# High-Harmonic Spectroscopy of Coherent Lattice Dynamics in Graphene

Navdeep Rana  Department of Physics, Indian Institute of Technology Bombay,
Powai, Mumbai 400076 India  M. S. Mrudul  Department of Physics, Indian
Institute of Technology Bombay, Powai, Mumbai 400076 India  Daniil Kartashov
Institute of Optics and Quantum Electronics, Friedrich-Schiller University
Jena, Max-Wien-Platz 1, 07743 Jena, Germany  Misha Ivanov  Max-Born Straße 2A,
D-12489 Berlin, Germany  Gopal Dixit<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076 India

###### Abstract

High-harmonic spectroscopy of solids is a powerful tool, which provides access
to both electronic structure and ultrafast electronic response of solids, from
their band structure and density of states, to phase transitions, including
the emergence of the topological edge states, to the PetaHertz electronic
response. However, in spite of these successes, high harmonic spectroscopy has
hardly been applied to analyse the role of coherent femtosecond lattice
vibrations in the attosecond electronic response. Here we study coherent
phonon excitations in monolayer graphene to show how high-harmonic
spectroscopy can be used to detect the influence of coherent lattice dynamics,
particularly longitudinal and transverse optical phonon modes, on the
electronic response. Coherent excitation of the in-plane phonon modes results
in the appearance of sidebands in the spectrum of the emitted harmonic
radiation. We show that the spectral positions and the polarisation of the
sideband emission offer a sensitive probe of the dynamical symmetries
associated with the excited phonon modes. Our work brings the key advantage of
high harmonic spectroscopy – the combination of sub-femtosecond to tens of
femtoseconds temporal resolution – to the problem of probing phonon-driven
electronic response and its dependence on the dynamical symmetries in solids.

##  I Introduction

Strong-field driven high-harmonic generation (HHG) is a nonlinear frequency
up-conversion process, which emits radiation at integer multiples of the
incident laser frequency Ferray  _et al._ (1988). Taking advantage of major
technical advances in mid-infrared sources, the pioneering experiments Ghimire
_et al._ (2011) have extended HHG from gases to solids, stimulating intense
research into probing electron dynamics in solids on the natural timescale.
Today, high-harmonic spectroscopy has been employed to probe different static
and dynamic properties of solids, such as band dispersion Vampa  _et al._
(2015); Luu  _et al._ (2015); Lanin  _et al._ (2017); Mrudul  _et al._ (2019),
density of states Tancogne-Dejean  _et al._ (2017), band defects Mrudul  _et
al._ (2020); Pattanayak  _et al._ (2020), valley pseudospin Mrudul  _et al._
(2021); Jiménez-Galán  _et al._ (2020); Langer  _et al._ (2018); Mrudul and
Dixit (2021a), Bloch oscillations Schubert  _et al._ (2014), topology and
light-driven phase transitions, including strongly correlated systems Bauer
and Hansen (2018); Silva  _et al._ (2018); Bai  _et al._ (2020); Imai  _et
al._ (2020); Borsch  _et al._ (2020); Baykusheva  _et al._ (2021); Bharti  _et
al._ (2022); Pattanayak  _et al._ (2022); Chacón  _et al._ (2020); Shao  _et
al._ (2022), and even combine attosecond temporal with pico-meter spatial
resolution of electron trajectories in lattices Lakhotia  _et al._ (2020).

Availability of mid-infrared light sources also enables coherent excitation of
a desired phonon mode by tuning the polarisation and frequency of the laser
pulse Mankowsky  _et al._ (2016). Yet, the analysis of the effect of coherent
lattice dynamics on high harmonic generation in solids appears lacking, apart
from a lone experiment Hollinger  _et al._ (2019). This situation stands in
stark contrast to molecular gases, where high-harmonic spectroscopy has been
extensively employed to probe nuclear motion in various molecules Patchkovskii
(2009); Wagner  _et al._ (2006); Le  _et al._ (2012); Baker  _et al._ (2006);
Lein (2005); Wörner  _et al._ (2011). Present work aims to fill this gap and
highlight some of the capabilities offered by high harmonic spectroscopy in
time-resolving the interplay of femtosecond lattice and attosecond electronic
motions. Such interplay is essential for many fundamental phenomena, including
thermal conductivity Niedziela  _et al._ (2019), optical reflectivity Dove
(1993); Katsuki  _et al._ (2013), structural phase transition Bansal  _et al._
(2020); Hase  _et al._ (2015), heat capacity Bansal  _et al._ (2018), and
optical properties Fultz (2010); Gambetta  _et al._ (2006).

Various spectroscopic methods have been developed to excite and probe phonons,
see e.g. Dhar  _et al._ (1994); Debnath  _et al._ (2021); Graf  _et al._
(2007); Virga  _et al._ (2019); Koivistoinen  _et al._ (2017); Rana  _et al._
(2021); Brown  _et al._ (2019); Flannigan (2018); Gierz  _et al._ (2015);
Moulet  _et al._ (2017); Géneaux  _et al._ (2020), but their temporal
resolution is limited by the length of the pulses used. Large coherent
bandwidth of high harmonic signals offers sub-laser-cycle temporal resolution
and the possibility to time-resolve the impact of lattice distortions on the
faster electronic response.

One difficulty in tracking lattice vibrations via highly nonlinear optical
response stems from their small amplitude. If the corresponding changes in
both the band structure and couplings are similarly small, the high-harmonic
response hardly changes. Yet, large distortions are not needed if the excited
phonon mode dynamically changes the symmetry of the unit cell. Here we show
how coherent phonon dynamics and the associated changes in the lattice
symmetry are encoded in the electronic response and the harmonic signal, and
how the sub-cycle temporal resolution inherent in the harmonic signal can be
used to track the interplay of electronic and lattice dynamics.

Figure 1: Hexagonal honeycomb structure of graphene and associated in-plane
phonon modes. (a) Real-space structure of graphene. (b) Brillouin zone in
momentum-space with $\mathsf{\Gamma,M,}$ and $\mathsf{K}$ as the high symmetry
points. (c) and (d) are the sketches of atomic vibrations associated with the
degenerate E2g phonon modes in real-space. Here, modes are labelled as (c) in-
plane longitudinal optical (iLO) phonon mode and (d) in-plane transverse
optical (iTO) phonon mode, respectively.

We analyse monolayer graphene, which belongs to $\textbf{D}_{6h}$ point group
symmetry, see Fig. 1(a). It exhibits six phonon branches: three optical and
three acoustic. Here we focus on the former. Out of the three optical phonon
modes, one is out-of-plane, the two others are in-plane modes. We will
consider only the in-plane modes. The lattice vibrations corresponding to the
in-plane Longitudinal Optical (iLO), and the in-plane Transverse Optical (iTO)
E2g modes are shown in Fig. 1(c) and 1(d), respectively. The two modes are
degenerate at the $\mathsf{\Gamma}$-point, with the phonon frequency equal to
194 meV (oscillation period $\sim$ 21 femtoseconds Kim  _et al._ (2013)); both
are Raman active and can be excited with a resonant pulse pair or impulsively
by a short pulse with bandwidth covering 194 meV. Moreover, it is possible to
selectively excite either iLO or iTO coherent phonon mode by tuning the
polarisation of the pump pulse either along $\mathsf{\Gamma-K}$ or
$\mathsf{\Gamma-M}$ direction, respectively.

Coherent lattice dynamics should in general introduce periodic modulations of
the system parameters and thus of its high-harmonic response. In the frequency
domain, such modulations add sidebands to the main peaks in the harmonic
spectrum. We shall see that their position and polarisation encode the
information about the frequency and the symmetry of the excited phonon mode,
respectively.

##  II Theoretical Method

Carbon atoms are arranged at the corners of a hexagon in the honeycomb lattice
of the graphene. The unit-cell of graphene has a two-atom basis, usually
denoted as A and B atoms. The corresponding Brillouin zone in momentum space
is shown in Fig. 1(b), where $\mathsf{\Gamma}$, $\mathsf{M}$, and $\mathsf{K}$
are the high-symmetry points. In our convention, the zigzag and armchair
directions of graphene are along $\mathsf{X}$-axis
($\mathsf{\Gamma}-\mathsf{K}$ direction) and $\mathsf{Y}$-axis
($\mathsf{\Gamma}-\mathsf{M}$ direction), respectively.

The electronic ground-state of the graphene is described by the nearest-
neighbour tight-binding approximation and the corresponding Hamiltonian is
written as

$\mathcal{\hat{H}}_{\textbf{k}}=-\gamma_{0}\sum_{i\in
nn}~{}e^{i\textbf{k}\cdot\textbf{d}_{i}}\hat{a}_{\textbf{k}}^{\dagger}\hat{b}_{\textbf{k}}+\textrm{H.
c.}$ (1)

Here, the summation is over the nearest neighbour atoms. $\gamma_{0}$ is the
nearest-neighbour hopping energy, which is chosen to be 2.7 eV.
$\textbf{d}_{i}$ is the separation vector between an atom with its nearest
neighbour, such that $|\textbf{d}_{i}|=a$ = 1.42 Å is the inter-atomic
distance, for a lattice parameter $a_{0}$ of 2.46 Å.
$\hat{a}_{k}^{\dagger}(\hat{b}_{k})$ is creation (annihilation) operator for
atom A (B) in the unit cell. The low-energy band-structure of graphene is
obtained by solving Eq. (1) and has zero band-gap and exhibits linear
dispersion at $\mathsf{K}$-points in the Brillouin zone.

We treat lattice dynamics classically, and assume that atoms perform harmonic
oscillations for short displacements from their equilibrium positions. The
displacement vector for a particular phonon mode is expressed as

$\textbf{q}(t)=\textrm{q}_{0}~{}\hat{\textbf{e}}~{}\rm{Re}\left(e^{i\omega_{ph}t}\right).$
(2)

Here, $\textrm{q}_{0}$ is the maximum displacement of an atom from its
equilibrium position, $\omega_{\textrm{ph}}=$ 194 meV is the energy of the E2g
phonon mode , and $\hat{\textbf{e}}$ is the normalised eigenvector for a
particular phonon mode. From Figs. 1(c) and 1(d), it is clear that
$\hat{\textbf{e}}_{\textsf{iLO}}$ = $[1,0,-1,0]/\sqrt{2}$ and
$\hat{\textbf{e}}_{\textsf{iTO}}$ = $[0,1,0,-1]/\sqrt{2}$, in which the first
(last) two elements are components of A (B) atom.

Due to coherent phonon excitations, lattice dynamics causes temporal
variations in the relative distance between atoms ($\textbf{d}_{i}$). In this
case, the corresponding time-dependent Hamiltonian within the tight-binding
approximation can be written as Mohanty and Heller (2019); Rodriguez-Vega  _et
al._ (2021); Wang and Fischer (2014)

$\mathcal{\hat{H}}_{\textbf{k}}(t)=-\gamma(t)\sum_{i\in
nn}e^{i\textbf{k}\cdot\textbf{d}_{i}(t)}\hat{a}_{\textbf{k}}^{\dagger}\hat{b}_{\textbf{k}}+\textrm{H.
c.}$ (3)

Here, the hopping energy is modelled as an exponentially decaying function of
the relative displacement between nearest-neighbour atoms as $\gamma(t)$ =
$\gamma_{0}~{}e^{-(|\textbf{d}_{i}(t)|-a)/\delta}$, in which $\delta$ is the
width of the decay function chosen to be 0.184$a_{0}$ Moon and Koshino (2013).

The interaction among laser, electrons and coherently excited phonon mode in
graphene is modelled by solving following equations of the single-particle
density matrix. By updating the modified Hamiltonian as a result of the
lattice dynamics, semiconductor Bloch equations in co-moving frame
$|n,\textbf{k}+\textbf{A}(t)\rangle$, is extended and equations of motion read
as

$\displaystyle\frac{d}{dt}\rho_{vv}^{\textbf{k}}$
$\displaystyle=i\textbf{E}(t)\cdot\textbf{d}_{vc}(\textbf{k}_{t},t)\rho_{cv}^{\textbf{k}}+\textrm{c.c.}$
(4a) $\displaystyle\frac{d}{dt}\rho_{cv}^{\textbf{k}}$
$\displaystyle=\left[-i\varepsilon_{cv}(\textbf{k}_{t},t)-\frac{1}{T_{2}}\right]\rho_{cv}^{\textbf{k}}+i\textbf{E}(t)\cdot\textbf{d}_{cv}(\textbf{k}_{t},t)\left[\rho_{vv}^{\textbf{k}}-\rho_{cc}^{\textbf{k}}\right].$
(4b)

Here, $\textbf{E}(t)$ and $\textbf{A}(t)$ are, respectively, the electric
field and the vector potential corresponding to the laser field, which are
related as $\textbf{E}(t)$ = $-d\textbf{A}(t)/dt$, and $\textbf{k}_{t}$ is the
shorthand notation for $\textbf{k}+\textbf{A}(t)$.
$\varepsilon_{cv}(\textbf{k})$ and $\textbf{d}_{cv}(\textbf{k})$ are,
respectively, the band-gap energy and dipole matrix elements between valence
and conduction bands at k. $\textbf{d}_{cv}(\textbf{k})$ is defined as
$\textbf{d}_{cv}(\textbf{k})=i\langle
c,\textbf{k}|\nabla_{\textbf{k}}|v,\textbf{k}\rangle$. Also,
$\rho_{cc}^{\textbf{k}}(t)=1-\rho_{vv}^{\textbf{k}}(t)$, and
$\rho^{\textbf{k}}_{vc}(t)=\rho^{\textbf{k}*}_{cv}(t)$.

A phenomenological term to take care of the interband decoherence is added
with a constant dephasing time $T_{2}$. We calculate the matrix elements at
each time-step during temporal evolution of the coherently excited phonon
mode, which results in the additional time-dependence in the matrix elements.
As long as the maximum displacement of the atoms are small and the time-step
is too small compared to the phonon time-period, the matrix elements at
consecutive time-steps are smoothly updated.

We solve the coupled differential equations described in Eq. (4) using the
fourth-order Runge-Kutta method with a time-step of 0.01 fs. We sampled the
Brillouin zone with 251$\times$251 grid. The current at any k point in the
Brillouin zone is defined as

$\textbf{J}(\textbf{k},t)=\sum_{m,n\in\\{c,v\\}}\rho_{mn}^{\textbf{k}}(t)\textbf{p}_{nm}(\textbf{k}_{t},t).$
(5)

Here, $\textbf{p}_{nm}$ are the momentum matrix-elements defined as
$\textbf{p}_{nm}(\textbf{k})=\langle
n,\textbf{k}|\nabla_{\textbf{{k}}}\mathcal{\hat{H}}_{\textbf{k}}|m,\textbf{k}\rangle$.
The total current, J(t) can be calculated by integrating
$\textbf{J}(\textbf{k},t)$ over the entire Brillouin zone.

The high-harmonic spectrum is simulated as

$\mathcal{I}(\omega)=\left|\mathcal{FT}\left(\dfrac{\rm{d}}{{\rm
d}t}\textbf{J}(t)\right)\right|^{2}.$ (6)

Here, $\mathcal{FT}$ stands for the Fourier transform.

High-order harmonics are generated from monolayer graphene, with or without
coherent lattice dynamics, using a linearly polarised pulse with a wavelength
of 2.0 $\mu$m and peak intensity of 1$\times 10^{11}$ W/cm2. The pulse is 100
fs long and has a sin-squared envelope. The laser parameters used in this work
are below the damage threshold of graphene Currie  _et al._ (2011). Similar
laser parameters have been used to investigate electron dynamics in graphene
via intense laser pulse Heide  _et al._ (2018); Higuchi  _et al._ (2017);
Yoshikawa  _et al._ (2017). The value of the dephasing time $T_{2}=$ 10 fs is
used throughout in this work Mrudul and Dixit (2021b). The observations we
made here are consistent for other values of $T_{2}$ in the range 5 - 30 fs.
Both in-plane E2g phonon modes are considered here. Results presented in this
work correspond to a maximum 0.03a0 displacement of atoms from their
equilibrium positions during coherent lattice dynamics. However, our findings
remain valid for displacements ranging from 0.01a0 to 0.05a0 with respect to
the equilibrium positions.

##  III Results and Discussion

High-harmonic spectra for monolayer graphene, with and without coherent
lattice dynamics, are presented in Fig. 2. The spectrum corresponding to the
graphene, without lattice dynamics, is shown by grey shaded area as a
reference. Owing to the inversion symmetry of the graphene, the reference
spectrum in grey colour exhibits only odd harmonics (consistent with earlier
reports, e.g. Refs. Mrudul and Dixit (2021b); Yoshikawa  _et al._ (2017); Al-
Naib  _et al._ (2014).)

We assume that coherent phonon dynamics is excited prior to a high harmonic
probe. When one of the E2g phonon modes in graphene is coherently excited, the
harmonic spectra display sidebands along with the main odd harmonic peaks as
reflected from Fig. 2. The energy difference between the adjacent sidebands
matches the phonon energy ($\omega_{\textrm{ph}}$). The sideband intensity is
sensitive to the phonon amplitude but is clearly visible already for
amplitudes above 0.01 of the lattice constant. Here we present the case of the
amplitude equal to 0.03 of the lattice constant.

Figure 2: High-harmonic spectra of monolayer graphene with and without
coherent lattice dynamics. (a) and (c) High-harmonic spectra corresponding to
the coherent iLO E2g phonon mode and the probe harmonic pulse is polarised
along $\mathsf{\Gamma K}$ and $\mathsf{\Gamma M}$ directions, respectively.
(b) and (d) Same as (a) and (c) except iTO E2g phonon mode is coherently
excited. In all the cases, sidebands corresponding to the first harmonic are
marked at frequencies ($\omega_{0}\pm m\omega_{\textrm{ph}}$), where
$\omega_{0}$ is the frequency of the harmonic generating probe pulse, and
$\omega_{\textrm{ph}}$ is phonon frequency. The harmonics with grey shaded
area are the reference spectra and represent the spectra of graphene without
phonon excitation. The unit cell of the graphene with the corresponding phonon
eigenvector and polarisation of the harmonic generating probe pulse are shown
in the respective insets. Red (blue) colour corresponds the polarisation of
emitted radiation parallel (perpendicular) to the polarisation of harmonic
generating probe pulse.

As E2g phonon modes preserve inversion centre, only odd harmonics are
generated. When the coherent iLO mode and the probe harmonic pulse (along
$\mathsf{\Gamma-K}$) are in the same direction, the even-oder sidebands are
polarised along $\mathsf{\Gamma-K}$ (red colour), whereas the odd-order
sidebands are polarised perpendicular to $\mathsf{\Gamma-K}$ (blue colour),
i.e., along $\mathsf{\Gamma-M}$ direction [see Fig. 2(a)]. When the
polarisation of the probe pulse changes from $\mathsf{\Gamma-K}$ to
$\mathsf{\Gamma-M}$ direction, the polarisation of the sidebands remains the
same with respect to the laser polarisation. In this case, the even-oder
sidebands are polarised along $\mathsf{\Gamma-M}$ (blue colour), whereas odd-
order sidebands are polarised along $\mathsf{\Gamma-K}$ (red colour) [see Fig.
2(c)]. In both the cases, the main harmonic peaks are always polarised along
the direction of the probe pulse.

The situation is simpler in the case of coherent iTO mode excitation. Both the
main harmonic peaks and the sidebands are polarised along the direction of the
probe pulse [see Figs. 2(b) and (d)]. Thus, we see that the polarisation of
the sidebands yields information about the symmetries of the excited phonon
modes.

We now investigate how the dynamical changes in symmetries differ from similar
static variation in the high-harmonic spectra. Consider the static case with
the maximum displacement of atoms, along a particular phonon mode direction,
3$\%$ of the lattice parameter from their equilibrium positions. Figure 3
compares high-harmonic spectra for the statically-deformed and undeformed
graphene (grey color). The probe polarisation is along $\mathsf{\Gamma-K}$ and
$\mathsf{\Gamma-M}$ directions in the top and bottom panels of Fig. 3,
respectively.

When the graphene is deformed along the iLO phonon mode, odd harmonics are
generated along parallel and perpendicular directions with respect to the
laser polarisation as shown in Figs. 3(a) and 3(c), respectively. However,
only odd harmonics, parallel to the laser polarisation, are generated when
graphene is deformed in accordance with iTO-mode [see Figs. 3(b) and (d)].

Figure 3: High-harmonic spectra of monolayer deformed graphene. (a) and (c)
When the atoms in graphene are maximally displaced, from their equilibrium
position, along iLO phonon mode. (b) and (d) Similar to (a) and (b) but atoms
are displaced along iTO phonon mode. The harmonic spectrum of undeformed
graphene is shown in grey shaded area for reference. The unit cell of the
deformed graphene lattice and the polarisation of the harmonic generating
probe pulse are shown in the respective insets. Red (blue) colour corresponds
the polarisation of emitted radiation parallel (perpendicular) to the
polarisation of harmonic generating probe pulse.

The emergence of parallel and the perpendicular components in the first case
and the parallel component in second case can be explained as follows: the
monolayer graphene has $\sigma_{x}$ and $\sigma_{y}$ symmetry planes, in
addition to the inversion centre. When the polarisation of the probe laser is
along the high symmetry direction ($\mathsf{\Gamma-K}$ or
$\mathsf{\Gamma-M}$), there is no perpendicular component of the current.
However, if the polarisation of the probe pulse is along any other than these
high-symmetry directions, symmetry constraints allow the generation of odd
harmonics perpendicular to the direction of the laser polarisation. Recently,
same symmetry concept is employed in twisted bilayer graphene to correlate the
twist angle with its high-harmonic spectrum Du  _et al._ (2021). It is
straightforward to see that the distortion due to the iLO phonon mode breaks
the symmetries of the reflection planes in monolayer graphene. The absence of
the reflection symmetry planes along X and Y directions guarantees the
generation of harmonics in both $\mathsf{\Gamma-K}$ and $\mathsf{\Gamma-M}$
directions as shown in Figs. 3(a) and (c). On the other hand, iTO phonon mode
preserves both the symmetry planes and as a results harmonics along the laser
polarisation are only allowed [Figs. 3(b) and (d)].

In short, the presence or absence of the perpendicular current is a result of
the transient breaking of the symmetry planes, which can be correlated to the
results in Fig. 2. To understand the mechanism behind the sideband generations
and associated polarisation properties during a coherent lattice dynamics, we
need to consider the changes in the symmetries dynamically during the probe
pulse.

To understand the symmetry constraint on the polarisation of sidebands, let us
consider dynamical symmetries (DSs) of the system, accounting for the coherent
lattice dynamics and the probe pulse. We apply the Floquet formalism to a
periodically driven system, represented by the Hamiltonian described by Eq.
(3), which satisfies $\hat{\mathcal{H}}_{\textbf{k}}$(t) =
$\hat{\mathcal{H}}_{\textbf{k}}(t+\tau_{\textrm{ph}})$, where
$\tau_{\textrm{ph}}$ is the time-period corresponds to $\omega_{\textrm{ph}}$.
The Hamiltonian obeys the time-dependent Schrödinger equation and its solution
is obtained in the basis of the Floquet states as $|\psi_{n}^{\rm
F}(t)\rangle=e^{-i\epsilon_{n}^{\rm F}t}|\phi_{n}^{\rm F}(t)\rangle$. Here,
$\epsilon_{n}^{\rm F}$ is the quasi-energy corresponds to the $n^{\rm th}$
Floquet state, and $|\phi_{n}^{\rm F}(t)\rangle$ is the time-periodic part of
the wave function, such that $|\phi_{n}^{\rm
F}(t+\tau_{\textrm{ph}})\rangle=|\phi_{n}^{\rm F}(t)\rangle$. The DSs in a
Floquet system are the combined spatio-temporal symmetries, which provide
different kinds of selection rules as discussed in Ref. Neufeld  _et al._
(2019); Nagai  _et al._ (2020).

In the presence of the probe pulse, the laser-graphene interaction within
tight-binding approximation can be modelled with the Peierls substitution as
$\hat{\mathcal{H}}_{\textbf{k}}(t)\rightarrow\hat{\mathcal{H}}_{\textbf{k}+\textbf{A}(t)}(t)$.
For the sake of simplicity, we employ a perturbative approach to understand
the polarisation of the sidebands as the strength of the sidebands is much
weaker in comparison to the main harmonic peaks. Let us expand
$\hat{\mathcal{H}}_{\textbf{k}+\textbf{A}(t)}(t)$ in terms of
$i\textbf{A}(t)\cdot\textbf{d}_{i}(t)$ as

$\hat{\mathcal{H}}_{\textbf{k}+\textbf{A}(t)}(t)\approx\hat{\mathcal{H}}_{\textbf{k}}(t)+\textbf{A}(t)\cdot\nabla_{\textbf{k}}\hat{\mathcal{H}}_{\textbf{k}}(t).$
(7)

The second term in the above equation can be treated as perturbation as
$\hat{\mathcal{H}}_{\textbf{k}}^{\prime}(t)=\textbf{A}(t)\cdot\hat{\textbf{J}}(t)$
with
$\hat{\textbf{J}}=\nabla_{\textbf{A}(t)}\hat{\mathcal{H}}_{\textbf{k}+\textbf{A}(t)}$
is the current operator in the Bloch basis. In Eq. (7), higher-order terms are
neglected.

By following Ref. Nagai  _et al._ (2020) and assuming the electron initially
is in the Floquet state $|\phi_{i}^{\rm F}\rangle$, we can solve time-
dependent Schrödinger equation within first-order perturbation theory and the
$\mu^{\rm th}$-component of the current can be written as

$\begin{split}J_{\mu}(t)&=\langle\phi_{i}^{\rm
F}(t)|\hat{J}_{\mu}(t)|\phi_{i}^{\rm F}(t)\rangle\\\ &-\sum_{e\neq
i}\int_{-\infty}^{t}idt^{\prime}e^{-i\omega_{ei}(t-t^{\prime})}\chi^{\rm
F}_{\mu\nu}(t,t^{\prime})A_{\nu}(t^{\prime})+\textrm{c.c.}\end{split}$ (8)

Here, $\chi^{\rm F}_{\mu\nu}(t,t^{\prime})=\langle\phi_{i}^{\rm
F}(t)|\hat{J}_{\mu}(t)|\phi_{e}^{\rm F}(t)\rangle\langle\phi_{e}^{\rm
F}(t^{\prime})|\hat{J}_{\nu}(t^{\prime})|\phi_{i}^{\rm F}(t^{\prime})\rangle$.
From the above equation, it is apparent that the second term correlates to the
generations of the sidebands via Raman process.

The symmetry constraint for the $m^{\rm th}$-order sideband can be written as
$\hat{X}^{t}\textbf{E}_{s,m}(t)[\hat{X}^{t}\textbf{E}(t)]^{\dagger}$ =
$\textbf{E}_{s,m}\textbf{E}^{\dagger}(t)$, provided spatial symmetries of
$\hat{X}^{t}$ and probe-pulse are same Nagai  _et al._ (2020). Here,
$\textbf{E}_{s,m}(t)$ and $\textbf{E}(t)$ are, respectively, the electric
fields associated with $m^{\rm th}$-order sideband and the probe laser; and
$\hat{X}^{t}$ is the dynamical symmetry operation. The quantity
$\textbf{E}_{s,m}(t)\textbf{E}(t)^{\dagger}$ is denoted by
$\mathcal{R}_{m}(t)$ and known as Raman tensor Nagai  _et al._ (2020). Thus
the selection rules for the sidebands depend on the invariance of the Raman
tensor under operation with the DSs of the Floquet system.

There are two DSs corresponding to the coherent iLO phonon mode as shown in
Fig. 4. We define $\tau_{n}$ as the time translation of
$\tau_{\textrm{ph}}/n$, $\hat{C}_{n\mu}$ is the rotation of 2$\pi/n$ with
respect to $\mu$-axis, $\hat{\sigma}_{\mu}$ is the reflection with respect to
$\mu$-axis, and $\hat{\mathcal{T}}$ is the time-reversal operator. The
symmetry operations $\mathcal{D}_{1}=\hat{\sigma}_{x}\cdot\tau_{2}$ [see Fig.
4(a)], and $\mathcal{D}_{2}=\hat{\sigma_{x}}$ [see Fig. 4(b)] leaves the
system invariant.

Figure 4: Schematic representations of the dynamical symmetries of the Floquet
Hamiltonian (a) $\hat{\mathcal{D}}_{1}$ =
$\hat{\sigma_{x}}\cdot\hat{\tau}_{2}$ (b) $\hat{\mathcal{D}}_{2}$ =
$\hat{\sigma_{x}}$. The arrows show the displacements of the atom for a
particular phonon mode.

The selection rules for the sidebands and its polarisation directions are
obtained from the DSs as shown in Fig. 4 and requires a condition as
$\hat{\mathcal{D}}\mathcal{R}_{m}(t)=\mathcal{R}_{m}(t)$. We assume that the
temporal part of the $m^{\rm th}$-order sideband as $e^{i(\omega_{0}\pm
m\omega_{\textrm{ph}})t+\phi_{0}}$, and of the probe laser pulse as
$e^{i\omega_{0}t}$. In such situation, the Raman tensor is explicitly written
as

$\mathcal{R}_{m}(t)=e^{i(\pm
m\omega_{\textrm{ph}}t+\phi_{0})}\begin{bmatrix}E_{s,m_{x}}E^{*}_{x}&E_{s,m_{x}}E^{*}_{y}\\\
E_{s,m_{y}}E^{*}_{x}&E_{s,m_{y}}E^{*}_{y}\end{bmatrix}.$ (9)

When the probe laser is polarised along X-axis, the invariance condition for
the Raman tensor $\hat{\mathcal{D}_{1}}\mathcal{R}_{m}(t)=\mathcal{R}_{m}(t)$
reduces to

$e^{i(\pm m\omega_{\textrm{ph}}t)}\begin{bmatrix}E_{s,m_{x}}\\\
E_{s,m_{y}}\end{bmatrix}=e^{i[\pm
m(\omega_{\textrm{ph}}t+\pi)]}\begin{bmatrix}E_{s,m_{x}}\\\
-E_{s,m_{y}}\end{bmatrix}.$ (10)

The selection rule for the $m^{\rm th}$-order sideband is as follows: when $m$
is odd (even), the polarisation of the sideband will be along the Y(X)
direction. Our observations in Fig. 2(a) are consistent with Eq. (10).

When the iLO phonon mode is excited and the probe pulse is along
$\mathsf{\Gamma-M}$ direction, $\hat{\sigma}_{y}\cdot\tau_{2}$, and
$\hat{C}_{2Z}$ are the DSs, which leave the Raman tensor invariant. It is
straightforward to see that the selection rules for the $m^{\rm th}$-order
sideband are deduced as: when $m$ is odd (even), the polarisation of the
sidebands will be along the X(Y) direction. On the other hand, when the iTO
phonon mode is excited and the probe pulse is along $\mathsf{\Gamma-K}$
($\mathsf{\Gamma-M}$) direction, $\hat{\sigma}_{x}$ ($\hat{\sigma}_{y}$) is
the DS, which yields Raman tensor invariant [see Fig. 4(b)]. This symmetry
restricts the polarisation of the sidebands to be along the direction of the
probe pulse. Our results are consistent with the observation made in Fig. 2.
With the increased intensity of the probe, higher-order harmonics and
sidebands will appear.

To summarise, we have established that high-harmonic spectroscopy is
responsive to the coherent lattice dynamics in solids. The high-harmonic
spectrum is modulated by the frequency of the excited phonon mode within the
solid. Both in-plane E2g Raman-active phonon modes of the monolayer graphene
lead to the generation of higher-order sidebands, along with the main harmonic
peaks. In the case of iLO phonon mode excitation, the even- and odd-order
sidebands are polarised parallel and perpendicular to the polarisation of
probe harmonic pulse, respectively. In the case of iTO phonon mode, all
sidebands are polarised along the probe harmonic pulse’s polarisation. The
polarisations of the sidebands are dictated by the dynamical symmetries of the
combined system, which includes the phonon modes and probe laser pulse.
Therefore, the polarisation properties are a sensitive probe of these
dynamical symmetries. The presence of high-harmonic signal perpendicular to
the polarisation of the probe pulse is a signature of lattice excitation-
driven symmetry breaking of the reflection plane. The present work is paving a
way for probing phonon-driven processes in solids and non-linear phononics
with sub-cycle temporal resolution.

## Acknowledgements

We acknowledge fruitful discussion with Sumiran Pujari (IIT Bombay), Dipanshu
Bansal (IIT Bombay) and Klaus Reimann (MBI Berlin). G. D. acknowledges support
from Science and Engineering Research Board (SERB) India (Project No.
MTR/2021/000138). D.K. acknowledges support from CRC 1375 “NOA–Nonlinear
optics down to atomic scales”, Project C4, funded by the Deutsche
Forschungsgemeinschaft (DFG).

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# Classical information and collapse in Wigner’s friend setups

Veronika Baumann<EMAIL_ADDRESS>Atominstitut, Technische
Universität Wien, 1020 Vienna, Austria  Institute for Quantum Optics and
Quantum Information (IQOQI), Boltzmanngasse 3, 1090 Vienna, Austria

###### Abstract

The famous Wigner’s friend experiment considers an observer – the friend– and
a superobserver –Wigner– who treats the friend as a quantum system and her
interaction with other quantum systems as unitary dynamics. This is at odds
with the friend describing this interaction via collapse dynamics, if she
interacts with the quantum system in a way that she would consider a
measurement. These different descriptions constitute the Wigner’s friend
paradox. Extended Wigner’s friend experiments combine the original thought
experiment with non-locality setups. This allows for deriving local
friendliness inequalities, similar to Bell’s theorem, which can be violated
for certain extended Wigner’s friend scenarios. A Wigner’s friend paradox and
the violation of local friendliness inequalities require that no classical
record exists, which reveals the result the friend observed during her
measurement. Otherwise Wigner agrees with his friend’s description and no
local friendliness inequality can be violated. In this article, I introduce
classical communication between Wigner and his friend and discuss its effects
on the simple as well as extended Wigner’s friend experiments. By controlling
the properties of a (quasi) classical communication channel between Wigner and
the friend one can regulate how much outcome information about the friend’s
measurement is revealed. This gives a smooth transition between the
paradoxical description and the possibility of violating local friendliness
inequalities, on the one hand, and the effectively collapsed case, on the
other hand.

##  I Introduction

The Wigner’s friend thought experiment was originally proposed in
wigner1963problem  to reason about the applicability of the two dynamics
featured by quantum mechanics. On the one hand sufficiently isolated quantum
systems evolve unitarily. On the other hand, quantum systems upon measurement
undergo a seemingly instantaneous collapse to the eigenstate corresponding to
the observed outcome. These two different dynamics and the lack of a clear
prescription of when to use one or the other is one of the main aspects of the
quantum measurement problem bub2010two ; busch1996quantum ; maudlin1995three .

$S$$\ket{\phi}_{S}$$\ket{0}_{S}\ket{\bm{0}}_{F}\,$$\ket{1}_{S}\ket{\bm{1}}_{F}\,$F$\ket{\Phi}_{SF}$W
Figure 1: Simple Wigner’s friend experiment: The source emits a quantum state
$\ket{\phi}_{S}$, which is measured by the friend in the computational basis
{$\ket{0}_{S},\ket{1}_{S}$}. The result observed by the friend is stored in
some memory register $\ket{\cdot}_{F}$ and she ascribes the respective product
$\ket{i}_{S}\ket{\bm{i}}_{F}$, with $i\in\\{0,1\\}$, to herself and the system
she measured. Wigner performs a measurement on both the system and the
friend’s memory register, which according to his unitary description is in
state $\ket{\Phi}_{SF}$ that is, in general, a superposition of
$\ket{0}_{S}\ket{\bm{0}}_{F}$ and $\ket{1}_{S}\ket{\bm{1}}_{F}$. The different
state assignments to the joint system $S+F$ will lead to different probability
assignments for Wigner’s measurement.

The original thought experiment features an observer –called Wigner’s friend–
who measures a quantum system, as well as a so-called superobserver –Wigner–
who performs a joint measurement on the quantum system S and the friend F.
Provided that the joint system S+F is sufficiently isolated, Wigner describes
the friend’s interaction with the system via unitary dynamics. To Wigner’s
friend, however, this interaction constitutes a measurement and she uses the
collapse postulate after observing an outcome. These disagreeing descriptions
of one and the same situation is called the Wigner’s friend paradox, see Fig.
1. Let the source emit a qubit in the state
$\ket{\phi}_{S}=\alpha\ket{0}_{S}+\beta\ket{1}_{S}$, which is measured by the
friend in the computational basis {$\ket{0},\ket{1}$}. The result she
observes, $f\in\\{\bm{0},\bm{1}\\}$ is stored in her memory register
$\ket{\cdot}_{F}$, which is supposed to correspond to the friend having a
perception of the outcome $f$. Wigner then performs a measurement on the qubit
and the friend’s memory given by the states
$\ket{1}_{SF}=a\ket{0,\bm{0}}_{SF}+b\ket{1,\bm{1}}_{SF}$ and
$\ket{2}_{SF}=b^{*}\ket{0,\bm{0}}_{SF}-a^{*}\ket{1,\bm{1}}_{SF}$ and their
orthogonal complement. Due to their different descriptions of the friend’s
measurement Wigner and his friend will assign different probabilities to the
outcomes of Wigner’s measurement. According to the friend after her
measurement the qubit and her memory are either in state
$\ket{0,\bm{0}}_{SF}$, which happens with probability $p(0)=|\alpha|^{2}$, or
in state $\ket{1,\bm{1}}_{SF}$, which happens with probability
$p(1)=|\beta|^{2}$. Hence, the friend will assign the following probability to
Wigner’s results

${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}p^{F}(w)}=p^{clps}(w)=|\alpha|^{2}|\braket{w}{0,\bm{0}}|^{2}+|\beta|^{2}|\braket{w}{1,\bm{1}}|^{2},$
(I.1)

where $\ket{w}$ is either $\ket{1}_{SF}$ or $\ket{2}_{SF}$. Wigner, however,
assigns the state
$\ket{\Phi}_{SF}=\alpha\ket{0,\bm{0}}_{SF}+\beta\ket{0,\bm{0}}_{SF}$ to the
qubit and his friend’s memory and, hence, probabilities

${\color[rgb]{0,0.38671875,0.65234375}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.38671875,0.65234375}p^{W}(w)}=p^{uni}(w)=|\braket{w}{\Phi}_{SF}|^{2}.$
(I.2)

More concretely, we obtain the following predictions for Wigner’s measurement
according to Wigner and his friend in the simple Wigner’s friend experiment

$\begin{array}[]{l c|c c l
c|c}\color[rgb]{0,0.38671875,0.65234375}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.38671875,0.65234375}p^{W}(w):&1&2&&\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}p^{F}(w):&1&2\\\
\cline{2-3}\cr\cline{6-7}\cr&(\alpha a^{*}+\beta b^{*})^{2}&(\beta a-\alpha
b)^{2}&&&|\alpha|^{2}|a|^{2}+|\beta|^{2}|b|^{2}&|\beta|^{2}|a|^{2}+|\alpha|^{2}|b|^{2}.\end{array}$
(I.3)

Wigner originally argued in favor of the friend’s description (and probability
assignment) claiming that at the level of an observer, at the latest, unitary
quantum theory must break down. Since then, however, the idea of observers in
superposition has become accepted deutsch1985quantum , which begs the question
of whether the disagreement between Wigner and his friend can be
experimentally verified. As already discussed in baumann2018formalisms ;
cavalcanti2021view ; baumann2020wigner , the unitary description of the
friend’s measurement is incompatible with the existence of a record revealing
which result the friend observed before Wigner performs his measurement. Such
a record destroys any coherences Wigner could reveal in his measurement.
However, there are persistent records, which do not contain any outcome
information of the friend’s measurement and can, therefore, be present without
destroying the coherence of an observer in superposition deutsch1985quantum ;
baumann2020wigner . In special cases the disagreement between Wigner and his
friend becomes manifest in terms of contradicting records that both Wigner and
his friend can access after the thought experiment has been performed. Note
that, in these cases there is no persistent record of which result the friend
observed at her measurement, since Wigner’s measurement will, in general,
alter the friend’s perception of her measurement result, i.e. the result
stored in the internal memory register allard2020no .

The newest versions of the thought experiment combine multiple Wigner’s friend
scenarios with various non-locality proofs haddara2022possibilistic ;
ormrod2022no ; leegwater2022greenberger ; Zukowski2020 ; Bong2020 ;
brukner2018no ; frauchiger2018quantum ; brukner2017quantum . Some of these
extended Wigner’s friend experiments rely on conflicting probability
assignments of observers and superobservers similar to the simple Wigner’s
friend experiment above. This lead to closer investigations of how agents in
these settings can make predictions and consistently reason about each other
vilasini2019multi ; vilasini2022general . Other extended Wigner’s friend
scenarios concern the joint probabilities of the results of observers and
superobservers, similar to Bell’s theorem. More concretely, a set of
assumptions called local friendliness – namely, that the superobservers’ and
observers’ results are both “objective facts”, locality, freedom of choice,
and universality of quantum theory, meaning observers can exist in
superpositions of different observation states – cannot all hold in extended
Wigner’s friend setups. In the simplest extended Wigner’s friend scenario, as
depicted in Fig. 2, a bipartite entangled state is shared between a Wigner’s
friend setup and one additional observer – Bob. The violation of a CHSH-like-
inequality between Wigner and Bob asserts that the local friendliness
assumptions cannot hold simultaneously.

$\ket{\Psi}_{12}$FWB Figure 2: Simplest extended Wigner’s friend experiment: A
bipartite state $\ket{\Psi}_{12}$ is emitted by the source. One subsystem is
measured by the friend – F – in a simple Wigner’s friend setup. Together with
the subsystem she interacted with the friend is then measured by Wigner – W.
The other subsystem is measured by an additional observer Bob – B.

Consider the setup in Fig. 2, where Wigner randomly chooses between the
measurement that reveals which result his friend observed in her measurement
$W_{z}=|0,\bm{0}\rangle\langle 0,\bm{0}|_{2F}-|1,\bm{1}\rangle\langle
1,\bm{1}|_{2F}$ and one projecting on the states
$\ket{\Phi^{\pm}}_{2F}=1/\sqrt{2}(\ket{0,\bm{0}}_{2F}\pm\ket{1,\bm{1}}_{2F})$,
i.e.
$W_{x}=\outerproduct{0,\bm{0}}{1,\bm{1}}_{2F}+\outerproduct{1,\bm{1}}{0,\bm{0}}_{2F}$.
Bob, on the other hand, performs measurements
$B_{z}=1\sqrt{2}(|0\rangle\langle
0|+\outerproduct{0}{1}+\outerproduct{1}{0}-|1\rangle\langle 1|)$ and
$B_{x}=1\sqrt{2}(|0\rangle\langle
0|-\outerproduct{0}{1}-\outerproduct{1}{0}-|1\rangle\langle 1|)$ on qubit 1.
One local friendliness inequality for this scenario is given by the following
CHSH-expression

$\langle B_{z}\otimes W_{z}\rangle+\langle B_{x}\otimes W_{z}\rangle-\langle
B_{z}\otimes W_{x}\rangle+\langle B_{x}\otimes W_{x}\rangle\leq 2.$ (I.4)

Other inequalities can be obtained by making an alternative choice for which
expectation value is subtracted on the left hand side. If the source emits the
singlet state $\ket{\psi^{-}}_{12}=1/\sqrt{2}(\ket{0,0}_{12}-\ket{1,1}_{12})$
and the friend measures qubit 2 in the computational basis, we obtain the
overall state

$\ket{\Psi}=\frac{1}{\sqrt{2}}\left(\ket{0}_{1}\ket{0,\bm{0}}_{2F}-\ket{1}_{1}\ket{1,\bm{1}}_{2F}\right)$
(I.5)

on which Wigner and Bob perform their measurements. This give the following
violation of the inequality in Eq. (I.4)

$\langle B_{z}\otimes W_{z}\rangle+\langle B_{x}\otimes W_{z}\rangle-\langle
B_{z}\otimes W_{x}\rangle+\langle B_{x}\otimes
W_{x}\rangle=4\cdot\frac{1}{\sqrt{2}}=2\sqrt{2}>2.$ (I.6)

The violations of local friendliness inequalities have been confirmed
experimentally in proof of principle experiments Bong2020 ;
proietti2019experimental , where an additional qubit played the role of
Wigner’s friend. Such experiments arguably do not constitute genuine Wigner’s
friend experiments, since the interaction between two qubits does not satisfy
many basic characteristics of an observation brukner2021qubits . In response
to that, an extended Wigner’s friend experiment involving a human level AI on
a quantum computer playing the role of the friend has been proposed
wiseman2022thoughtful . Such a friend would satisfy most qualitative features
of an observer while operating fully unitarily by construction.

The rest of this article is structured as follows. In the main Section II I
consider communication between Wigner and his friend first by incorporating
record systems that are not subject to Wigner’s measurement in II.1. Second, I
introduce a (quasi) classical communication channel into the simple Wigner’s
friend experiment in II.2. This allows to recover probabilities in agreement
with collapse dynamics as well as with unitary dynamics and anything in
between depending on the properties of the communication channel. In Section
II.3 I then consider the records as well as the communication channel in an
extended Wigner’s friend setup and discuss their implications on the violation
of the local friendliness inequality presented above. Finally, the conclusions
are summarize in Section III.

##  II Classical information and collapse

As discussed in deutsch1985quantum ; zurek2018quantum ; cavalcanti2021view
unitary dynamics is incompatible with simultaneously preserving a classical
record of a measurement result. For Wigner’s friend experiments this means
that there cannot be a record revealing the friend’s observed outcome. In
general, records or messages that are not subject to Wigner’s measurement
influence the probabilities for Wigner’s results according to a unitary
description of the setup.

###  II.1 Effective collapse

If the friend produces a classical record revealing her observed result, also
Wigner’s description of the setup gives rise to the probabilities induced by
collapse dynamics. Consider a record Hilbert space $\mathcal{H}_{R}$ with a
fixed basis {$\ket{r_{i}}$}, which encodes the (quasi) classical messages
Wigner receives from his friend. For example, in the simplest Wigner’s friend
experiment in Fig. 1 these messages could correspond to
$\ket{r_{0}}=\ket{\text{``I saw $0$.''}}$ and $\ket{r_{1}}=\ket{\text{``I saw
$1$.''}}$. A unitary description of the setup, then, leads to the following
over all state

$\ket{\Psi^{r}}=\alpha\ket{0,\bm{0}}_{SF}\ket{r_{0}}_{R}+\beta\ket{1,\bm{1}}_{SF}\ket{r_{1}}_{R},$
(II.1)

upon which Wigner performs his measurement. The probabilities for Wigner’s
measurement result are then given by

${\color[rgb]{0,0.38671875,0.65234375}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.38671875,0.65234375}p^{W}(w)}=\Tr\left(|w\rangle\langle
w|_{SF}|\Psi^{r}\rangle\langle\Psi^{r}|\right),$ (II.2)

which is equal the ones assigned by the friend who uses collapse dynamics

$\begin{array}[]{l
c|c}{\color[rgb]{0,0.38671875,0.65234375}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.38671875,0.65234375}p^{W}(w)}={\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}p^{F}(w)}:&1&2\\\
\cline{2-3}\cr&|\alpha|^{2}|a|^{2}+|\beta|^{2}|b|^{2}&|\beta|^{2}|a|^{2}+|\alpha|^{2}|b|^{2}.\end{array}$
(II.3)

Note that, tracing out the record system can be understood as Wigner ignoring
the record of the friend’s result. Yet the existence of such a message alone,
even if Wigner does not know what it reads, effectively collapses the state of
the system and the friend. When taking into account which result the friend
observed we need to consider conditional probabilities. In the collapse
description employed by the friend, this means either using the state
$\ket{0,\bm{0}}$ or $\ket{1,\bm{1}}$ when calculating
${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}p^{F}(w]f)}=|\braket{w}{f,\bm{f}}]^{2}$.
In Wigner’s unitary description we condition on the record by evaluating
probabilities

$\displaystyle{\color[rgb]{0,0.38671875,0.65234375}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.38671875,0.65234375}p^{W}(w,j)}=\Tr{|w\rangle\langle
w|_{SF}\otimes|r_{j}\rangle\langle
r_{j}|_{R}|\Psi^{r}\rangle\langle\Psi^{r}|}=p(j)\cdot{\color[rgb]{0,0.38671875,0.65234375}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.38671875,0.65234375}p^{W}(w|j)}$
(II.4)

where $p(j)=\Tr(\mathds{1}\otimes|r_{j}\rangle\langle
r_{j}|_{R}|\Psi^{r}\rangle\langle\Psi^{r}|)$ is the probability of message
$r_{j}$ and we define $p(w|j)=0$ if $p(j)=0$. This gives

$\begin{array}[]{l c|c c l
c|c}{\color[rgb]{0,0.38671875,0.65234375}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.38671875,0.65234375}p^{W}(w|0)}={\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}p^{F}(w]f=0)}:&1&2&&{\color[rgb]{0,0.38671875,0.65234375}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.38671875,0.65234375}p^{W}(w|1)}={\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}p^{F}(w]f=1)}:&1&2\\\
\cline{2-3}\cr\cline{6-7}\cr&|a|^{2}&|b|^{2}&&&|b|^{2}&|a|^{2}.\end{array}$
(II.5)

if the record reveals which result the friend observed. Conversely, if the
record space is only one dimensional, for example
$\ket{r^{\prime}}=\ket{\text{``I saw a definite outcome''}}$, the record
factors out and

$\ket{\Psi^{r^{\prime}}}=(\alpha\ket{0,\bm{0}}_{SF}+\beta\ket{1,\bm{1}}_{SF})\ket{r^{\prime}}_{R},$
(II.6)

which preserves the disagreement between Wigner and his friend. Conditioning
on the record is obsolete in this case and the probabilities
${\color[rgb]{0,0.38671875,0.65234375}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.38671875,0.65234375}p^{W}(w)}=\Tr\left(|w\rangle\langle
w|_{SF}|\Psi^{r^{\prime}}\rangle\langle\Psi^{r^{\prime}}|\right)\neq{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}p^{F}(w)}$
are again those in Eq. (I.3). This is due to the fact that such a one
dimensional record cannot reveal any outcome information of the friend’s two-
outcome measurement.

###  II.2 Partial collapse

$S$F$\ket{\phi}_{S}$$\mathcal{C}$$\ket{r}_{R}$$\ket{m}_{R}$$\ket{\Phi}_{SF}$W
Figure 3: Communication channel $\mathcal{C}$ between Wigner and his friend:
The friend can encode a message via a fixed basis {$\ket{r}$} of some record
Hilbert space $\mathcal{H}_{R}$. The channel $\mathcal{C}$ takes these record
states as input and produces a classical message, which can in principle be
encoded in another basis { $\ket{m}$} of the record Hilbert space. This
freedom to choose different bases for the incoming and the outgoing messages
allows for controlling how much outcome information the friend can send to
Wigner via this channel.

We now consider a more general scenario, where instead of directly exchanging
messages, there is a (quasi) classical communication channel holevo1998quantum
between Wigner and his friend, see Fig. 3. Such a channel measures the
incoming state and prepares a corresponding outcome in some fixed basis
{$\ket{n}$} of the record Hilbert space $\mathcal{H}_{R}$

$\mathcal{C}[\sigma]:=\sum_{m}\bra{m}\sigma\ket{m}|m\rangle\langle m|.$ (II.7)

Other than in Section II.1 the messages $\ket{m}$ sent to Wigner can now be
encoded in a different basis of $\mathcal{H}_{R}$ than the records $\ket{r}$
the friend produces. This allows for these messages to only partially reveal
which outcome the friend observed. The state Wigner performs his measurement
on is now

$\displaystyle\rho_{SFR}=$
$\displaystyle(\mathds{1}\otimes\mathcal{C})|\Psi^{r}\rangle\langle\Psi^{r}|=\sum_{m}|\alpha|^{2}|\braket{m}{r_{0}}|^{2}\rho^{00}_{SF}\otimes|m\rangle\langle
m|_{R}+|\beta|^{2}|\braket{m}{r_{1}}|^{2}\rho^{11}_{SF}\otimes|m\rangle\langle
m|$
$\displaystyle+\alpha\beta^{*}\braket{m}{r_{0}}\braket{r_{1}}{m}\rho^{01}_{SF}\otimes|m\rangle\langle
m|+\alpha^{*}\beta\braket{m}{r_{1}}\braket{r_{0}}{m}\rho^{10}_{SF}\otimes|m\rangle\langle
m|,$ (II.8)

where

$\displaystyle\rho^{00}_{SF}$ $\displaystyle=|a|^{2}|1\rangle\langle
1|_{SF}+|b|^{2}|2\rangle\langle
2|_{SF}+a^{*}b^{*}\outerproduct{1}{2}_{SF}+ab\outerproduct{2}{1}_{SF},$
$\displaystyle\rho^{11}_{SF}$ $\displaystyle=|b|^{2}|1\rangle\langle
1|_{SF}+|a|^{2}|2\rangle\langle
2|_{SF}-a^{*}b^{*}\outerproduct{1}{2}_{SF}-ab\outerproduct{2}{1}_{SF},$
$\displaystyle\rho^{01}_{SF}$ $\displaystyle=a^{*}b(|1\rangle\langle
1|_{SF}-|2\rangle\langle
2|_{SF})-a^{*}a^{*}\outerproduct{1}{2}_{SF}+bb\outerproduct{2}{1}_{SF},$
$\displaystyle\rho^{10}_{SF}$ $\displaystyle=b^{*}a(|1\rangle\langle
1|_{SF}-|2\rangle\langle
2|_{SF})+b^{*}b^{*}\outerproduct{1}{2}_{SF}-aa\outerproduct{2}{1}_{SF},$

with $\ket{1}_{SF}=a\ket{0,\bm{0}}_{SF}+b\ket{1,\bm{1}}_{SF}$,
$\ket{2}_{SF}=b^{*}\ket{0,\bm{0}}_{SF}-a^{*}\ket{1,\bm{1}}_{SF}$ being the
eigenstates corresponding to Wigner’s measurement results “1” and “2”
respectively. Analogous to the Eq. (II.4) in section II.1 we consider the
conditional probabilities of Wigner’s result $w$ given message $n$, i.e.
$p(n)p(w|n)=\Tr(|w\rangle\langle w|_{SF}\otimes|n\rangle\langle
n|_{R}\rho_{SFR})$, obtaining the following joint probabilities

$\begin{array}[]{l c|c}&p(w=1,n)&p(w=2,n)\\\
\cline{2-3}\cr&|\alpha|^{2}|a|^{2}|\braket{n}{r_{0}}|^{2}+|\beta|^{2}|b|^{2}|\braket{n}{r_{1}}|^{2}&|\beta|^{2}|a|^{2}|\braket{n}{r_{0}}|^{2}+|\alpha|^{2}|b|^{2}|\braket{n}{r_{1}}|^{2}\\\
&+\alpha\beta^{*}a^{*}b\braket{n}{r_{0}}\braket{r_{1}}{n}+\alpha^{*}\beta
ab^{*}\braket{n}{r_{1}}\braket{r_{0}}{n}&-\alpha\beta^{*}a^{*}b\braket{n}{r_{0}}\braket{r_{1}}{n}-\alpha^{*}\beta
ab^{*}\braket{n}{r_{1}}\braket{r_{0}}{n}.\end{array}$ (II.9)

The overlaps $\braket{n}{r_{i}}$ indicate how much outcome information Wigner
can obtain from the channel output message $n$. If the basis {$\ket{n}$} is
the same as {$\ket{r_{i}}$}, i.e.$\braket{n}{r_{i}}=\delta_{ni}$, the message
perfectly reveals which result the friend observed and $p(n=0)=|\alpha|^{2}$,
$p(n=1)=|\beta|^{2}$. In this case we recover the collapse behavior in
Eq.(II.5), as discussed in Section II.1. However, if the two bases are
mutually unbiased, for example
$\braket{0}{r_{i}}=\braket{1}{r_{0}}=1/\sqrt{2}=-\braket{1}{r_{1}}$, the
records reveal no outcome information about the friend’s measurement and
$p(n)=1/2$ for both $n$. In this case, we obtain probabilities in accordance
with a unitary description for each of the records

$\begin{array}[]{l c|c c l c|c}p(w|0):&1&2&&p(w|1):&1&2\\\
\cline{2-3}\cr\cline{6-7}\cr&(\alpha a^{*}+\beta b^{*})^{2}&(\beta a-\alpha
b)^{2}&&&(\alpha a^{*}-\beta b^{*})^{2}&(\beta a+\alpha b)^{2}.\end{array}$
(II.10)

Note that the phase shift between the expressions for the two messages does
not allow for simply adding them when wanting to preserve the resemblance to
the unitary description without records. Evaluating $p(w|0)+p(w|1)$
corresponds to tracing out the record system and, as already mentioned in
Section II.1, gives the collapse probabilities in Eq. (II.3). In general, we
can express the messages Wigner receives in the basis of the records generated
by the friend as follows

$\displaystyle\ket{0}$
$\displaystyle=\cos{\theta}\ket{r_{0}}+e^{i\phi}\sin{\theta}\ket{r_{1}}$
(II.11) $\displaystyle\ket{1}$
$\displaystyle=e^{-i\phi}\sin{\theta}\ket{r_{0}}-\cos{\theta}\ket{r_{1}},$
(II.12)

where one can think of $\theta,\phi$ as variable parameters of the
communication channel $\mathcal{C}$. Hence, the joint probabilities for
Wigner’s measurement result and message $n$ are given by

$\begin{array}[]{l c|c}p(w,0):&1&2\\\
\cline{2-3}\cr&|\alpha|^{2}|a|^{2}\cos^{2}(\theta)+|\beta|^{2}|b|^{2}\sin^{2}(\theta)&|\beta|^{2}|a|^{2}\cos^{2}(\theta)+|\alpha|^{2}|b|^{2}\sin^{2}(\theta)\\\
&+\sin(2\theta)(\alpha\beta^{*}a^{*}be^{i\phi}+\alpha^{*}\beta
ab^{*}e^{-i\phi})&-\sin(2\theta)(\alpha\beta^{*}a^{*}be^{i\phi}+\alpha^{*}\beta
ab^{*}e^{-i\phi})\end{array}$ (II.13)

and

$\begin{array}[]{l c|c}p(w,1):&1&2\\\
\cline{2-3}\cr&|\alpha|^{2}|a|^{2}\sin^{2}(\theta)+|\beta|^{2}|b|^{2}\cos^{2}(\theta)&|\beta|^{2}|a|^{2}\sin^{2}(\theta)+|\alpha|^{2}|b|^{2}\cos^{2}(\theta)\\\
&-\sin(2\theta)(\alpha\beta^{*}a^{*}be^{i\phi}+\alpha^{*}\beta
ab^{*}e^{-i\phi})&+\sin(2\theta)(\alpha\beta^{*}a^{*}be^{i\phi}+\alpha^{*}\beta
ab^{*}e^{-i\phi}).\end{array}$ (II.14)

Varying $\theta$ and $\phi$ then allows for smoothly changing between the
effectively collapsed case in Eq. (II.5) and the effectively unitary case in
Eq. (II.10), see Fig. 4 for a simple example. Note that, for the partially
collapsed scenarios there is still a paradoxical situation, since the friend
when describing her measurement via collapse dynamics will always assign the
probabilities in Eq. (II.5). Wigner’s probability assignments, namely those
derived from Eqs. (II.13)-(II.14), will always differ from his friend’s unless
the messages fully reveal which outcome the friend observed.

Figure 4: Partial collapse for communication between Wigner and his friend: If
the source in the setup depicted in Fig.3 emits the state
$\ket{\phi}=1/\sqrt{2}(\ket{0}+\ket{1})$ and Wigner measures in the Bell-basis
{$\ket{\phi^{+}},\ket{\phi^{-}},\ket{\psi^{+}},\ket{\psi^{-}}$} he will obtain
results “$+$” and “$-$” corresponding to states
$\ket{\phi^{+}}=1/\sqrt{2}(\ket{0,\bm{0}}+\ket{1,\bm{1}}$ and
$\ket{\phi^{-}}1/\sqrt{2}(\ket{0,\bm{0}}-\ket{1,\bm{1}}$ respectively. For a
communication channel $\mathcal{C}(\theta,\phi)$, and with $\phi=0$, the
probabilities for Wigner’s outcomes according to the unitary description of
the setup depend on parameter $\theta$ as shown above. These probabilities
(violet) vary between those corresponding to the effectively collapsed state
(green) and the effectively unitary case (blue).

###  II.3 Local friendliness inequalities and communication

Analogous to the simple Wigner’s friend experiment the presence of classical
records of the friend’s observed result effectively collapses the state Wigner
performs his measurement on also in the extended Wigner’s friend setup
depicted in Fig. 2. This, in turn, prevents the violation of the local
friendliness inequality presented in Section I. More concretely, using state

$\ket{\Psi^{r}}=\frac{1}{\sqrt{2}}\left(\ket{0}_{1}\ket{0,\bm{0}}_{2F}\ket{r_{0}}_{R}-\ket{1}_{1}\ket{1,\bm{1}}_{2F}\ket{r_{1}}_{R}\right)$
(II.15)

instead of the one in Eq. (I.5) leads to the following expression for the
CHSH-like local friendliness inequality

$\langle B_{z}\otimes W_{z}\rangle+\langle B_{x}\otimes W_{z}\rangle-\langle
B_{z}\otimes W_{x}\rangle+\langle B_{x}\otimes
W_{x}\rangle=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}+0+0=\sqrt{2}<2,$ (II.16)

which means that none of the local friendliness assumptions needs to be
rejected. This is due to the fact that the presence of the records, revealing
which result the friend observed, effectively collapse the state in Eq.
(II.15) to

$\Tr_{R}(|\Psi^{r}\rangle\langle\Psi^{r}|)=\frac{1}{2}\left(|0\rangle\langle
0|_{1}|0,\bm{0}\rangle\langle 0,\bm{0}|_{2F}+|1\rangle\langle
1|_{1}|1,\bm{1}\rangle\langle 1,\bm{1}|_{2F}\right),$ (II.17)

which means that any expectation value containing Wigner’s $W_{x}$-measurement
vanishes. Note that, this is also true when we condition on the friend’s
observed result, meaning that we use either $|0\rangle\langle
0|_{1}|0,\bm{0}\rangle\langle 0,\bm{0}|_{2F}$ or $|1\rangle\langle
1|_{1}|1,\bm{1}\rangle\langle 1,\bm{1}|_{2F}$ as the effective state.

We now, again, consider a general communication channel $\mathcal{C}$ between
the friend and Wigner, as depicted in Fig. 3, also for this extended setup.
Starting from the state in Eq. (II.15) we now let the channel act on the
register space and obtain the state

$\displaystyle\rho^{\prime}_{12FR}$
$\displaystyle=(\mathds{1}\otimes\mathcal{C})\left(|\Psi^{r}\rangle\langle\Psi^{r}|\right)$
(II.18) $\displaystyle=\frac{1}{2}\Big{(}|0\rangle\langle
0|_{1}\otimes|0,\bm{0}\rangle\langle
0,\bm{0}|_{2F}\otimes\mathcal{C}(|r_{0}\rangle\langle
r_{0}|)-\outerproduct{0}{1}_{1}\otimes\outerproduct{0,\bm{0}}{1,\bm{1}}_{2F}\mathcal{C}(\outerproduct{r_{0}}{r_{1}})$
$\displaystyle\quad-\outerproduct{1}{0}_{1}\otimes\outerproduct{1,\bm{1}}{0,\bm{0}}_{2F}\mathcal{C}(\outerproduct{r_{1}}{r_{0}})+|1\rangle\langle
1|_{1}\otimes|1,\bm{1}\rangle\langle
1,\bm{1}|_{2F}\otimes\mathcal{C}(|r_{1}\rangle\langle r_{1}|)\Big{)},$

upon which Wigner and Bob perform their measurements. The action of the
classical channel on the records, again, gives terms of the form

$\displaystyle\sum_{m}\braket{m}{r_{i}}\braket{r_{j}}{m}|m\rangle\langle m|,$
(II.19)

which, if there is no conditioning on the message lead to the effective
collapse discussed above regardless of the properties of the channel, i.e.
parameters $\phi,\theta$, since

$\displaystyle\Tr\left(\sum_{m}\braket{m}{r_{i}}\braket{r_{j}}{m}|m\rangle\langle
m|\right)=\sum_{m}\braket{m}{r_{i}}\braket{r_{j}}{m}=\braket{r_{j}}{r_{i}}=\delta_{ij}.$
(II.20)

Similar to the probabilities for Wigner’s outcome in the simple Wigner’s
friend setup, we can now define the expectation values for the measurements of
Bob and Wigner, conditioned on the message $n$ put out by the classical
channel $\mathcal{C}$ as follows

$\langle B\otimes W\rangle^{|n}:=\begin{cases}\frac{1}{p(n)}\Tr\left(B\otimes
W\otimes|n\rangle\langle n|\cdot\rho\right)&\text{for }p(n)>0\\\
\qquad\qquad\qquad\qquad 0&\text{for }p(n)=0,\end{cases}$ (II.21)

where the probability for the messages is now given by
$p(n)=\Tr\left(\mathds{1}\otimes|n\rangle\langle
n|\cdot\rho^{\prime}_{12FR}\right)=1/2(\cos^{2}(\theta)+\sin^{2}(\theta))=1/2$.
Plugging the state in Eq. (II.18) into this expression, then gives

$\displaystyle\langle B\otimes W\rangle^{|n}=$
$\displaystyle\Big{(}\braket{0}{B}{0}\braket{0,\bm{0}}{W}{0,\bm{0}}|\braket{n}{r_{0}}|^{2}+\braket{1}{B}{1}\braket{1,\bm{1}}{W}{1,\bm{1}}|\braket{n}{r_{1}}|^{2}$
(II.22)
$\displaystyle\quad-\braket{1}{B}{0}\braket{1,\bm{1}}{W}{0,\bm{0}}\braket{n}{r_{0}}\braket{r_{1}}{n}-\braket{0}{B}{1}\braket{0,\bm{0}}{W}{1,\bm{1}}\braket{n}{r_{1}}\braket{r_{0}}{n}\Big{)}.$

For the settings of Bob and Wigner presented in Section I we obtain the
conditional expectation values

$\displaystyle\langle B_{z}\otimes W_{z}\rangle^{|n}$
$\displaystyle=\frac{1}{\sqrt{2}}(|\braket{n}{r_{0}}|^{2}+|\braket{n}{r_{1}}|^{2})=\frac{1}{\sqrt{2}}=\langle
B_{x}\otimes W_{z}\rangle^{|n},$ $\displaystyle\langle B_{z}\otimes
W_{x}\rangle^{|n}=$
$\displaystyle-\frac{1}{\sqrt{2}}(\braket{n}{r_{0}}\braket{r_{1}}{n}+\braket{n}{r_{1}}\braket{r_{0}}{n})=-\langle
B_{x}\otimes W_{x}\rangle^{|n}.$

Hence, when conditioned on the message $n$ the local friendliness inequality
in Eq. (II.16) becomes

$\displaystyle\langle B_{z}\otimes W_{z}\rangle^{|0}+\langle B_{x}\otimes
W_{z}\rangle^{|0}-\langle B_{z}\otimes W_{x}\rangle^{|0}+\langle B_{x}\otimes
W_{x}\rangle^{|0}$
$\displaystyle=\sqrt{2}+\sqrt{2}\cdot\cos(\phi)\sin(2\theta),$ (II.23)
$\displaystyle\langle B_{z}\otimes W_{z}\rangle^{|1}+\langle B_{x}\otimes
W_{z}\rangle^{|1}-\langle B_{z}\otimes W_{x}\rangle^{|1}+\langle B_{x}\otimes
W_{x}\rangle^{|1}$
$\displaystyle=\sqrt{2}-\sqrt{2}\cdot\cos(\phi)\sin(2\theta).$ (II.24)

where the term $\cos(\phi)\sin(2\theta)$ is determined by the properties of
the communication channel. If the messages perfectly reveal which result the
friend observed, i.e. $\phi=k\cdot\pi$ and $\theta=l\cdot\pi/2$, the channel
dependent term, which corresponds to the expectation values $\langle B\otimes
W_{x}\rangle^{|n}$, vanishes and we obtain the expression in Eq. (II.16) for
both messages. If the messages reveal no outcome information about the
friend’s measurement, i.e. $\theta=l\cdot\pi/4$ and $\phi=k\cdot\pi/2$,
conditioning on one of the two messages gives the maximal violation of
$2\sqrt{2}$. Since the channel dependent term smoothly varies in the interval
$[-1,1]$, one can obtain all values from $0$ to $2\sqrt{2}$ for the CHSH-like
local friendliness inequality by controlling the channel parameters $\phi$ and
$\theta$. Note that, due to the different signs for the two messages, whenever
the conditional expectation values for one message violate local friendliness,
the conditional expectation values for the other message do not violate the
inequality, compare Fig. 5. This is similar to the conditional probabilities
$p(w|n)$ for the simple Wigner’s friend experiment discussed in Section II.2.
There due to the different signs in $p(w|0)$ and $p(w|1)$ these probabilites
can exactly reproduce those according to unitary dynamics only for one of the
two messages.

Figure 5: CHSH-values for the extended Wigner’s friend setup with
communication: The conditional expectation values $\langle B\otimes
W\rangle^{|n}$ for the extended Wigner’s friend experiment in Fig. 2 give
CHSH-values depend on the communication channel parameters $\phi$ and
$\theta$. For $\phi=0$ and $\theta\in[0,\pi]$, the expression given by
conditioning on message “$0$” are shown in blue, the one corresponding to
message “$1$” is depicted in green. There are values of $\theta$ where neither
of the conditional CHSH-values lies above the local friendliness threshold of
$2$, meaning no violation occurs. Whenever the CHSH-like local friendliness
inequality is violated for one of the messages, when conditioning on the other
message the local friendliness inequality is satisfied.

##  III Conclusions

We considered the possibility of communication between Wigner and his friend
in both the simple Wigner’s friend scenario and the simplest extended Wigner’s
friend setup that allows for the violation of local friendliness inequalities.
As we showed explicitly a classical message revealing which result the friend
observed during her measurement effectively collapses the state of her and the
system she measured also in the unitary description employed by Wigner. For
the simple Wigner’s friend experiment, this means that the probabilities for
Wigner’s measurement result according to the unitary description of the setup
are the same as those corresponding to collapse dynamics. For the extended
Wigner’s friend setup such records and the corresponding effective collapse
prevent the violation of local friendliness inequalities.
We further considered the more general scenario of a (quasi) classical
communication channel between Wigner and his friend. In that case, the records
the friend produces are the input to the channel while the messages Wigner
receives are the output of that channel. Provided that the friend’s records
encode which outcome she observed, how much of that which-outcome information
is revealed by the output now depends on the properties of the channel.
Controling the channel parameters then allows for gradually changing between
collapse and unitary behavior for Wigner’s friend experiments. In case of the
simple Wigner’s friend experiment this means that the probabilities based on
the unitary description of the setup, associated with Wigner, will gradually
approach those assigned by the friend based on the collapse description. For
the extended Wigner’s friend scenario, the channel properties determine
whether and to what extent local friendliness inequalities can be violated.
Both the recovery of probabilities corresponding to unitary dynamics without
records and the maximum violation of local friendliness inequalities not only
occur when the messages Wigner receives contain no information about which
outcome the friend observed, but also require conditioning on the messages.
Simply ignoring the messages put out by the channel always leads to a full
effective collapse. This can be understood as showing that Wigner’s unitary
description does not just signify his ignorance about which result his friend
observed.

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