# Thermometry by correlated dephasing of impurities in a 1D Fermi gas

Sindre Brattegard<EMAIL_ADDRESS>School of Physics, Trinity College Dublin,
College Green, Dublin 2, Ireland  Mark Mitchison<EMAIL_ADDRESS>School
of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland  Trinity
Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse
Street, Dublin 2, D02YN67

###### Abstract

We theoretically investigate the pure dephasing dynamics of two static
impurity qubits embedded within a common environment of ultracold fermionic
atoms, which are confined to one spatial dimension. Our goal is to understand
how bath-mediated interactions between impurities affect their performance as
nonequilibrium quantum thermometers. By solving the dynamics exactly using a
functional determinant approach, we show that the impurities become correlated
via retarded interactions of the Ruderman–Kittel–Kasuya–Yosida type. Moreover,
we demonstrate that these correlations can provide a metrological advantage,
enhancing the sensitivity of the two-qubit thermometer beyond that of two
independent impurities. This enhancement is most prominent in the limit of low
temperature and weak collisional coupling between the impurities and the gas.
We show that this precision advantage can be exploited using standard Ramsey
interferometry, with no need to prepare correlated initial states nor to
individually manipulate or measure the impurities. We also quantitatively
assess the impact of ignoring these correlations when constructing a
temperature estimate, finding that acceptable precision can still be achieved
in some parameter regimes using a simplified model of independent impurities.
Our results demonstrate the rich nonequilibrium physics of impurities
dephasing in a common Fermi gas, and may help to provide better temperature
estimates at ultralow temperatures.

##  I Introduction

The last decades have seen impressive developments in experimental techniques
for ultracold atomic gases. It is now possible to tune the geometry and
dimensionality of the system, the nature and strength of interparticle
interactions, and even the exchange statistics of the constituent particles,
enabling exploration of a wide range of fascinating physical phenomena [1, 2,
3, 4]. A particularly promising development in this regard is the creation of
homogeneous ultracold gases [5, 6, 7, 8, 9, 10, 11]: these are especially
useful for the quantum simulation of condensed-matter and high-energy physics
models, for which translation invariance is crucial. However, measuring the
temperature of these systems is challenging. Standard methods like time-of-
flight measurements completely obliterate the system, do not give access to
spatially resolved temperature information, and may suffer from a loss of
precision at ultra-low temperatures [12].

An interesting alternative is to work with ultracold atomic mixtures
comprising more than one species of atom [13, 14]. If the density of one
species is much lower than the others, they may be considered as impurities
embedded in a fluid of majority atoms. Such impurities a exhibit range of
interesting behaviour straddling the interface between open quantum systems
and condensed-matter physics. For example, mobile impurities form polarons in
bosonic [15, 16] and fermionic [17, 18] environments, while a static impurity
exhibits universal dynamics manifesting the Anderson orthogonality catastrophe
[19, 20, 21, 22, 23]. By measuring static and dynamic properties of the
impurities, numerous experiments have been able to accurately infer the
temperature of the host gas [24, 25, 26, 27, 28, 29, 30, 31]. This opens up
the tantalising prospect of exploiting the quantum mechanical behaviour of
ultracold impurities for improved thermometry, in a way that is also local and
minimally destructive, in principle.

The quest to understand the fundamental capabillities and limits of quantum
thermometry has inspired a substantial theoretical literature (see Ref. [32]
for a review). Seminal early work established the optimum sensitivity and
level structure of fully equilibrated probes [33], while numerous proposals
have put forward the possibility of using nonequilibrium impurity dynamics for
thermometry, especially in the context of ultracold gases [34, 35, 36, 37, 38,
39, 40, 41, 42]. More recently, it has been established how temperature
estimation is affected by informational constraints such as limited
measurement data [43, 44, 45, 46, 47, 48] or coarse-grained measurements [49,
50]. A particularly important issue of current interest is to understand how
relevant physical effects such as strong system-bath correlations [51, 52, 53,
54] may affect thermometry protocols in real impurity systems.

Another important physical effect is the interaction between several probes
induced by their mutual interaction with a common thermal environment. This is
important because experiments typically operate in a regime with several
impurities — perhaps even hundreds or thousands — embedded in a single copy of
the gas. Naively, increasing the number of impurities is helpful to increase
the signal-to-noise ratio, which scales by a factor of $\sqrt{M}$ for $M$
independent impurities. Yet independence is generally spoiled by bath-induced
interactions, which arise naturally in ultracold mixtures [55, 56, 57, 58, 59,
60, 61, 62, 63, 64] and have been observed in recent experiments [65, 66].
Previous theoretical work has shown that, in some settings, trapped impurities
can be configured to suppress bath-mediated interactions [36, 67], but in
general these interactions can give rise to classical and quantum correlations
between impurities [68]. It is well known that quantum correlations can yield
a metrological advantage in some scenarios [69, 70] and thermometry is no
exception [71]. Indeed, recent works have shown that bath-mediated
interactions can improve temperature estimation for impurities embedded in a
bosonic environment [72, 73, 74].

In this work, we take the first steps towards understanding how bath-induced
interactions affect thermometry for impurities embedded in a fermionic
environment. Following previous work by one of us [39], we focus on dephasing
probes [38, 40, 41, 42], where information about the temperature is imprinted
on quantum coherences that can be measured experimentally by Ramsey
interferometry [22, 75, 16, 31]. Specifically, we consider a system of two
static impurities, each possessing two internal states, which are coupled to a
one-dimensional, homogeneous Fermi gas. Homogeneous systems are particularly
challenging for thermometry, since absorption imaging of the spatial density
provides no information on temperature, while time-of-flight imaging of the
momentum distribution yields diminishing sensitivity at low temperature
because only a small number of atoms near the Fermi energy yield useful
information. In this context, we show that correlations induced by bath-
mediated interactions can yield a collective enhancement for thermometry at
low temperatures. Remarkably, this advantage survives even when one is limited
to local observables that are accessible via Ramsey interferometry.

In the following, we describe our two-impurity setup in detail (Sec. II.1) and
explain how to solve the quantum dynamics of the impurities exactly using a
functional determinant approach [76, 77, 78] (Sec. II.2). We then analyse the
dephasing dynamics in detail (Sec. III), focusing especially on the effect
that the impurities have on each other via the bath. Unlike the dipole-dipole
interactions that typically arise in bosonic baths, itinerant fermions induce
Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions between localised spins [79,
80] with a non-trivial oscillatory spatial dependence whose period is set by
the Fermi wavelength. Interestingly, we show that monitoring the two-impurity
dynamics allows one to observe the effect of RKKY-like interactions developing
in real time.

Next, we describe how temperature can be optimally inferred from measurements
on the impurities (Sec. IV.1), and propose a Ramsey protocol that is generally
sub-optimal but experimentally feasible (Sec. IV.2). Our approach is based on
local temperature estimation theory [81, 32], where the quantum Fisher
information sets the ultimate limit for precision. Using these tools, we
analyse the temperature sensitivity of the two-impurity system (Sec. IV.3),
finding that bath-induced correlations enhance precision even under local
measurements, while using entangled initial states yields no advantage.

Finally, we ask under what conditions the impurities can be modelled as
independent (Sec. IV.4). This approximation may be useful to simplify the
construction of the temperature estimator, especially when scaling to a larger
number of impurities $M>2$. For two impurities at fixed positions, we find
that the validity of this approximation depends quite strongly on the impurity
separation, but works best at higher temperatures. Apart from their potential
relevance for current experiments on multiple impurities, our results open up
several interesting future directions for theoretical research that we discuss
in Sec. IV.4.

##  II Setup

###  II.1 Description of the system

Figure 1: A sketch of the setup we are considering. Two impurities separated
by distance $2x_{0}$ (gray balls) embedded in a 1D Fermi gas confined by a box
potential of length $L$ (blue background).

We consider a system $S$ of two impurity atoms coupled to an environment $E$
comprising a gas of spin-polarized fermions with homogeneous density, which is
confined to a one-dimensional (1D) box of length $L$. We are interested in the
ultra-low temperature regime where $s$-wave scattering is dominant. Because of
the anti-symmetry of the fermionic wavefunction there is no $s$-wave
scattering between identical fermions, so the environment can be treated as a
non-interacting gas to a good approximation.

The impurity atoms are modelled as two-level systems with energy eigenstates
$\ket{\uparrow}_{i}$, $\ket{\downarrow}_{i}$, with $i=1,2$ labelling the two
impurity qubits. We take the impurities to be stationary and strongly
localized at fixed positions $x_{1}$ and $x_{2}$, which can be achieved using
a species-selective optical lattice that only affects the impurities [82, 31,
83]. We work in the pseudopotential approximation, which models $s$-wave
collisions between the impurities and the surrounding gas atoms as a contact
interaction. The total Hamiltonian is then

$\displaystyle\hat{H}=\hat{H}_{S}+\hat{H}_{E}+\hat{H}_{I},$ (1)
$\displaystyle\hat{H}_{S}=\sum_{i=1}^{2}\varepsilon_{i}\ket{\uparrow}_{i}\\!\bra{\uparrow},$
(2)
$\displaystyle\hat{H}_{E}=-\frac{\hbar^{2}}{2m}\int_{-L/2}^{L/2}\differential
x\,\hat{\Psi}^{\dagger}(x)\nabla^{2}\hat{\Psi}(x),$ (3)
$\displaystyle\hat{H}_{I}=-\frac{\hbar^{2}}{ma}\sum_{i=1}^{2}\ket{\uparrow}_{i}\\!\bra{\uparrow}\otimes\hat{\Psi}^{\dagger}(x_{i})\hat{\Psi}(x_{i}),$
(4)

where $\varepsilon_{i}$ is the local energy splitting between the impurity
eigenstates, $\hat{\Psi}^{\dagger}(x)$ is the field operator that creates a
fermion of mass $m$ at position $x$, such that
$\\{\hat{\Psi}(x),\hat{\Psi}^{\dagger}(x^{\prime})\\}=\delta(x-x^{\prime})$,
and $a$ is the scattering length describing impurity-fermion collisions. We
have assumed that the states $\ket{\downarrow}_{i}$ effectively do not
interact with the gas. This can be achieved, for example, by using a spin-
dependent Feshbach resonance [84] to tune the corresponding scattering length
to a very large value. Note that in 1D the interaction strength is inversely
proportional to the scattering length [85].

We assume that the environment is initially in thermal equilibrium at
temperature $T$, with $\hat{\rho}_{T}\propto e^{-\hat{H}_{E}/k_{B}T}$ the
corresponding thermal state. The initial state of the system and environment
is taken to be a tensor product

$\hat{\rho}(0)=\hat{\rho}_{S}(0)\otimes\hat{\rho}_{T},$ (5)

which can be prepared since the $\ket{\downarrow}_{i}$ states do not perturb
the gas. A specific experimental protocol to realise this are discussed in
Sec. IV.2.

We want to infer the temperature of the gas by observing the dynamics of the
probes. The latter are described by their reduced density matrix

$\hat{\rho}_{S}(t)=\tr_{E}\left[e^{-i\hat{H}t/\hbar}\hat{\rho}(0)e^{i\hat{H}t/\hbar}\right],$
(6)

obtained by tracing over the environment. Since $[\hat{H}_{I},\hat{H}_{S}]=0$,
the system Hamiltonian merely generates trivial phase factors that are
irrelevant for the system-environment dynamics. From here on we remove these
by working in a rotating frame via the transformation $\hat{\rho}_{S}(t)\to
e^{i\hat{H}_{S}t/\hbar}\hat{\rho}_{S}(t)e^{-i\hat{H}_{S}t/\hbar}$, which is
tantamount to setting $\varepsilon_{i}=0$.

Let $\sigma$ label the different internal states of $S$, taking the values
$\nolinebreak{\sigma\in\\{\uparrow\uparrow,\uparrow\downarrow,\downarrow\uparrow,\downarrow\downarrow\\}}$.
It is straightforward to show that

$\matrixelement{\sigma}{\hat{\rho}_{S}(t)}{\sigma^{\prime}}=\nu_{\sigma,\sigma^{\prime}}(t)\matrixelement{\sigma}{\hat{\rho}_{S}(0)}{\sigma^{\prime}},$
(7)

where we have defined the complex functions

$\nu_{\sigma,\sigma^{\prime}}(t)=\tr_{E}\left[e^{i\hat{H}_{\sigma}t/\hbar}e^{-i\hat{H}_{\sigma^{\prime}}t/\hbar}\hat{\rho}_{T}\right],$
(8)

with $\hat{H}_{\sigma}$ denoting the Hamiltonian of the environment
conditioned on the internal state of the system:

$\hat{H}_{\sigma}=\matrixelement{\sigma}{\hat{H}_{E}+\hat{H}_{I}}{\sigma}.$
(9)

The diagonal elements of $\hat{\rho}_{S}(t)$, with $\sigma=\sigma^{\prime}$,
are constant because the energy of the probes is conserved. The off-diagonal
terms with $\sigma\neq\sigma^{\prime}$ will evolve and generally decay in time
according to $\nu_{\sigma,\sigma^{\prime}}(t)$, which we refer to as the
decoherence functions of our system.

###  II.2 Calculation of the decoherence functions

To calculate the decoherence functions in Eq. (8), we employ the Levitov
formula or functional determinant approach (FDA) [76, 77, 78], which has been
widely used to study nonequilibrium impurity systems [86]. The FDA is a
numerically exact method that maps a many-body expectation value into a
determinant in single-particle space:

$\nu_{\sigma,\sigma^{\prime}}(t)=\det[1-\hat{n}+\hat{n}e^{i\hat{h}_{\sigma^{\prime}}t/\hbar}e^{-i\hat{h}_{\sigma}t/\hbar}],$
(10)

This equation holds because Eq. (8) involves only exponentials of quadratic
fermionic operators. Here, $\hat{h}_{\sigma}$ is the single-particle
equivalent of $\hat{H}_{\sigma}$, i.e. the Hamiltonian of a single particle in
the gas conditioned on the impurities being in state $\sigma$. Meanwhile,
$\hat{n}$ is an operator describing the initial Fermi-Dirac distribution of
the gas atoms

$\hat{n}=\left[1+e^{(\hat{h}_{\downarrow\downarrow}-\mu)/k_{B}T}\right]^{-1}.$
(11)

We work in the grand canonical ensemble and use the chemical potential $\mu$
to fix the total number of atoms $N=\bar{n}L$, where $\bar{n}$ is the number
density of the atoms.

The box is modelled as an infinite square well with hard-wall boundary
conditions at $x=\pm L/2$. The single-particle Hamiltonians $\hat{h}_{\sigma}$
are then differential operators of the form

$\hat{h}_{\sigma}=-\frac{\hbar^{2}\nabla^{2}}{2m}+V_{\sigma}(x),$ (12)

defined on the interval $x\in[-L/2,L/2]$. Here, $V_{\sigma}(x)$ is the
effective potential felt by the fermions when the impurities are in state
$\sigma$. In the pseudopotential approximation, we have

$\displaystyle V_{\downarrow\downarrow}(x)=0,$ (13) $\displaystyle
V_{\uparrow\downarrow}(x)=-\frac{\hbar^{2}}{ma}\delta(x-x_{1})$ (14)
$\displaystyle
V_{\downarrow\uparrow}(x)=-\frac{\hbar^{2}}{ma}\delta(x-x_{2}),$ (15)
$\displaystyle
V_{\uparrow\uparrow}(x)=V_{\uparrow\downarrow}(x)+V_{\downarrow\uparrow}(x).$
(16)

From here on, we take the impurities to be placed symmetrically around the
centre of the box, $x_{1}=-x_{0}$ and $x_{2}=+x_{0}$. This assumption can be
made without loss of generality because we always take $L$ to be large enough
to avoid boundary effects, so that only the impurity separation $\Delta
x=x_{2}-x_{1}=2x_{0}$ is relevant. Under this assumption, there are four
independent decoherence functions that we need to fully describe the dynamics
of the system. Further details on how we evaluate Eq. (10) numerically can be
found in Appendix B.

##  III Dynamics of two impurities dephasing in a 1D Fermi gas

In this section we systematically investigate the dephasing dynamics of two
impurities in a 1D Fermi gas. The physical scales of the gas are fully
determined by the fermion number density, $\bar{n}=N/L$, and are given by the
Fermi wavevector $k_{F}=\bar{n}\pi$, the Fermi energy
$E_{F}=\hbar^{2}k_{F}^{2}/2m$,the Fermi time $\tau_{F}=\hbar/E_{F}$, the Fermi
velocity $v_{F}=\hbar k_{F}/m$, and the Fermi temperature $T_{F}=E_{F}/k_{B}$.
The interaction between the impurities and the gas is parameterized by the
dimensionless parameter $k_{F}a$. We also define the interaction time between
the impurities $\tau_{i}=\Delta x/v_{F}$. This can be understood as the time
for excitations to travel from one impurity to the other, i.e. the time after
which bath-mediated interactions begin to play a role.

In Fig. 2 we plot all four independent decoherence functions for both
$k_{F}a=-0.1$ and $k_{F}a=-1$, which correspond to relatively strong and weak
system-environment interaction strengths, respectively. For strong system-
environment coupling we also plot the decoherence for temperatures
$T/T_{F}=0.1$, $0.05$ and $0.0001$. Here we focus exclusively on negative
scattering lengths, in order to avoid the additional complication of the bound
state that appears for $a>0$. While in general all four decoherence functions
contribute to the dynamics, each one can be tied to a distinct initial state
$\hat{\rho}_{S}(0)$, which is helpful to interpret their features.

Figure 2: (a-d) The full dynamics of two impurity qubits with separation
$k_{F}\Delta x/2\pi=3$ coupled to a 1D fermionic bath at temperature
$T=0.0001T_{F}$ (solid), $T=0.05T_{F}$ (dashdot) and $T=0.1T_{F}$ (dotted)
with coupling strength $k_{F}a=-1$ (blue) and $k_{F}a=-0.1$ (black). The Gray
dotted line denotes the interaction time $\tau_{i}=\Delta x/v_{F}$. (e-f) The
power law behaviour of $\nu_{\uparrow\uparrow,\downarrow\downarrow}$ and
$\nu_{\uparrow\downarrow,\downarrow\uparrow}$ for $T=0.0001T_{F}$ with an
interaction strength of $k_{F}a=-5$. The dotted lines are the power laws
discussed in the main text valid for low temperature and weak coupling.

We begin by examining the decoherence function
$\nu_{\uparrow\downarrow,\downarrow\downarrow}$ in Fig. 2(a). This describes
the situation where only a single impurity interacts with the gas. For
example, given the initial condition
$\hat{\rho}_{S}(0)=\ket{+}_{1}\\!\bra{+}\otimes\ket{\downarrow}_{2}\\!\bra{\downarrow}$
with

$\ket{+}_{i}=\frac{1}{\sqrt{2}}\left(\ket{\uparrow}_{i}+\ket{\downarrow}_{i}\right),$
(17)

the evolution of $\hat{\rho}_{S}(t)$ is determined purely by
$\nu_{\uparrow\downarrow,\downarrow\downarrow}(t)$ and all other matrix
elements are constant or zero. The dynamics of a single heavy impurity qubit
interacting with a Fermi gas has been extensively studied in the literature,
e.g. see [23] for a comprehensive review of the three-dimensional case and the
Supplemental Material of Ref. [39] for analysis of the 1D case.

Fig. 2(a) reproduces the behaviour expected from these previous studies. For
intermediate times between the Fermi time and the thermal timescale,
$\tau_{F}\ll t\ll\hbar\beta$, the decoherence function has a universal power-
law decay, with an exponent that depends on the interaction between the
impurity and Fermi gas. For later times, thermal effects dominate and the
decoherence function decays exponentially in time with a rate proportional to
temperature. In order to elucidate this behaviour, in Appendix C we derive an
expression for the decoherence functions valid in the weak-coupling and low-
temperature regime, using a cumulant expansion method adapted from Ref. [39].
The result is that

$|\nu_{\uparrow\downarrow,\downarrow\downarrow}(t)|\sim\begin{cases}t^{-\alpha}\quad&(\tau_{F}\ll
t\ll\hbar\beta)\\\ e^{-\alpha t/\hbar\beta}\quad&(\hbar\beta\ll
t),\end{cases}$ (18)

where the exponents are determined by the dimensionless coupling strength

$\alpha=\left(\pi k_{F}a\right)^{-2}.$ (19)

The decoherence functions $\nu_{\uparrow\downarrow,\downarrow\uparrow}$ and
$\nu_{\uparrow\uparrow,\downarrow\downarrow}$ are shown in Fig. 2(b) and (c).
These are the relevant decoherence functions for initial Bell state
preparations $\hat{\rho}_{S}(0)=\outerproduct{\Psi^{\pm}}{\Psi^{\pm}}$ and
$\hat{\rho}_{S}(0)=\outerproduct{\Phi^{\pm}}{\Phi^{\pm}}$, respectively, where

$\begin{split}|\Psi^{\pm}\rangle&=\frac{1}{\sqrt{2}}\left(\ket{\uparrow_{1}\downarrow_{2}}\pm\ket{\downarrow_{1}\uparrow_{2}}\right)\\\
|\Phi^{\pm}\rangle&=\frac{1}{\sqrt{2}}\left(\ket{\uparrow_{1}\uparrow_{2}}\pm\ket{\downarrow_{1}\downarrow_{2}}\right).\end{split}$
(20)

For short times, $t\ll\tau_{i}$, both decoherence functions have the same
power-law behavior as two impurities in independent baths, i.e.
$\nu_{\sigma,\sigma^{\prime}}\sim t^{-2\alpha}$. This reflects the fact that,
for times much less than the bath-mediated interaction time, the two
impurities should evolve independently. This implies that, for $t<\tau_{i}$,
we have

$\hat{\rho}_{S}(t)=\mathcal{E}\hat{\rho}_{S}(0)\approx\left(\mathcal{E}_{1}\otimes\mathcal{E}_{1}\right)\hat{\rho}_{S}(0),$
(21)

where $\mathcal{E}$ is the quantum channel describing the exact two-impurity
evolution and $\mathcal{E}_{1}$ is the channel describing a single impurity
immersed in a Fermi gas. Eq. (21) immediately implies that
$\nu_{\uparrow\downarrow,\downarrow\uparrow}\approx|\nu_{\uparrow\downarrow,\downarrow\downarrow}|^{2}$
and
$\nu_{\uparrow\uparrow,\downarrow\downarrow}\approx\nu_{\uparrow\downarrow,\downarrow\downarrow}^{2}$.
Therefore, the phase of $\nu_{\uparrow\downarrow,\downarrow\uparrow}$ vanishes
while the phase of $\nu_{\uparrow\uparrow,\downarrow\downarrow}$ is twice that
of a single impurity, and their magnitudes are identical. However, around the
interaction time $\tau_{i}$ these two decoherence functions begin to show
drastically different behaviour: the decay of
$\nu_{\uparrow\downarrow,\downarrow\uparrow}$ slows down completely, while
$\nu_{\uparrow\uparrow,\downarrow\downarrow}$ begins to show marked
oscillations with approximate period $\tau_{i}$. These oscillations represent
a non-Markovian effect, where excitations from one impurity travel through the
gas and hit the other one, representing backflow of information from
environment into the system [87, 88].

These dynamical features for $t\lesssim\tau_{i}$ are qualitatively captured by
the cumulant expansion derived in Appendix C. For short times $t\ll\tau_{i}$,
our analytical theory predicts that both
$\nu_{\uparrow\downarrow,\downarrow\uparrow}$ and
$\nu_{\uparrow\uparrow,\downarrow\downarrow}$ behave as
$\nu_{\sigma,\sigma^{\prime}}(t)\sim t^{-2\alpha}$, as expected for
independent impurities. After the impurity interaction time $t=\tau_{i}$, we
find that the algebraic decay changes exponent as

$\alpha\to\alpha\left(1\pm\cos^{2}(2k_{F}\Delta x)\right),$ (22)

where the plus and minus signs correspond to
$\nu_{\uparrow\uparrow,\downarrow\downarrow}$ and
$\nu_{\uparrow\downarrow,\downarrow\uparrow}$, respectively. A comparison
between this analytical prediction and the exact numerics can be seen in Fig.
2 (e) and (f), demonstrating excellent agreement within the weak-coupling and
low-temperature regime where the cumulant expansion is valid.

The change of exponent at $t=\tau_{i}$ manifests the well-known phenomenon of
super- and sub-decoherence [89, 90, 91, 92]. In Appendix C we provide an
explanation for this effect: the Bell states $\ket{\Phi^{\pm}}$ and
$\ket{\Psi^{\pm}}$ couple respectively to density fluctuations at the impurity
positions that are in phase or anti-phase, respectively, which have very
different low-frequency properties. Moreover, in Eq. (22) we recognise the
sinusoidal spatial dependence that typically arises in perturbed Fermi gases,
with the same period of $\pi/k_{F}$ that characterises both Friedel
oscillations and the RKKY interaction. Therefore, the transition from normal
to sub- or super-decoherent dynamics manifests the time-retarded effect of
RKKY-like correlations developing in real time. Note that the sub- and super-
decoherent behaviour persists only up to the thermal time,
$t\simeq\hbar\beta$, after which both decoherence functions decay
exponentially as $\sim e^{-2\alpha t/\hbar\beta}$. This is the decay expected
for two independent impurities, meaning that thermal noise washes out the
dynamical effect of RKKY interactions on the decoherence signal at long times.
We note that RKKY-like interactions have been observed in ultracold atomic
gases recently [65].

Finally, Fig. 2 d shows the decoherence function
$|\nu_{\uparrow\uparrow,\uparrow\downarrow}|$. This is the relevant
decoherence function for the initial state
$\hat{\rho}_{S}(0)=\ket{\uparrow}_{1}\\!\bra{\uparrow}\otimes\ket{+}_{2}\\!\bra{+}$,
which describes a situation where the pseudospin of the first impurity is
flipped at the same time that the second impurity is prepared in a
superposition. Thus, for short times,
$\nu_{\uparrow\uparrow,\uparrow\downarrow}$ is indistinguishable from
$\nu_{\uparrow\downarrow,\downarrow\downarrow}$. For times $t\gtrsim\tau_{i}$
the perturbations from the first impurity hit the second one, causing the
decoherence to slow down. The cumulant expansion at second order fails to
capture this behaviour, indicating that this is a non-perturbative effect.

Aside from the temperature dependence of the long-time exponential decay, the
interaction effects discussed above depend strongly on both temperature and
coupling strength. In particular, non-Markovian effects due to exchange of
excitations are most prominent at low temperatures, and disappear as the
temperature is increased. Observing these features thus yields useful
information on temperature, as we show in the next section.

##  IV Thermometry with correlated dephasing probes

###  IV.1 Local temperature estimation theory

Since the impurity dynamics depends on the temperature of the gas, we can
infer the temperature from measurements of many identical preparations of the
probes. We first consider a general “prepare and measure” scenario: a given
probe state $\hat{\rho}_{S}(0)$ is prepared as in Eq. (5), the system
decoheres for a time $t$, and then a measurement in a given basis is made on
the resulting state $\hat{\rho}_{S}(t)$. Repeating this procedure for $M$
identical preparations yields a sequence of measurement outcomes
$\boldsymbol{x}=\\{x_{1},x_{2},\dots,x_{M}\\}$. In the most general case, the
measurement may be a positive operator-valued measure (POVM): a collection of
positive operators $\hat{\Pi}(x)>0$ that are normalised as
$\sum_{x}\hat{\Pi}(x)=1$.

Using our knowledge of the temperature-dependence of $\hat{\rho}_{S}(t)$, a
temperature estimate $\check{T}(\textbf{x})$ can be constructed from the
measurement data x, e.g. using maximum-likelihood estimation. Any such
estimate will carry uncertainty due to the randomness inherent to quantum
measurements and the finite number of samples $M$. The achievable precision
depends not only on the probe state $\hat{\rho}_{S}(T)$ but also the specific
choice of POVM $\\{\hat{\Pi}(x)\\}$.

To quantify the precision attainable in our setup, we use the theory of local
quantum parameter estimation [81]. Here, the central quantity is the quantum
Fisher information (QFI), $\mathcal{F}_{T}$, which provides a lower bound on
the variance of any unbiased estimator. In particular, an unbiased estimator
obeys $\mathbb{E}[\check{T}(\textbf{x})]=T$, while its variance obeys the
(quantum) Cramér-Rao bound [93]

$\mathbb{E}\left[\left(\check{T}(\textbf{x})-T\right)^{2}\right]\geq\frac{1}{MF_{T}}\geq\frac{1}{M\mathcal{F}_{T}}.$
(23)

Here, $F_{T}$ is the Fisher information for the measurement
$\\{\hat{\Pi}(x)\\}$,

$F_{T}\left(\hat{\rho}_{S}(t),\\{\hat{\Pi}(x)\\}\right)=\sum_{x}p(x)\left(\partialderivative{\ln{p(x)}}{T}\right)^{2},$
(24)

where $p(x)=\tr\left[\hat{\rho}_{S}(t)\hat{\Pi}(x)\right]$ is the probability
of obtaining outcome $x$. The QFI is defined as maximum of the Fisher
information over all POVMs [93],

$\mathcal{F}_{T}\left(\hat{\rho}_{S}(t)\right)=\max_{\\{\hat{{\Pi}(x)}\\}}F_{T}\left(\hat{\rho}_{S}(t),\\{\hat{\Pi}(x)\\}\right).$
(25)

Therefore, the QFI quantifies the maximum information on temperature that can
be obtained from repeated measurements on the state $\hat{\rho}_{S}(t)$.

The maximum in Eq. (25) is achieved by projective measurements of the
symmetric logarithmic derivative (SLD), $\hat{\Lambda}_{T}$. Writing the probe
state in its eigenbasis as
$\hat{\rho}_{S}(t)=\sum_{n}r_{n}\ket{r_{n}}\bra{r_{n}}$, the QFI and SLD are
given explicitly by [81]

$\displaystyle\mathcal{F}_{T}=2\sum_{m,n}\frac{\lvert\bra{r_{m}}\partial_{T}\hat{\rho}_{S}\ket{r_{n}}\rvert^{2}}{r_{m}+r_{n}},$
(26)
$\displaystyle\hat{\Lambda}_{T}=2\sum_{m,n}\frac{\bra{r_{m}}\partial_{T}\hat{\rho}_{S}\ket{r_{n}}}{r_{m}+r_{n}}\ket{r_{m}}\bra{r_{n}}.$
(27)

Note that the SLD is explicitly dependent on temperature, and the quantum
Cramér-Rao bound (23) can only be saturated in the asymptotic limit of $M\gg
1$. Therefore, local estimation theory is most relevant when some coarse
information on temperature is already known and a large quantity of
measurement data is available. Alternative approaches to temperature
estimation based on Bayesian statistics [43, 44, 45, 46, 47, 48] and single-
shot information theory [94] have recently been developed, allowing
thermometry even with few measurements and no prior information. Nevertheless,
the QFI is very useful as a quantitative benchmark for our impurity
thermometer in different parameter regimes.

###  IV.2 Ramsey interferometry protocol

Since the impurities become correlated through their interaction with the gas,
the SLD is generally a non-local observable that may be difficult to measure.
We therefore consider the following thermometry protocol based on Ramsey
interferometry, which is feasible in cold-atom systems [22, 31]. To generate
the initial product state (5), both of the impurities start in the non-
interacting state $\ket{\downarrow}_{i}$ while the gas is prepared in a
thermal state. The impurities may either be left in situ during this
preparation phase [22], or transported into the Fermi gas by a moving trap
potential [83, 31]. At $t=0$, a $\pi/2$ pulse is applied to the impurities,
generating the initial state

$\hat{\rho}_{S}(0)=\ket{+}_{1}\\!\bra{+}\otimes\ket{+}_{2}\\!\bra{+}.$ (28)

The system is then left to decohere in contact with the gas for a time $t$,
giving rise to a state $\hat{\rho}_{S}(t)$ which is a function of all four
decoherence functions discussed in Sec. III. Finally, another $\pi/2$ pulse is
applied with a phase $\phi$ relative to the first pulse, and then the
population of the qubit energy eigenstates is immediately measured
projectively, e.g. by using laser pumping to induce state-dependent
flourescence.

In experiments with multiple impurities embedded in ultracold atomic gases, it
is typically not possible to address the individual impurities during the
measurement. Instead, only the total measurement signal is accessible, e.g. in
Ref. [31] the total number of impurities left in the ground state after the
final $\pi/2$-pulse is counted. This is equivalent to measuring the
expectation value of the observable

$\hat{O}(\phi)=\sum_{i=1}^{2}\left(\hat{\sigma}_{i}^{\|}(\phi)+\hat{\sigma}_{i}^{\perp}(\phi)\right),$
(29)

with
$\hat{\sigma}_{i}^{\|}=\cos\phi\hat{\sigma}_{i}^{x}+\sin\phi\hat{\sigma}_{i}^{y}$
and
$\hat{\sigma}_{i}^{\perp}=\sin\phi\hat{\sigma}_{i}^{x}-\cos\phi\hat{\sigma}_{i}^{y}$.
The temperature uncertainty $\Delta T$ expected for $M\gg 1$ measurements of
this observable can be quantified by the error propagation formula [95]

$\Delta T=\frac{\Delta\hat{O}}{\sqrt{M}\partial_{T}\langle\hat{O}\rangle},$
(30)

where $\Delta\hat{O}=\langle\hat{O}^{2}\rangle-\langle\hat{O}\rangle^{2}$ is
the operator variance and $\langle\hat{O}\rangle$ is the expectation value in
the state $\hat{\rho}_{S}(t)$. The signal-to-noise ratio is then

$\frac{T}{\Delta
T}=\sqrt{M}\frac{T\partial_{T}\langle\hat{O}\rangle}{\Delta\hat{O}}\equiv\sqrt{M}\mathcal{S}_{T}({\hat{O}})$
(31)

which defines the effective temperature sensitivity
$\mathcal{S}_{T}({\hat{O}})$ for measurements of the observable $\hat{O}$.

Figure 3: The signal-to-noise ratio of different measurements for the two
impurity qubits separated by $k_{F}\Delta x/2\pi=5$, temperature
$T=0.05T_{F}$, and interaction strength $k_{F}a=-1$ (a) and $k_{F}a=-0.1$ (b).
Measurement of the SLD including bath-induced interactions (blue), the SLD of
two independent impurities (orange), the operator of Eq. (30) with angle
$\phi$ that gives the maximum information (green). We also investigated the
effect of starting in one of the Bell states
$\outerproduct*{\Psi^{+}}{\Psi^{+}}$ or $\outerproduct*{\Phi^{+}}{\Phi^{+}}$
and doing measurement on their respective SLD (red and purple).

The sensitivity $\mathcal{S}_{T}$ will be our metric of performance in the
following. It can be shown that the sensitivity is maximised by measurements
of the SLD because [96]

$\mathcal{S}_{T}(\hat{\Lambda}_{T})=T\sqrt{\mathcal{F}_{T}},$ (32)

in which case, according to Eq. (31), the temperature error $\Delta T$
asymptotically saturates the quantum Cramér-Rao bound (23). For local
observables of the form in Eq. (29), the sensitivity is generally smaller;
however, for any state $\hat{\rho}_{S}(t)$ we can find the optimal operator
$\hat{O}(\phi)$ to measure by maximising $\mathcal{S}_{T}$ over $\phi$. Like
the SLD, the optimal $\phi$ depends on $T$ and thus some prior knowledge about
the temperature is needed for this to work in practice.

###  IV.3 Thermometric performance and collective advantages

We now discuss how the dynamical features explored in Sec. III affect the
performance of our two-qubit thermometer. The blue curves in Fig. 3 show the
optimal sensitivity,
$\mathcal{S}_{T}(\hat{\Lambda}_{T})=T\sqrt{\mathcal{F}_{T}}$, as a function of
time in various different scenarios for temperature $T=0.05T_{F}$ and two
interaction strengths: $k_{F}a=-1$ [Fig. 3(a)] and $k_{F}a=-0.1$ [Fig. 3(b)].
The sensitivity peaks at a particular moment in time, which defines the
optimal time to perform the measurement. We see that the peak sensitivity is
generally larger at weak system-environment coupling but also occurs at a
later time. A similar tradeoff between sensitivity and measurement time was
found in previous work on thermometry with single impurities [39]. At stronger
coupling, the optimal sensitivity exhibits complicated oscillations. These are
a consequence of non-Markovian effects induced by the impurities exchanging
excitations via the gas, as discussed in Sec. III.

An interesting question is whether bath-induced interactions improve the
precision of our two-impurity thermometer. In order to assess this, we compare
to the case of two independent impurities with
$\hat{\rho}_{S}(t)=[\hat{\rho}_{1}(t)]^{\otimes 2}$, where $\hat{\rho}_{1}(t)$
is the state of a single impurity dephasing in a 1D Fermi gas. That is, the
diagonal elements of $\hat{\rho}_{1}(t)$ are constant while the off-diagonal
elements are propotional to
$\nu_{\uparrow\downarrow,\downarrow\downarrow}(t)$. This is equivalent to the
situation where the impurity separation is very large, so that
$\tau_{i}\gg\hbar\beta$ and thermal decoherence kicks in before interactions
can play any role. The corresponding QFI is additive,
$\mathcal{F}_{T}([\hat{\rho}_{1}(t)]^{\otimes
2})=2\mathcal{F}_{T}(\hat{\rho}_{1})$. Fig. 3(a) shows that the optimal
sensitivity of two impurities in a common bath can be markedly larger than the
corresponding value for two independent impurities (yellow curves in Fig. 3).
This effect, which only arises for sufficiently weak coupling and low
temperature, indicates that bath-induced interactions yield an advantage for
thermometry.

Remarkably, this advantage persists even when restricted to more realistic
local observables as in Eq. (29). To show this, we numerically optimize the
angle $\phi$ to find the observable that gives the largest sensitivity
$\mathcal{S}_{T}$ at each point in time. The results, shown by the green curve
in Fig. 3, demonstrate that optimal local measurements may yield almost as
much temperature information as measurements of the SLD. This is true for a
wide range of parameters. Similar effects have been observed previously in the
context of equilibrium thermometry [97].

Figure 4: The mutual information between the two impurity probes as a function
of time, with $\Delta x=0.2L$.

To further elucidate the role of bath-induced interactions, we quantify the
correlations that develop between the two impurity probes using the quantum
mutual information

$I(1:2)=S(\hat{\rho}_{1})+S(\hat{\rho}_{2})-S(\hat{\rho}_{S}),$ (33)

where $\hat{\rho}_{i}$ is the reduced density matrix for impurity $i$, e.g.
$\hat{\rho}_{1}=\tr_{2}(\hat{\rho}_{S})$, and $S(\hat{\rho})$ is the von
Neumann entropy

$S(\hat{\rho})=-\tr(\hat{\rho}\ln\hat{\rho}).$ (34)

Fig. 4 shows the mutual information of the two probes as a function of time.
We see that correlations are strongest in the weak coupling and low
temperature regime, which is precisely where the collective advantage to
thermometry is greatest. Conversely, for high temperature or strong coupling
the correlations quickly vanish.

However, correlations alone are not sufficient to obtain a precision
advantage, as the following example demonstrates. Consider preparing the
impurities in the maximally entangled Bell states
$\hat{\rho}_{S}(0)=\outerproduct{\Phi^{+}}{\Phi^{+}}$ or
$\hat{\rho}_{S}(0)=\outerproduct{\Psi^{+}}{\Psi^{+}}$. Entangled initial
states are known to give a precision boost in many metrological settings, e.g.
for phase estimation in cold atom systems [98, 99]. For our system, however,
this is not the case, as shown by the purple and red curves in Fig. 3. For the
$\ket{\Psi^{+}}$ state, this can be understood because the relevant
decoherence function is almost purely real, while much of the temperature
information at weak coupling is known to be contained in bath-induced phase
shifts [39]. For the initial $|\Phi^{+}\rangle$ state, phase information is
amplified but the super-decoherence effect discussed in Sec. III causes the
matrix elements of $\hat{\rho}_{S}$ to decay too quickly to take advantage of
this effect.

In summary, correlations between the impurities are generated by bath-mediated
interactions, and these correlations can increase the temperature sensitivity.
This effect is strongest at weak coupling and low temperature and, most
interestingly, can be exploited using only product-state preparations and
local measurements, e.g. via Ramsey interferometry. However, since these bath-
mediated interactions vary rapidly with the distance between the impurities,
the impurity’s positions must be controlled with a precision comparable to the
Fermi wavelength [c.f. Eq. (22)] to take advantage of these collective
effects. Moreover, finding the optimal measurement and constructing the
estimator requires solving the dynamical problem for two impurities, which is
significantly more complicated than the single-impurity case. In the following
section, therefore, we consider whether one may ignore bath-induced
correlations and still obtain a reasonable temperature estimate.

###  IV.4 When can we assume uncorrelated probes?

In principle, our functional determinant approach can be scaled to describe
the dynamics of an arbitrary number of probes, $M$. In practice, however, this
becomes impractical due to the large number of different decoherence functions
that must be evaluated, as well as numerical instabilities and convergence
issues that must be carefully addressed (see Appendix B). This motivates us to
ask: under what conditions the bath-induced correlations can be ignored
without sacrificing the quality of the temperature estimate? This would enable
one to treat the impurities as independent, so that only the dynamics of a
single impurity need be solved, which is far simpler.

To investigate the error arising in the temperature estimate by assuming the
two impurity probes are uncorrelated, we adopt the following procedure:

  1. 1.

Numerically find the operator $\hat{O}_{\rm max}(\phi)$ that gives the most
information on temperature as well as the optimal time $t_{\rm max}$ to
perform the measurement, assuming the impurities are evolving in independent
Fermi gases of temperature $T$. This yields the optimum measurement that an
experimenter who is unaware of bath-induced interactions would perform.

  2. 2.

Evaluate the reduced density matrix for two impurities in a shared bath of
temperature $T$ and evaluate the expectation value $\langle\hat{O}_{\rm
max}\rangle=\tr\left(\hat{\rho}_{S}(t_{\rm max})\hat{O}_{\rm max}\right)$.
This yields the expected result that the unaware experimentalist would
measure.

  3. 3.

Find $T_{\rm est}$, the temperature that the unaware experimenter would infer
from the result $\langle\hat{O}_{\rm max}\rangle$ (assuming $M\to\infty$
measurements), using the model of independent impurities.

  4. 4.

Compare $T_{\rm est}$ with $T$ to infer the error that would be incurred.

The error in assuming uncorrelated probes depends crucially on the dynamics of
the system at the measurement time $t_{\rm max}$. If $t_{\rm max}\ll\tau_{i}$,
the impurities remain uncorrelated and $T_{\rm est}=T$ to an excellent
approximation. Therefore, we instead focus on impurities with a separation
such that the interaction time is less than or similar to the $t_{max}$, where
the effect of bath-induced interactions on the temperature estimate is most
significant.

Figure 5: The relative error when assuming uncorrelated impurity probes as a
function of temperature for coupling strength $k_{F}a=-1$. The blue and orange
points are for separations such that $k_{F}\Delta x=n\pi$ and $k_{F}\Delta
x=n\pi/2$, respectively.

Fig. 5 shows the error in assuming uncorrelated probes for different
temperatures for two separations of the impurities. From Section III, we know
the effect of the bath-induced impurity interaction is quantified by
$\cos^{2}(2k_{F}\Delta x)$. We consider the two extremal cases $2k_{F}\Delta
x=n\pi$ and $2k_{F}\Delta x=n\pi/2$ ($n\in\mathbb{Z}$), where this interaction
is maximal or minimal, respectively. We find that in both cases, and for most
temperatures considered, approximating the impurities as independent yields a
relative error $|T-T_{\rm est}|/T\lesssim 10\%$. However, the error depends in
a complicated way on the temperature, because this determines the optimal
measurement time $t_{\rm max}$. For example, at temperature $T=0.175T_{F}$ the
error for the maximally interacting probes is large because $t_{\rm max}$ is
close to an integer multiple of $\tau_{i}$.

These results suggest that, at least for two fixed impurities, achieving the
best precision requires an estimator that explicitly accounts for bath-induced
correlations. However, one may hope that, when averaging over the signal from
many impurities at widely varying positions, the effect of correlations
averages out. This presumably depends on the impurities’ spatial distribution,
but we leave a careful analysis of this problem for $M>2$ to future work.

##  V Discussion & conclusions

In this work, we have proposed and analysed a thermometry protocol based on
the dynamics of two qubit impurities dephasing in a 1D gas of ultracold
fermions. We have solved the quantum evolution of the two impurities exactly,
including the effect of bath-induced interactions. We have also gained
valuable physical intuition into the observed behaviour at different
timescales by means of a perturbative cumulant expansion. We have found that
certain impurity decoherence functions manifest a retarded RKKY-like
interaction, giving rise to sub- and super-decoherent behaviour depending on
the initial state, which persists for intermediate times
$\tau_{i}<t<\hbar\beta$ and is eventually washed away by thermal fluctuations.

To understand how the bath-mediated interactions affect the achievable
precision, we have compared the thermal sensitivity of our two-qubit
thermometer to a pair of impurities interacting with independent environments.
We have found that, at low temperatures and weak coupling, bath-induced
correlations between the impurities can enhance precision. These results
reinforce other recent work showing that bath-mediated interactions can be
helpful in the context of low-temperature thermometry [72, 73, 74]. This
conclusion is by no means obvious, since correlations between the impurities
could also arise from redundant encoding of temperature information, thus
reducing the signal-to-noise ratio relative to truly independent measurements
[100, 67].

In order to exploit these correlations to the full, one would need to measure
the non-local SLD observable. However, we have shown that a simple Ramsey
protocol can approach the optimal precision, without the need to individually
address the impurities or perform entangling operations. Moreover, we have
quantified the systematic error incurred by neglecting correlations
altogether. Our results show that this error depends strongly on the impurity
separation, but remains is on the order of a few percent for most system
parameters at not-too-low temperatures. This suggests that it may be an
acceptable approximation to neglect bath-induced correlations in most
situations, potentially simplifying future thermometry experiments with many
impurities embedded in a single copy of the gas.

It is possible to generalize the techniques presented in this paper to
investigate other systems, for example higher-dimensional systems. It has
recently been suggested that the precision one is able to obtain depends
crucially on the spatial dimensionality of the system [101] and that
thermometry in higher dimensions could be more effective. However, we also
expect bath-induced interactions to be less prominent in higher dimensions
because the excitations from the impurity-bath interactions will spread out
more. It would be interesting to investigate how the dynamics, as well as the
thermometric performance, would change for several impurities decohering in a
higher-dimensional Fermi gas. Our approach could also be adapted to explore
thermometry in the presence of different confining potentials [21], charged
impurities [40], and more exotic environments such as Fermi superfluids [102,
103]. Other properties of these systems beyond temperature, such as transport
coefficients [104], could also be extracted using correlated impurities.
Moreover, it has been suggested that using multi-dimensional spectroscopy
instead of the Ramsey protocol considered in this text could yield extra
information [105, 106], and the procedure presented here could be generalised
to such scenarios as well.

Finally, it would be interesting to see how thermometric performance scales
with the number of impurities $M$, especially for $M\gg 1$. A closely related
question is how disturbing such a temperature measurement would be in terms of
heat absorbed by the environment [107]. Intuitively, one may expect that
improved precision comes at the cost of increased measurement backaction,
especially if the number of impurities scales extensively with system size.
This could be quantified using the recently developed thermodynamic
description of decoherence [108, 109].

###### Acknowledgements.

We are grateful to S. Campbell, G. Mihailescu, and A. K. Mitchell for
insightful discussions, and we also thank S. Campbell, R. Onofrio, and A.
Purkayastha for useful comments on the manuscript. We acknowledge financial
support from a Royal Society-Science Foundation Ireland University Research
Fellowship (URF\R1\221571).

##  Appendix A Solutions to the single particle Schrödinger equations

In this section, we solve the single-particle Schrödinger equation for a
particle in a box, with no, one, or two delta potentials present. We will
solve the equations

$\hat{h}_{\sigma}\psi_{n}^{(\sigma)}=E_{n}^{(\sigma)}\psi_{n}^{(\sigma)}.$
(35)

for each $\sigma$.

For $\hat{h}_{\downarrow\downarrow}$, this is simply given by

$\psi_{n}^{(\downarrow\downarrow)}\equiv\psi_{n}=\sqrt{\frac{2}{L}}\sin\left(k_{n}(x+L/2)\right),$
(36)

and $E_{n}=\hbar^{2}k_{n}^{2}/2m$, with $k_{n}=n\pi/L$.

We next consider the case of $\sigma=\downarrow\uparrow$. The eigenfunctions
in this case are given by

$\psi_{n}^{(\downarrow\uparrow)}=\begin{cases}A_{n}\sin{\left(k_{n}^{\prime}(x+L/2)\right)},\;-L/2<x<x_{0}\\\
B_{n}\sin{\left(k_{n}^{\prime}(x-L/2)\right)},\quad L/2>x>x_{0}\end{cases}$
(37)

with energy

$E_{n}^{(\downarrow\uparrow)}\equiv
E_{n}^{\prime}=\frac{\hbar^{2}k_{n}^{\prime 2}}{2m}$ (38)

and $k_{n}^{\prime}$ is given by the quantization condition

$\cot\left(k_{n}^{\prime}(x_{0}+L/2\right)-\cot\left(k_{n}^{\prime}(x_{0}-L/2)\right)=\frac{1}{k_{n}^{\prime}a}.$
(39)

The coefficients $A_{n}$ and $B_{n}$ can be found by the normalization
requirement.

The solution for $\sigma=\uparrow\downarrow$ is similar to the one above. We
can note immediately that the eigenfunctions should be mirrored versions of
Eq. (37) around the center of the box. Indeed we find that

$\psi_{n}^{(\uparrow\downarrow)}=\begin{cases}(-1)^{-n}B_{n}\sin{\left(k_{n}^{\prime}(x+L/2)\right)},\;-L/2<x<-x_{0}\\\
(-1)^{-n}A_{n}\sin{\left(k_{n}^{\prime}(x-L/2)\right)},\quad
L/2>x>-x_{0}\end{cases}$ (40)

with the same eigenenergies as above.

For $\sigma=\uparrow\uparrow$ we treat the even and odd eigenfunctions
separately. For $n$ even we find that the solution can be written

$\psi_{n}^{(\uparrow\uparrow)}=\begin{cases}C_{n}\sin{\left(k_{n}^{\prime\prime}x-\delta_{n}\right)},\;-L/2<x<-x_{0}\\\
D_{n}\sin{\left(k_{n}^{\prime\prime}x\right)},\quad-x_{0}<x<x_{0}\\\
C_{n}\sin{\left(k_{n}^{\prime\prime}x+\delta_{n}\right)},\;L/2>x>x_{0}\end{cases}$
(41)

where $k_{n}^{\prime\prime}=k_{n}-2\delta_{n}/L$ and the quantization
condition takes the form

$\cot(k_{n}^{\prime\prime}x_{0})-\cot(k_{n}^{\prime\prime}x_{0}+\delta_{n})=\frac{2}{k_{n}^{\prime\prime}a}.$
(42)

For $n$ odd, the eigenfunctions are given by

$\psi_{n}^{(\uparrow\uparrow)}=\begin{cases}C_{n}\cos{\left(k_{n}^{\prime\prime}x-\delta_{n}\right)},&\quad-L/2<x<-x_{0}\\\
D_{n}\cos{\left(k_{n}^{\prime\prime}x\right)},&\quad-x_{0}<x<x_{0}\\\
C_{n}\cos{\left(k_{n}^{\prime\prime}x+\delta_{n}\right)},&\quad
L/2>x>x_{0}.\end{cases}$ (43)

with $k_{n}^{\prime\prime}$ the same as above. The quantization condition this
time takes the form

$\tan(k_{n}^{\prime\prime}x_{0}+\delta_{n})-\tan(k_{n}^{\prime\prime}x_{0})=\frac{2}{k_{n}^{\prime\prime}a}.$
(44)

The eigenenergy is in both cases given by

$E_{n}^{(\uparrow\uparrow)}\equiv
E_{n}^{\prime\prime}=\frac{\hbar^{2}k_{n}^{\prime\prime 2}}{2m},$ (45)

and the coefficients $C_{n}$ and $D_{n}$ can be found by requiring normalized
states. It is worth noting that one may run into numerical instabilities when
solving for $k_{n}^{\prime\prime}$ and $\delta_{n}$. We implement some safety
procedures to check the validity of our solutions. The phase shift should
never be higher than $\pi$, and the magnitude of $\delta_{n}$ can oscillate
but should be contained within a decaying envelope function going to zero as
$n$ increases. If these conditions are not satisfied for any given pair
$\delta_{n}$, $k_{n}^{\prime\prime}$, we find that rewriting the quantization
conditions may help in solving the equations.

##  Appendix B Computational details for the decoherence functions

In this section, we give some computational details for evaluating eq. (10).

For each temperature $T$, we need to determine the chemical potential $\mu$.
We do this by solving the equation

$\tr\hat{n}=N_{s},$ (46)

with $N_{s}$ the number of particles in our gas. The trace is computed over a
large basis set. In our calculations, we used $\sim 10^{4}$ basis states.

We go to the thermodynamic limit by increasing the number of particles in the
gas $N_{s}$ while keeping the density $\bar{n}=N_{s}/L$ fixed until
convergence is reached. For the timescales we are considering, we find that
using $N_{s}$ in the range of $500-1000$ gives good convergence avoiding
finite size effects in the time scale we are interested in.

We need to calculate the matrix elements of operators like

$\hat{A}=1-\hat{n}+\hat{n}e^{i\hat{h}_{\uparrow\downarrow}t/\hbar}e^{-i\hat{h}_{\downarrow\uparrow}t/\hbar}.$
(47)

This matrix is in principle infinite-dimensional, but we can get a good
approximation by using a finite basis set of size $N$. We determine $N$ by
requiring

$\big{|}\sum_{i=1}^{N}f(E_{i})-N_{S}\big{|}<\epsilon,$ (48)

with $f(E)$ is the Fermi-Dirac distribution of the unperturbed gas. We find
that using $\epsilon=10^{-3}$ gives good convergence. In Appendix A we
calculated the eigenfunctions and energies of the single particle operators.
We insert the resolution of identity and get the matrix elements of the
$N\crossproduct N$ matrix

$A_{nm}=(1-f(E_{n}))\delta_{nm}+f(E_{n})\sum_{i=1}^{N^{\prime}}\sum_{j=1}^{N^{\prime}}\sum_{k=1}^{N^{\prime\prime}}e^{i(E_{i}^{\prime}-E_{j}^{\prime})t/\hbar}\innerproduct*{\psi_{n}}{\psi_{i}^{(\uparrow\downarrow)}}\innerproduct*{\psi_{i}^{(\uparrow\downarrow)}}{\psi_{k}}\innerproduct*{\psi_{k}}{\psi_{j}^{(\downarrow\uparrow)}}\innerproduct*{\psi_{j}^{(\downarrow\uparrow)}}{\psi_{m}}.$
(49)

We have to fix the size of the perturbed basis set $N^{\prime}$. We do this by
the requirement of unitarity

$\sum_{n=1}^{N^{\prime}}\big{|}\innerproduct*{\psi_{m}}{\psi_{n}^{(\uparrow\downarrow)}}\big{|}^{2}>1-\epsilon,$
(50)

for any $m\leq N$. We then use this to determine the size of the unperturbed
basis set we also insert $N^{\prime\prime}$ by requiring

$\sum_{n=1}^{N^{\prime\prime}}\big{|}\innerproduct*{\psi_{m}^{(\uparrow\downarrow)}}{\psi_{n}}\big{|}^{2}>1-\epsilon,$
(51)

for any $m\leq N^{\prime}$. We find that using $\epsilon\sim 10^{-4}$ yields
excellent convergence. In a completely analogous way, we can calculate the
other 3 decoherence functions.

##  Appendix C Cumulant expansion for weak coupling and low temperature

In this section, we derive the behavior of the decoherence functions, both in
the power-law and thermal regimes. We explicitly go through the derivation for
$\nu_{\uparrow\downarrow,\downarrow\uparrow}(t)$ and state the results for the
other three. They can be found in a completely analogous way. Our results are
valid in the weak coupling $k_{F}a\ll 1$ and low temperature $T/T_{F}\ll 1$
regime. This calculation follows closely to the one discussed in the
supplemental material of [39].

Figure 6: The contour $\gamma$ on which the time-ordering needs to be
performed. In the time-ordering, operators on the lower branch are later than
operators on the upper branch.

The starting point of our derivation is the many-body expectation value of eq.
(8) of the main text which for the decoherence function we consider reads

$\nu_{\uparrow\downarrow,\downarrow\uparrow}(t)=\Big{\langle}e^{i\hat{H}_{\uparrow\downarrow}t/\hbar}e^{-i\hat{H}_{\downarrow\uparrow}t/\hbar}\Big{\rangle}.$
(52)

We might write the Hamiltonians appearing as
$\hat{H}_{\sigma}=\hat{H}_{0}+\hat{V}_{\sigma}$ with

$\hat{H}_{0}=\sum_{k}E_{k}c_{k}^{\dagger}c_{k}$ (53)
$\hat{H}_{\sigma}=\sum_{n,m}V_{nm}^{(\sigma)}c_{n}^{\dagger}c_{m},$ (54)

where $c_{n}^{\dagger}$ creates a fermion in the state $\psi_{n}$ given in eq.
(36). The interaction matrix $V_{nm}^{(\sigma)}$ has elements given by

$\begin{split}V_{nm}^{(\sigma)}&=\int_{-L/2}^{L/2}\differential
x\psi_{n}(x)\lambda\delta(x\pm x_{0})\psi_{m}(x)\\\
&=\frac{2\lambda}{L}\sin\left(k_{n}x_{0}\pm\frac{\pi}{2}\right)\sin\left(k_{m}x_{0}\pm\frac{\pi}{2}\right),\end{split}$
(55)

where $\lambda=-\hbar^{2}/ma$, $k_{n}=n\pi/L$, and the plus (minus) sign is
valid for $\sigma=\uparrow\downarrow$ ($\downarrow\uparrow$).

To simplify notation we will from now denote
$\nu_{\uparrow\downarrow,\downarrow\uparrow}=|\nu|e^{i\phi}$. We can write
this as a time-ordered exponential, which can be expanded in terms of time-
ordered cumulants [110]

$\begin{split}|\nu|e^{i\phi}&=\expectationvalue{\mathcal{T}\exp[\int_{\gamma}\differential
t^{\prime}\frac{\hat{V}(t^{\prime})}{i\hbar}\bigg{]}}\\\
&\approx\exp\left[\bigg{\langle}\int_{\gamma}\differential
t^{\prime}\frac{\hat{V}(t^{\prime})}{i\hbar}\bigg{\rangle}_{c}+\frac{\mathcal{T}}{2}\bigg{\langle}\bigg{(}\int_{\gamma}\differential
t^{\prime}\frac{\hat{V}(t^{\prime})}{i\hbar}\bigg{)}^{2}\bigg{\rangle}_{c}\right],\end{split}$
(56)

where the symbol $\mathcal{T}$ denotes time ordering on the curve $\gamma$
shown in Fig. 6 such that operators on the branch $t_{+}$ occurs at a later
time than operators on the $t_{-}$ branch. The integration is performed over
this same curve. $\expectationvalue{\bullet}$ denotes the thermal expectation
value with respect to the initial thermal state, and
$\expectationvalue{\bullet}_{c}$ is the corresponding cumulant. The operator
$\hat{V}(t)$ in the equation above takes the form

$\hat{V}(t)=\begin{cases}\hat{V}_{\uparrow\downarrow}(t),\quad t\in t_{-}\\\
\hat{V}_{\downarrow\uparrow}(t),\quad t\in t_{+}\end{cases}$ (57)

on the two branches of the curve $\gamma$. The operators are written in the
interaction picture such that
$\hat{V}_{\sigma}(t)=e^{i\hat{H}_{0}t/\hbar}\hat{V}_{\sigma}e^{-i\hat{H}_{0}t/\hbar}$.
In the second line of Eq. (56) we have neglected terms of order
$\mathcal{O}(\hat{V}^{3})$.

By using the thermal expectation value
$\expectationvalue{c_{n}^{\dagger}c_{m}}=f(E_{n})\delta_{nm}$, with $f(E)$
being the Fermi-Dirac distribution, we find that the first cumulant is

$\begin{split}&\int_{\gamma}\differential
t^{\prime}\expectationvalue*{\hat{V(t)}}_{c}=\int_{0}^{t}\differential
t^{\prime}\frac{\expectationvalue*{\hat{V}_{\uparrow\downarrow}(t^{\prime})}_{c}}{i\hbar}+\int_{t}^{0}\differential
t^{\prime}\frac{\expectationvalue*{\hat{V}_{\downarrow\uparrow}(t^{\prime})}_{c}}{i\hbar}\\\
&=\frac{t}{i\hbar}\sum_{n}f(E_{n})\left(V_{nn}^{(\uparrow\downarrow)}-V_{nn}^{(\downarrow\uparrow)}\right)=0.\end{split}$
(58)

To capture the decay of the decoherence function we calculate the second
cumulant. We use the standard relation for Gaussian fermionic states
$\expectationvalue*{c_{n}^{\dagger}c_{m}c_{k}^{\dagger}c_{l}}_{c}=f(E_{n})[1-f(E_{m})]\delta_{nl}\delta_{mk}$
to get

$\begin{split}&\expectationvalue{\hat{V}_{\sigma}(t)\hat{V}_{\sigma^{\prime}}(t^{\prime})}_{c}\\\
&=\sum_{n,m}V_{nm}^{(\sigma)}V_{mn}^{(\sigma^{\prime})}e^{i(E_{n}-E_{m})(t-t^{\prime})/\hbar}f(E_{n})[1-f(E_{m})].\end{split}$
(59)

The terms with $n=m$ will contribute a term proportional to $t^{2}$, and will
go to zero as $L^{-1}$, and can be neglected in the thermodynamic limit. We
then use this result to compute the second cumulant. After integrating over
the contour $\gamma$, being careful of the time ordering, we find that

$\begin{split}&-\Gamma(t)\equiv-\frac{\mathcal{T}}{2}\bigg{\langle}\bigg{(}\int_{\gamma}\differential
t^{\prime}\frac{\hat{V}(t^{\prime})}{i\hbar}\bigg{)}^{2}\bigg{\rangle}_{c}\\\
&=\sum_{n\neq
m}f(E_{n})[1-f(E_{m})]\frac{1-\cos[(E_{n}-E_{m})t/\hbar]}{(E_{n}-E_{m})^{2}}V_{nm}^{2},\end{split}$
(60)

where we have defined
$V_{nm}=V_{nm}^{(\uparrow\downarrow)}-V_{nm}^{(\downarrow\uparrow)}$. Note
that Eq. (60) is a real number, so the decoherence function is real at least
to second order in $\hat{V}$. This is unique for the
$\nu_{\uparrow\downarrow,\downarrow\uparrow}$, and will not hold true for the
other decoherence functions.

To simplify Eq. (60), we introduce the spectral density

$\begin{split}J(\omega)&=\frac{1}{\hbar}\sum_{n,m}V_{nm}^{2}f(E_{n})[1-f(E_{m})]\delta(\hbar\omega+E_{n}-E_{m})\\\
&=\frac{1}{\hbar}\int_{0}^{\infty}\differential Ef(E)[1-f(E+\hbar\omega)]\\\
&\crossproduct\sum_{n\neq
m}V_{nm}^{2}\delta(E-E_{n})\delta(\hbar\omega+E-E_{m})\end{split}$ (61)

representing the coupling strength to particle-hole excitations weighted by
the finite temperature density of states. The decoherence rate then takes the
form

$\Gamma(t)=-\fint_{-\infty}^{\infty}\differential\omega\frac{1-\cos(\omega
t)}{\omega^{2}}J(\omega),$ (62)

where $\fint$ denotes the principal value integral excluding $\omega=0$. To
make further progress we take the continuum limit, introducing the s-wave
density of states

$D_{s}(E)=\frac{1}{L}\sum_{n}\delta(E-E_{n})=\frac{1}{\pi\hbar}\sqrt{\frac{m}{2E}}$
(63)

valid when $L\rightarrow\infty$. We replace $V_{nm}$ by $V(E)$ by letting

$k_{n}\rightarrow k(E)=\sqrt{\frac{2mE}{\hbar^{2}}}.$ (64)

Doing this simplifies the spectral density to take the form

$\begin{split}J(\omega)=\frac{2\lambda^{2}}{\hbar}\int_{0}^{\infty}&\differential
Ef(E)[1-f(E+\hbar\omega)]\\\ &\crossproduct
D_{s}(E)D_{s}(E+\hbar\omega)g(E,\omega).\end{split}$ (65)

where we have introduced the function $g$ capturing the effect of the
impurities on the gas. We can write it as

$g(E,\omega)=1-\cos(2k(E)x_{0})\cos(2k(E+\hbar\omega)x_{0}).$ (66)

We are interested in low temperatures such that $T\ll T_{F}$ and $\mu\approx
E_{F}$. For $\hbar\omega\leq-E_{F}$, $J(\omega)$ is exponentially suppressed.
If we for a second ignore $g(E,\omega)$ in Eq. (65), we see that
$J(\omega)\sim\sqrt{\omega\tau_{F}}$ for $\hbar\omega\geq E_{F}$. Going back
to Eq. (62), such contributions can be neglected in $\Gamma(t)$. Including
$g(E,\omega)$, will only make $J(\omega)$ grow slower as a function of
$\gamma$, and thus we can restrict ourselves to low frequencies
$\hbar|\omega|\ll E_{F}$. In this regime, the function
$f(E)[1-f(E+\hbar\omega)$ is sharply peaked around $E=E_{F}$, and we may
replace
$\sqrt{E(E+\hbar\omega}\rightarrow\sqrt{E_{F}(E_{F}+\hbar\omega}\approx
E_{F}$. In $g(E,\omega)$ we make the replacement $E\rightarrow E_{F}$ and make
a series expansion in $\hbar\omega/E_{F}$ in $k(E+\hbar\omega)$. We also
introduce the interaction time $\tau_{i}=2x_{0}/v_{F}$ as discussed in the
main text to get the expression

$g(\omega)=1-\cos(2\tau_{i}/\tau_{F})\cos(2\tau_{i}/\tau_{F}+\tau_{i}\omega).$
(67)

The remaining integral in Eq. (65) can then be computed to yield

$J(\omega)=\frac{1}{2}\alpha\omega
g(\omega)\left[1+\coth(\hbar\beta\omega/2)\right],$ (68)

which is valid for $|\omega|<\Lambda$, where $\Lambda\sim E_{F}/\hbar$ is a
cutoff-frequency. Here, we have introduced the dimensionless coupling strength
$\alpha=(\pi k_{F}a)^{-2}$.

Going back to Eq. (62), we have to calculate the following integral

$\begin{split}\Gamma(t)=&-\frac{\alpha}{2}\fint_{-\Lambda}^{\Lambda}\differential\omega\frac{1-\cos\omega
t}{\omega}g(\omega)[1+\coth(\hbar\beta\omega/2)]\\\
=&-\frac{\alpha}{2}\fint_{-\infty}^{\infty}\differential\omega\frac{1-\cos\omega
t}{\omega}g(\omega)[1+\coth(\hbar\beta\omega/2)]\\\
&+\alpha\int_{\Lambda}^{\infty}\differential\omega\frac{1-\cos\omega
t}{\omega}g(\omega),\end{split}$ (69)

where in the third line we have used that
$\coth(\hbar\beta\omega/2)\approx\text{sign}(\omega)$ for $|\omega|>\Lambda$,
which is valid as long as $\hbar\beta\Lambda\gg 1$. The two integrals in the
second equality of Eq. (69) can be evaluated by writing the cosines as complex
exponentials and we end up with 9 terms of the form

$-\frac{\alpha}{2}\fint_{\infty}^{\infty}\differential\omega
e^{iy\omega}[1+\coth(\hbar\beta\omega/2)]/\omega+\alpha\int_{\Lambda}^{\infty}\differential\omega
e^{iy\omega}/\omega.$ (70)

For the first of these integrals, we extend $\omega$ to the complex plane, and
depending on the sign of $y$ we close the contour in either the upper or lower
plane, being careful going around the origin. We then employ the residue
theorem, and end up with a geometric series over the residues at the poles at
$\omega=i\omega_{n}$ with $\omega_{n}=2n\pi/\hbar\beta$ the bosonic Matsubara
frequencies. $n$ runs over $n=\pm 1,\pm 2,\dots$ where the + (-) sign is valid
if $y>0$ ($y<0$) such that the integrand vanish on the edge of the contour.

The second integral in Eq. (70) can be explicitly evaluated by adding a small
imaginary part to $y\rightarrow y+i\epsilon$ and taking the limit
$\epsilon\rightarrow 0$. The integral will be proportional to the incomplete
gamma function $\Gamma(0,iy)$. Combining this we obtain the following
expression for the decoherence rate

$\begin{split}&\Gamma(t)=-2\alpha\left[\ln{\frac{\hbar\Lambda\beta}{\pi}\sinh\left(\frac{\pi
t}{\hbar\beta}\right)}-\text{Ci}(\Lambda t)+\gamma_{E}\right]\\\
&-\alpha\cos^{2}\frac{2\tau_{i}}{\tau_{F}}\ln{\frac{\left(1-e^{-\frac{2\pi\tau_{i}}{\hbar\beta}}\right)^{2}}{\left(1-e^{-\frac{2\pi(t+\tau_{i})}{\hbar\beta}}\right)\left(1-e^{-\frac{2\pi|t-\tau_{i}|}{\hbar\beta}}\right)}}\\\
&+\alpha\cos^{2}\frac{2\tau_{i}}{\tau_{F}}\big{[}2\text{Ci}(\tau_{i}\Lambda)-\text{Ci}((t+\tau_{i})\Lambda)-\text{Ci}(|t-\tau_{i}|\Lambda)\big{]}\\\
&+\alpha\cos\frac{2\tau_{i}}{\tau_{F}}\sin\frac{2\tau_{i}}{\tau_{F}}\big{[}2\text{Si}(\tau_{i}\Lambda)-\text{Si}((t+\tau_{i})\Lambda)\\\
&\qquad\qquad\qquad\qquad-\text{Si}(|t-\tau_{i}|\Lambda)\big{]},\end{split}$
(71)

where Ci and Si denote the cosine and sine integrals, $\gamma_{E}$ is the
Euler constant and we have used the relation

$\begin{split}&e^{i\theta}\Gamma(0,ix)+e^{-i\theta}\Gamma(0,-ix)\\\
&=-2\cos\theta\text{Ci}(x)-2\sin\theta(\text{Si}(x)-\pi/2).\end{split}$ (72)

We are interested in how the decoherence behaves in time, so all constant
terms will be neglected. The terms proportional to $\text{Ci}(\Lambda t)$,
$\text{Ci}(\Lambda(t+\tau_{i}))$ and $\text{Si}(\Lambda(t+\tau_{i}))$ are
oscillating, but will quickly become negligible compared to the other terms.
The terms proportional to $\text{Ci}(|t-\tau_{i}|\Lambda)$ and
$\text{Si}(|t-\tau_{i}|\Lambda)$ regulate the solution at $t=\tau_{i}$. If
these terms did not appear, the decoherence function would go to zero at this
point. We are left with a decoherence function of the following form

$\begin{split}&\nu_{\uparrow\downarrow,\downarrow\uparrow}(t)\propto\left[\frac{\hbar\beta}{\pi\tau_{F}}\sinh\frac{\pi
t}{\hbar\beta}\right]^{-2\alpha}\\\
&\crossproduct\left[\frac{\left(1-e^{-\frac{2\pi\tau_{i}}{\hbar\beta}}\right)^{2}}{\left(1-e^{-\frac{2\pi(t+\tau_{i})}{\hbar\beta}}\right)\left(1-e^{-\frac{2\pi|t-\tau_{i}|}{\hbar\beta}}\right)}\right]^{-\alpha\cos^{2}\frac{2\tau_{i}}{\tau_{F}}}.\end{split}$
(73)

Here, we got rid of the cut-off dependence by using that
$(\Lambda/E_{F})^{-2\alpha}\sim 1$.

We find three regimes of the decoherence function. Firstly, if
$t\ll\tau_{i}\ll\hbar\beta$, we have algebraic decoherence with an exponent
$2\alpha$. This is the same as two independent impurities decohering. If we
have $\tau_{i}\ll t\ll\hbar\beta$ the exponent changes to
$-2\alpha(1-\cos(4k_{F}x_{0})$. Thus the bath-induced interaction leads to
suppressed decoherence. This effect is known as sub-decoherence [89, 90], and
will be discussed more in the next paragraph. As the impurities are only able
to produce low-energy, long-wavelength excitations, the decoherence slows
down. Finally, for $t\gg\hbar\beta$ the decoherence is exponential
$\nu_{\uparrow\downarrow,\downarrow\uparrow}(t)\propto e^{-\gamma t}$ with
$\gamma=\frac{2\alpha\pi}{\hbar\beta}$.

To see why the sub-decoherence occurs for
$\nu_{\uparrow\downarrow,\downarrow\uparrow}$ it is useful to rewrite the
interaction Hamiltonian in the Bell-basis. It can easily be checked that the
interaction Hamiltonian can be written as

$\begin{split}\frac{\hat{H}_{I}}{\lambda}&=\frac{1}{2}\hat{I}\otimes(\hat{n}_{1}+\hat{n}_{2})\\\
&+\frac{1}{2}\left(\outerproduct{\Phi^{+}}{\Phi^{-}}+h.c.\right)\otimes(\hat{n}_{1}+\hat{n}_{2})\\\
&+\frac{1}{2}\left(\outerproduct{\Psi^{+}}{\Psi^{-}}+h.c.\right)\otimes(\hat{n}_{1}-\hat{n}_{2}),\end{split}$
(74)

where $\hat{n}_{1}$ and $\hat{n}_{2}$ are density operators for the gas at
positions $\pm x_{0}$. It is clear that decoherence functions like
$\nu_{\uparrow\downarrow,\downarrow\uparrow}$ couple to density differences in
the gas. At long times, only low-frequency excitations are relevant for the
decoherence function: in particular, at time $t$ only excitations with
frequencies $\omega\lesssim 1/t$ contribute significantly. Low frequency
generally entails long wavelength, and density excitations with wavelengths
much greater than the impurity separation are not able to create an
appreciable density difference at the positions of the impurities. In
particular, excitations close to the Fermi surface have a wavelength
$\lambda\sim 2\pi v_{F}/\omega$, and therefore the decoherence signal slows
down significantly when $t\gtrsim\tau_{i}$ after which time all relevant
wavelengths exceed $\Delta x$.

Following the approach outlined in this section, we can also compute the
behavior of the other decoherence functions. Here, we just state the results.
For the decoherence function $\nu_{\uparrow\downarrow,\downarrow\downarrow}$,
the behavior is that of a single qubit in a thermal bath. For short times
$t\ll\hbar\beta$ we have algebraic decay with exponent $\alpha$, while for
long times $t\gg\hbar\beta$ we have exponential decay $\sim e^{-\alpha\pi
t/\hbar\beta}$.

For the decoherence function $\nu_{\uparrow\uparrow,\downarrow\downarrow}$, we
have the same behavior as $\nu_{\uparrow\downarrow,\downarrow\uparrow}$ for
short times, that is algebraic decay with exponent $2\alpha$, but for the
intermediate regime, $\tau_{i}\ll t\ll\hbar\beta$ the exponent changes to
$-2\alpha(1+\cos(4k_{F}x_{0})$. As can be seen from Eq. (74), this decoherence
function couples to low-frequency excitations, leading to the phenomena of
super-decoherence.Finally, in the limit $t\gg\hbar\beta$ the decoherence is
exponential $\sim e^{-2\alpha\pi t/\hbar\beta}$.

Finally, for $\nu_{\uparrow\uparrow,\uparrow\downarrow}$, the low-$T$ weak
coupling expansion breaks down. The derivation of the cumulant expansion of
$\nu_{\uparrow\uparrow,\uparrow\downarrow}$ will be almost identical to that
given above, the only change being that
$\hat{V}^{(\downarrow\uparrow)}\rightarrow\hat{V}^{(\uparrow\uparrow)}$, where

$\displaystyle\hat{V}^{(\uparrow\uparrow)}_{mn}=\hat{V}^{(\uparrow\downarrow)}_{mn}+\hat{V}^{(\downarrow\uparrow)}_{mn}$
(75)

are the matrix elements of the interaction Hamiltonian when both impurities
are in the state $\ket{\uparrow}$. Looking at Eq. (60), this means that the
$V_{nm}$ that appears will change into
$V_{nm}=V_{nm}^{(\uparrow\downarrow)}-V_{nm}^{(\uparrow\uparrow)}=-V^{(\downarrow\uparrow)}_{mn}$.
This in turn means that the effect of the first impurity interacting with the
gas cancels out, and we end up with essentially the same behavior for
$\nu_{\uparrow\uparrow,\uparrow\downarrow}$ as for
$\nu_{\uparrow\downarrow,\downarrow\downarrow}$, i.e. a single impurity
interacting with a bath of fermions. In order to capture the slowing down of
the decoherence observed in Fig. 2 of the main text, we need to consider
higher-order cumulants.

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