# Universal energy fluctuations in inelastic scattering processes

Samuel L. Jacob<EMAIL_ADDRESS>School of Physics, Trinity College Dublin,
Dublin 2, Ireland  John Goold<EMAIL_ADDRESS>School of Physics, Trinity
College Dublin, Dublin 2, Ireland  Trinity Quantum Alliance, Unit 16, Trinity
Technology and Enterprise Centre, Pearse Street, Dublin 2, D02 YN67, Ireland
Algorithmiq Limited, Kanavakatu 3C 00160 Helsinki, Finland  Gabriel T. Landi
Department of Physics and Astronomy, University of Rochester, Rochester, New
York 14627, USA<EMAIL_ADDRESS>Felipe Barra
<EMAIL_ADDRESS>Departamento de Física, Facultad de Ciencias Físicas y
Matemáticas, Universidad de Chile, 837.0415 Santiago, Chile

###### Abstract

Quantum scattering is used ubiquitously in both experimental and theoretical
physics across a wide range of disciplines, from high-energy physics to
mesoscopic physics. In this work, we uncover universal relations for the
energy fluctuations of a quantum system scattering inelastically with a
particle at arbitrary kinetic energies. In particular, we prove a fluctuation
relation describing an asymmetry between energy absorbing and releasing
processes which relies on the non-unital nature of the underlying quantum map.
This allows us to derive a bound on the average energy exchanged. We find that
energy releasing processes are dominant when the kinetic energy of the
particle is comparable to the system energies, but are forbidden at very high
kinetic energies where well known fluctuation relations are recovered. Our
work provides a unified view of energy fluctuations when the source driving
the system is not macroscopic but rather an auxiliary quantum particle in a
scattering process.

Introduction. — Scattering is a mechanism of interaction between physical
systems that is pervasive across nature and experiment, from low to high
energies [1, 2]. It is an essential tool in the characterization of materials
and quantum phenomena in condensed matter [3, 4], in describing the transport
properties of quantum systems [5, 6, 7, 8, 9, 10] and the properties of
ultracold gases [11, 12, 13, 14, 15, 16, 17, 18, 19]. Quantum scattering
theory describes how two (or more) quantum systems change their state after
they collide, which entails an energy exchange between them when the
scattering process is inelastic [1, 2, 3]. Such energy exchanges have been
recently analysed from a thermodynamic viewpoint [20, 21, 22, 23, 24], but a
more general scattering treatment at the level of energy fluctuations is still
not available.

In thermodynamics, energy fluctuations are usually studied for small –
classical or quantum – systems interacting with macroscopic sources. The
assumption of a macroscopic source allows us to define some Hamiltonian for
the system with time dependent parameter that we imagine is operated in a
classical way [25]. Within this paradigm, some of the most famous results of
stochastic thermodynamics have been derived, for example the so-called
fluctuation relations [26, 27, 28, 29, 30, 31]. As an example, consider a
system of any size prepared in thermal equilibrium with its environment
characterized by $\beta=1/k_{B}T$, where $k_{B}$ is the Boltzmann constant and
$T$ is the temperature. When the system is driven out of equilibrium by a
macroscopic source in a cyclic fashion (so that the system Hamiltonian is the
same before and after the interaction), then the fluctuation relation reads
$e^{-\beta W}p_{W}=\tilde{p}_{-W}$, where $p_{W}$ is the probability
distribution for an energy change $W$ during the process and $\tilde{p}_{-W}$
is the probability distribution for an energy change $-W$ in the time-reversed
process [29, 30]. Jarzynski’s equality $\langle e^{-\beta W}\rangle=1$ [32]
follows by a simple average over $W$ which, through Jensen’s inequality,
implies (on average) the impossibility of energy extraction in a cyclic
process $\langle W\rangle\geq 0$. Since the macroscopic source is considered
to behave deterministically, i.e. as a work source, the energy consumed is
interpreted as work done on the system. Through the use of the two-point
measurement scheme, fluctuation relations have been extended to closed quantum
systems [33, 34, 35, 36, 37, 38], derived for macroscopic heat sources [39,
29] and experimentally verified across different platforms [40, 41, 42, 43,
44].

A valid framework to go beyond the macroscopic source paradigm is that of open
quantum systems [45, 46], where the system dynamics is described by a
dynamical map [47, 48, 49, 50] obtained after the interaction with another
quantum system of arbitrary size. It is known that fluctuation relations can
be derived within the two-point measurement scheme when the map is unital,
i.e. if the maximally-mixed state is an invariant state [51, 52, 41]. If the
map is not unital, it has been shown that Jarzynski’s equality is modified to
$\langle e^{-\beta W}\rangle=1+\eta$, where $\eta$ can be positive or negative
[53, 54, 55]; similar modifications have been found when studying fluctuations
with generalized measurements [56] and in the presence of feedback control
[57, 58]. Since Jensen’s inequality then implies $\langle
W\rangle\geq-\log(1+\eta)$, this suggests that $\eta>0$ allows for energy
releasing processes. However, the physical significance of non-unital maps
remains poorly understood and appreciated. Quantum scattering theory, by being
realistic and treating the interacting systems as quantum systems in their own
right, could provide a unified view of energy fluctuations beyond the
macroscopic source limit.

In this Letter, we provide such a unified view on energy fluctuations by
studying a quantum system scattering inelastically with a particle at an
arbitrary kinetic energy. We show that the dynamical map for the system
naturally encodes its energy fluctuations without relying on any measurement
scheme. Our main result [Eq. (8)] describes a universal fluctuation relation
obeyed for a system driven out of equilibrium by the colliding particle and
reflects the non-unital nature of the scattering process. From this result, we
derive an exact bound for the energy exchanged [Eqs. (10) and (Universal
energy fluctuations in inelastic scattering processes)] as a function of the
particle’s kinetic energy. We show that non-unitality dominates when the
particle’s kinetic energy is comparable to the energy gaps of the system,
allowing energy extraction from the system; while at very high kinetic
energies we recover unitality and the standard fluctuation relation.

Setup and energy fluctuations. — We consider a quantum scattering process
between a system $S$ and a particle $P$. In a reference frame co-moving with
the center of mass, only the reduced mass plays a role, but we simplify the
treatment by fixing the position of system $S$ and consider the particle $P$
to be travelling in one direction with associated momentum $\hat{p}$ and
position $\hat{x}$ operators. The total Hamiltonian is
$\hat{H}=\hat{H}_{0}+\hat{V}(\hat{x})$ where
$\hat{H}_{0}=\hat{H}_{S}\otimes\hat{\mathbb{I}}_{P}+\hat{\mathbb{I}}_{S}\otimes\hat{p}^{2}/2m$
is the bare Hamiltonian. The energy of the system is defined by
$\hat{H}_{S}\ket{j}=e_{j}\ket{j}$, where $\\{\ket{j}\\}$ is a basis of
eigenvectors associated to its discrete energy spectrum $\\{e_{j}\\}$. The
energy of the particle is described by $\hat{p}^{2}/2m\ket{p}=E_{p}\ket{p}$,
where $\\{\ket{p}\\}$ are improper (non-normalizable) eigenvectors whose
position representation are plane waves
$\braket{x}{p}=\exp(ipx/\hbar)/\sqrt{2\pi\hbar}$ and $E_{p}=p^{2}/2m\geq 0$ is
the kinetic energy. The interaction operator $\hat{V}(\hat{x})$ is assumed to
vanish sufficiently far away from the scattering region where the system is
located, so that the unitary scattering operator
$\hat{S}=\lim_{t\to+\infty}e^{\frac{it}{\hbar}\hat{H}_{0}}e^{-\frac{i2t}{\hbar}\hat{H}}e^{\frac{it}{\hbar}\hat{H}_{0}}$
exists and satisfies energy conservation $[\hat{S},\hat{H}_{0}]=0$ [1, 2].
Considering the initial state of the system $\hat{\rho}_{S}$ and particle
$\hat{\rho}_{P}$ to be uncorrelated before the collision, the state of the
system after the collision is

$\displaystyle\Phi(\hat{\rho}_{S})=\mathrm{Tr}_{P}[\hat{S}(\hat{\rho}_{S}\otimes\hat{\rho}_{P})\hat{S}^{\dagger}]\;,$
(1)

where $\mathrm{Tr}_{P}$ is the partial trace over the particle and $\Phi$ is a
completely positive and trace preserving map [45, 46, 47, 48, 50].

The explicit evaluation of Eq. (1) can be performed in the following kinetic
energy eigenstates $\ket{E^{\alpha}_{p}}\equiv\sqrt{m/|p|}\ket{p}$, where
$\alpha=\mathrm{sign}(p)$ accounts for the initial direction of the incoming
particle, which can be travelling to the left ($\alpha=+$) or right
($\alpha=-$). First, we need the representation of the scattering operator in
this basis which reads
$\braket{E^{\alpha^{\prime}}_{p^{\prime}}}{\hat{S}}{E^{\alpha}_{p}}=\sum_{j^{\prime},j}\ket{j^{\prime}}\bra{j}\braket{E^{\alpha^{\prime}}_{p^{\prime}},j^{\prime}}{\hat{S}}{E^{\alpha}_{p},j}$
where $\ket{E^{\alpha}_{p},j}$ is the eigenbasis of $\hat{H}_{0}$ and
$\braket{E^{\alpha^{\prime}}_{p^{\prime}},j^{\prime}}{\hat{S}}{E^{\alpha}_{p},j}=\delta(E_{p^{\prime}}+e_{j^{\prime}}-E_{p}-e_{j})s^{\alpha^{\prime}\alpha}_{j^{\prime}j}(E_{p}+e_{j})$.
In the last expression, the $\delta$ function ensures energy conservation for
the collision and $s_{j^{\prime}j}^{\alpha^{\prime}\alpha}(E)$ is the
scattering matrix encoding the transition amplitudes from
$\ket{E^{\alpha}_{p},j}\rightarrow\ket{E^{\alpha^{\prime}}_{p^{\prime}},j^{\prime}}$
at total energy $E=E_{p}+e_{j}$ [1, 2]. Rewriting the sum over $j^{\prime},j$
as a sum over energy differences $\Delta$ then yields simply
$\braket{E^{\alpha^{\prime}}_{p^{\prime}}}{\hat{S}}{E^{\alpha}_{p}}=\sum_{\Delta}\delta(E_{p^{\prime}}-E_{p}+\Delta)\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})$
where

$\displaystyle\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})=\sum_{\begin{subarray}{c}j^{\prime},j:\\\
e_{j^{\prime}}-e_{j}=\Delta\end{subarray}}s_{j^{\prime}j}^{\alpha^{\prime}\alpha}(E_{p}+e_{j})\ket{j^{\prime}}\bra{j}\;,$
(2)

are eigenoperators of $\hat{H}_{S}$ and thus obey
$[\hat{H}_{S},\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})]=\Delta\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})$.
Second, we need the representation of the particle’s state in the same basis
$\rho^{\alpha\beta}_{P}(E_{p},E_{q})\equiv\braket{E^{\alpha}_{p}}{\hat{\rho}_{P}}{E^{\beta}_{q}}$
and we can carry out the trace in Eq. (1). After integrating the $\delta$
functions, we find that the particle’s state becomes dependent on the energy
differences as $\rho^{\alpha\beta}_{P}(E_{p},E_{p}-\Delta+\Delta^{\prime})$
(see Appendix B for more details). However, as shown in Ref. [20], if the
particle has a well-defined direction before the collision and is sufficiently
narrow in kinetic energy with respect to the energy differences, then we can
write

$\displaystyle\rho^{\alpha\beta}_{P}(E_{p},E_{p}-\Delta+\Delta^{\prime})\simeq\delta_{\alpha\beta}\delta_{\Delta,\Delta^{\prime}}~{}\rho^{\alpha}_{P}(E_{p})\;,$
(3)

where $\rho^{\alpha}_{P}(E_{p})\equiv\rho^{\alpha\alpha}_{P}(E_{p},E_{p})$ is
the kinetic energy distribution for a particle travelling with direction
$\alpha$. In this case, Eq. (1) can be written as

$\displaystyle\Phi(\hat{\rho}_{S})=\int
dE_{p}~{}\sum_{\alpha=\pm}\rho^{\alpha}_{P}(E_{p})~{}\Phi^{\alpha}(E_{p})(\hat{\rho}_{S})\;,$
(4)

where $\Phi^{\alpha}(E_{p})$ is a completely positive and trace preserving map
conditioned on the particle’s kinetic energy $E_{p}$ and direction $\alpha$
given by

$\displaystyle\Phi^{\alpha}(E_{p})(\cdot)$ $\displaystyle=\int
dW~{}\Phi^{\alpha}(E_{p},W)(\cdot)$ (5)
$\displaystyle\Phi^{\alpha}(E_{p},W)(\cdot)$
$\displaystyle=\sum_{\Delta}\delta(W-\Delta)\sum_{\alpha^{\prime}}\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})\cdot\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})^{\dagger}\;.$
(6)

Eqs. (4), (5) and (6) define the dynamical map. Note that the Kraus operators
in Eq. (6) are system eigenoperators due to condition (3), inducing a
transition with energy change $\Delta$. Indeed, assuming that $\hat{H}_{S}$
has a non-degenerate spectrum, it is easy to see that the quantum operation in
Eq. (6) defines a probability distribution for the energy changes through

$\displaystyle P^{\alpha}(E_{p},W)$
$\displaystyle=\mathrm{Tr}_{S}\big{[}\Phi^{\alpha}(E_{p},W)(\hat{\rho}_{S})\big{]}$
$\displaystyle=\sum_{j^{\prime},j}\delta(W-e_{j^{\prime}}+e_{j})P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})p_{j}\;,$
(7)

where $p_{j}\equiv\braket{j}{\hat{\rho}_{S}}{j}$ and
$P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})=\sum_{\alpha^{\prime}}|s^{\alpha^{\prime}\alpha}_{j^{\prime}j}(E_{p}+e_{j})|^{2}$
is the transition probability. Note that Eq. (Universal energy fluctuations in
inelastic scattering processes) has the same form as the distribution for
energy changes induced by a unitary operator $U$ on the system in a two-point
measurement scheme [34, 33, 36, 29, 30], with two crucial differences. First,
there is no need for a two-point measurement scheme as a consequence of
condition (3): a particle with a well-defined kinetic energy effectively
measures the energy changes in the system [23]. Second, the transition
probabilities are dictated by $P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})$ instead
of $|\braket{}{j^{\prime}}{U|j}|^{2}$, thus becoming dependent on both the
system and the particle’s energy. The normalization $\int
P(E_{p},W)dW=\sum_{j^{\prime},j}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})p_{j}=1$
holds since the map in Eq. (5) is trace preserving by construction. Indeed,
the property $\sum_{j^{\prime}}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})=1$ can be
proven independently from the unitarity of the scattering operator and holds
for any fixed total energy $E$ (see Appendix C).

Main result. — We now take our system to be in a thermal state
$\hat{\rho}_{S}=e^{-\beta\hat{H}_{S}}/Z$ where
$Z=\mathrm{Tr}_{S}[e^{-\beta\hat{H}_{S}}]$ is the partition function
111Strictly speaking, we do not require the system to be in a thermal state,
but only that its populations are thermally distributed.. Using the property
of the eigenoperators
$\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})e^{-\beta\hat{H}_{S}}=e^{\beta\Delta}e^{-\beta\hat{H}_{S}}\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})$,
it is easy to see that the quantum operation satisfies $e^{-\beta
W}\Phi^{\alpha}(E_{p},W)(\hat{\rho}_{S})=\hat{\rho}_{S}\Phi^{\alpha}(E_{p},W)(\hat{\mathbb{I}}_{S})$
and thus the distribution in Eq. (Universal energy fluctuations in inelastic
scattering processes) obeys

$\displaystyle e^{-\beta W}P^{\alpha}(E_{p},W)$
$\displaystyle=\mathbb{P}^{\alpha}(E_{p},-W)\;,$ (8)

where $\mathbb{P}^{\alpha}(E_{p},-W)$ is dual probability distribution given
by

$\displaystyle\mathbb{P}^{\alpha}(E_{p},-W)$
$\displaystyle=\mathrm{Tr}_{S}\big{[}\Phi^{\alpha}(E_{p},W)^{\dagger}(\hat{\rho}_{S})\big{]}$
$\displaystyle=\sum_{j^{\prime},j}\delta(-W-e_{j}+e_{j^{\prime}})p_{j^{\prime}}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})\;,$
(9)

with the dual operation defined by
$\mathrm{Tr}_{S}\big{[}\hat{\rho}_{S}\Phi^{\alpha}(E_{p},W)(\hat{\mathbb{I}}_{S})\big{]}=\mathrm{Tr}_{S}\big{[}\Phi^{\alpha}(E_{p},W)^{\dagger}(\hat{\rho}_{S})\big{]}$.
Eq. (Universal energy fluctuations in inelastic scattering processes) has the
same form as the distribution for energy changes induced by a time-reversed
unitary operator $U^{\dagger}=\Theta U\Theta^{\dagger}$ on the system in a
two-point measurement scheme, where $\Theta$ is the (anti-unitary) time
reversal operator [34, 33, 36, 29, 30]. In this sense, the dual operation
$\Phi^{\alpha}(E_{p},W)^{\dagger}$ reverses the energy change induced by
$\Phi^{\alpha}(E_{p},W)$ 222Note that the reverse energy change induced by the
dual operation is generally not equivalent to time-reversing the collision as
a whole. The latter requires a reversal of the particle direction and
scattering matrix (see Appendix G).. However, a crucial point is that the dual
distribution in Eq. (Universal energy fluctuations in inelastic scattering
processes) is generally not normalized
$\gamma^{\alpha}(E_{p})\equiv\int\mathbb{P}^{\alpha}(E_{p},-W)dW=\sum_{j^{\prime},j}p_{j^{\prime}}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})\neq
1$. This reflects the fact that the map in Eq. (5) is non-unital, or
equivalently that its dual is not trace preserving [51, 52, 53, 55]; unitality
would require $\sum_{j}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})=1$ which is
generally not obeyed in quantum scattering theory. Below, we show that both
non-unitality and unitality are general features of the scattering process and
discuss the physical conditions where each arises.

From our main result in Eq. (8) we can obtain an integral fluctuation relation
$\int e^{-\beta W}P^{\alpha}(E_{p},W)dW=\gamma^{\alpha}(E_{p})$ and, through
Jensen’s inequality, derive the following lower bound for the average energy
change

$\displaystyle\langle
W\rangle^{\alpha}(E_{p})\geq-\beta^{-1}\log[1+\eta^{\alpha}(E_{p})]\;,$ (10)

where $\langle W\rangle^{\alpha}(E_{p})\equiv\int WP^{\alpha}(E_{p},W)dW$ and
$\eta^{\alpha}(E_{p})\equiv\gamma^{\alpha}(E_{p})-1$. When
$\eta^{\alpha}(E_{p})>0$ the lower bound in Eq. (10) becomes negative and an
initially thermal system can release energy in the collision; while when
$\eta^{\alpha}(E_{p})\leq 0$ it is impossible to extract energy from the
system, with the equality holding for unital dynamics. We show in Appendix F
that $\eta^{\alpha}(E_{p})$ can be written in the exact form

$\displaystyle\eta^{\alpha}(E_{p})$
$\displaystyle=\sum_{\Delta>0}\tanh{\bigg{(}\frac{\beta\Delta}{2}\bigg{)}}\sum_{\begin{subarray}{c}j^{\prime},j:\\\
e_{j^{\prime}}-e_{j}=\Delta\end{subarray}}\Bigg{(}\frac{Z_{j^{\prime}j}}{Z}\Bigg{)}$
$\displaystyle\times[P^{\alpha}_{jj^{\prime}}(E_{p}+e_{j^{\prime}})-P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})]\;.$
(11)

The first sum in the last expression is over all the energy gaps of the
system. For a given energy gap $\Delta>0$, the second sum is over all pairs of
energy levels whose difference is $\Delta$ and $Z_{j^{\prime}j}=e^{-\beta
e_{j^{\prime}}}+e^{-\beta e_{j}}$ is the partition function of one of these
pairs. The last term describes the imbalance between relaxation and excitation
probabilities of the pair, being positive (negative) when the former are
higher (lower) than the latter.

Discussion and example. — In general, it is difficult to predict the behaviour
of $\langle W\rangle^{\alpha}(E_{p})$ and $\eta^{\alpha}(E_{p})$, since they
depends strongly on the scattering matrix, which in turns depends on the
system Hamiltonian $\hat{H}_{S}$ and scattering potential $\hat{V}(\hat{x})$.
However, we can study their behaviour based on universal scattering features
in two regimes: when the kinetic energy is comparable to the minimum energy
gap of the system or when it is much larger than the maximum energy gap. As an
example, consider a particle colliding with a two-level system with energy gap
$\Delta>0$. For simplicity, we consider a spatially symmetric potential
$\hat{V}(\hat{x})=V(-\hat{x})$ in which case the scattering process is
independent of the initial direction of the particle $\alpha$ [2, 20] and we
omit this label. The relevant quantities in Eq. (10) then read exactly
$\langle
W\rangle(E_{p})=(\Delta/2)\cosh^{-1}(\beta\Delta/2)[P_{10}(E_{p}+e_{0})e^{\beta\Delta/2}-P_{01}(E_{p}+e_{1})e^{-\beta\Delta/2}]$
and
$\eta(E_{p})=\tanh(\beta\Delta/2)[P_{01}(E_{p}+e_{1})-P_{10}(E_{p}+e_{0})]$,
where $\ket{1}$ and $\ket{0}$ are the excited and ground state.

Figure 1:  Average system energy change and lower bound in Eq. (10) at low and
high kinetic energies (upper and lower panel, respectively). We consider a
two-level system $\hat{H}_{S}=(\Delta/2)\hat{\sigma}_{z}$ and scattering
potential
$\hat{V}(\hat{x})=(V_{0}\pi/2)\hat{\sigma}_{x}\otimes\cos(\pi\hat{x}/a)$,
where $\Delta$ is the energy gap, $\hat{\sigma}_{z,x}$ are Pauli matrices and
$V_{0},a$ are the energy and length of the potential. The scattering matrix is
found by solving numerically the multichannel scattering equations [61, 20].
The parameters shown are $\Delta=m=a=\beta=1$, $V_{0}=100$ and the vertical
dashed line in the upper panel indicates $E_{p}=\Delta=1$.

At low kinetic energies $0\leq E_{p}<\Delta$ we see that Eq. (10) allows for
energy extraction from the system (Fig. 1, upper panel). This is because
system excitation is forbidden when the particle has an initial kinetic energy
lower than the gap: the excitation channel is closed, i.e.
$P_{10}(E_{p}+e_{0})=0$ for $0\leq E_{p}<\Delta$. In contrast, a system
initially at finite temperature has a non-zero probability to be excited and
then relax in the collision: the relaxation channel is always open i.e.
$P_{01}(E_{p}+e_{1})\geq 0$ for $E_{p}\geq 0$. In this regime, we can then
write $\langle W\rangle(E_{p})=-\Delta f(\beta\Delta)P_{01}(E_{p}+e_{1})\leq
0$ where $f(x)=(1+e^{x})^{-1}$ is the Fermi function and
$\eta(E_{p})=\tanh(\beta\Delta/2)P_{01}(E_{p}+e_{1})\geq 0$. Thus, the maximum
energy that can be extracted from a two-level system in any scattering process
is $\langle W\rangle_{\mathrm{ext}}^{\mathrm{max}}=\Delta f(\beta\Delta)$.
Note when both channels are open, energy can still be extracted in the range
$\Delta\leq E_{p}\leq E^{\mathrm{max}}_{p}$ where $E^{\mathrm{max}}_{p}$ is
implicitly defined by $\langle W\rangle(E^{\mathrm{max}}_{p})=0$. Similar
conclusions can be drawn for an $N$-level system at low kinetic energies, with
different expressions for $\langle W\rangle(E_{p})$ and $\eta(E_{p})$,
provided that we consider the minimum energy gap of the system. These were
confirmed numerically, but a systematic study of energy extraction for larger
systems is left for future work.

At high kinetic energies $E_{p}\gg\Delta$ energy extraction becomes impossible
and we recover unital dynamics (Fig. 1, lower panel). Since in this regime
$\eta(E_{p})\rightarrow 0$, the signature of unital dynamics is
$P_{01}(E_{p}+e_{1})=P_{10}(E_{p}+e_{0})$ and we can write $\langle
W\rangle(E_{p})=\Delta\tanh(\beta\Delta/2)P_{10}(E_{p}+e_{0})\geq 0$. The
maximum energy consumed by the two-level system in any scattering process can
never exceed $\langle
W\rangle_{\mathrm{cons}}^{\mathrm{max}}=\Delta\tanh(\beta\Delta/2)$. In fact,
the convergence towards unitality at high kinetic energies is a universal
feature of the scattering process, where the behaviour of the scattering
matrix is mainly determined by the kinetic energy and depends weakly on the
system energy gaps $P_{10}(E_{p}+e_{0})=P_{10}(E_{p}+e_{1}-\Delta)\simeq
P_{10}(E_{p}+e_{1})=P_{01}(E_{p}+e_{1})$, where the last equality follows from
the time-reversal symmetry of the scattering matrix [2, 20, 23] (see also
Appendix C). Similar conclusions hold for an $N$-level system, with a
different expression for $\langle W\rangle(E_{p})$, provided that kinetic
energy is much larger than the maximum energy gap. These conclusions also hold
for non-symmetric potentials, since at sufficiently high kinetic energies the
precise shape of the potential $\hat{V}(\hat{x})$ is not captured by the
scattering matrix [21]. We confirmed numerically these predictions for non-
symmetric potentials and larger systems sizes.

Note that for a two-level system, the maximum energy that can be extracted
$\langle W\rangle_{\mathrm{ext}}^{\mathrm{max}}=\Delta f(\beta\Delta)$ is
maximal $\Delta/2$ at $\beta=0$ (infinite temperature) and decreases
monotonically to zero as $\beta\rightarrow\infty$ (zero temperature), while
the maximum energy consumed $\langle
W\rangle_{\mathrm{cons}}^{\mathrm{max}}=\Delta\tanh(\beta\Delta/2)$ is zero at
$\beta=0$ and increases monotonically to $\Delta$ at $\beta\rightarrow\infty$.
Curiously, there is a temperature above which extraction supersedes
consumption $0\leq\beta\leq\beta_{0}$, where $\beta_{0}=\Delta^{-1}\log(2)$ is
determined by the intersection of both functions. At this threshold
temperature we have $\langle W\rangle_{\mathrm{ext}}^{\mathrm{max}}=\langle
W\rangle_{\mathrm{cons}}^{\mathrm{max}}=\Delta/3$.

Conclusions. — We have shown how energy fluctuations of a quantum system can
be studied within scattering theory beyond the macroscopic source limit. When
a collision with a particle pushes the system away from thermal equilibrium,
the probability distribution for the energy changes obeys a universal
fluctuation relation (8) which allows for energy releasing processes as
dictated by non-unital dynamics. Such processes are particularly important if
the kinetic energy of the particle is of the order of the energy fluctuations,
highlighting the importance of non-unital maps in describing interactions with
microscopic sources. At high kinetic energies, unitality is recovered,
together with the standard fluctuation theorems for unital dynamics.

Our results may surprise readers familiar with the second law of
thermodynamics. As stated by Thomson and Planck: ”There is no physical process
whose sole effect is energy extraction from a thermal bath.” However, we have
to note that the state of the particle – generally described by a wavepacket –
will be distorted in the scattering process [20, 23]. In this regard, we show
in Appendix E that the entropy production, defined as the average log-ratio of
the probability for the forward process and the backward process, is always
positive – even at low kinetic energies, where energy extraction from a
thermal system is possible. In Appendices H and I we prove that heat
fluctuation theorems also follow from (8) when the kinetic energy of the
particle is thermally distributed. Our work provides a unifying perspective on
thermodynamics of quantum systems within a realistic scattering setup.

##  Appendix A Fluctuations for quantum maps

We review here, in full generality, how to define energy fluctuations for
dynamical maps, highlighting the importance of non-unital and unital maps.

A quantum (or dynamical) map $\Lambda$ is a completely positive and trace
preserving map, taking quantum states to quantum states
$\hat{\rho}^{\prime}=\Lambda(\hat{\rho})$ [47, 48, 49, 50]. The dual (or
adjoint) map $\Lambda^{\dagger}$ associated to $\Lambda$ is defined through
$\mathrm{Tr}[\hat{O}\Lambda(\hat{\rho})]=\mathrm{Tr}[\Lambda^{\dagger}(\hat{O})\hat{\rho}]$,
where $\hat{O}$ is an arbitrary linear operator. When
$\hat{O}=\hat{\mathbb{I}}$ we get
$\mathrm{Tr}\big{[}\Lambda(\hat{\rho})\big{]}=\mathrm{Tr}\big{[}\Lambda^{\dagger}(\hat{\mathbb{I}})\hat{\rho}\big{]}=\mathrm{Tr}[\hat{\rho}]$
which follows from the fact that $\Lambda$ is trace preserving. Thus, we
conclude that $\Lambda$ is trace preserving if and only if its dual is a
unital map $\Lambda^{\dagger}(\hat{\mathbb{I}})=\hat{\mathbb{I}}$, i.e. if its
dual preserves the identity. However, note that $\Lambda$ itself is generally
non-unital which means that its dual is not trace preserving:
$\mathrm{Tr}[\Lambda^{\dagger}(\hat{\rho})]=\mathrm{Tr}[\Lambda(\hat{\mathbb{I}})\hat{\rho}]\neq\mathrm{Tr}[\hat{\rho}]$.

A dynamical map represents the most general type of evolution for an open
quantum system [45, 46]. The map $\Lambda$ and its dual $\Lambda^{\dagger}$
can always be written as

$\displaystyle\Lambda(\cdot)=\sum_{l}\hat{K}_{l}\cdot\hat{K}_{l}^{\dagger}$
(12)
$\displaystyle\Lambda^{\dagger}(\cdot)=\sum_{l}\hat{K}_{l}^{\dagger}\cdot\hat{K}_{l}$
(13)

where $\\{\hat{K}_{l}\\}$ are called Kraus operators and the trace preserving
property reads
$\Lambda^{\dagger}(\hat{\mathbb{I}})=\sum_{l}\hat{K}^{\dagger}_{l}\hat{K}_{l}=\hat{\mathbb{I}}$.
In order to define energy fluctuations for a general dynamical map, we
consider two (non-degenerate) Hamiltonians
$\hat{H}=\sum_{n}E_{n}\hat{\Pi}_{n}$ and
$\hat{H}^{\prime}=\sum_{m}E^{\prime}_{m}\hat{\Pi}^{\prime}_{m}$ describing the
energy of the quantum system before and after the open system evolution
induced by Eq. (12), with $\hat{\Pi}_{n}=\ket{E_{n}}\bra{E_{n}}$ and
$\hat{\Pi}^{\prime}_{m}=\ket{E^{\prime}_{m}}\bra{E^{\prime}_{m}}$ being
projectors onto the energy eigenbasis. According to the two-point measurement
scheme [36, 29, 30], we measure the energy of the system before and after the
evolution. We can then define the following probability distribution for the
energy changes [52, 53, 55]

$\displaystyle P(W)$
$\displaystyle=\sum_{l,m,n}\delta(W-E^{\prime}_{m}+E_{n})\mathrm{Tr}[\hat{\Pi}^{\prime}_{m}\hat{K}_{l}\hat{\Pi}_{n}\hat{\rho}\hat{\Pi}_{n}\hat{K}_{l}^{\dagger}\hat{\Pi}^{\prime}_{m}]$
$\displaystyle=\sum_{l,m,n}\delta(W-E^{\prime}_{m}+E_{n})|\braket{E^{\prime}_{m}}{K_{l}}{E_{n}}|^{2}p_{n}\;,$
(14)

where $\rho$ is the initial quantum state of the system and
$p_{n}\equiv\braket{E_{n}}{\rho}{E_{n}}$. Since the dynamical map is trace
preserving, this distribution can be easily shown to be normalized $\int
P(W)dW=\mathrm{Tr}[\sum_{l}\hat{K}^{\dagger}_{l}\hat{K}_{l}\hat{\rho}]=1$.
Consider now that the system is in a thermal state with respect to the initial
Hamiltonian $\rho=e^{-\beta H}/Z$ where $Z=\mathrm{Tr}[e^{-\beta H}]$ is the
partition function of the initial Hamiltonian. Then
$p_{n}=(p_{n}/p^{\prime}_{m})p^{\prime}_{m}=e^{\beta(E^{\prime}_{m}-E_{n})}e^{-\beta\Delta
F}p^{\prime}_{m}$, where
$p^{\prime}_{m}\equiv\braket{E^{\prime}_{m}}{\rho^{\prime}}{E^{\prime}_{m}}$
and $\rho^{\prime}=e^{-\beta H^{\prime}}/Z^{\prime}$ is the thermal state with
respect to the final Hamiltonian, $Z^{\prime}=\mathrm{Tr}[e^{-\beta
H^{\prime}}]$ is the associated partition function and $\Delta
F=-\beta^{-1}\log(Z^{\prime}/Z)$ is the free energy difference between the two
thermal states. Using this in Eq. (A) leads immediately to the relation

$\displaystyle e^{-\beta W}P(W)=e^{-\beta\Delta F}\mathbb{P}(-W)\;,$ (15)

where the dual distribution is defined by

$\displaystyle\mathbb{P}(-W)$
$\displaystyle=\sum_{l,m,n}\delta(-W-E_{n}+E^{\prime}_{m})\mathrm{Tr}[\hat{\Pi}_{n}\hat{K}_{l}^{\dagger}\hat{\Pi}^{\prime}_{m}\hat{\rho}^{\prime}\hat{\Pi}^{\prime}_{m}\hat{K}_{l}\hat{\Pi}_{n}]$
$\displaystyle=\sum_{l,m,n}\delta(-W-E_{n}+E^{\prime}_{m})p^{\prime}_{m}|\braket{E^{\prime}_{m}}{K_{l}}{E_{n}}|^{2}\;.$
(16)

Note that this distribution encodes the energy changes in the reverse process,
where the evolution is now induced by the dual map. However, the dual
distribution is not normalized
$\int\mathbb{P}(-W)dW=\mathrm{Tr}[\sum_{l}\hat{K}_{l}\hat{K}^{\dagger}_{l}\hat{\rho}^{\prime}]=\mathrm{Tr}[\Lambda^{\dagger}(\hat{\rho}^{\prime})]\equiv\gamma\neq
1$ since the dynamical map is not unital. Integrating Eq. (15) leads to the
modified Jarzynski equality [55]

$\displaystyle\int e^{-\beta W}P(W)dW=e^{-\beta[\Delta
F-\beta^{-1}\log\gamma]}\;.$ (17)

An application of Jensen’s inequality for exponential functions $\langle
e^{-X}\rangle\geq e^{-\langle X\rangle}$, where $\langle\cdot\rangle$ denotes
the average with respect to some probability distribution, leads to the bound

$\displaystyle\langle W\rangle\geq\Delta F-\beta^{-1}\log\gamma\;.$ (18)

Note that in a cyclic process, the initial and final Hamiltonians are the same
and thus $\Delta F=0$. The scattering process studied here constitutes a
cyclic process.

##  Appendix B Explicit form of Eq. 1

The explicit evaluation of Eq. (1) is best performed by changing from momentum
eigenstates to kinetic energy eigenstates. We can do this through the
resolution of identity in the Hilbert space of the particle

$\displaystyle\hat{\mathbb{I}}_{P}=\int^{+\infty}_{-\infty}dp\ket{p}\bra{p}=\int^{+\infty}_{0}dE_{p}\sum_{\alpha=\pm}\ket{E^{\alpha}_{p}}\bra{E^{\alpha}_{p}}\;,$
(19)

where we separated the integral in momentum into its positive ($\alpha=+$) and
negative ($\alpha=-$) contributions and then changed variables from momentum
to kinetic energy. The kinetic energy eigenstates are then defined by
$\ket{E^{\alpha}_{p}}\equiv\sqrt{|dp/dE_{p}|}\ket{\alpha|p|}$ where
$|p|=\sqrt{2mE_{p}}$ and $|dp/dE_{p}|=\sqrt{m/|p|}$. They are orthogonal and
obey
$\braket{E^{\alpha}_{p}}{E^{\beta}_{q}}=\delta_{\alpha\beta}\delta(E_{p}-E_{q})$.
Using these eigenstates, the map in Eq. (1) can be written explicitly

$\displaystyle\Phi(\hat{\rho}_{S})$ $\displaystyle=\int dE_{p}\int dE_{q}\int
dE_{q^{\prime}}\sum_{\beta,\beta^{\prime}}\braket{E^{\beta}_{q}}{\hat{\rho}_{P}}{E^{\beta^{\prime}}_{q^{\prime}}}$
$\displaystyle\sum_{\alpha}\braket{E^{\alpha}_{p}}{\hat{S}}{E^{\beta}_{q}}\hat{\rho}_{S}\braket{E^{\alpha}_{p}}{\hat{S}}{E^{\beta^{\prime}}_{q^{\prime}}}^{\dagger}\;,$
(20)

where we omit the integration limits and summation values for simplicity. The
scattering operator in this representation can now be written as

$\displaystyle\braket{E^{\alpha}_{p}}{\hat{S}}{E^{\beta}_{q}}$
$\displaystyle=\sum_{j^{\prime},j}\ket{j^{\prime}}\bra{j}\braket{E^{\alpha}_{p},j^{\prime}}{\hat{S}}{E^{\beta}_{q},j}$
$\displaystyle=\sum_{j^{\prime},j}\ket{j^{\prime}}\bra{j}\delta(E_{p}+e_{j^{\prime}}-E_{q}-e_{j})s^{\alpha\beta}_{j^{\prime}j}(E_{q}+e_{j})$
$\displaystyle=\sum_{\Delta}\delta(E_{p}-E_{q}+\Delta)\sum_{\begin{subarray}{c}j^{\prime},j:\\\
e_{j^{\prime}}-e_{j}=\Delta\end{subarray}}s_{j^{\prime}j}^{\alpha\beta}(E_{q}+e_{j})\ket{j^{\prime}}\bra{j}$
$\displaystyle=\sum_{\Delta}\delta(E_{p}-E_{q}+\Delta)\hat{S}^{\alpha\beta}_{\Delta}(E_{q})\;.$
(21)

In the first line, we inserted two resolutions of identity in the Hilbert
space of the system $\hat{\mathbb{I}}_{S}=\sum_{j}\ket{j}\bra{j}$; in the
second line, we used the representation of the scattering operator in the
eigenbasis of $\hat{H}_{0}$ [1, 2]; in the third line, we split the sum over
system eigenstates into energy differences; in the last line, we defined the
eigenoperators of the system which appear in Eq. (2). Plugging this
representation into Eq. (B) and performing the integrals over $E_{p}$ and
$E_{q^{\prime}}$ yields

$\displaystyle\Phi(\hat{\rho}_{S})$ $\displaystyle=\int
dE_{q}\sum_{\beta,\beta^{\prime}}\sum_{\Delta,\Delta^{\prime}}\rho^{\beta\beta^{\prime}}_{P}(E_{q},E_{q}+\Delta^{\prime}-\Delta)$
$\displaystyle\sum_{\alpha}\hat{S}^{\alpha\beta}_{\Delta}(E_{q})\hat{\rho}_{S}\hat{S}^{\alpha\beta^{\prime}}_{\Delta^{\prime}}(E_{q}+\Delta^{\prime}-\Delta)^{\dagger}\;,$
(22)

where we used the notation
$\rho^{\alpha\beta}_{P}(E_{p},E_{q})\equiv\braket{E^{\alpha}_{p}}{\hat{\rho}_{P}}{E^{\beta}_{q}}$
for the state of the particle. If this state obeys condition (3), i.e.
$\rho^{\beta\beta^{\prime}}_{P}(E_{q},E_{q}-\Delta+\Delta^{\prime})\simeq\delta_{\beta\beta^{\prime}}\delta_{\Delta,\Delta^{\prime}}~{}\rho^{\beta}_{P}(E_{q})$,
where $\rho^{\beta}_{P}(E_{q})\equiv\rho^{\beta\beta}_{P}(E_{q},E_{q})$ is the
kinetic energy distribution for a particle traveling with direction $\beta$,
then the map simplifies to

$\displaystyle\Phi(\hat{\rho}_{S})$ $\displaystyle=\int
dE_{q}\sum_{\beta}\rho^{\beta}_{P}(E_{q})\sum_{\alpha,\Delta}\hat{S}^{\alpha\beta}_{\Delta}(E_{q})\hat{\rho}_{S}\hat{S}^{\alpha\beta}_{\Delta}(E_{q})^{\dagger}\;,$
(23)

which is equivalent to Eq. (4). Eqs. (5) and (6) are easily obtained by
introducing a new integration over a continuous variable $W$ with the help of
a $\delta$ function.

##  Appendix C Properties of the scattering matrix

###  C.1 Unitarity

The unitarity of the scattering operator
$\hat{S}^{\dagger}\hat{S}=\hat{S}\hat{S}^{\dagger}=\hat{\mathbb{I}}$, where
$\hat{\mathbb{I}}=\hat{\mathbb{I}}_{S}\otimes\hat{\mathbb{I}}_{P}$, enforces
two properties on the scattering matrix which we now briefly review, but can
be consulted in more detail elsewhere [1, 2, 20]. The two properties can be
obtained through the expressions

$\displaystyle\braket{E^{\alpha^{\prime}}_{p^{\prime}},j^{\prime}}{\hat{S}^{\dagger}\hat{S}}{E^{\alpha}_{p},j}=\delta_{\alpha\alpha^{\prime}}\delta_{j^{\prime}j}\delta(E_{p}-E_{p^{\prime}})$
(24)
$\displaystyle\braket{E^{\alpha^{\prime}}_{p^{\prime}},j^{\prime}}{\hat{S}\hat{S}^{\dagger}}{E^{\alpha}_{p},j}=\delta_{\alpha\alpha^{\prime}}\delta_{j^{\prime}j}\delta(E_{p}-E_{p^{\prime}})\;.$
(25)

By inserting resolutions of identity in the full Hilbert space between the two
operators, using the representation of the scattering operator
$\braket{E^{\alpha^{\prime}}_{p^{\prime}},j^{\prime}}{\hat{S}}{E^{\alpha}_{p},j}=\delta(E_{p^{\prime}}+e_{j^{\prime}}-E_{p}-e_{j})s^{\alpha^{\prime}\alpha}_{j^{\prime}j}(E_{p}+e_{j})$
and integrating out one of the $\delta$ functions, we find that the scattering
matrix obeys

$\displaystyle\sum_{\beta,k}[s^{\beta\alpha^{\prime}}_{kj^{\prime}}(E)]^{*}s^{\beta\alpha}_{kj}(E)=\delta_{\alpha\alpha^{\prime}}\delta_{j^{\prime}j}$
(26)
$\displaystyle\sum_{\beta,k}s^{\alpha^{\prime}\beta}_{j^{\prime}k}(E)[s^{\alpha\beta}_{jk}(E)]^{*}=\delta_{\alpha\alpha^{\prime}}\delta_{j^{\prime}j}\;,$
(27)

where $E=E_{p}+e_{j}$ is the total energy. These are the most important
properties of the scattering matrix holding for any fixed total energy $E$.
The transition probability
$P^{\alpha^{\prime}\alpha}_{j^{\prime}j}(E)\equiv|s^{\alpha^{\prime}\alpha}_{j^{\prime}j}(E)|^{2}$
then obeys
$\sum_{j^{\prime},\alpha^{\prime}}P^{\alpha^{\prime}\alpha}_{j^{\prime}j}(E)=\sum_{j,\alpha}P^{\alpha^{\prime}\alpha}_{j^{\prime}j}(E)=1$.
Thus $P^{\alpha^{\prime}\alpha}_{j^{\prime}j}(E)$ is a bistochastic matrix at
any fixed total energy $E$.

###  C.2 Time-reversal symmetry

Another important property of the scattering matrix is time-reversal symmetry.
Namely, if $H$ and $\hat{H}_{0}$ commute with the time-reversal operator
$\Theta$, then scattering is invariant under time-reversal and the scattering
matrix obeys

$\displaystyle
s^{\alpha^{\prime}\alpha}_{j^{\prime}j}(E)=s^{-\alpha-\alpha^{\prime}}_{jj^{\prime}}(E)\;,$
(28)

where we assume for simplicity of notation that the eigenstates of
$\hat{H}_{S}$ are time-reversal invariant $\Theta\ket{j}=\ket{j}$ [1, 2, 20].
As a consequence of this symmetry the transition probabilities obey
$P^{\alpha^{\prime}\alpha}_{j^{\prime}j}(E)=P^{-\alpha-\alpha^{\prime}}_{jj^{\prime}}(E)$.

##  Appendix D Properties of the Kraus operators

###  D.1 Eigenoperators

As we have discussed in the main text, the Kraus operators in Eq. (6) are
eigenoperators of the system. Using their definition in Eq. (2), it is easy to
see that they obey

$\displaystyle\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})^{\dagger}\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})$
$\displaystyle=\sum_{\begin{subarray}{c}k,j,j^{\prime}:\\\
e_{k}-e_{j^{\prime}}=e_{k}-e_{j}=\Delta\end{subarray}}\ket{j^{\prime}}\bra{j}[s^{\alpha^{\prime}\alpha}_{kj^{\prime}}(E_{p}+e_{j^{\prime}})]^{*}s^{\alpha^{\prime}\alpha}_{kj}(E_{p}+e_{j})$
(29)
$\displaystyle\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})^{\dagger}$
$\displaystyle=\sum_{\begin{subarray}{c}k^{\prime},k,j:\\\
e_{k^{\prime}}-e_{j}=e_{k}-e_{j}=\Delta\end{subarray}}\ket{k^{\prime}}\bra{k}s^{\alpha^{\prime}\alpha}_{k^{\prime}j}(E_{p}+e_{j})[s^{\alpha^{\prime}\alpha}_{kj}(E_{p}+e_{j})]^{*}\;.$
(30)

However, assuming that $\hat{H}_{S}$ is non-degenerate, we have
$e_{k}-e_{j^{\prime}}=e_{k}-e_{j}\Leftrightarrow
e_{j^{\prime}}=e_{j}\Rightarrow j=j^{\prime}$. Therefore (29) and (30) become

$\displaystyle\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})^{\dagger}\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})=\sum_{\begin{subarray}{c}k,j:\\\
e_{k}-e_{j}=\Delta\end{subarray}}P^{\alpha^{\prime}\alpha}_{kj}(E_{p}+e_{j})\ket{j}\bra{j}$
(31)
$\displaystyle\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})^{\dagger}=\sum_{\begin{subarray}{c}k,j:\\\
e_{k}-e_{j}=\Delta\end{subarray}}\ket{k}\bra{k}P^{\alpha^{\prime}\alpha}_{kj}(E_{p}+e_{j})\;.$
(32)

These last two expressions are diagonal in the eigenbasis of $\hat{H}_{S}$.
When they are used to compute the probability distribution and its dual [Eqs.
(Universal energy fluctuations in inelastic scattering processes) and
(Universal energy fluctuations in inelastic scattering processes)], they imply
that only on the populations of the system are relevant. Summing over all
energy differences and final particle directions $\alpha^{\prime}$ yields

$\displaystyle\sum_{\Delta,\alpha^{\prime}}\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})^{\dagger}\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})=\mathbb{I}_{S}\;,$
(33)
$\displaystyle\sum_{\Delta,\alpha^{\prime}}\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})^{\dagger}=\sum_{k}\ket{k}\bra{k}\sum_{j,\alpha^{\prime}}P^{\alpha^{\prime}\alpha}_{kj}(E_{p}+e_{j})\;.$
(34)

The first line expression follows from the properties of the scattering matrix
in Eqs. (26) and (27) and expresses the trace preserving property of the
dynamical map. The second line cannot be further simplified since in general
$\sum_{j,\alpha^{\prime}}P^{\alpha^{\prime}\alpha}_{kj}(E_{p}+e_{j})\neq 1$
[see discussion below Eqs. (26) and (27)]. Therefore, the dynamical map is
generally non-unital.

###  D.2 Time-reversal symmetry

In addition, the time-reversal symmetry of scattering matrix enforces a time-
reversal symmetry of the eigenoperators. Using Eq. (28) in Eq. (2) immediately
leads to

$\displaystyle\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})^{\dagger}=\hat{S}^{-\alpha-\alpha^{\prime}}_{-\Delta}(E_{p}-\Delta)\;.$
(35)

The physical interpretation is clear: the operator
$\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})$ induces a transition in the
system with energy $\Delta$ – the particle having initial kinetic energy
$E_{p}$ and direction $\alpha$, and final kinetic energy $E_{p}-\Delta$ and
direction $\alpha^{\prime}$ – while its adjoint
$\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})^{\dagger}$ induces the time-
reversed transition – the particle having initial kinetic energy
$E_{p}-\Delta$ and direction $-\alpha^{\prime}$, and final kinetic energy
$E_{p}$ and direction $-\alpha$. Note that the initial kinetic of the time-
reversed process $E_{p}-\Delta$ depends on the energy jump $\Delta$ induced in
the forward process. Although our main results presented in the paper do not
require (28) or (35), we show in Appendix G that they shed light on the
connection between the dual map and time-reversibility when our scattering
setup is microscopically reversible.

##  Appendix E Entropy production

The inequality in Eq. (10) can be interpreted as an expression of the entropy
production for the scattering process. Indeed, we can write

$\displaystyle\beta\langle W\rangle^{\alpha}(E_{p})$ $\displaystyle=\int\beta
WP^{\alpha}(E_{p},W)dW$
$\displaystyle=\int\log\Bigg{[}\frac{P^{\alpha}(E_{p},W)}{\mathbb{P}^{\alpha}(E_{p},-W)}\Bigg{]}P^{\alpha}(E_{p},W)dW$
$\displaystyle=\int\log\Bigg{[}\frac{P^{\alpha}(E_{p},W)}{\tilde{\mathbb{P}}^{\alpha}(E_{p},-W)}\Bigg{]}P^{\alpha}(E_{p},W)dW$
$\displaystyle-\log\gamma^{\alpha}(E_{p})\;,$ (36)

where in the last line we introduced the normalized dual distribution
$\tilde{\mathbb{P}}^{\alpha}(E_{p},-W)\equiv\mathbb{P}^{\alpha}(E_{p},-W)/\gamma^{\alpha}(E_{p})$
and used the normalization of $P^{\alpha}(E_{p})$. Therefore we conclude

$\displaystyle\Sigma^{\alpha}(E_{p})$
$\displaystyle=\int\log\Bigg{[}\frac{P^{\alpha}(E_{p},W)}{\tilde{\mathbb{P}}^{\alpha}(E_{p},-W)}\Bigg{]}P^{\alpha}(E_{p},W)dW$
$\displaystyle=\beta\langle
W\rangle^{\alpha}(E_{p})+\log\gamma^{\alpha}(E_{p})\geq 0\;.$ (37)

The entropy production is a relative entropy [49] between two distributions:
the energy distribution induced by the map and the (normalized) energy
distribution induced by the dual map.

##  Appendix F Derivation of Eq. 11

In order to derive Eq. (Universal energy fluctuations in inelastic scattering
processes), we start from its definition

$\displaystyle\eta^{\alpha}(E_{p})$
$\displaystyle=\gamma^{\alpha}(E_{p})-1=\int\mathbb{P}^{\alpha}(E_{p},-W)dW-1$
$\displaystyle=\sum_{j^{\prime},j}\frac{e^{-\beta
e_{j^{\prime}}}}{Z}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})-1\;.$ (38)

Expressing unity as
$1=\sum_{j^{\prime}j}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})e^{-\beta
e_{j}}Z^{-1}$ and splitting the sum into energy changes yields

$\displaystyle\eta^{\alpha}(E_{p})$
$\displaystyle=\frac{1}{Z}\sum_{\Delta}\sum_{\begin{subarray}{c}j^{\prime},j:\\\
e_{j^{\prime}}-e_{j}=\Delta\end{subarray}}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})(e^{-\beta
e_{j^{\prime}}}-e^{-\beta e_{j}})$
$\displaystyle=-\frac{2}{Z}\sum_{\Delta}\sum_{\begin{subarray}{c}j^{\prime},j:\\\
e_{j^{\prime}}-e_{j}=\Delta\end{subarray}}e^{-\beta\Delta/2}\sinh{\Bigg{(}\frac{\beta\Delta}{2}\Bigg{)}}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})e^{-\beta
e_{j}}\;.$

In the last expression we expressed the difference in exponentials as a
hyperbolic sine function. Now we split the sum over energy differences
$\Delta$ into positive and negative contributions, corresponding to system
excitation and relaxation, rewriting the expression as

$\displaystyle\eta^{\alpha}(E_{p})$
$\displaystyle=-\frac{2}{Z}\sum_{\Delta>0}\sinh{\Bigg{(}\frac{\beta\Delta}{2}\Bigg{)}}$
$\displaystyle\times\Bigg{[}\sum_{\begin{subarray}{c}j^{\prime},j:\\\
e_{j^{\prime}}-e_{j}=\Delta\end{subarray}}e^{-\beta\Delta/2}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})e^{-\beta
e_{j}}$ $\displaystyle-\sum_{\begin{subarray}{c}j^{\prime},j:\\\
e_{j^{\prime}}-e_{j}=-\Delta\end{subarray}}e^{\beta\Delta/2}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})e^{-\beta
e_{j}}\Bigg{]}\;.$

The first sum is now over energy gaps (positive by definition), the first term
in parenthesis corresponding to excitation and the last one to relaxation.
Permuting the labels in the last expression, it can be further is simplified
to

$\displaystyle\eta^{\alpha}(E_{p})$
$\displaystyle=\frac{2}{Z}\sum_{\Delta>0}\sinh{\Bigg{(}\frac{\beta\Delta}{2}\Bigg{)}}\sum_{\begin{subarray}{c}j^{\prime},j:\\\
e_{j^{\prime}}-e_{j}=\Delta\end{subarray}}$
$\displaystyle[P^{\alpha}_{jj^{\prime}}(E_{p}+e_{j^{\prime}})e^{-\beta
e_{j^{\prime}}}e^{\beta\Delta/2}-P^{\alpha}_{j^{\prime}j}(E_{p}+e_{j})e^{-\beta
e_{j}}e^{-\beta\Delta/2}]\;.$

The last step involves noting that we can write the quantity
$Z_{j^{\prime}j}=e^{-\beta e_{j^{\prime}}}+e^{-\beta e_{j}}$ in two equivalent
ways, namely $Z_{j^{\prime}j}=2\cosh(\beta\Delta/2)e^{-\beta
e_{j}}e^{-\beta\Delta/2}=2\cosh(\beta\Delta/2)e^{-\beta
e_{j^{\prime}}}e^{\beta\Delta/2}$ which leads directly to Eq. (Universal
energy fluctuations in inelastic scattering processes).

##  Appendix G Microscopic reversibility

The results presented in the main text are independent of whether or not our
scattering setup is microscopically reversible; nevertheless, we show here the
impact of microscopic reversibility on our results. In addition to the
scattering matrix obeying time-reversal symmetry in Eq. (28), microscopic
reversibility holds when the potential is spatially symmetric or when the
particle approaches the scattering region from the left or right with equal
probability [20]. We will assume the latter, which amounts to considering that
the distribution of the particle introduced in Eq. (3) is given by
$\rho^{\alpha}_{P}(E_{p})=\rho_{P}(E_{p})/2$. The map in Eq. (4) then reads
simply $\Phi(\hat{\rho}_{S})=\int
dE_{p}~{}\rho_{P}(E_{p})~{}\Phi(E_{p})(\hat{\rho}_{S})$, where $\Phi(E_{p})$
is now only conditioned on the kinetic energy of the particle. It can be
expressed as an integral over quantum operations, just like in Eq. (5), with
the expression

$\displaystyle\Phi(E_{p},W)(\cdot)=\sum_{\Delta}\delta(W-\Delta)\frac{1}{2}\sum_{\alpha^{\prime},\alpha}\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})\cdot\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})^{\dagger}\;$
(39)

and Eq. (Universal energy fluctuations in inelastic scattering processes) now
becomes

$\displaystyle P(E_{p},W)$
$\displaystyle=\mathrm{Tr}_{S}\big{[}\Phi(E_{p},W)(\hat{\rho}_{S})\big{]}$
$\displaystyle=\sum_{j^{\prime},j}\delta(W-e_{j^{\prime}}+e_{j})P_{j^{\prime}j}(E_{p}+e_{j})p_{j}\;,$
(40)

where
$P_{j^{\prime}j}(E_{p}+e_{j})=1/2\sum_{\alpha,\alpha^{\prime}}|s^{\alpha^{\prime}\alpha}_{j^{\prime}j}(E_{p}+e_{j})|^{2}$
are the transition probabilities. On the other hand, using Eq. (35) the dual
quantum operation can now be written as

$\displaystyle\Phi(E_{p},W)^{\dagger}(\cdot)=\sum_{\Delta}\delta(W-\Delta)\frac{1}{2}\sum_{\alpha^{\prime},\alpha}\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})^{\dagger}\cdot\hat{S}^{\alpha^{\prime}\alpha}_{\Delta}(E_{p})$
$\displaystyle=\sum_{\Delta}\delta(W-\Delta)\frac{1}{2}\sum_{\alpha^{\prime},\alpha}\hat{S}^{\alpha^{\prime}\alpha}_{-\Delta}(E_{p}-\Delta)\cdot\hat{S}^{\alpha^{\prime}\alpha}_{-\Delta}(E_{p}-\Delta)^{\dagger}$
$\displaystyle=\Phi(E_{p}-W,-W)(\cdot)\;.$ (41)

We see that if our setup is microscopically reversible, the dual operation
$\Phi(E_{p},W)^{\dagger}$ is given by $\Phi(E_{p}-W,-W)$, where the kinetic
energy in the argument is now conditioned on the energy change $W$ [see
discussion below Eq. (35)]. The dual distribution in Eq. (Universal energy
fluctuations in inelastic scattering processes) now becomes

$\displaystyle P(E_{p}-W,-W)$
$\displaystyle=\mathrm{Tr}_{S}\big{[}\Phi(E_{p}-W,-W)(\hat{\rho}_{S})\big{]}$
$\displaystyle=\sum_{j^{\prime},j}\delta(-W-e_{j^{\prime}}+e_{j})P_{j^{\prime}j}(E_{p}+e_{j}-W)p_{j}\;,$
(42)

and our main result in Eq. (8) takes the form

$\displaystyle e^{-\beta W}P(E_{p},W)=P(E_{p}-W,-W)\;.$ (43)

In this microscopically reversible formulation, it is clear that the map
becomes unital when $\Phi(E_{p}-W,-W)\simeq\Phi(E_{p},-W)$. As we have already
discussed, a sufficient condition for this to happen is that the scattering
matrix depends very weakly on the energy gaps of the quantum system as is the
case at very high kinetic energies.

##  Appendix H The unconditioned map

The map in Eq. (4) can also be written as $\Phi(\hat{\rho}_{S})=\int
dW\Phi(W)(\hat{\rho}_{S})$ where

$\displaystyle\Phi(W)(\hat{\rho}_{S})=\int
dE_{p}~{}\sum_{\alpha}\rho^{\alpha}_{P}(E_{p})~{}\Phi^{\alpha}(E_{p},W)(\hat{\rho}_{S})$
(44)

is the quantum operation in Eq. (6) integrated over the initial state of the
particle. It gives rise to an unconditioned form of the distribution in Eq.
(Universal energy fluctuations in inelastic scattering processes) reading

$\displaystyle P(W)$
$\displaystyle=\mathrm{Tr}_{S}\big{[}\Phi(W)(\hat{\rho}_{S})\big{]}=\sum_{j^{\prime},j}\delta(W-e_{j^{\prime}}+e_{j})\mathbb{S}_{j^{\prime}j}p_{j}\;,$
(45)

where

$\displaystyle\mathbb{S}_{j^{\prime}j}=\int
dE_{p}~{}\sum_{\alpha}P^{\alpha}_{j^{\prime}j}(E_{p}+e_{i})\rho^{\alpha}_{P}(E_{p})$
(46)

is a stochastic matrix ruling the transition probabilities for the system. For
an initially thermal system, the fluctuation relation in Eq. (8) now reads

$\displaystyle e^{-\beta W}P(W)=\mathbb{P}(-W)$ (47)

where
$\mathbb{P}(-W)=\mathrm{Tr}_{S}\big{[}\Phi(W)^{\dagger}(\hat{\rho}_{S})\big{]}=\sum_{j^{\prime},j}\delta(-W-e_{j}+e_{j^{\prime}})p_{j^{\prime}}\mathbb{S}_{j^{\prime}j}$
is the dual distribution integrated over the initial state of the particle.
From this relation, we can derive an inequality analogous to (10) but
unconditioned on the initial state of the particle. In summary, our results
carry over for the unconditioned map in Eq. (5).

##  Appendix I Heat fluctuation theorems

The properties of the stochastic matrix $\mathbb{S}_{j^{\prime}j}$ in Eq. (46)
have been studied in Ref. [20] in the context of thermalization. Namely, it
was shown if the kinetic energy of the particle is thermally distributed
$\rho^{\alpha}_{P}(E_{p})\sim e^{-\tilde{\beta}E_{p}}$ with inverse
temperature $\tilde{\beta}$, then detailed balance

$\displaystyle\mathbb{S}_{j^{\prime}j}=e^{-\tilde{\beta}(e_{j^{\prime}}-e_{j})}\mathbb{S}_{jj^{\prime}}$
(48)

holds, provided the scattering setup is microscopically reversible (see
Appendix G). In this case, applying detailed balance to Eq. (45) now yields

$\displaystyle P(W)=e^{-\tilde{\beta}W}\mathbb{P}(W)\;.$ (49)

This new result expresses a detailed balance symmetry between the energy
distribution and its dual, being independent of the state of the system.
However, when the system is thermal then Eqs. (47) and (49) together imply

$\displaystyle e^{-(\beta-\tilde{\beta})W}P(W)=P(-W)\;,$ (50)

which are the heat exchange fluctuation theorems [39].

## References

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# State Estimation for Vision-based Localization under Uncertain Conditions

Prashant V. Patil  Pranav Thakkar  Leena Vachhani  IITB-Monash Research
Academy, Mumbai, India (e-mail<EMAIL_ADDRESS> Dept. of
Aerospace Engineering, IIT Bombay, Mumbai, India (e-mail:
<EMAIL_ADDRESS> Systems and Control Engineering, IIT Bombay, Mumbai,
India (e-mail<EMAIL_ADDRESS>
###### Abstract

Vision based localization is a popular approach to carry out manoeuvres
particularly in GPS-restricted indoor environments, because vision can
complement other activities performed by the robot. The objective is to
estimate the current location with respect to a known location by matching the
bearings. The problem is challenging as the known location information is in
terms of the bearings of landmarks extracted from an image. We address the
problem under more challenging scenario when landmarks are semi-static. In
this work, an observer formulation is presented which enables to incorporate
the effect of change in landmark position as parameters. The efficacy of two
estimators: Augmented Extended Kalman Filter (A-EKF) and a Proportional-
Integral EKF (PI-EKF) is tested under the cases where there are changes in
some of the landmark positions. Morever, it is likely that not all landmarks
are visible to the robot at all instants of time. A multi-rate estimation
framework is proposed to mitigate this issue. Observability analysis is
carried out to arrive upon a minimum number of landmarks required for such a
formulation. Simulation studies are presented to test the efficacy of the
formulations.

###### keywords:

Vision Based localization, Navigation, Visual Homing, State and Parameter
Estimation, Robust State Estimation, Non-linear Observers and Filter Design,
Autonomous Robotic System

††thanks: Both the authors contributed equally for this submission

##  1 Introduction

Navigation using bearing only information from the landmarks, is a popular
research area, with applications ranging from manoeuvres to docking as well as
pick-and-place tasks. Vision based localization and navigation is a prominent
approach in GPS restricted indoor environments (Liu et al. (2010), Corke
(2003),Bekris et al. (2006)). Vision based sensors can complement in dealing
with the other activities performed by the robot, and this is what makes these
sensors appealing for low cost applications. Bearing information can be easily
and reliably extracted from an image and a significant research progress has
been achieved in this area. For the control formulation used in this exercise,
it can be shown that this choice of measurements also eliminates the need to
measure the position of the robot with respect to the home position
instantaneously (Arunkumar et al. (2018)). However, an environment like a
warehouse, or in a pick and place application, the landmarks could be
typically be the products and items in the environment. It may happen that
these landmarks are slightly disturbed from their position. Most of the
popular localization methods consider the features of the landmarks as
parameters. Thus, a change in the position of landmarks change could propagate
a parametric change in the models. Rosen et al. (2016), Meyer-Delius et al.
(2010) have proposed ways to mitigate the problems that arise in a semi-static
environment. However, in our case, as the landmarks could be slightly
disturbed from their positions, the approaches could be an overkill.
Frameworks which may increase the computational burden on a low cost platform
are also not desirable. Also, knowingly or otherwise, uncertainty seeps in the
localization problem for a wheeled mobile robot, from sources other than those
resulting from the disturbed landmarks.
The Bayesian state estimation approach handles the effect of random noise in
the states and the output of the system systematically, and hence is a widely
used framework in this area(Sebastian Thrun (2005)). The popular Bayesian
estimation algorithms work satisfactorily provided the model parameters values
are fairly accurately known. (Patwardhan et al. (2012)). The accuracy of these
Bayesian algorithms are leveraged on the accuracy of the model that is used to
carry out the predictions. Using an inaccurate model or a model without
correct parameter values could result in biased state estimates (Bavdekar et
al. (2013)). This can subsequently have ramifications on applications which
use these estimates directly such as in state-feedback control formulations.
It therefore becomes necessary to systematically take the parametric model-
plant mismatch into account and modify the standard estimation algorithms.
Some of the widely popular approaches to compensate for this mismatch are the
joint state and parameter estimation approaches, dual estimation approaches,
adaptive estimation based approaches. The dual and the joint state-parameter
estimation based approaches deal with the errors arsing from this mismatch by
re-identifying the model parameters. The Proportional-Integral Kalman
Filter(PI-KF) uses the idea of compensating the error caused due to the model-
plant mismatch or by linearization, by a bias term, which is integrated over
time based on the innovation sequence (Bavdekar et al. (2013)). We propose a
simple extension to this filter to suit the non-linear state transition map,
and call it the Proportional-Integral Extended Kalman Filter(PI-EKF). The PI-
EKF implementation is similar to the conventional Extended Kalman Filter(EKF).
This nonlinear filter appears to be a promising, computationally efficient way
of dealing with uncertainties and we use this filter in our studies and
benchmark its results against the conventional EKF, and the Augmented State-
Parameter estimation based EKF(A-EKF) (Bavdekar et al. (2013)).
Occlusion of landmarks is unavoidable in vision based methods. Several
features may not be visible for a given pose due to occlusion, or due to a
limited field of view. If the initial position of the robot is far apart from
the target position, it may happen that the neighbourhood of these two
positions do not share any common visual feature. Therefore, the robot must
know if it can initiate the vision based localization with respect to the
given known location or not. We present a multi-rate feedback based estimation
approach based on the filters above to mitigate these issues. (Armesto et al.
(2004)) The highlights of the paper can be summarized as

  * •

A state space formulation based approach is presented to suit the multi-rate
homing problem in presence of uncertainties

  * •

Observability analysis is carried out to arrive at minimum number of landmarks
required for this formulation

  * •

The efficacy of the EKF, A-EKF and PI-EKF is tested through open loop
simulation studies.

  * •

An exercise is carried out to reach the known location(homing) using a
controller that uses these estimates (Arunkumar et al. (2018)) and the closed
loop performance of the scheme is evaluated.

This work builds upon the formulation presented by Gupta et al. (2017). The
organization of this paper is as follows: The next section of this paper
formulates the state space realization for the estimation exercise. The
observability analysis, which gives us a sense of minimum number of landmarks
to be used for the formulation, is presented in the section 3. The estimators
used in this exercise are briefly described in the section 4. The simulation
studies are presented in the section 5. The section 6 concludes this paper.

##  2 State Space Formulation

Consider a unicycle robot moving on a 2-D plane as shown in the figure 1. Let
O denotes the known target position, which, without loss of generality, is
considered as the origin of the X-Y reference frame. Let P denote the
instantaneous position of the unicycle. Polar coordinates are used to denote
the position of the robot and the landmarks. In this work, $R$ denotes the
range of the robot and $\alpha$ denotes the orientation of the robot. Let
$L_{i}$ be the position of the $i^{th}$ landmark as shown in the figure 1. The
unicycle is equipped with a monocular panoramic vision, and, in our exercise,
the landmarks typically correspond to a distinct visual feature in the
environment. There are $q$ landmarks in the environment. Bearing angles of all
the landmarks w.r.t the positive X-axis, at the home position $O$, are
calculated and stored a-priori, and are denoted by $\beta_{1}^{*}$,
$\beta_{2}^{*}$, …$\beta_{q}^{*}$ respectively. The distance of the $i^{th}$
landmark from the target position $D_{i}^{*}$ is not necessarily known
(particularly for the case with monocular vision), and, can be easily
estimated using relations provided later in this paper. At every time instant,
the robot measures the bearing angles from each landmarks as shown in the
figure 1. They are calculated with reference to the velocity vector $\vec{v}$
and are given by $\beta_{1}$, $\beta_{2}$, …$\beta_{q}$ respectively. The
convention used here is to consider all the angles positive in an anti-
clockwise direction. In our applications, the robot has to visit the landmark
positions. In this paper, an augmented state space formulation is used by
defining the state vector as :

Figure 1: Planar view describing the system

$X=\begin{bmatrix}R\ \ \ \ \theta\ \ \ \ \alpha\ \ \ \ \beta_{1}\ \ \ \
\beta_{2}\ \ \ \ \ldots\ \ \ \ \beta_{q}\end{bmatrix}^{\textsf{T}}$

Typically, the bearing angles of the landmarks, $\beta_{i}$, which are
measured, are used in the control laws and such a formulation arrives at more
accurate estimates of these angles. The unicycle model, in polar co-ordinates
is described by:

$\begin{split}\dfrac{dR}{dt}=v\cos(\alpha-\theta)\\\
\dfrac{d\theta}{dt}=\frac{v\sin(\alpha-\theta)}{R}\end{split}$ (1)

where $v$ is linear velocity of the robot. Using the geometry in the figure 1
( Gupta et al. (2017)), the following relation is established :

$\displaystyle\dfrac{R}{\sin(\beta_{i}^{*}-(\beta_{i}+\alpha))}$
$\displaystyle=\dfrac{D_{i}^{*}}{\sin(\pi-(\theta-(\beta_{i}+\alpha)))},$ (2)
$\displaystyle D_{i}^{*}$
$\displaystyle=R\dfrac{\sin(\theta-(\beta_{i}+\alpha))}{\sin(\beta_{i}^{*}-(\beta_{i}+\alpha))}$
(3)

for $i=1,2,3\ldots q$. Note that $D_{i}^{*}$ for $i=1,2,3\ldots q$ are
constants,and are not measured. Typically, $\beta_{i}$, $i=1,2,3\ldots q$ are
calculated as a part of an offline exercise and these relations help in
estimating $D_{i}^{*}$. The rate of change of the bearing angle with respect
to time $t$ can be expressed using (1) and (3) as:

$\dot{\beta_{i}}=\dfrac{-v\sin(\beta_{i})}{R\cos(\theta-(\beta_{i}+\alpha))-D_{i}^{*}\cos(\beta_{i}^{*}-(\beta_{i}+\alpha))}-\dot{\alpha}$
(4)

where $D_{i}^{*}$ is given by (3). The systems of equations (1) and (4) can be
used to arrive at a state space representation of the form :

$\begin{split}\dfrac{dx}{dt}=g(X,U)\\\ \vskip 3.0pt plus 1.0pt minus
1.0pty=h(x)\end{split}$

where $U=[v$ $\omega]^{\textsf{T}}$, $v$ is the linear velocity of the robot
and $\omega$ is its angular velocity.

$\displaystyle g(X,U)$
$\displaystyle=\left(\begin{array}[]{cc}\cos(\alpha-\theta)&0\\\
\dfrac{\sin(\alpha-\theta)}{R}&0\\\ 0&1\\\
\dfrac{-\sin(\beta_{1})}{RC_{1}-D_{1}^{*}C_{1}^{*}}&-1\\\
\dfrac{-\sin(\beta_{2})}{RC_{2}-D_{2}^{*}C_{2}^{*}}&-1\\\ \vdots&\vdots\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{-\sin(\beta_{q})}{RC_{q}-D_{q}^{*}C_{q}^{*}}&-1\\\
\end{array}\right)U$ (12) $\displaystyle=\ \ \ \ \ \ \ \ \ \ [g_{1}\ \ \ \ \ \
\ \ \ g_{2}]\ \ U\dots say$ (13)

where, for brevity, $\cos(\theta-(\beta_{i}+\alpha))$,
$\cos(\beta_{i}^{*}-(\beta_{i}+\alpha))$, $\cos(\beta_{i})$,
$\sin(\theta-(\beta_{i}+\alpha))$, $\sin(\beta_{i}^{*}-(\beta_{i}+\alpha))$,
$\sin(\beta_{i})$ are concisely denoted as $C_{i}$, $C_{i}^{*}$, $c_{i}$,
$S_{i}$, $S_{i}^{*}$, $s_{i}$, respectively. The measurement equation is given
by:

$Y=h(x)\begin{bmatrix}\beta_{1}\\\ \beta_{2}\\\ \vdots\\\
\beta_{q}\end{bmatrix}=HX$ (14) $H=[\bar{0}_{q\times 3}\ \ I_{q}]$ (15)

where $\bar{0}_{q\times 3}\in\mathbb{R}^{3\times q}$ is the null matrix and
$I_{q}\in\mathbb{R}^{q\times q}$ is the identity matrix. In this exercise, we
consider that the robot is moving with a constant linear velocity. The robot
is controlled by changing its angular velocity $\omega$.

##  3 Observability Analysis

Our state space realization (12)-(14) consists of $3+q$ states but only the
last $q$ of these states: $\beta_{1},\beta_{2},\dots\beta_{q}$ are obtained
from the images. We are interested in estimating the robot’s configuration
along with estimating the angles, $\beta_{i},i\in\\{1,2,\ldots,q\\}$ using
these measurements. We are also interested in arriving at the value of the
minimum number of landmarks required in this formulation such that it is fully
observable. Our state space realization (12,14) is of the form:

$\dot{X}=g(X,U)\qquad\qquad Y=h(X)=HX$ (16)

We are interested in investigating the local observability of the system. We
use the Lie derivative approach for this investigation (Hermann and Krener
(1977), Gupta et al. (2017)). For notational simplicity, let
$\scriptstyle(RC_{i}-D_{i}^{*}C_{i}^{*})$
,$\frac{s_{i}(C_{i}-\frac{C_{i}^{*}D_{i}^{*}}{R})}{d_{i}^{2}}$,
$\frac{Rs_{i}(S_{i}+\frac{C_{i}C_{i}^{*}}{S_{i}^{*}})}{d_{i}^{2}}$,
$\frac{-s_{i}C_{i}^{*}}{S_{i}^{*}d_{i}}$,
$\frac{-c_{i}}{d_{i}}-\frac{s_{i}C_{i}^{*}}{S_{i}^{*}d_{i}}$ be denoted as
$d_{i}$, $J_{i}$, $K_{i}$, $P_{i}$, $Q_{i}$ respectively for
$i\in\\{1,2,3,\ldots q\\}$. The $zero^{th}$ order lie derivative of $h$ along
$g$ is

$L^{0}h=h(X)=\left(\begin{array}[]{cccc}\beta_{1}&\beta_{2}&\ldots&\beta_{q}\end{array}\right)^{\textsf{T}}$
(17) $\nabla L^{0}h=\nabla h(X)=[\bar{0}_{q\times 3}\ \ I_{q}]$ (18)

The first order lie derivative of h along $g_{1}$ is

$\nabla
L_{g_{1}}^{1}h=\left(\begin{array}[]{lllllll}J_{1}&K_{1}&P_{1}&Q_{1}&0&\ldots&0\\\
J_{2}&K_{2}&P_{2}&0&Q_{2}&\ldots&0\\\
&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
J_{q}&K_{q}&P_{q}&0&0&\ldots&Q_{q}\end{array}\right)$ (19)
$L_{g_{2}}^{1}h=\frac{\partial h(X)}{\partial X}.g_{2}\\\
=\left(\begin{array}[]{cccc}-1&-1&\ldots&-1\end{array}\right)^{\textsf{T}}$
(20) $\nabla L_{g_{2}}^{1}h=\bar{0}_{q\times(q+3)}$ (21)

Now, consider the first three rows of the observability matrix $\mathcal{O}$
(Hermann and Krener (1977)):

$\mathcal{O}=\left(\begin{array}[]{c}\nabla L^{0}h\\\ \nabla L^{1}_{g_{1}}h\\\
\nabla L^{1}_{g_{2}}h\\\ \vdots\end{array}\right)$ (22)

The observability matrix $\mathcal{O}$ in (22) has rank $n=q+3$,(equal to our
number of states), provided $q\geq 2$. Thus, our state space realization is
locally observable provided that there are at-least two landmarks available.

##  4 Estimators

We now present the formulations of the estimators used in this work.

###  4.1 Multi-Rate Extended Kalman Filter Formulation

The conventional EKF as an observer is modified to incorporate the
measurements not being available at all instants of time. At every instant
only $p\geq 2$ of the $q$ landmarks are visible to the robot. Hence, the state
space formulation is reformulated as:

$Y_{p}=h_{p}(x)\begin{bmatrix}\beta_{1}\\\ \beta_{2}\\\ \vdots\\\
\beta_{p}\end{bmatrix}=H(k)X$ (23)

Since the number of landmarks keep on changing at each iteration, therefore
dimension of H matrix and R matrix in EKF change. Prediction and the updates
are carried out as:
Prediction:-

$\displaystyle\hat{X}(k\mid k-1)$ $\displaystyle=F(\hat{X}(k-1\mid
k-1),U(k-1))$ $\displaystyle P(k\mid k-1)$ $\displaystyle=G(k)P(k-1\mid
k-1)G(k)^{\textsf{T}}+\Gamma_{u}Q\Gamma_{u}^{\textsf{T}}$

Update :-

$\displaystyle L_{p}(k)=$ $\displaystyle
P(k|k-1)H_{p}(k)^{\textsf{T}}[H_{p}(k)P(k|k-1)H_{p}(k)^{\textsf{T}}$
$\displaystyle+R_{p}]^{-1}$ $\displaystyle e_{p}(k)=$
$\displaystyle[Y_{p}(k)-H_{p}(k)\hat{X}(k\mid k-1)]$
$\displaystyle\hat{X}(k\mid k)=$ $\displaystyle\hat{X}(k\mid
k-1)+L_{p}(k)e_{p}(k)$ $\displaystyle P(k\mid k)=$
$\displaystyle[I-L_{p}(k)H_{p}(k)]P(k\mid k-1)$

where

${G}(k){=}\left[\frac{\partial
f}{\partial{X}}\right]\bigg{|}_{({\hat{X}}(k-1|k-1),{U}(k))}$

${\Gamma_{u}}(k){=}\left[\frac{\partial
f}{\partial{U}}\right]\bigg{|}_{({\hat{X}}(k-1|k-1),{U}(k))}$

If H is measurement matrix at initial position and home position then,
$H_{p}(k)$ is formed with those rows of H which are measured at the instant
$k$. $R_{p}$ is the covariance matrix of the measured noises, corresponding to
these $p$ measurements. Note that the size of matrices and vectors with
subindex $p$ varies depending on the measured features in the output vector.
In the case where landmarks are visible, there are no measurements that are
obtained for that time instant. Matrices denoted with the subindex $p$ are
void and therefore no correction is performed, the state is simply predicted,
providing an inter-sampling behavior (Armesto et al. (2004)).
The augmented EKF (A-EKF) formulation is a simple extension of this filter the
state matrix is augmented as

$X=[X\ \ \theta_{p}]^{\textsf{T}}$ (24)

The covariance matrix $P$ is also block diagonally appended. The variation of
the parameter vector, $\theta_{p}$ is assumed to be governed by the random
walk model (Rangegowda et al. (2018)).

###  4.2 Proportional Integral-EKF

The proportional-integral Kalman filter (PI-KF)(Bodizs et al. (2011), Bavdekar
et al. (2013)) tries to incorporate the model-plant mismatch or the errors due
to linearization to carry out predictions. The original formulation is
presented for linear systems and is extended in this exercise for the case
wherein the state-transition map is nonlinear. The prediction step for this
filter is of the form:

$\displaystyle\hat{X}(k\mid k-1)$ $\displaystyle=F(\hat{X}(k-1\mid
k-1),U(k-1))+\mathcal{P}{\kappa_{i-1}}$

where $\mathcal{P}\in\mathbb{R}^{n\times n}$ is equivalent to the gain of a PI
controller. ${\kappa_{i-1}}\in\mathbb{R}^{n}$ is the mismatch term obtained as

${\kappa_{i}}={\kappa_{i-1}}+\mathcal{M}[Y(k-1)-H\hat{X}(k-1|k-2)]$ (25)

where $\mathcal{M}$ can be seen as a “forgetting factor"(Bavdekar et al.
(2013)). It incorporates the impact of the past measurements on the
accumulated errors. This extension of the (PI-KF) to non-linear systems is
termed as (PI-EKF). The rest of the algorithm is identical to that of the EKF.
The multi-rate formulation of this estimator is also along the lines of the
filter described in the subsection 4.1.

##  5 Simulation Studies

In the simulation studies, homing was used as the navigation problem which
uses the estimates generated by the localization exercise. The simulation
studies were carried out for the following cases:

  1. 1.

Homing using exactly known landmark information

  2. 2.

Homing using inaccurate landmark information (happens when landmark is semi-
static in nature), in a multi-rate scenario.

  3. 3.

Simultaneous homing and landmark parameter estimation

  4. 4.

Open loop analysis with constant linear and angular velocity commanded: to
investigate the asymptotic properties of the estimators.

The simulation studies are carried out in two cases: 1. Using ideal conditions
where the measurements are generated using the equations (12,14) using an ode
solver, and 2. Using an in-house image data-set used to generate measurements
for the filter. A sample panoramic image with extracted features is shown in
Figure 2

Figure 2: Representational image from the dataset (Gupta et al. (2017))

A proportional control law is used to steer the robot to the home position.
The control law is

$\omega=\pi-(\hat{\alpha}(k)-\hat{\theta}(k))$ (26)

where $\hat{\theta(k)}$ and $\hat{\alpha}(k)$ are obtained from the
estimators.
Case 2 Homing in the case where the position of the landmarks is not
accurately known: In all the homing exercises, the robot commanded to move
with a forward velocity of 12.5cm/sec. The robot is controlled using the
angular velocity alone. The table (1) gives the simulation conditions used in
this exercise. The configuration of the robot is denoted by $X_{r}(k)=[R(k)\ \
\theta(k)\ \ \alpha(k)]^{\textsf{T}}$

Table 1: Simulation Parameters Parameter | Value
---|---
Sampling Time | 0.05 sec
$X_{r}(0)$ | $[75cm\ \ -0.872rad\ \ 0.349rad]^{\textsf{T}}$
$P(0|0)$ | $\mathbf{I_{n}}$
$R$ | $diag(0.05)_{p\times p}$
$Q$ | $diag(0.01)_{n\times n}$
$Q_{d}$ | $diag(0.0025)n_{ip}\times n_{ip}$
$\mathcal{P}$ | 0.05 $\mathbf{I_{n}}$
$\mathcal{M}$ | $\bar{1}_{n\times p}$
q | 7

. The measure of performance used for comparison is: Root Mean Squared
Error(RMSE). Let the state estimation error be defined as

$\varepsilon(k|k)={X}(k)-\hat{{X}}(k|k)$ (27)

Root Mean Squared Error$(RMSE)_{i}$ is calculated for each state variable,
where $(RMSE)_{i}$ is calculated as:

$(RMSE)_{i}=\sqrt{\frac{1}{N}\sum_{k=1}^{N}[{x_{i}}(k)-\hat{{x}_{i}}(k|k)]^{2}}$
(28)

The RMSE values for the three filters have been tabulated below, while the
evolution of pose error with time is depicted in Figure 3. The PI-EKF results
in a superior performance than the other two, however, a trade-off needs to be
achieved in tuning of its parameters for guaranteeing satisfactory transient
as well as steady state performance.

Table 2: Closed Loop Performance Filters States | $\hat{R}(cm)$ | $\hat{\theta}(rad)$ | $\hat{\alpha}(rad)$
---|---|---|---
EKF | 1.9091 | 0.3840 | 0.0589
A-EKF | 1.8076 | 0.2324 | 0.0616
PI-EKF | 0.3778 | 0.0285 | 0.0271

Note that Case 1 is a trivial case of homing with inaccurate home information
available to the robot. Hence it is encompassed by the above results. The
estimator is seen to converge to true state when precise information is
provided, however those results are not included for the sake of brevity.

Figure 3: Estimation error (Closed Loop)

Case 3: The A-EKF enables parameter estimation in conjunction with recursively
estimating robot pose and bearing angle states at each time instant. This
allows for the possibility to correct inaccurate landmark estimates during the
navigation itself. However, performance pertaining to parameter estimation is
highly sensitive to tuning. For estimation of just one parameter
$\beta_{1}^{*}$, tuning is a relatively easy task. The state as well as
parameter convergence can be shown in Figure 4 and in Figure 5, in the case of
$30\%$ error in parameter value.The initial position is marked with an arrow
as shown in the figure (4) .

Figure 4: 2-D Homing for A-EKF in presence of 1 fault Figure 5: Estimation for
a Single Parameter using A-EKF

As more landmark parameters are estimated (regardless of whether the initial
estimate is accurate or not), the problem becomes cumbersome and sensitive to
the choice of tuning and leads to inconsistent performance, both for state and
parameter estimation.

Case 4: Open loop analysis of the estimators is crucial despite homing being
guaranteed under uncertain conditions, as the homing problem is a finite time
problem and estimator convergence is itself a sufficient condition for homing.
A comparison is done between the three filters for robot performing circular
motion, with landmark information merely being used for state prediction as
part of the filter. To observe the asymptotic properties of each estimator, no
faults in the knowledge of $\beta_{i}^{*}$, $i\in\\{1,2,\dots p\\}$ is
assumed. Table 3 shows the RMSE values for an open-loop trajectory of the
robot, in the absence of any faults.

Table 3: Closed Loop Performance (in absence of faults) Filters States | $\hat{R}(cm)$ | $\hat{\theta}(rad)$ | $\hat{\alpha}(rad)$
---|---|---|---
EKF | 0.4172 | 0.0174 | 0.0431
A-EKF | 0.5624 | 0.0457 | 0.0509
PI-EKF | 0.2739 | 0.0307 | 0.0386

Despite the significantly better performance of the PI-EKF in terms of
estimation error, the interplay between the ’forgetting factor’ and
proportional gain for the parameter can lead to oscillatory behaviour in the
estimates, as is visible in Figure 6. Tuning can help achieve a balance
between minimal bias and subdued oscillatory behaviour.

Figure 6: Performance of PI-EKF in absence of faults

##  6 Conclusions

In this work, a state space formulation is presented to suit the multi-rate
estimation problem under uncertain conditions for a homing application.
Observability analysis for the realization was carried out which shows that
the choice of measurements are sufficient to estimate the configuration of the
robot in the 2-d plane as well as to estimate the bearing angles of the
landmarks with respect to the robot, provided that at-least two landmarks are
visible to the robot. This choice of measurement also eliminates the need to
measure the position of the robot with respect to the home position.
Simulation studies were carried out under 4 cases. Performance of three
filters were evaluated: EKF, A-EKF and the PI- EKF. The conventional EKF
failed perform satisfactorily for the case where there was an uncertainty in
the parameter values. PI-EKF tends to show an aggressive behaviour around the
steady state, and the estimation error eventually converges asymptotically
even in the presence of large perturbations in the parameters. Though PI-EKF
is a simple formulation, the interplay between the forgetting factor and the
proportional grain plays a key role in deciding the transient performance of
the filter. Tuning of the augmented EKF(A-EKF) becomes a challenging task in
the multi-rate case. Although this approach appears to be simple
theoretically, actual implementation of the filter strongly depends on the
manner in which the perturbation in the parameter affects the states. For the
A-EKF formulation, the information transferred from the observed variables to
the parameters is governed by the cross-covariances and therefore accurate
information about these is crucial for a satisfactory joint performance. In
reality, the true error statics for the parameters would not be known and we
have to best approximate using a simple structure like the random walk model,
which poses challenges for this approach. The extension of this work could
involve experimenting the mobile robots navigating through series of way
points. The authors acknowledge Ms. Misha Gupta for her support

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