (2018).
  * Scully  _et al._ [1991] M. O. Scully, B.-G. Englert, and H. Walther, Quantum optical tests of complementarity, Nature 351, 111 (1991).
  * Zeilinger  _et al._ [1996] A. Zeilinger, T. Herzog, M. A. Horne, P. G. Kwiat, K. Mattle, and H. Weinfurter, Path information in quantum interferometry, in  _Coherence and Quantum Optics VII_ (Springer, Boston, 1996) pp. 305–312.
  * Scully and Zubairy [1997] M. O. Scully and M. S. Zubairy,  _Quantum Optics_ (Cambridge University Press, Cambridge, 1997).
  * Schlosshauer [2007] M. Schlosshauer,  _Decoherence and the Quantum-To-Classical Transition_ (Springer, Berlin, Heidelberg, 2007).
  * Raimond  _et al._ [2001] J. M. Raimond, M. Brune, and S. Haroche, Manipulating quantum entanglement with atoms and photons in a cavity, Rev. Mod. Phys. 73, 565 (2001).
  * Haroche and Raimond [2006] S. Haroche and J.-M. Raimond,  _Exploring the Quantum: Atoms, Cavities, and Photons_ (Oxford University Press, Oxford, 2006).