# Unconventional Berezinskii-Kosterlitz-Thouless Transition in the
Multicomponent Polariton System

G. Dagvadorj  Department of Physics, University of Warwick, Coventry, CV4 7AL,
UK  Department of Physics and Astronomy, University College London, Gower
Street, London, WC1E 6BT, UK  P. Comaron<EMAIL_ADDRESS>Department of
Physics and Astronomy, University College London, Gower Street, London, WC1E
6BT, UK  Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46,
02-668 Warsaw, Poland  M. H. Szymańska<EMAIL_ADDRESS>Department of
Physics and Astronomy, University College London, Gower Street, London, WC1E
6BT, UK

###### Abstract

We study a four-component polariton system in the optical parametric
oscillator regime consisting of exciton/photon and signal/idler modes across
the Berezinskii-Kosterlitz-Thouless (BKT) transition. We show that all four
components share the same BKT critical point, and algebraic decay of spatial
coherence with the same critical exponent. However, while the collective
excitations in different components are strongly locked, both close to and far
from criticality, the spontaneous creation of topological defects in the
vicinity of the phase transition is found to be largely independent of the
inter-component mode locking, and instead strongly dependent on the density
within a given mode. This peculiar characteristic allows us to reveal a novel
state of matter, characterised by configurations of topological defects
proliferating on top of a superfluid with algebraic decay of coherence,
observation of which is demonstrated to be within reach of current
experiments.

The study of nonequilibrium phase transitions in driven-dissipative quantum
systems became of particular interest in recent years due to the unprecedented
experimental progress in realising quantum fluids of light. In two dimensions,
the onset of order is described by the celebrated Berezinskii-Kosterlitz-
Thouless mechanism Berezinskii (1971); Kosterlitz and Thouless (1973), where
the quasi-condensate phase energy is minimised by the pairing of topological
defects with opposite charge. Such a phenomenon has been investigated in a
wide class of conservative systems, from 4He Bishop and Reppy (1978) to
ultracold atoms Fletcher  _et al._ (2015), and in different types of
confinements Hadzibabic  _et al._ (2006). In the context of driven-dissipative
polariton fluids Kasprzak  _et al._ (2006); Carusotto and Ciuti (2013),
numerical studies of incoherently driven Comaron  _et al._ (2021); Mei  _et
al._ (2021), as well as coherently driven systems in the OPO regime Dagvadorj
_et al._ (2015), have predicted a non-equilibrium-type BKT phase transition,
recently confirmed in an experiment Caputo  _et al._ (2017). Finally, the
presence of further qualitative corrections to the standard BKT picture
introduced by higher-order phase fluctuations treated within the
Kardar–Parisi–Zhang framework has been proposed Kardar  _et al._ (1986);
Altman  _et al._ (2015); Fontaine  _et al._ (2021); Ferrier  _et al._ (2022).

The polariton fluids are, however, intrinsically multicomponent. They consist
of excitonic, photonic and, in the case of the OPO regime, the signal, idler
and pump modes. Within the OPO picture, the polariton coherence has been
previously studied theoretically Ciuti  _et al._ (2001, 2003); Carusotto and
Ciuti (2005); Wouters and Carusotto (2007); Dunnett and Szymańska (2016);
Dunnett  _et al._ (2018) and experimentally Krizhanovskii  _et al._ (2006);
Baumberg  _et al._ (2000). At a mean-field level the sum of the signal and
idler phases is locked to the spatially-coherent pump phase Carusotto and
Ciuti (2013). However, including fluctuations such a phase locking does not
hold or is weak, especially in the vicinity of the critical point Dagvadorj
_et al._ (2015). Despite that, the investigations to date were limited to the
signal mode of the photonic component Dagvadorj  _et al._ (2015) only, while
the interplay of critical features in different components are yet unrevealed.
Therefore, questions regarding the multicomponent nature of the phase
transition remains open: (i) How the spatial coherence differs in the
different modes, and how this affects the BKT phase transition? (ii) How
topological defects in distinct components are correlated?

In this work, we investigate numerically the OPO thresholds showing that the
nonequilibrium BKT transition occurs simultaneously (for the same pump
strength) in all four components (photonic, excitonic, signal and idler)
despite marked differences in the density of the different components,
Moreover, we demonstrate that coherence is characterised by the same algebraic
power-law exponent, indicating a strong phase locking between collective
excitations in all modes. Interestingly, such a phase-locking takes place also
in the vicinity of the critical point, where strong phase fluctuations are
expected to modify the mean-field picture Dagvadorj  _et al._ (2015). In
contrast, we discover that, unlike the collective excitations (the sound
modes), the topological defects (the vortices) in different components are not
strongly correlated close to the phase transition, where the density of the
fluid is small, nor further from the transition for the component with the
weakest density i.e. the photonic idler mode. Remarkably, the photonic idler
component is found to possess an algebraic order, characteristic of two-
dimensional superfluids, in the presence of multi-vortex configurations. This
is possible only due to the coupling of the low-density idler mode to higher
density components. To our knowledge, such a peculiar state has never been
seen before. This raises questions about the importance of the interplay
between sound modes and vortices in superfluid phase transitions.

##  System and theoretical modelling. —

We use stochastic simulations based on the truncated Wigner approximation
Carusotto and Ciuti (2005). Our method considers the full two dimensional
multimode polariton field, which includes fluctuations in both density and
phase, represented by complex number fields $\psi_{\chi}({\mathbf{r}},t)$,
where $\chi=\mathrm{X,C}$ denotes the single excitonic and photonic component,
respectively. The stochastic differential equation for trajectories of these
fields takes the following form Carusotto and Ciuti (2013, 2005); Dagvadorj
_et al._ (2015) ($\hbar=1$)

$i\text{d}\begin{pmatrix}\psi_{\text{X}}\\\
\psi_{\text{C}}\end{pmatrix}=\left[\hat{H}_{0}\begin{pmatrix}\psi_{\text{X}}\\\
\psi_{\text{C}}\end{pmatrix}+\begin{pmatrix}0\\\
F\end{pmatrix}\right]\text{d}t\\\
+i\begin{pmatrix}\sqrt{\kappa_{\text{X}}}\text{d}W_{\text{X}}\\\
\sqrt{\kappa_{\text{C}}}\text{d}W_{\text{C}}\end{pmatrix}\;.$ (1)

In this notation, the Hamiltonian operator

$\hat{H}_{0}=\begin{pmatrix}-\frac{\nabla^{2}}{2m_{\text{X}}}-g_{\text{X}}|\psi_{\text{X}}|_{-}^{2}-i\kappa_{\text{X}}&\Omega_{\text{R}}/2\\\
\Omega_{\text{R}}/2&-\frac{\nabla^{2}}{2m_{\text{C}}}-i\kappa_{\text{C}}\end{pmatrix}\;,$
(2)

is written in terms of the Rabi splitting $\Omega_{\text{R}}$, the exciton-
exciton interaction strength $g_{\text{X}}$, the mass $m_{\chi}$ and the
damping rates $\kappa_{\chi}$, where $\chi=$ X,C indicates excitons and
photons, respectively. Here, we consider a homogeneous monochromatic
continuous-wave pump
$F({\mathbf{r}},t)=f_{p}\exp{i({\mathbf{k}}_{p}\cdot{\mathbf{r}}-\omega_{p}t)}$
with momentum ${\mathbf{k}}_{p}$, strength $f_{p}$ and frequency $\omega_{p}$,
resonant with the bare lower-polariton dispersion, so that, as shown in Fig.
1(a), polaritons undergo parametric scattering into the signal and idler
states Ciuti  _et al._ (2003) for a certain range of pump strengths $f_{p}$.
$\text{d}W_{\chi}$ are independent white complex Gaussian noise terms, with
zero mean and local correlations in time and space:
$\langle\text{d}W_{\chi}({\mathbf{r}},t)\text{d}W_{\chi^{\prime}}^{*}({\mathbf{r}},t)\rangle={2}\delta_{\chi,\chi^{\prime}}{\delta_{{\mathbf{r}},{\mathbf{r}}^{\prime}}}\text{d}t/{\text{d}A}$,
and
$\langle\text{d}W_{\chi}({\mathbf{r}},t)\text{d}W_{\chi^{\prime}}({\mathbf{r}},t)\rangle=0$.
The reduced Wigner density reads
$|\psi_{\text{X}}|_{-}^{2}=\left(|\psi_{\text{X}}|^{2}-1/dV\right)$. Physical
observables can be calculated by appropriate averages over stochastic
realisations.

For the numerics, we have considered specific system parameters, relevant for
current experiments: $\Omega_{\text{R}}=4.4$ meV, $m_{\text{C}}=2.3\times
10^{-5}m_{0}$, where $m_{0}$ denotes the electron mass, $\kappa_{\chi}=0.1$
meV, and $g_{\text{X}}=2\times 10^{-3}$ meV$\mu$m2. We have ignored the
exciton dispersion as $m_{\text{X}}\gg m_{\text{C}}$. Eqs. (1) are numerically
implemented on a $N^{2}$ grid with $N=256$ points and length $L=N\times a$,
where $a=0.87\mathrm{\mu m}$ is the uniform grid spacing. The pump is injected
at finite momentum ${\mathbf{k}}_{p}=(k_{p},0)$ in the $x$-direction, with
$k_{p}=1.6$ $\mu$m-1.

Figure 1: Polariton OPO regime. (a) Typical spectrum of the system within the
OPO window showing signal, pump and idler modes on the lower polariton branch.
(b) Signal, pump and idler momentum $k_{x}$ as a function of the pump strength
$f_{p}$, exhibiting a double-threshold. The ordered (disordered) phase lies in
the pump range within (outside) a lower (solid green) and an upper (solid
orange line) critical point.

We evolve the stochastic equations (1) in a two-dimensional geometry with
periodic boundary conditions until the steady-state. While mean-field
wavefunctions are used as initial conditions, the Weiner noise terms are
adiabatically switched on along the dynamics. As explained in Ref. SM , once
the steady-state is achieved, for each $\chi$-component a filtering process
allows us to extract the different fields $\psi_{\chi,n}({\mathbf{r}},t)$, and
the momenta $\textbf{k}_{\chi,n}$ at which they are peaked in the momentum
distribution, where $n={s,p,i}$ labels the signal, pump and idler mode,
respectively. We ascertain that at a given pump value, the set of modes $n$
have the same momentum structure in both components $\chi$ SM . In Fig. 1(b)
we plot the behaviour of the extracted momenta $k_{x}$ for the photonic
component, where $\textbf{k}_{\text{C},n}=(k_{x},0)$, as a function of the
pump strength, for each mode $n$. Note that the OPO quasi-ordered phase
possesses two different thresholds, easily distinguishable from Fig. 1; we
extract a lower threshold (LT) at $f_{p}=1.419$ and an upper threshold (UT) at
$f_{p}=5.149$. Further details on the filtering process, the resulting steady-
state diagram and identification of the critical points are reported in Ref.
SM .

As discussed earlier, at the mean-field level the sum of the OPO signal and
idler phases is locked to that of the external pump $F$. In the next sections
we show how fluctuations can alter such phase locking Dagvadorj  _et al._
(2015), and its implications on the observation of unconventional order in a
multicomponent phase transition.

##  Onset of multicomponent order. —

Figure 2:  Vortices near the OPO thresholds. Top: averaged number of
vortices/antivortices in all four components, around the (a) lower and (b)
upper thresholds. The quasi-condensate phase is shown as a shaded coloured
area. Bottom: Vortex-antivortex (V-AV) correlations at the upper threshold for
the photonic component in the signal (c) and the idler (d) modes.

We proceed by investigating the nonequilibrium phase diagram of our
multicomponent system. As in previous works Dagvadorj  _et al._ (2015);
Comaron  _et al._ (2018, 2021), we calculate the average number of vortices
(Fig. 2) and spatial correlation functions (Fig. 3) in the steady-state close
to the lower (LT) and upper (UT) phase transitions. We see that, for all
components, while in the disordered phases, below (above) the LT (UT), the
total vortex/antivortex number is large and almost constant. When the pump
strength passes the critical point, the number of topological defects abruptly
decrease. This is in agreement with previous resultsComaron  _et al._ (2018):
in the late dynamics, the system approaches a steady-state number of vortices,
which becomes lower as one goes deeper into the ordered phase.

While we observe an abrupt change in the steady-state number of vortices at
the same pump power for all components, we note that they host a different
number of vortices at a given pump power — the inter-component phase locking
is not perfect in the vortex channel beyond the mean-field level. In
particular, the photonic idler mode contains a number of topological defects
which is, in the ordered phase, orders of magnitude larger than in the case of
the other components. This can be understood by noting the difference in the
particle densities (reported in Fig. 1b of Ref. SM ): at small densities,
characteristic of the photonic-idler, critical fluctuations are relatively
stronger and the creation of random vortex-antivortex pairs is more effective
as compared to other higher density components. Such a behaviour is in line
with the observations in Refs. Comaron  _et al._ (2021); Mei  _et al._ (2021).

Figure 3: Inter-components phase-locking. First order spatial correlation
function along the $x-$direction at the lower
$f_{\mathrm{th}}^{\mathrm{low}}=1.419$ (a) and the upper
$f_{\mathrm{th}}^{\mathrm{up}}=5.149$ (b) threshold. The power-law exponents
$\alpha$ (insets) show long-range phase-locking between collective modes in
different components. (c) Zoomed-in view of the real-space phase-profiles at
the lower threshold. Note, the presence (absence) of vortex configurations in
the photonic idler (signal) , while away from the vortices the phase still
exhibits correlation between the two components.

The first order spatial correlation function in an isotropic system at a given
time $t$, reads:

$g^{(1)}({\mathbf{r}})=\displaystyle\frac{\langle\psi_{s}^{*}({\mathbf{r}}+{\mathbf{R}})\psi_{s}({\mathbf{R}})\rangle_{{\mathbf{R}},\mathcal{N}}}{\sqrt{\langle\psi_{s}^{*}({\mathbf{R}})\psi_{s}({\mathbf{R}})\rangle_{{\mathbf{R}},\mathcal{N}}\langle\psi_{s}^{*}({\mathbf{r}}+{\mathbf{R}})\psi_{s}({\mathbf{r}}+{\mathbf{R}})\rangle_{{\mathbf{R}},\mathcal{N}}}}\;,$
(3)

where the average $\langle\cdots\rangle$ is performed over the spatial
position R and a large number $\mathcal{N}$ of stochastic realisations. Our
numerics show that all components exhibit a phase transition from exponential
to power-law decay of correlations at the same pump power, similarly to the
abrupt decrease of the number of topological defects analysed in Fig. 2. Such
a transition point has been quantitatively estimated by comparing different
fitting functions and choosing the best fit, as in the methods reported in
Refs. Dagvadorj  _et al._ (2015); Comaron  _et al._ (2021). In Fig. 3 we plot
$g^{(1)}({\mathbf{r}})$ (3) at the critical pump powers for the LT (a) and UT
(b). Although the four modes contain very different numbers of vortices, we
extract (within error bars) the same algebraic exponents, characterising the
long-range decay of $g^{(1)}({\mathbf{r}})$ (3). A detailed discussion is
reported in Ref. SM .

Motivated by the unusual observation of a high density of defects in the
algebraic phase of the photonic idler, even far from the critical point, we
investigate further their spatial distributions. An example of a single-
realisation of the photonic signal and photonic idler phases is shown in Fig.
3(c). In the conventional BKT transition, away from the critical point on the
quasi-ordered side, vortices are expected to be either fully paired or absent.
On the contrary, the real space vortex configurations in the photonic idler,
especially close to the upper threshold, show presence of multi-vortex
configurations even at pump powers deep in the ordered phase (see Figs. S9-S10
and discussion in Ref. SM ). This visual observation is corroborated by the
vortex-antivortex space correlations, reported in Fig. 2 (c,d) and discussed
in more detail in the next section.

##  Vortex correlations and pairing in single component. —

In order to gain a better understanding of correlations between vortices in
different components, as well as the pairing of vortices and antivortices
known to underly the BKT phase transition, we proceed by computing various
vortex correlation functions. We define the spatial number correlators between
vortex-vortex/antivortex-antivortex (hereafter denotes as V-V), and vortex-
antivortex/antivortex-vortex (V-AV) as:

$\eta^{(1)}_{\\{\text{V:}\alpha\beta\gamma,\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\\}}({\mathbf{r}})=\displaystyle\frac{\langle
n_{\text{V}}^{\alpha\beta\gamma}({\mathbf{r}},t)n_{\text{V}}^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}({\mathbf{r}}+{\mathbf{R}},t)\rangle_{{\mathbf{R}},{\mathcal{N}}}}{\sqrt{\langle
N_{\text{V}}^{\alpha\beta\gamma}\rangle_{\mathcal{N}}\langle
N_{\text{V}}^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}\rangle_{\mathcal{N}}}},$
(4)

where $n_{\text{V}}({\mathbf{r}})$ and $\langle N_{\text{V}}\rangle$ are the
spatial distribution and average number of vortices with components
$\alpha\in\\{\text{Vrtx},\text{AntV}\\},\quad\beta\in\\{\text{Sig},\text{Idl}\\}\quad\text{and}\quad\gamma\in\\{\text{Ph},\text{Ex}\\}.$

Let us first focus on V-AV correlations. An example is shown in panel (c) and
(d) of Fig. 2, where the 3D surfaces show the correlator (4) as a function of
the relative distance $r$ as the pump-power is varied across the phase
transition. The threshold point is indicated as a dashed red curve on top of
the 3D surface. In all cases, the correlator assumes a double peak structure
with a minimum at the origin indicating V-AV attraction leading to
annihilation events. In the disordered phase, characterised by a high number
of free vortices, we find that V-AV correlations asymptotically approach a
finite (non-zero) value as $r\to\infty$, indicating presence of vortices,
whose binding nature is undetermined. In the ordered phase of the photonic
signal (Fig. 2 c), V-AV correlations plateau to a null value, indicating that
on average only one tightly-bound V-AV pair is left in the system. On the
other hand, the ordered phase of the photonic idler (Fig. 2(d)), in agreement
with Fig. 2(a), still contains a large number of vortices as the correlator
plateaus to a smaller but finite value. By visually inspecting the spatial
distribution of vortices in Figs. S9-S10 of Ref. SM , we find multi-vortex
configurations. Note, that this ensemble of vortices is mainly composed of
vortex pairs; interestingly however, we also find indication of the presence
of rare free vortices. Furthermore, in Figs. S5-S6 of Ref. SM  we investigate
and compare V-AV correlations in different components. We find that all
components exhibit the same pairing crossover from a “randomly distributed”
disordered phase to a “single-pair” ordered phase, except for the photonic-
idler. For pump strengths both above LT and below UT, we find a non-zero
plateau indicating presence of a high number of vortices. The UT case is
depicted in Fig. 2(d).

We also compute V-V correlations, which are shown and discussed in Figs. S2-
S4 of Ref. SM . Similarly to the V-AV case, for all components other than the
idler, we find a finite-valued plateau at large distances (many vortices),
which crossovers to zero (single-pair) as one approaches the ordered phase.
Close to the origin the autocorrelations drop to zero, indicating that
vortices repel each other. Again, similarly to the V-AV, the V-V correlations
also show that the photonic-idler component, even deep in the ordered phase,
still shows a (lower than the one in the disordered phase, but still
noticeable) finite-valued plateau.

Figure 4:  Correlations between vortices in different components at the upper
threshold. Top: Correlation between vortices in signal and anti-vortices in
idler for the photonic (a) and the excitonic (b) component. Bottom:
Correlation between vortices in photonic and excitonic components for signal
(c) and idler (d) modes.

##  Correlations between vortices in different components. —

Considering the processes underlying the symmetry-breaking mechanism of
parametric regimes — corresponding to an opposite rotation of the phases of
the signal and idler components — it is expected that vortices in different
modes would be “position-locked” while possessing opposite circulation
Carusotto and Ciuti (2005). Vice-versa, when considering vortices in the same
mode (signal or idler) but different component (excitonic and photonic), the
locking would take place between two vortices with same sign-circulation.
Following this reasoning, it is straightforward to chose the V-AV (V-V)
correlations to quantify same-component (same-mode) vortex-locking.

First, let us discuss the former case: signal-idler V-AV correlations are
shown in Fig. 4(a),(b) for the photonic and excitonic components,
respectively. Here, vortex-locking is easily quantified by how smooth (or
peaked) the V-AV correlation profiles are across the transition: while a
homogeneous profile along the $r$-axis would indicate a similar probability of
finding the V or AV partner everywhere, a peaked distribution would rather
show that it is more probable to find such objects in the same position.
Results shown in Fig. 4(a),(b) reveal that this inter-mode spatial pairing
mechanism takes place only for the excitonic fraction, but not for the
photonic one.

A similar analysis is carried out for investigating the single-mode vortex-
coupling between the excitonic and photonic components. The V-V correlator is
shown in Fig. 4(c) for the signal and in Fig. 4(d) for the idler modes. The
results reveal a strongly-peaked distribution for the signal case, while the
idler mode shows a smooth profile. We conclude that the spatial vortex-locking
between components and/or modes is always strong apart from the cases with
photonic-idler mode; this is a further confirmation of a weak vortex-locking
mechanism observed for the photonic-idler component.

Figure 5: Correlations between vortices and antivortices in the signal
photonic component along $x$ (red curves) and $y$ (blue curves) directions for
different pump powers across the lower threshold.

##  Vortex-antivortex correlation anisotropy. —

Finally, we discuss the presence of anisotropic features in the pairing
mechanism, which we observe in the vortex correlation analysis discussed
above. Let us focus on the signal mode in the photonic component near the two
thresholds. Fig. 5 shows the V-AV correlation function in the photonic signal
across the LT, along two orthogonal directions $x,y$. Here we see that the
magnitude of pairing is different along the x and y directions. Moreover, the
dominant pairing direction switches across the BKT phase transition. Near the
UT, vortex correlations are instead isotropic (see Fig. S6 in Ref. SM ). To
explain this behaviour, we consider the possibility for the V-AV pairs to feel
anisotropic forces, possibly inherited from the anisotropic pump, which is
injected in the $x-$direction. We proceed by checking the diffusion
coefficients of the quantum fluid, which can be extracted by mapping the
exciton-polariton model to the Kardar–Parisi–Zhang (KPZ) equation Kardar  _et
al._ (1986) for the phase. Note, that analytical estimates for the vortex-
vortex interactions in this problem exist only for the isotropic case Wachtel
_et al._ (2016) allowing only approximate analysis. The results obtained are
not conclusive (see a detailed discussion in Ref. SM ), leaving this as an
exciting open question to be addressed in future works.

##  Conclusions. —

In this work, we have investigated theoretically the multicomponent BKT phase
transition of a two-dimensional driven-dissipative bosonic system in the OPO
regime. Clear signatures of the transition from a disordered to an ordered
state are inferred by computing the first order coherence, the number of
vortices and vortex correlations. We find, that the critical point occurs at
the same pump power with the same exponent of the algebraic decay of the first
order spatial coherence function for all components. However, collective phase
fluctuations (the sound modes) and the vortices are found to behave quite
differently: while collective excitations are locked across all components,
vortices are not perfectly spatially correlated, especially those involving
the photonic idler at the upper threshold. This results in a new phase, where
an algebraically ordered state exists in the presence of multi-vortex
configurations, composed of V-AV pairs and occasionally free vortices. Such a
peculiar phase is, to our knowledge, unique in the realm of condensed matter
systems, and sets the basis for a novel form of superfluidity, where a low
density quantum fluid is populated by topological defects, but remains in the
algebraically ordered phase due to locking of the collective excitation
channel to other higher density components.

##  Acknowledgements. —

We thank I. Carusotto, F. M. Marchetti and A. Zamora for stimulating
discussions. We thank A. Ferrier for proofreading the manuscript. MHS, GD and
PC gratefully acknowledge financial support from QuantERA InterPol and EPSRC
(Grant No. EP/S019669/1, EP/R04399X/1 and No. EP/K003623/2).

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# Relativistic time-of-arrival measurements: predictions, post-selection and
causality problems

Charis<EMAIL_ADDRESS>
Department of Physics, University of Patras, 26500 Greece
and
Maria-Electra Plakitsi<EMAIL_ADDRESS>
The Moraitis School, Psichiko, 15452, Greece

###### Abstract

We analyze time-of-arrival probability distributions for relativistic
particles in the context of quantum field theory (QFT). We show that QFT leads
to a unique prediction, modulo post-selection that incorporates properties of
the apparatus into the initial state. We also show that an experimental
distinction of different probability assigments is possible especially in
near- field measurements. We also analyze causality in relativistic
measurements. We consider a quantum state obtained by a spacetime-localized
operation on the vacuum, and we show that detection probabilities are
typically characterized by small transient non-causal terms. We explain that
these terms originate from Feynman-propagation of the initial operation,
because the Feynman propagator does not vanish outside the light-cone. We
discuss possible ways to restore causality, and we argue that this may not be
possible in measurement models that involve switching the field-apparatus
coupling on and off.

##  1 Introduction

Finding a consistent quantum description for time-of-arrival measurements is a
classic problem in the foundations of quantum theory. In the simplest time-of-
arrival measurement, a particle is prepared on an initial state
$|\psi_{0}\rangle$ with positive momentum and localized around $x=0$. A
particle detector is placed at $x=L$. The issue is to determine the
probability $P(t,L)dt$ that the detector clicks at some moment between $t$ and
$t+\delta t$. Despite the apparent simplicity of the problem, no unique time-
of-arrival probability exists [1, 2]. Fundamentally, this is due to the fact
that time is not a quantum observable. There is no self-adjoint operator for
time [3], hence, we cannot rely on Born’s rule for a unique answer.

The time-of-arrival problem is exacerbated for relativistic particles, because
it becomes entangled with another foundational problem, the issue of
localization. Particle localization is essential to any description of
measurements, because all particle-detection records are localized in space
and in time. However, the existence of observables associated to spatial
localization is in conflict with the requirement of causality, as evidenced by
several theorems [4, 5, 6].

The most well known set-up where this conflict appears is Fermi’s two-atom
problem. Fermi studied information transmission through a quantum field in a
system of two localized atoms at distance $r$ [7]. He assumed that at time
$t=0$, atom A is in an excited state and atom B is in the ground state. He
asked when the influence of A will cause B to leave its ground state. In
accordance with the locality principle that spacelike separated events cannot
influence each other, he found that this happens only at time greater than
$r$. However, Fermi’s result was shown to be an artefact of an approximation
[8]. Later studies of the problem came to different results that were heavily
dependent on approximations. Eventually, Hegerfeldt showed that the conflict
of localization and causality is generic in relativistic systems [9, 10], it
only requires energy positivity and the treatment of atoms as localized in
disjoint spatial regions——see, also the clarification of this result in [11].

In this paper, we analyze a broad class of probability distributions for the
time of arrival of relativistic particles. This class was identified in Ref.
[12] through a analysis of measurements in QFT. Part of our motivation is the
possibility of experimentally distinguishing between different proposals for
the time of arrival. We also analyse the structure of small apparently super-
luminal transient terms—analogous to the ones in the Fermi-atom problem—that
appear in the time-of-arrival probability distribution. We identify their
origin and examine their implications for relativistic quantum theory of
measurement.

An important reason for studying the time of arrival is that it forces us to
re-conceptualize the description of quantum measurements. Ever since von
Neumann [13], measurements have been described as almost instantaneous
processes that occur at a single moment of time $t$. In von Neumann
measurements, the interaction of the apparatus with the measured quantum
system is switched on for a pre-determined time interval—the exact timing of
the measurement is determined by the shape of the switching function. Hence,
the timing of the measurement is an external parameter of the measurement
scheme, and not a random variable of the experiment. This logic has recently
been extended to QFT measurements [14, 15]: the measured field and the
apparatus are initially uncorrelated, and they interact only in a finite,
predetermined spacetime region.

Time-of-arrival measurements challenge this measurement paradigm. Actual
particle detectors (e.g., photographic plates, silicon strips) have a fixed
location in space and they are made sensitive for a long time interval during
which particles may be detected. Therefore, the location of a detection event
is a fixed parameter of the experiment; the actual random variable is the
detection time. Von Neumann type measurements are fundamentally incapable of
describing time as a random variable, but it turns out that they can mimic
some aspects of time-of-arrival measurements. However, imitations have
limitations: they work only to lowest order in perturbation theory and that it
eventually leads to causality problems.

Our results are the following.

First, we re-derive the relativistic time-of-arrival probabilities of Ref.
[12] in a von Neumann measurement scheme for quantum fields. The original
derivation involved the Quantum Temporal Probabilities (QTP) method [16, 17,
12, 18, 19] that explicitly constructed a probability density with respect to
time. In the von Neumann measurement scheme, we use a switching function that
is localized in a compact spacetime region, and we reinterpret the
probabilities in order to define a probability density. This works only to
leading order in perturbation theory. We use this alternative derivation in
order to identify the problems that persist in this treatment of measurements.

Time-of-arrival probability distributions are post-selected, i.e., they refer
only to the fraction of particles that has been detected. They depend on the
apparatus through a specific operator $\hat{S}$ that describes the
localization of the detection records. We show that this operator can be
absorbed into a post-selection of the initial state. A unique time-of-arrival
probability measure ensues, modulo post-selection. This measure was first
derived by Leon [20] and then rederived from a QFT analysis in [16]. In the
non-relativistic limit, it coincides with Kijowski’s time-of-arrival
distribution [21].

The measured time of arrival probability distribution does depend on the
properties of the apparatus. We analyze the probability measure for an
operationally meaningful class of initial states that was recently proposed,
in relation to experiments for distinguishing between different time-of-
arrival proposals. We find that the different distributions are in principle
distinguishable in near-field experiments.

Then, we analyse locality in the time-of-arrival probabilities. We consider an
initial state that is generated by a localized external source acting on the
quantum field vacuum. By ”localized” we mean that the source has support in a
compact spacetime region. We find that the time-of-arrival probability has a
small but non-zero contribution outside the light-cone of the source’s
support. We analyze the origin of this term, and we find that it originates
from the fact that in QFT sources evolve with the Feynman propagator, and
Feynman propagator is non-zero outside the light-cone. We argue that von-
Neumann type models, that rely on switching on the interaction, may not be
able to resolve this problem.

Finally, we briefly revisit the QTP description of relativistic measurements,
and present possible strategies through which the super-luminal transient
terms can be consistently removed.

##  2  Relativistic time-of-arrival probabilities

In this section, we first re-derive the time-of-arrival probabilities of Ref.
[12] using a von Neumann type of measurement. We show that these probabilities
are unique modulo post-selection, and that the effect of the detector—as
determined by its localization properties—can be distinguished in near-field
measurements.

###  2.1 Detection probability from a von Neumann type measurement

We consider the measurement of a free scalar field $\hat{\phi}(x)$ on a
Hilbert space ${\cal F}$, interacting with an apparatus described by a Hilbert
space ${\cal H}$. As our focus is time-of-arrival measurements, we will work
only in one spatial dimension. The reason is that only particles that
propagate along the axis that connects the source with the detector contribute
to the total probability.

The Hamiltonian of the total system is

$\displaystyle\hat{H}=\hat{H}_{\phi}\otimes\hat{I}+\hat{I}\otimes\hat{H}_{A}+\hat{H}_{I},$
(1)

where $\hat{H}_{\phi}$ is the quantum field Hamiltonian and $\hat{H}_{A}$ the
Hamiltonian of the apparatus. We consider a von-Neumann type of measurement,
in which the interaction is switched on for a finite time. Then, we choose an
interaction Hamiltonian

$\displaystyle\hat{H}_{I}=\int
dxF_{\bar{t},\bar{x}}(t,x)\hat{\phi}(x)\otimes\hat{J}(x),$ (2)

where $F_{\bar{t},\bar{x}}(t,x)$ is a switching function centered around the
spacetime point $(\bar{t},\bar{x})$, and $\hat{J}(x)$ is a current operator on
${\cal F}$333Certainly, the idea of a switching interaction is not realistic
in QFT. Fields interact with the apparatuses through terms defined by the
Standard Model of particle physics, and in a Poincaré covariant theory, the
interaction terms are always present..

We assume an initial state $|\psi\rangle$ for the field and an initial state
$|\Omega\rangle$ for the detector. It is convenient to identify
$|\Omega\rangle$ with the ground state of the Hamiltonian $\hat{H}_{A}$,
$\hat{H}_{A}|\Omega\rangle=0$, and also to be annihilated by the generator of
space translations $\hat{P}_{A}$ of the apparatus. Then, the probability that
the detector is found in an excited state, when measured after the interaction
has been switched off, is given by

$\displaystyle\mbox{Prob}(\bar{t},\bar{x})=\langle\psi_{0},\Omega|\hat{S}^{\dagger}_{\bar{t},\bar{x}}(\hat{I}\otimes\hat{I}-|\Omega\rangle\langle\Omega|)\hat{S}_{\bar{t},\bar{x}}|\psi_{0},\Omega\rangle,$
(3)

expressed in terms of the S-matrix $\hat{S}_{\bar{t},\bar{x}}={\cal
T}\exp\left[-i\int
dtdxF_{\bar{t},\bar{x}}(t,x)\hat{\phi}(t,x)\otimes\hat{J}(t,x)\right]$;
$\hat{\phi}(t,x)$ and $\hat{J}(t,x)$ are Heisenberg-picture operators and
${\cal T}$ stands for time-ordering.

To leading order in perturbation theory,

$\displaystyle\mbox{Prob}(\bar{t},\bar{x})=\int
dt_{1}dx_{1}dt_{2}dx_{2}F_{\bar{t},\bar{x}}(t_{1},x_{1})F_{\bar{t},\bar{x}}(t_{2},x_{2})G(t_{x},t_{1};t_{2},x_{2})\langle\Omega|\hat{J}(t_{1},x_{1})\hat{J}(t_{2},x_{2})|\Omega\rangle,$
(4)

where

$\displaystyle
G(t_{1},x_{1};t_{2},x_{2})=\langle\psi_{0}|\hat{\phi}(t_{1},x_{1})\hat{\phi}(t_{2},x_{2})|\psi_{0}\rangle$
(5)

is a two-point correlation function for the field.

We take a reference point $x_{0}=0$ on the apparatus, and we define
$|\omega\rangle=\hat{J}(0)|\Omega\rangle$. Then, we can write
$\Omega|\hat{J}(t_{1},x_{1})\hat{J}(t_{2},x_{2})|\Omega\rangle=R(t_{2}-t_{2},x_{2}-x_{1})$,
where

$\displaystyle
R(t,x)=\langle\omega|e^{i\hat{H}_{A}t-i\hat{P}_{A}x}|\omega\rangle$

is the detector kernel. The key property of $R(t,x)$ is that by energy
positivity, its Fourier transform

$\displaystyle\tilde{R}(E,K):=\int
dtdxR(t,x)e^{-iEt+iKx}=2\pi\langle\omega|\delta(E-\hat{H}_{A})\delta(K-\hat{P}_{A})$
(6)

vanishes for $E<0$.

In Eq. (4) $\bar{x}$ and $\bar{t}$ appear as parameters, not as random
variables. However, Eq. (4) has a natural interpretation in terms of a
probability density. Consider a homogeneous switching function
$F_{\bar{t},\bar{x}}(t,x)=f(t-\bar{t},x-\bar{x})$, where
$f(t,x)=\exp\left[-\frac{t^{2}}{2\delta_{t}^{2}}-\frac{x^{2}}{2\delta_{x}^{2}}\right]$
for space and time spreads $\delta_{x}$ and $\delta_{t}$, respectively.
Gaussians satisfy the identity

$\displaystyle
f(t,x)f(t^{\prime},x^{\prime})=f^{2}\left(\frac{t+t^{\prime}}{2},\frac{x+x^{\prime}}{2}\right)\sqrt{f}(t-t^{\prime},x-x^{\prime}).$
(7)

Then, Eq. (4) takes the form

$\displaystyle\mbox{Prob}(\bar{t},\bar{x})=\int
dtdxf^{2}\left(\bar{t}-t,\bar{x}-x\right)P(t,x),$ (8)

i.e., it is a convolution of the probability density

$\displaystyle P(t,x)=\int
dsdyR(s,y)\sqrt{f}(s,y)G(t-\frac{s}{2},x-\frac{1}{2}y;t+\frac{1}{2}s,x+\frac{1}{2}y).$
(9)

Typically, the detector kernel decays to zero for $|t|$ larger than a temporal
scale $\hat{\sigma}_{T}$ or $|x|$ larger than a length scale $\sigma_{x}$.
Both scales depend on properties of the apparatus. Taking
$\delta_{t}>>\sigma_{t}$ and $\delta_{x}>>\sigma_{x}$, the contribution of
$\sqrt{f}$ can be dropped from Eq. (9), so that

$\displaystyle P(t,x)=\int
dsdyR(s,y)G(t-\frac{s}{2},x-\frac{1}{2}y;t+\frac{1}{2}s,x+\frac{1}{2}y).$ (10)

It is important to emphasize that the probability density (10) is meaningful
only to leading order in perturbation theory. We cannot construct a spacetime
density out of higher orders terms in this procedure. This is to be
constructed with the QTP method, briefly described in Sec. 3, that leads to
probability densities at all orders of perturbation theory.

Eq. (10) holds for any scalar field theory. For a free field of mass $m$,

$\displaystyle\hat{\phi}(t,x)=\int\frac{dp}{2\pi\sqrt{2\epsilon_{p}}}[\hat{a}(k)e^{ipx-i\epsilon_{p}t}+\hat{a}^{\dagger}(k)e^{-ipx+i\epsilon-
pt}],$ (11)

where $\epsilon_{p}=\sqrt{p^{2}+m^{2}}$.

Eq. (10) becomes

$\displaystyle
P(t,x)=P_{0}+\int\frac{dpdp^{\prime}}{2\pi}\frac{\rho(p,p^{\prime})}{2\sqrt{\epsilon_{p}\epsilon_{p^{\prime}}}}\;\tilde{R}\left(\frac{p+p^{\prime}}{2},\frac{\epsilon_{p}+\epsilon_{p^{\prime}}}{2}\right)e^{i(p-p^{\prime})x-i(\epsilon_{p}-\epsilon_{p^{\prime}})t},$
(12)

where
$\hat{\rho}(p,p^{\prime})=\langle\psi|\hat{a}^{\dagger}(p^{\prime})\hat{a}(p)|\psi\rangle$
is the single -particle reduced density matrix. The term

$\displaystyle P_{0}=\int\frac{dp}{4\pi\epsilon_{p}}\tilde{R}(p,\epsilon_{p})$
(13)

is constant and state-independent. It corresponds to vacuum noise, i.e., a
background rate of false alarms. Hence, a detection signal exists only as long
as $P(t,x)$ is larger than $P_{0}$. We will ignore $P_{0}$ in what follows,
except for checking whether transient terms that appear in $P(t,x)$ are actual
detection signals.

###  2.2 Post-selection with respect to recorded events

Eq. (12) is an unnormalized probability density. To normalize, we consider a
set-up where the particle source is in the vicinity of $x=0$, and the detector
at $x=L$, where $L>0$ is a macroscopic distance. In this setup, the
contribution of negative momenta to $P(t,L)$ is negligible. It is therefore
convenient to normalize over initial states with support only over positive
momenta. Then, integrating over $t\in\mathbb{R}$, we obtain the total
detection probability for constant $L$,

$\displaystyle P_{tot}=\int
dp\rho(p,p)\;\frac{\tilde{R}(p,\epsilon_{p})}{2p}.$ (14)

This means that

$\displaystyle\alpha(p)=\frac{\tilde{R}(p,\epsilon_{p})}{2p}$ (15)

is the absorption coefficient of the detector, i.e., it records the fraction
of particles of momentum $p$ that are absorbed, and hence detected. Note that
we integrated $t$ over the full real axis, because the contribution of $t<0$
is very small if the initial state has no negative momenta. This is an
approximation, as only positive values of $t$ are physically meaningful.

The conditional probability distribution $P_{c}(t,L):=P(t,L)/P_{tot}$ is
normalized to unity, as long as the initial state contains positive momenta.
Then, we define the post-selected density matrix

$\displaystyle\rho_{ps}(p,p^{\prime}):=\frac{\sqrt{\alpha(p)}\sqrt{\alpha(p^{\prime})}}{P_{tot}}\rho(p,p^{\prime}).$
(16)

Note, that for an initial pure state, the post-selected state remains pure.
Then,

$\displaystyle
P_{c}(t,L)=\int\frac{dpdp^{\prime}}{2\pi}\rho_{ps}(p,p^{\prime})\sqrt{v_{p}v_{p^{\prime}}}S(p,p^{\prime})e^{i(p-p^{\prime})L-i(\epsilon_{p}-\epsilon_{p^{\prime}})t},$
(17)

where $v_{p}=p/\epsilon_{p}$ is the particle velocity. $S(p,p^{\prime})$ are
the matrix elements $\langle p|\hat{S}|p^{\prime}\rangle$ of the localization
operator $\hat{S}$, defined by

$\displaystyle\langle
p|\hat{S}|p^{\prime}\rangle:=\frac{\tilde{R}\left(\frac{p+p^{\prime}}{2},\frac{\epsilon_{p}+\epsilon_{p^{\prime}}}{2}\right)}{\sqrt{\tilde{R}(p,\epsilon_{p})\tilde{R}(p^{\prime},\epsilon_{p^{\prime}})}}.$
(18)

By definition, $\langle p|\hat{S}|\hat{p^{\prime}}\rangle\geq 0$ and
$S(p,p)=1$. Its name originates from the fact that $\hat{S}$ describes the
localization of an elementary measurement event. $\hat{S}$ is a positive
operator if $\ln\tilde{R}(p,\epsilon_{p})$ is a convex function of $p$. Then,
the Cauchy-Schwarz inequality applies,

$\displaystyle\langle p|\hat{S}|p^{\prime}\rangle\leq\sqrt{\langle
p|\hat{S}|p\rangle\langle p^{\prime}|\hat{S}|p^{\prime}\rangle}=1.$ (19)

Maximal localization is achieved when Eq. (19) is saturated, i.e., for
$\langle p|\hat{S}|p^{\prime}\rangle=1$. For a pure initial state

$\displaystyle
P_{c}(t,L)=\left|\int_{0}^{\infty}dp\psi_{ps}(p)\sqrt{v_{p}}e^{ipL-i\epsilon_{p}t}\right|^{2}.$
(20)

Eq. (20) was first derived by [20]. In the non-relativistic limit, it
coincides with Kijowski’s formula [21]. Note that these expressions differ
from those obtained by the current operator
$\hat{J}=\frac{1}{2}\left[\hat{v}\delta(\hat{x}-L)+\delta(\hat{x}-L)\hat{v}\right]$,

$\displaystyle
P_{J}(t,L)=\int\frac{dpdp^{\prime}}{2\pi}\rho_{ps}(p,p^{\prime})\frac{1}{2}(v_{p}+v_{p^{\prime}})e^{i(p-p^{\prime})L-i(\epsilon_{p}-\epsilon_{p^{\prime}})t}.$
(21)

Note that Eq. (21) does not, in general guarantee, the positivity of
probabilities.

To understand the meaning of the localization operator, we evaluate its
Wigner-Weyl transform.

$\displaystyle\tilde{S}(x,p):=\int\frac{d\xi}{2\pi}S\left(p-\frac{\xi}{2},p+\frac{\xi}{2}\right)e^{-i\xi
x},$ (22)

It is straightforward to show that $\int dx\tilde{S}(x,p)=S(p,p)=1$.
Furthermore, by Bochner’s theorem, $\tilde{S}(x,p)\geq 0$. Hence,
$\tilde{S}(x,p)$ is a $p$-dependent probability distribution with respect to
$x$, which we will denote by $u_{p}(x)$. This probability distribution defines
the irreducible spread in determining the location of the measurement record.
Maximum localization corresponds to $u_{p}(x)=\delta(x)$ [12]. We note that
the general form (17) is fixed by the requirements of Poincaré covariance
[22]; however, this requirement does not fix the properties of the
localization operator or its physical interpretation in terms of properties of
the apparatus.

We can absorb the localization operator $\hat{S}$ through further post-
selection of the initial state,

$\displaystyle\tilde{\rho}_{ps}(p,p^{\prime})=S(p,p^{\prime})\rho_{ps}(p,p^{\prime}).$
(23)

It is straightforward to show that $\tilde{\rho}_{ps}$ is a density matrix.
Hence, all time-of-arrival probabilities can be brought in the form (19)
modulo post-selection of the initial state.

However, this uniqueness result is primarily structural, and it is not
reflected in the measured probability distributions, which depend strongly on
the properties of the detector. The latter are incorporated in two quantities,
the absorption coefficient $\alpha(p)$ and the localization operator
$\hat{S}$. In principle, we can determine both for a given detector:
$\alpha(p)$ is essentially determined by the attenuation coefficient of the
detecting medium, while $\hat{S}$ can be determined through time-of-arrival
experiments. Hence, both redefinitions (23) and (16) make operational sense.

###  2.3 Wigner representation

We gain some insight into the structure of the time of arrival probabilities
by expressing them in terms of the Wigner function associated to the density
matrix $\rho_{ps}(p,p^{\prime})$,

$\displaystyle
W(x,p)=\int\frac{d\xi}{2\pi}\rho_{ps}\left(p-\frac{\xi}{2},p+\frac{\xi}{2}\right)e^{-i\xi
x}.$ (24)

With these definitions, Eq. (17) becomes

$\displaystyle P_{c}(t,L)=\int
dpdx_{0}dx_{f}W(x_{0},p)u_{p}(x_{f})F_{t}(L-x_{f}-x_{0},t),$ (25)

where

$\displaystyle
F_{t}(x,p)=\int_{-2p}^{2p}d\xi\sqrt{|v_{p+\xi/2}v_{p-\xi/2}|}e^{ix\xi-i(\epsilon_{p+\xi/2}-\epsilon_{p-\xi/2})t}$
(26)

is a function on the classical phase space that represents the quantum time-
of-arrival observable.

Changing variables to $x:=x_{f}-x_{0}$, we can write Eq. (25) as

$\displaystyle P_{c}(t,L)=\int dpdx\tilde{W}(x,p)F_{t}(x,p),$ (27)

where $\tilde{W}(x,p)=\int dx_{f}u_{p}(x_{f})W(x-x_{f},p)$ is the Wigner
function associated to the post-selected quantum state $\tilde{\rho}_{ps}$.
Hence, the localization operator implements a position smearing of the
particle’s Wigner function.

In the non-relativistic limit,

$\displaystyle
F_{t}(x,p)=m^{-1}\int_{-2p}^{2p}d\xi\sqrt{p^{2}-\frac{\xi^{2}}{4}}e^{ix\xi-
ip\xi t/m}=v_{p}\frac{\pi J_{1}(2p|x-v_{p}t|)}{|x-v_{p}t|},$ (28)

where $J_{1}$ is the Bessel function. Note that $F_{t}$ takes negative values.
This is fully compatible with the positivity of the time-of-arrival
probabilities, because probability distributions concentrated in the regions
where $F_{t}$ is negative are not admissible Wigner functions for quantum
states.

In the ultra-relativistic limit ($m=0$),

$\displaystyle F_{t}(x,p)=\int_{-2p}^{2p}d\xi e^{ix\xi-i\xi
t}=\frac{\sin[2p(t-x)]}{p(t-x)}$ (29)

These expressions are to be compared with the corresponding classical
expression for the time-of-arrival observable,

$\displaystyle F^{cl}_{t}(x,p)=\delta(t-x/v_{p}),$ (30)

and the expression obtained from the non-relativistic current operator

$\displaystyle
F^{J}_{t}(x,p)=\int_{-2p}^{2p}d\xi\frac{1}{2}(v_{p+\xi/2}+v_{p-\xi/2})e^{ix\xi-i(\epsilon_{p+\xi/2}-\epsilon_{p-\xi/2})t}=v_{p}\frac{2\sin[2p(x-v_{p}t)]}{(x-v_{p}t)}$
(31)

The different versions of the time-of-arrival functions in the non-
relativistic regime are plotted in Fig. 1 as a function of $s:=p(x-v_{p}t)$.
The classical time of arrival observable is a delta function at $s=0$. Eqs.
(28) and (31) have finite spread around $s=0$, and they differ strongly near
$s=0$.

Figure 1:  The time-of-arrival functions (28) (solid) and (31) (dashed)
divided by $2E=p^{2}/m$, as a function of $s:=p(x-v_{p}t)$.

###  2.4 Probability dependence on the detector kernel

One motivation of this paper is the possibility of experimental comparison
between different proposals for the time-of-arrival probabilities. To do so,
we need to specify an operationally meaningful initial condition. Such a
condition was proposed in [23], where it was suggested that a strong
divergence of the prediction by Bohmian mechanics from the predictions of the
semi-classical theory for the time of arrival.

The idea is to prepare a number of particles in a box of width $a$, and we
remove one wall of the box (or both) at time $t=0$. Then, the initial state is
an eigenfunction of the Hamiltonian for a particle in a box,
$\psi_{n}(x)=\sqrt{\frac{2}{a}}\sin(\pi x/a_{n})$ for $x\in[0,a]$ and zero
otherwise; here $a_{n}=a/n$ and $n=1,2,\ldots$. Equivalently, in the momentum
representation

$\displaystyle\psi_{n}(p)=\pi\sqrt{\frac{2}{a}}a_{n}\frac{1+e^{-ia_{n}p}}{\pi^{2}-a_{n}^{2}p^{2}}.$
(32)

To understand the dependence of the probability density $P_{c}(t)$ on the
detection kernel, we consider a simple model, where
$S(p,p^{\prime})=\exp[-\sigma^{2}(p-p^{\prime})^{2}]$, i.e., a detection
kernel that is determined by the localization length $\sigma$. In Fig. 2, we
plot $P_{c}(t)$ for different values of $\sigma$; we also plot the pseudo-
probability density (21). We see that the probability densities differ
significantly at early times, and they have the same asymptotic fall-off
behavior.

We also find that if $\sigma/\alpha<<1$, $P_{c}(t)$ essentially coincides with
the maximal-localization form (20). The distribution (21) becomes negative at
early times, hence, it has no operational significance. We also note that the
differences are more pronounced for short $L$, i.e., in the near-field regime.
At large $L$, only distributions with $\sigma/\alpha$ of order one or larger
are distinguishable.

Figure 2:  The probability density $P_{c}(\tau)$ as a function of
$\tau:=t/(2ma^{2})$, for different values of the localization length $\sigma$
and for different distances, and for the pseudo-probability density (21). The
plots correspond to the non-relativistic regime. The insets show the early-
time behavior of $P_{c}(t)$.

##  3 Causality issues

In this section, we show that the time-of-arrival probabilities suffer from
small transient super-luminal terms. These terms originate from the fact that
unitary evolution implies Feynman-propagation of sources, and the Feynman
propagator does not vanish outside the light-cone. We discuss possible
strategies for removing such terms from the theoretical description, and we
argue that this may not be possible in von Neumann-type models of measurement.

###  3.1 Apparent causality violation

Any theory of relativistic measurements for quantum systems must respect
causality, in the sense that if the system is initially prepared within a
localized spacetime region $A$, then there should be no detection signal in
any region $B$ spacelike to $A$.

To implement this condition, we must first identify initial states that are
localized in a spacetime region. This is a difficult task. First, we know that
spatial localization at a moment of time does not work. Any notion of spatial
localization in the single-time quantum state (even approximate ones) leads to
faster-than-light signals [5, 6]. This conclusion is not an artefact of
specific models, but a consequence of fundamental properties of relativistic
quantum systems, namely, Poincaré covariance and energy positivity.

Rather than spatially localized states, we consider spacetime localized
operations. We assume that the field is initially in the vacuum state
$|0\rangle$, and that we prepare an initial state for the time-of-arrival
measurement by an external intervention. This intervention has support in a
compact spacetime region that lies wholly before the Cauchy surface $t=0$.
Then, we take the resulting state as $|\psi_{0}\rangle$ in Eq. (5).

The simplest type of action on a field involves the switching on of a source,
i.e., including a time dependent term $\int dx\hat{C}(x)J(x,t)$, where
$\hat{C}(x)$ is a local composite operator and $J(t,x)$ a source with support
in a compact spacetime region. Then, $|\psi_{0}\rangle={\cal T}\exp[-i\int
dtdx\hat{C}(x,t)J(x,t)]|0\rangle$. For $\hat{C}(t,x)=\hat{\phi}(t,x)$,
$|\psi_{0}\rangle$ coincides with the field coherent state $e^{-i\int
dtdxJ(x,t)\hat{\phi}(t,x)}|0\rangle$, modulo a phase. Hence, the single-
particle wave function is identified with

$\displaystyle\psi_{0}(p)=\frac{1}{\sqrt{2\epsilon_{p}}}\int
dtdxJ(t,x)e^{-ipx+i\epsilon_{p}t}.$ (33)

Note that states of this form cannot have support only on positive momenta,
hence, the probability density $P_{c}(t)$ is not normalized to unity444Any
state with only positive momenta, has a tail in position that extends up to
the location of the detector, and is therefore not appropriate for testing
causality.. We consider an source that vanishes outside the spatial interval
$[-a,0]$ and $[-T,0]$, where $a>0$ and $T>0$. We work in the regime $a<<T$,
i.e., a spatially localized interaction of long duration. For concreteness, we
take

$\displaystyle J(t,x)=\left\\{\begin{array}[]{cc}\delta(x)\sin(\pi
t/T),&t\in[-T,0]\\\ 0,&\mbox{otherwise}\end{array}\right..$ (36)

Then,

$\displaystyle\psi_{0}(p)=-\frac{\pi
T(1+e^{-i\epsilon_{p}T})}{\pi^{2}-\epsilon_{p}^{2}T^{2}}.$ (37)

Causality requires that the detection probability at $x=L$ vanishes for all
$t<L-T$, so that the detector is only influenced by events that happen after
we started our intervention. This is not the case. Fig. 3 shows that
$P_{c}(t)$ starts becoming appreciably larger than zero slightly before
$t=L-T$, and in fact, it is non-zero for all times $t>0$. The non-causal
component is more pronounced in the near field regime, i.e., small $L$.

Figure 3:  The probability density $P_{c}(t)$ for maximal localization as a
function of $\tau:=t/T$, for $L/T=10$ and $mT=10^{3}$. The non-causal behavior
is manifested in the jump of the probability density prior to $t=L-T$.

This result is remarkable, but, given past results on apparent causality
violation in localized relativistic systems, not unexpected. The specific
context of this apparent causality violation is rather different. Past results
focus on the evolution of spatially localized quantum states at a single
moment of time. Here, we considered a time-extended external localized
intervention. Here, the problem is not an artefact of using idealizations from
non-relativistic theory in describing a fundamentally relativistic system. It
is also not an artefact of perturbation theory. The conditional probability
density $P_{c}(t)$ that manifests the causality violating terms is of order
$\lambda^{0}$ with respect to the field-apparatus coupling $\lambda$, higher
perturbative corrections are negligible for sufficiently weak coupling.

This result does not indicate a fundamental failure of causality, rather it
points at the limitations of von Neumann’s description of measurements, i.e.,
the description of measurements in terms of an interaction that is switched
on. If we take the quantum formalism literally, the probability density (3)
does not describe a measurement that takes place at time $\bar{t}$; $\bar{t}$
specifies the moment when the field-apparatus is on. Rather, Eq. (3) refers to
a measurement that takes place after the interaction has been switched-off.
This distinction may be unimportant when the interaction time is very small.
In this formulation, there is no lower limit on the width of the switching
function, we can consider a delta function that is proportional to
$\delta(t-\bar{t})$. If we take the duration of the interaction larger than
the distance between source and detector, no causality problem arises.
However, in a measurement theory, the switching function is supposed to model
a defining feature of the apparatus, it cannot change arbitrarily if we decide
to place the source at a different distance.

In S-matrix theory the coupling is assumed to be switched on and off
adiabatically. This means that the effective duration of the interaction is
arbitrarily large, and measurement takes place effectively at
$t\rightarrow\infty$. In this description, transient terms disappear. However,
this is not a solution of the problem, merely a sidestep. In any particle
detector, particles are recorded at specific moments of time, i.e., particle
detection is a temporally localized process. An adiabatic switching of the
interaction fails to localize the measurement event in time. Hence, unless we
accept causality violations, we must admit that measurement models that rely
on switching interactions cannot account for a fundamental observed feature of
the particle detection process, namely, that it is localized in time.

One possible way of restoring causality is by showing that the causality
violating terms are impossible to measure, as a matter of principle. This may
be possible, because QFT measurements are invariably characterized by false
alarms, i.e., spurious detection events. Such events essentially define an
irreducible background noise that persists even if the field is in the vacuum
state. A detection signal exists only if it leads to a significant spike of
recorded events over the noise background. This means that the super-luminal
transient terms may be ignored if they contribute to probability less than the
noise term due to vacuum.

In the present model, this noise is expressed by the constant background
$P_{0}$ of Eq. (13). This term is quite strong, in fact, it drowns a large
part of the signal in the ultra-relativistic regime, and not only the
transient terms. However, the strength of this term is an artefact of coupling
switching. A realistic model for the apparatus must take into account that the
initial state of the apparatus is correlated with the field vacuum, i.e., it
is a dressed state. The noise $P_{0}$ incorporates also the equilibration of
the apparatus with the field vacuum when the coupling is switched on. This
equilibration is a spurious process. In the full description, the initial
state of the apparatus is already equilibrated with the field. This implies
that the restoration of causality through a signal/noise analysis requires a
different approach to measurements. In Sec. 3.1., we argue that such an
analysis is feasible in the QTP approach.

###  3.2 Retarded propagator versus Feynman propagator

For the initial state (33), the detection probability becomes

$\displaystyle P(t,x)=P_{0}+\int
dsdyR(s,y)\left[2\mbox{Re}\left[(\Delta_{F}J)(t-\frac{s}{2},x-\frac{1}{2}y)(\Delta_{F}J)(t+\frac{1}{2}s,x+\frac{1}{2}y)\right]\right.$
$\displaystyle\left.+(\Delta_{F}J)(t-\frac{s}{2},x-\frac{1}{2}y)(\Delta_{F}J)^{*}(t-\frac{s}{2},x-\frac{1}{2}y)\right].$
(38)

Here $\Delta_{F}J(t,x)$ stands for $\int
dt^{\prime}dx^{\prime}\Delta_{F}(t-t^{\prime},x-x^{\prime})J(t^{\prime},x^{\prime})$,
where

$\displaystyle\Delta_{F}(t,x)=\int_{-\infty}^{\infty}\frac{dp}{4\pi\epsilon_{p}}e^{ipx-i\epsilon_{p}|t|},$
(39)

is the Feynman propagator. The reason for the non-causal terms in the
probability assignment is that the Feynman propagator does not vanish outside
the light-cone.

The causal solutions to the Klein-Gordon equation with a source,
$\phi-m^{2}\phi=J$ are retarded solutions, i.e., they are generated from $J$
through the retarded propagator, $\phi(t,x)=\int
dx^{\prime}\Delta_{ret}(t^{\prime},x^{\prime})J(t,x)$. The retarded propagator
$\Delta_{ret}$ is simply the imaginary part of the Feynman propagator

$\displaystyle\Delta_{ret}(t,x)=\int_{-\infty}^{\infty}\frac{dp}{4\pi\epsilon_{p}}e^{ipx}\sin(\epsilon_{p}|t|),$
(40)

and it vanishes outside the light-cone, i.e., for $|x|>|t|$.

In classical field theory, the restriction to retarded solutions is
implemented as a boundary condition. This condition conflicts the time-
reversal invariance of the evolution equation. Fundamentally, it is justified
either cosmologically, or from thermodynamic time asymmetry —see Chap. 2 of
[24].

Certainly, if we substitute $\Delta_{F}$ with $\Delta_{ret}$ in Eq. (38), the
problem of causality is resolved555Analogous substitutions have been proposed
in the context of photo-detection theory, in order to cure Glauber’s model
from the transient super-luminal terms [25, 26].. This amounts to a
substitution of $e^{-i(\epsilon_{p}-\epsilon_{p^{\prime}})t}$ with
$\sin[(\epsilon_{p}-\epsilon_{p^{\prime}})|t|]$ in Eq. (12). We cannot
normalize by integrating $t$ along the whole real axis, because the error is
exactly equal with that from the substitution of $\Delta_{F}$ with
$\Delta_{ret}$. (They do not cancel.) )Still, for a large class of initial
states, the correction in the total probability is small, so it is meaningful
to define the post-selected density matrices (14) and (16), and write the
causally corrected version of Eq. (18),

$\displaystyle
P_{c}(t,L)=\left|\int_{0}^{\infty}dp\psi_{ps}(p)\sqrt{v_{p}}e^{ipL-i\epsilon_{p}t}\right|^{2}-\left|\int_{0}^{\infty}dp\psi_{ps}(p)\sqrt{v_{p}}e^{ipL+i\epsilon_{p}t}\right|^{2},$
(41)

modulo the fact that the probability density (41) is not normalized to unity.

However, the substitution of $\Delta_{F}$ with $\Delta_{ret}$, is equivalent
to the substitution of the evolution operator $e^{-i\hat{H}_{\phi}t}$ for the
field with the operator
$\hat{K}_{t}=e^{-i\hat{H}_{\phi}t}-e^{i\hat{H}_{\phi}t}$ which is non-unitary.
Not only is such a substitution completely ad hoc, it also contradicts the
basic rules of quantum theory. Furthermore, $\hat{K}_{t}$ makes no sense as
the evolution of an initial state, because $\hat{K}_{0}=0$. As long as we use
a measurement model in which the instant of detection is determined by the
Hamiltonian, the sources are Feynman-propagated, and probabilities cannot be
cured from super-luminal transients. A resolution requires at the very least,
an incorporation of time-irreversibility in the description of measurement, in
order to obtain some version of causal propagation.

###  3.3 Causal propagation versus restricted propagation

Next, we will describe the QTP approach to quantum field measurements [16, 12,
19], in which the space-time coordinates are genuine densities and the
interaction between system and apparatus is always present. The key point here
is that the propagator for an initial source is not the Feynman propagator. We
will explain the benefits of this construction, and its potential for
resolving the spurious superluminality problem from first principles. The
resolution would require an explicit modeling of the macroscopic apparatus in
a way consistent with thermodynamic irreversibility. This is taken up in a
different publication.

In QTP, we describe measurement events as transition between two complementary
subspaces ${\cal H}_{+}$ and ${\cal H}_{-}$ of a Hilbert space ${\cal H}={\cal
H}_{+}\oplus{\cal H}_{-}$. The subspace ${\cal H}_{+}$ describes the states of
the system that are compatible with the realization of the event under
consideration is realized. If the event is a detection of a microscopic
particle by a measuring apparatus, then the subspace ${\cal H}_{+}$
corresponds to all states of the apparatus compatible with a macroscopic
detection record. We denote the projection operator onto ${\cal H}_{+}$ as
$\hat{P}$ and the projector onto ${\cal H}_{-}$ as $\hat{Q}:=1-\hat{P}$.

Transitions that are correlated with the emergence of a macroscopic record of
observation are logically irreversible. Once they occur, and a measurement
outcome has been recorded, further time evolution of the system does not
affect our knowledge.

After the transition has occurred, a pointer variable $\lambda$ of the
measurement apparatus takes a definite value. Let $\hat{\Pi}(\lambda)$ be
positive operators that correspond to the different values of $\lambda$. For
example, when considering transitions associated with particle detection, the
projectors $\hat{\Pi}(\lambda)$ may be correlated to the position, or to the
momentum of the microscopic particle. Since $\lambda$ has a value only under
the assumption that a detection event has occurred, the alternatives
$\hat{\Pi}(\lambda)$ span the subspace ${\cal H}_{+}$ and not the full Hilbert
space ${\cal H}$. Hence, $\sum_{\lambda}\hat{\Pi}(\lambda)=\hat{P}$.

Assuming that the systems is prepared in the state $|\Psi_{0}\rangle\in{\cal
H}_{-}$, and that the Hamiltonian is $\hat{H}$, the probability distribution
$P(\lambda,t)$ that the transition took place at time $t$ and an outcome
$\lambda$ was recorded is

$\displaystyle P(\lambda,t)=\int
d\tau\langle\Psi_{0}|\hat{C}^{\dagger}(\lambda,t-\frac{\tau}{2})\hat{C}(\lambda,t+\frac{\tau}{2})|\Psi_{0}\rangle,$
(42)

where the class operator

$\displaystyle\hat{C}(\lambda,t):=e^{i\hat{H}t}\sqrt{\hat{\Pi}}(\lambda)\hat{H}\hat{S}_{t}$
(43)

is defined in terms of the restricted evolution operator $\hat{S}_{t}$ which
is the continuous limit of the product
$\hat{Q}e^{-i\hat{H}t/N}\hat{Q}e^{-i\hat{H}t/N}\hat{Q}\ldots\hat{Q}e^{-i\hat{H}t/N}\hat{Q}$
as the number of steps $N$ goes to infinity. By the Mishra-Sudarshan theorem
[27], $\hat{S}_{t}$ is a unitary operator on ${\cal H}_{-}$ in this limit.
However, one may modify the definition, for example, by regularizing the
product so that there is an effective minimal time $\tau$. Then, $\hat{S}_{t}$
may be non-unitary.

It is important to emphasize Eq. (42) is a genuine probability density with
respect to both $\lambda$ and $t$. The derivation of Eq. (42) requires the
assumption of decoherence in the measuring apparatus.

We note that $P(\lambda,t)$ vanishes if for $[\hat{P},\hat{H}]=0$. We consider
a Hamiltonian $\hat{H}=\hat{H_{0}}+\hat{H_{I}}$, where
$[\hat{H}_{0},\hat{P}]=0$, and $H_{I}$ a perturbing interaction. Since all
transitions are due to $\hat{H}_{I}$, to leading order in the perturbation,

$\displaystyle\hat{C}(\lambda,t)=e^{i\hat{H}_{0}t}\sqrt{\hat{\Pi}}(\lambda)\hat{H}_{I}e^{-i\hat{H}_{0}t},$
(44)

When we use Eq. (44) for the measurement model of Sec. 2.1, and we take
$\lambda$ to coincide with the position $x$ of a record, we obtain Eq. (12).
The derivation is conceptually more rigorous here, because Eq. (42) is a
genuine probability density to all orders of perturbation theory. Furthermore,
the interaction is always present, no switching is necessary in order to
specify the instant of detection.

There are two important benefits in the QTP approach to measurements. We can
employ use a dressed state for the apparatus, i.e., take a non factorized
initial state $|\Psi_{0}\rangle$. In perturbation theory,
$|\Psi_{0}\rangle=|\Omega,\psi_{0}\rangle+|\chi\rangle$, where $|\chi\rangle$
is a correction to leading order in the coupling. A first-principles
identification of $|\chi\rangle$ will allow us to identify the vacuum noise
for the measurement. Thus, we will be able to demonstrate whether the super-
luminal transients are strongly dominated by noise, and hence, are
unobservable.

It is highly unlikely that the non-factorized initial state will make the
transients vanish. However, when taking an initial state Still, the lowest
order contribution suffers from the super-luminal transients. However,
fundamentally the initial state is propagated by a restricted propagator
$\hat{S}_{t}$. For states derived by a localized source, this means that the
source $J(x,t)$ is Feynman-propagated only at the tree level; higher order
terms incorporate contributions from the apparatus through the projector
$\hat{Q}$. We conjecture that these contributions will introduce desired time-
asymmetry in the evolution of $J(x,t)$, that will lead to the suppression of
the transient terms.

##  4 Conclusions

We presented our motivation and our results in the introduction. Here, we want
to iterate the importance of a consistent formulation of a QFT measurement
theory, in order to provide a first-principles construction of relativistic
quantum information [18, 19]. Such formulations are also important for the
consistent description of proposed quantum optics experiments in space [28,
29]. These are characterized by very long baselines, and they can explore
fundamental issues of relativistic locality and causality at a fundamental
level.

The issues analyzed in this paper are challenges to any QFT measurement
theory. Any such theory must provide specific predictions for time-of-arrival
experiments, because in most of our measurements, the location of the detector
is fixed and detection time is a random variable. It must also be consistent
with locality and causality, meaning that it must provide an explicit recipe
for constructing initial states, so that superluminal transients are either
removed or they are drowned by vacuum noise. The first challenge is met by the
QTP approach, the second is work in progress.

## Acknowledgements

CA acknowledges support by grant JSF-19-07-0001 from the Julian Schwinger
Foundation; MEP acknowledges a studentship grant by the Moraitis School.

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