# Unveiling the pairing Symmetry of the superconducting Sn/Si(111) via angle-
resolved THz pump spectroscopy

Mattia Iannetti1 Tommaso Cea1 Cesare Tresca2 Lara Benfatto3 Gianni Profeta1,2
1Dipartimento di Scienze Fisiche e Chimiche, Universitá degli Studi
dell’Aquila, Via Vetoio 10, I-67100 L’Aquila, Italy
2CNR-SPIN C/o Dipartimento di Scienze Fisiche e Chimiche, Universitá Degli
Studi dell’Aquila, Via Vetoio 10, I-67100 L’Aquila, Italy
3Department of Physics, Sapienza University of Rome, P.le A. Moro 2, 00185
Rome, Italy

###### Abstract

Doping tin surface epitaxially grown on silicon, Sn/Si(111), with boron atoms
yields the appearance of a superconducting (SC) phase below $T_{c}\sim 4-5$K.
Even though the pairing mechanism remains unknown, experimental evidence of
chiral $d-$wave superconductivity has been recently reported, then ruling out
a phonon-mediated pairing. Here we study theoretically the SC phase and
symmetries of the doped Sn/Si(111) within a $t-J$ model. We analyze the photo-
excitation of the system by intense THz pulses and show that the polarization
dependence of the induced current can distinguish between different symmetries
of the SC gap, thus providing a novel experimental mean to investigate the
spectroscopic features of the Sn/Si(111) across the SC transition.

##  I Introduction

The $\alpha$ phase of monolayer group-IV elements epitaxially grown on
semiconducting surfaces provides a surprisingly rich platform to study the
competition between strongly correlated phenomena, spin transport and lattice
reconstruction. 1/3 monolayer coverage of heavy atoms, like Sn and Pb,
displays an isoelectronic $\sqrt{3}\times\sqrt{3}R30^{\circ}$ reconstruction,
where the host atoms, the adatoms, occupy the $T_{4}$ site in an triangular
lattice[1], which is called the $\alpha$ phase. The three dangling bonds at
the surface of the substrate saturate with the adatoms, leaving a free
electron at each $T_{4}$ site to form a half-filled surface band within the
substrate’s band gap. Such a surface band gets strongly affected by the
electronic interactions. Some materials, like Pb/Ge(111), Sn/Ge(111) and
Pb/Si(111) exhibit a low temperature $3\times 3$ lattice reconstruction,
yielding a charge-density-wave (CDW), which is often metallic[2, 3, 4, 5, 6,
7, 8, 9]. In contrast, Sn/Si(111) displays a low temperature Mott insulating
phase[2, 10, 4], with the possible development of a $2\sqrt{3}\times\sqrt{3}$
collinear antiferromagnetic (AFM) phase[11, 12], not yet clearly observed
experimentally. A renewed interest in the study of these systems has been
recently provided by the discovery of superconductivity in the Sn/Si(111)
hole-doped with boron[13], with critical temperatures of the order of:
$T_{c}\sim 4-5$K. Even though the pairing mechanism is still an open question,
it is believed that Sn/Si(111) displays unconventional superconductivity of
electronic origin, driven by the non-local Coulomb interactions[14, 15]. The
Ref. [16] recently found experimental evidence of chiral $d-$wave
superconductivity, ruling out i) a conventional phonon-mediated pairing and
ii) the presence of a spin-triplet superconducting (SC) order parameter (OP).
The present state of the art challenges a deeper understanding of the nature
of the SC pairing observed in the Sn/Si(111) and the engineering of new
experimental protocols to determine the symmetry of the SC gap. In the last
years, the use of intense THz light pulses either in pump-probe protocols or
for high-harmonic generation offered a powerful tool to address the physics of
several SC systems[17, 18, 19]. Exploiting the non-linear light-matter
interaction, this technique allows one to resonantly excite the low energy
modes of the superconductor and track their dynamics in the time domain. The
lowest-order non-linear effect induced by the incident light is the third
harmonic generation (THG), that can be strongly enhanced in the SC phase when
the frequency of the light lies in the neighboring of the SC gap $\Delta$[20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. In addition, the
measured dependence of the non-linear response on the light polarization with
respect to the main crystallographic axes[21, 22, 27, 26] can provide
additional information on the form of the pairing interaction[35], on the
disorder level[36, 37, 38] and on the symmetry of the SC OP[39]. So far, non-
linear THz photo-excitation has been intensively used to probe both
conventional BCS superconductors, like NbN[20, 21, 30], Nb3Sn[23, 25] and
MgB2[24, 29, 31], and unconventional high$-T_{c}$ cuprate[22, 26, 40, 33, 34]
and pnictide[28, 32] superconductors.

In this letter we study theoretically the superconductivity in the doped
Sn/Si(111), assuming that the interactions responsible for the pairing are of
AFM nature. At the meanfield level, we find two almost degenerate ground
states (GS’s), one with chiral $d_{x^{2}-y^{2}}+id_{xy}$-wave symmetry and the
other with pure $d$-wave, the latter breaking the $C_{3}$ symmetry out of
$C_{6}$. By computing the non-linear current induced by a THz light pulse as a
function of the polarization of the incident light, we show that the THG is
$\pi/3$-periodic for the $d_{x^{2}-y^{2}}+id_{xy}$-wave and $\pi$-periodic for
the pure $d$-wave. Thus, our work proposes the angle-resolved THz pump
spectroscopy as an experimental tool to unveil the symmetry of the SC OP in
the Sn/Si(111), alternatively to other spectroscopic techniques based on the
quasi-particle interference[41].

##  II The model for superconductivity

Figure 1:  Non-interacting band structure, $\xi^{0}_{\boldsymbol{k}}$.

As the superconductivity in Sn/Si(111) emerges by doping an half-filled AFM
Mott insulator, we model the Hamiltonian of the free bond of the Sn atoms on
the $T_{4}$ sites within the $t-J$ model:

$\displaystyle H=H_{tb}+H_{J},$ (1)

where:

$\displaystyle
H_{tb}=\sum_{ij\sigma}\left(t_{ij}-\mu\delta_{ij}\right)c^{\dagger}_{i\sigma}c_{j\sigma}$
(2a) is the tight binding Hamiltonian, $i,j$ label the sites of the triangular
network, $t_{ij}$ are the hopping amplitudes, $\mu$ is the chemical potential,
$c_{i\sigma}$ is the operator for the annihilation of one electron in the site
$i$ with spin $\sigma$. We consider hopping up to the $6^{\mathrm{th}}$
nearest neighbor, with the corresponding amplitudes fitted from first
principles calculations. We model the additional interaction Hamiltonian,
$H_{J}$, as an AFM exchange term: $\displaystyle H_{J}=J\sum_{\langle
i,j\rangle}\left(\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{j}-\frac{1}{4}\hat{n}_{i}\hat{n}_{j}\right),$
(2b) with $J>0$. The sum runs over all the pairs of nearest neighbor sites,
$\boldsymbol{S}_{i}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}c^{\dagger}_{i\sigma}\boldsymbol{\tau}_{\sigma\sigma^{\prime}}c_{i\sigma^{\prime}}$
is the spin operator, $\boldsymbol{\tau}=(\tau^{x},\tau^{y},\tau^{z})$ are the
Pauli’s matrices, $\hbar=1$ and
$\hat{n}_{i}=\sum_{\sigma}c^{\dagger}_{i\sigma}c_{i\sigma}$ is the local
number operator.

The $t-J$ Hamiltonian (1) has been intensively studied in the last decades as
a model for superconductivity in the high$-T_{c}$ cuprate superconductors[42,
43, 44, 45, 46, 47, 48], that paradigmatically feature the competition between
the Mott insulating and the SC phase. Indeed, besides providing a natural
model for AFM ordering it can also describe unconventional superconductivity
emerging out of repulsive electronic interactions. Recently, extended $t-J$
models have been exploited to describe topological SC states in triangular
lattices[49, 50].

Figure 2:  $T_{c}$ as a function of $J$, obtained for hole-doping:
$\delta=10\%$. The dashed lines identify the value: $J=29$meV, which gives
$T_{c}\simeq 4.8$K.

Figure 3:  Maps of $|\Delta_{\boldsymbol{k}}|$ at $T=0$K, for the solution
with $d_{x^{2}-y^{2}}+id_{xy}$-wave symmetry(a) and with
$d_{x^{2}-y^{2}}$-wave(b).

In the absence of the electromagnetic perturbation, we perform the meanfield
approximation to $H$ in both the particle-hole and the particle-particle
(pairing) channels, as detailed in the supplementary information [51]. This
procedure leads to the following quadratic Hamiltonian, expressed in the
reciprocal space:

$\displaystyle
H^{MF}_{tJ}=\sum_{\boldsymbol{k}}\left[\sum_{\sigma}\xi_{\boldsymbol{k}}c^{\dagger}_{\boldsymbol{k}\sigma}c_{\boldsymbol{k}\sigma}+\left(\Delta_{\boldsymbol{k}}c^{\dagger}_{\boldsymbol{k}\uparrow}c^{\dagger}_{-\boldsymbol{k}\downarrow}+h.c.\right)\right]+E_{0},$
(3)

where $\boldsymbol{k}$ is the wave vector in the first Brillouin zone (BZ),
$\xi_{\boldsymbol{k}}=\xi^{0}_{\boldsymbol{k}}+\delta\xi_{\boldsymbol{k}}$,
$\xi^{0}_{\boldsymbol{k}}$ is the non-interacting band dispersion computed
from the chemical potential, $\delta\xi_{\boldsymbol{k}}$ is the meanfield
correction in the particle-hole channel, $\Delta_{\boldsymbol{k}}$ is the
pairing amplitude and $E_{0}$ is an energy constant[51]. Both
$\delta\xi_{\boldsymbol{k}}$ and $\Delta_{\boldsymbol{k}}$ can be expressed in
terms of the form factors corresponding to the $s,d_{x^{2}-y^{2}},d_{xy}$
symmetries, which are the only ones allowed by the nearest neighbors AFM
interaction of the Eq. (2b): $\delta\xi_{\boldsymbol{k}}=\sum_{\alpha}\delta
t_{\alpha}\gamma_{\alpha}(\boldsymbol{k})$,
$\Delta_{\boldsymbol{k}}=\sum_{\alpha}\Delta_{\alpha}\gamma_{\alpha}(\boldsymbol{k})$,
where
$\gamma_{s}(\boldsymbol{k})=\left[\cos(k_{x}a)+2\cos(k_{x}a/2)\cos(k_{y}\sqrt{3}a/2)\right]/\sqrt{3}$,
$\gamma_{d_{x^{2}-y^{2}}}(\boldsymbol{k})=\left[\cos(k_{x}a)-\cos(k_{x}a/2)\cos(k_{y}\sqrt{3}a/2)\right]\sqrt{2/3}$,
$\gamma_{d_{xy}}(\boldsymbol{k})=\sqrt{2}\sin(k_{x}a/2)\sin(k_{y}\sqrt{3}a/2)$,
$a=7$Å is the lattice constant and $\delta t_{\alpha}$, $\Delta_{\alpha}$ are
the self-consistent solutions of (see [51]):

$\displaystyle\delta t_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{J}{2N}\sum_{\boldsymbol{k}}\gamma_{\alpha}(\boldsymbol{k})\xi_{\boldsymbol{k}}\tanh\left(E_{\boldsymbol{k}}/2T\right)/E_{\boldsymbol{k}}$
(4a) $\displaystyle\Delta_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{J}{N}\sum_{\boldsymbol{k}}\gamma_{\alpha}(\boldsymbol{k})\Delta_{\boldsymbol{k}}\tanh\left(E_{\boldsymbol{k}}/2T\right)/E_{\boldsymbol{k}},$
(4b)

where $N$ is the number of lattice sites,
$E_{\boldsymbol{k}}=\sqrt{\xi^{2}_{\boldsymbol{k}}+|\Delta_{\boldsymbol{k}}|^{2}}$,
$T$ is the temperature and we set $k_{B}=1$. The non-interacting band
structure, $\xi^{0}_{\boldsymbol{k}}$, is shown in the Fig. 1 along the high
symmetry path of the BZ specified in the inset panel. Having fixed
$\xi^{0}_{\boldsymbol{k}}$, $T_{c}$ is determined by the interaction strength
$J$. In order to reproduce the experimental results of the Ref. [13], we fix
the value of $J$ by imposing: $T_{c}=4.8$K and hole-doping: $\delta=10\%$.
This is illustrated by the Fig. 2, showing $T_{c}$ as a function of $J$. As
outlined by the dashed lines, we finally obtain: $J=29$meV, wich is of the
same order of magnitude as that obtained by first principles calculations
($J_{fp}=10$meV, see also [51]). We find two distinct SC solutions[51], one
featuring chiral $d_{x^{2}-y^{2}}+id_{xy}$-wave symmetry, as claimed in the
Ref. [16], and the other with pure $d$-wave symmetry. The maps of
$|\Delta_{\boldsymbol{k}}|$ at $T=0K$ are shown in the Fig. 3(a)-(b) for the
two cases, emphasizing that the $C_{6}$ symmetry is preserved by the chiral
solution, while it is broken by the pure $d$-wave, which only features
$C_{2}$. It’s worth noting that, for the pure $d$-wave case, the system is
symmetric under continuum $U(1)$ rotations in the
$\Delta_{d_{x^{2}-y^{2}}}-\Delta_{d_{xy}}$ plane. Therefore, we have chosen
the initial conditions of the meanfield calculation in order to select the
pure $d_{x^{2}-y^{2}}$-wave symmetry, which is the one shown in the Fig. 3(b).
As we show in [51], the chiral symmetry is energetically favored by a very
small amount: $\sim 10^{-7}$eV per site, allowing us to safely consider the
two solutions as almost degenerate.

##  III Tracking the evolution of the GS in a time-dependent electric field

We introduce a uniform vector potential, $\boldsymbol{A}(t)$, in the
Hamiltonian (3) via the Peierls substitution:
$\xi^{0}_{\boldsymbol{k}}\rightarrow\xi^{0}_{\boldsymbol{k}-e\boldsymbol{A}(t)}$,
where $e$ is the electron charge and $c=1$. It is convenient to describe the
dynamics of the meanfield GS in terms of the quantum average of the Anderson’s
pseudo-spin:
$\boldsymbol{\sigma}_{\boldsymbol{k}}=\left\langle\Psi^{\dagger}_{\boldsymbol{k}}\boldsymbol{\tau}\Psi_{\boldsymbol{k}}\right\rangle$,
where:
$\Psi_{\boldsymbol{k}}=\left(c_{\boldsymbol{k}\uparrow},c^{\dagger}_{-\boldsymbol{k}\downarrow}\right)^{T}$.
At the meanfield level, $\boldsymbol{\sigma}_{\boldsymbol{k}}$ describes a
precession motion given by the equations:

$\displaystyle\frac{\,d\boldsymbol{\sigma}_{\boldsymbol{k}}}{\,dt}(t)=2\boldsymbol{b}_{\boldsymbol{k}}(t)\times\boldsymbol{\sigma}_{\boldsymbol{k}}(t),$
(5)

with:
$\boldsymbol{b}_{\boldsymbol{k}}=\left(\Delta^{\prime}_{\boldsymbol{k}},-\Delta^{\prime\prime}_{\boldsymbol{k}},\xi_{\boldsymbol{k}}\right)$.
Imposing:
$\Delta^{\prime}_{\alpha}=-\frac{J}{N}\sum_{\boldsymbol{k}}\gamma_{\alpha}(\boldsymbol{k})\sigma^{x}_{\boldsymbol{k}}$,
$\Delta^{\prime\prime}_{\alpha}=\frac{J}{N}\sum_{\boldsymbol{k}}\gamma_{\alpha}(\boldsymbol{k})\sigma^{y}_{\boldsymbol{k}}$
and $\delta
t_{\alpha}=-\frac{J}{2N}\sum_{\boldsymbol{k}}\gamma_{\alpha}(\boldsymbol{k})\sigma^{z}_{\boldsymbol{k}}$,
one has to solve the Eqs. (5) self-consistently at any time, which is
equivalent to solving the Bloch’s equations. In this way one can track the
dynamics induced by the incident field on both the gap’s function,
$\Delta_{\boldsymbol{k}}(t)$, and the occupation number,
$n_{\boldsymbol{k}}(t)$, including all the non-linear effects emerging for
very intense fields. While $\Delta_{\boldsymbol{k}}(t)$ describes the
evolution of the SC condensate, $n_{\boldsymbol{k}}(t)$ defines the
electromagnetic current flowing through the sample, according to:

$\boldsymbol{J}(t)=-\sum_{\boldsymbol{k}}\nabla_{\boldsymbol{A}}\xi^{0}_{\boldsymbol{k}-e\boldsymbol{A}(t)}n_{\boldsymbol{k}}(t).$
(6)

Experimentally, $\boldsymbol{J}$ can be measured either directly by the
voltage difference at the edge of the sample, or indirectly by the electric
field reflected by the sample. Transmission measurements are not allowed
because of the Si substrate.

##  IV Numerical results: the angle-resolved THG

The current (6) contains explicitly the response of the system at all orders
in ${\bf A}$, since the vector potential enters both in the velocity term
(gradient of the band dispersion) and in the computation of the time evolution
of $n_{\bf k}(t)$. For weak field one recovers the usual linear-response
result, $J^{L}\sim\chi^{(1)}A$, that for a purely monochromatic field at
frequency $\omega_{0}$ implies an induced current at the same frequency
$\omega_{0}$. On the other hand, for increasing strength of the driving field
one can generate a non-linear current whose lowest order scales as
$J^{NL}\sim\chi^{(3)}A^{3}$, so that in the induced current one observes
higher harmonics of the driving field, the lowest one being at $3\omega_{0}$.
Since the THG measurements are usually performed with multicycle light
pulses[20, 21, 26, 28, 29, 31, 33, 34], that are not perfectly monochromatic,
in the numerical calculations we model the vector potential as an oscillatory
function at the central frequency $\omega_{0}$ convoluted with gaussian
envelope:
$\boldsymbol{A}(t)=A_{0}e^{-(t/\tau)^{2}}\sin(\omega_{0}t)\boldsymbol{n}$,
where $A_{0}$ defines the intensity of the field, $\tau$ the duration of the
pulse, and $\boldsymbol{n}$ is the polarization vector, with
$|\boldsymbol{n}|=1$. Since we are interested to exploit a possible resonance
of $\omega_{0}$ with the gap value, we choose $\omega_{0}=1$meV, which is of
the order of $\Delta_{\boldsymbol{k}}$, and $\tau=11$ps, so that the frequency
resolution of the pulse is $\Delta\omega/\omega_{0}\simeq 0.5$. The electric
field is given by: $\boldsymbol{E}(t)=-\partial_{t}\boldsymbol{A}(t)$. The
Fig. 4 shows the power spectrum, $|\boldsymbol{J}(\omega)|^{2}$, as a function
of the frequency, obtained in the chiral solution, at $T=0$, for three
different intensities of the incident field, as specified in the legend,
$E_{peak}$ being the maximum of $|\boldsymbol{E}(t)|$. Here the incident field
is polarized along one of the crystallografic direction, the $x$ axis by
convention. The dashed gray line shows the power spectrum of the incident
field, $|\boldsymbol{E}(\omega)|^{2}$. As expected, the intensity of the THG
at $\omega=3\omega_{0}$ increases with $E_{peak}$ much faster than the first
harmonic, the latter being dominated mostly by the linear response.

Figure 4:  Power spectrum as a function of the frequency, obtained in the
chiral solution, at $T=0$, for three different values of $E_{peak}$. The
dashed gray line shows the power spectrum of the incident field,
$|\boldsymbol{E}(\omega)|^{2}$. The incident field is polarized along one of
the crystallografic direction.

Next, we study the THG by varying the polarization of the incident field:
$\boldsymbol{n}=\cos\theta\hat{x}+\sin\theta\hat{y}$. In particular, we focus
on the two quantities:
$I^{\|}_{THG}=\left|\boldsymbol{n}\cdot\boldsymbol{J}(3\omega_{0})\right|$ and
$I^{\perp}_{THG}=\left|\boldsymbol{n}^{\perp}\cdot\boldsymbol{J}(3\omega_{0})\right|$,
representing the intensity of the THG in the direction parallel and orthogonal
to the incident field, respectively. The Fig. 5 shows $I^{\|}_{THG}$(a) and
$I^{\perp}_{THG}$(b) as a function of $\theta$, corresponding to
$E_{peak}=0.66$kVcm-1 and obtained for the chiral
$d_{x^{2}-y^{2}}+id_{xy}$-wave solution (red line) and for the pure
$d_{x^{2}-y^{2}}$-wave (blue line).

Figure 5:  $I^{\|}_{THG}$(a) and $I^{\perp}_{THG}$(b) as a function of
$\theta$, corresponding to $E_{peak}=0.66$kVcm-1 and obtained for the chiral
$d_{x^{2}-y^{2}}+id_{xy}$-wave solution (red line) and for the pure
$d_{x^{2}-y^{2}}$-wave (blue line).

As is evident, while $I^{\|}_{THG}$ is basically constant, $I^{\perp}_{THG}$
displays typical oscillations. We expect that the isotropic behavior of
$I^{\|}_{THG}$ is mainly due to the leading contribution of the instantaneous
electronic response, which is polarization-independent[35]. If we then focus
on $I^{\perp}_{THG}$, we note a marked difference between the two gap’s
symmetries: in the chiral symmetry, $I^{\perp}_{THG}$ has a period of $\pi/3$,
following from the $C_{6}$ symmetry of the crystal; in contrast, in the pure
$d$-wave symmetry, the period is $\pi$, as a consequence of the $C_{3}$
symmetry breaking which preserves only $C_{2}$. Therefore, the behavior of the
THG as a function of the polarization of the incident field provides an
experimental tool to identify the symmetry of the SC gap in the boron-doped
Sn/Si(111), distinguishing between the chiral $d_{x^{2}-y^{2}}+id_{xy}$\- and
the pure $d$-wave. It’s worth noting that the periodicity of $\pi/3$ cannot in
principle distinguish the chiral $d_{x^{2}-y^{2}}+id_{xy}$\- from the
$s$-wave, which preserves $C_{6}$ too. However, the $s$-wave can be easily
ruled out by the STM spectra reported in the Ref. [16]. Finally, in the
experiments the polarization dependence of $I_{THG}$ might be sensitive to the
effects of disorder[36, 38], not considered here, challenging the use of
ultra-clean samples. Indeed, when disorder is present paramagnetic-like
processes mediating the coupling of the light to SC excitations become
possible, providing additional channels to generate a non-linear response[52,
53, 36, 38]. So far numerical studies for square lattices have shown that
these additional contributions triggered by disorder tend to soften the
polarization dependence of the $I^{\parallel}_{THG}$ signal, irrespective of
the pairing symmetry[36, 37, 38]. On the other hand, the perpendicular
component is expected to still retain the angular dependence of the clean
case, at least in the limit where paramagnetic effects are not quantitatively
dominant. On this respect, the present calculations performed in the clean
limit should remain valid for what concerns the dependence of
$I^{\perp}_{THG}$ on the gap symmetry even when one includes realistic
disorder effects, that are beyond the scope of the present manuscript.

##  V Conclusions

We study the superconductivity reported recently in the Sn/Si(111) ad-layers
doped with boron. Assuming that the interactions responsible for the pairing
are of AFM nature, we describe the system by the $t-J$ model. Our meanfield
analysis well reproduces the value of the AFM coupling, $J$, predicted by
first principle calculations and the $d$-wave symmetry of the SC OP claimed by
recent experimental findings[16]. By perturbing the system with an intense THz
electromagnetic field, we focus on the leading non-linear effects, the THG,
showing that its dependence on the polarization of the incident field can
distinguish between the chiral $d_{x^{2}-y^{2}}+id_{xy}$\- and the pure
$d$-wave symmetry of the SC gap. Therefore, our work i) provides a minimal
efficient theoretical model for describing the pairing in the emerging
superconductor Sn/Si(111) and ii) paves the way for unveiling the pairing
symmetry therein by means of the THz pump protocol, a method that has
succeeded so far in investigating both conventional and high-$T_{c}$ cuprate
superconductors.

## Acknowledgements

We thank Andrea Marini for helpful and interesting conversations.

T. C., C. T. and G. P. acknowledge support from CINECA for computational
resources through ISCRA projects. Research at SPIN-CNR has been funded by the
European Union - NextGenerationEU under the Italian Ministry of University and
Research (MUR) National Innovation Ecosystem grant ECS00000041 - VITALITY, C.
T. acknowledges Università degli Studi di Perugia and MUR for support within
the project Vitality. C. T. acknowledges financial support from the Italian
Ministry for Research and Education through PRIN-2022 project “DARk-mattEr-
DEVIces-for-Low-energy-detection - DAREDEVIL” (IT-MIUR Grant No. 2022Z4RARB).
L.B. acknowledges financial support by Sapienza University under projects
Ateneo (RM12117A4A7FD11B and RP1221816662A977) and by ICSC–Centro Nazionale di
Ricerca in High Performance Computing, Big Data and Quantum Computing, funded
by European Union – NextGenerationEU.

G. P. acknowledges financial support from the Italian Ministry for Research
and Education through PRIN-2017 project “Tuning and understanding Quantum
phases in 2D materials - Quantum 2D” (IT-MIUR Grant No. 2017Z8TS5B) and
fundings from the European Union - NextGenerationEU under the Italian Ministry
of University and Research (MUR) National Innovation Ecosystem grant
ECS00000041 - VITALITY - CUP E13C22001060006.

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Supplementary information for:
Unveiling the pairing Symmetry of the superconducting Sn/Si(111) via angle-
resolved THz pump spectroscopy

##  I The meanfield approximation for the $t-J$ model

We first consider the tight binding Hamiltonian:

$\displaystyle
H_{tb}=\sum_{ij\sigma}\left(t_{ij}-\mu\delta_{ij}\right)c^{\dagger}_{i\sigma}c_{j\sigma}=\sum_{\boldsymbol{k}\sigma}\xi^{0}_{\boldsymbol{k}}c^{\dagger}_{\boldsymbol{k}\sigma}c_{\boldsymbol{k}\sigma},$
(S1)

where:

$\displaystyle\xi^{0}_{\boldsymbol{k}}=\sum_{i}t_{ij}e^{-i\boldsymbol{k}\cdot\left(\boldsymbol{R}_{i}-\boldsymbol{R}_{j}\right)}-\mu,$
(S2)

is the non-interacting band dispersion computed from $\mu$ and the
$\boldsymbol{R}_{i}$’s are the lattice coordinates. In the numerical
calculations, we compute $\xi^{0}_{\boldsymbol{k}}$ by truncating the sum over
the coordinates at the 6th nearest neighbor.

Next, we consider the AFM Hamiltonian:

$\displaystyle H_{J}=J\sum_{\langle
i,j\rangle}\left(\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{j}-\frac{1}{4}\hat{n}_{i}\hat{n}_{j}\right).$
(S3)

In the momentum space, it can be written as:

$\displaystyle
H_{J}=\frac{J}{N}\sum_{\boldsymbol{q}}\sum_{\alpha=1}^{3}\cos(\boldsymbol{q}\cdot\boldsymbol{a}_{\alpha})\left(\boldsymbol{S}_{\boldsymbol{q}}\cdot\boldsymbol{S}_{-\boldsymbol{q}}-\frac{1}{4}\hat{n}_{\boldsymbol{q}}\hat{n}_{-\boldsymbol{q}}\right),$
(S4)

where:

$\displaystyle\boldsymbol{S}_{\boldsymbol{q}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\boldsymbol{k}\sigma\sigma^{\prime}}c^{\dagger}_{\boldsymbol{k}-\boldsymbol{q}/2,\sigma}\boldsymbol{\tau}_{\sigma\sigma^{\prime}}c_{\boldsymbol{k}+\boldsymbol{q}/2,\sigma^{\prime}},$
(S5a) $\displaystyle\hat{n}_{\boldsymbol{q}}$ $\displaystyle=$
$\displaystyle\sum_{\boldsymbol{k}\sigma}c^{\dagger}_{\boldsymbol{k}-\boldsymbol{q}/2,\sigma}c_{\boldsymbol{k}+\boldsymbol{q}/2,\sigma}$
(S5b)

and
$\boldsymbol{a}_{1}=a(1,0),\boldsymbol{a}_{2}=a\left(-1/2,\sqrt{3}/2\right),\boldsymbol{a}_{3}=a\left(-1/2,-\sqrt{3}/2\right)$
are unit lattice vectors. After a lengthy calculations, one can easily rewrite
the Eq. (S4) as:

$\displaystyle
H_{J}=\frac{J}{N}\sum_{\boldsymbol{q}\boldsymbol{k}\boldsymbol{p}}\sum_{\alpha=1}^{3}\cos(\boldsymbol{q}\cdot\boldsymbol{a}_{\alpha})\left(c^{\dagger}_{\boldsymbol{k}-\boldsymbol{q}/2,\uparrow}c_{\boldsymbol{k}+\boldsymbol{q}/2,\downarrow}c^{\dagger}_{\boldsymbol{p}+\boldsymbol{q}/2,\downarrow}c_{\boldsymbol{p}-\boldsymbol{q}/2,\uparrow}-c^{\dagger}_{\boldsymbol{k}-\boldsymbol{q}/2,\uparrow}c_{\boldsymbol{k}+\boldsymbol{q}/2,\uparrow}c^{\dagger}_{\boldsymbol{p}+\boldsymbol{q}/2,\downarrow}c_{\boldsymbol{p}-\boldsymbol{q}/2,\downarrow}\right).$
(S6)

To perform the meanfield approximation of $H_{J}$, we consider all the quantum
averages preserving spin and momentum, we neglect the quantum fluctuations and
we require the time-reversal symmetry. This yields:

$\displaystyle
H^{MF}_{J}=-\frac{J}{N}\sum_{\boldsymbol{k}\boldsymbol{p}}\sum_{\alpha=1}^{3}\gamma_{\alpha}(\boldsymbol{k})\gamma_{\alpha}(\boldsymbol{p})\left[2\left(\left\langle
c_{-\boldsymbol{p}\downarrow}c_{\boldsymbol{p}\uparrow}\right\rangle
c^{\dagger}_{\boldsymbol{k}\uparrow}c^{\dagger}_{-\boldsymbol{k}\downarrow}+h.c.\right)+\frac{n_{\boldsymbol{p}}}{2}\sum_{\sigma}c^{\dagger}_{\boldsymbol{k}\sigma}c_{\boldsymbol{k}\sigma}\right]+E_{0},$
(S7)

where:

$\displaystyle
E_{0}=\frac{2J}{N}\sum_{\alpha}\left|\sum_{\boldsymbol{k}}\gamma_{\alpha}(\boldsymbol{k})\left\langle
c_{-\boldsymbol{k}\downarrow}c_{\boldsymbol{k}\uparrow}\right\rangle\right|^{2}+\frac{J}{4N}\sum_{\alpha}\left[\sum_{\boldsymbol{k}}\gamma_{\alpha}(\boldsymbol{k})n_{\boldsymbol{k}}\right]^{2}$
(S8)

is an energy constant and $n_{\boldsymbol{k}}=\sum_{\sigma}\left\langle
c^{\dagger}_{\boldsymbol{k}\sigma}c_{\boldsymbol{k}\sigma}\right\rangle$. Note
that we have neglected the terms involving only a constant shift of the
chemical potential in the Eq. (S7). The meanfield Hamiltonian is the sum:

$\displaystyle H_{tJ}^{MF}=H_{tb}+H^{MF}_{J}.$ (S9)

Now we define:

$\displaystyle\Delta_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{2J}{N}\sum_{\boldsymbol{k}}\gamma_{\alpha}(\boldsymbol{k})\left\langle
c_{-\boldsymbol{k}\downarrow}c_{\boldsymbol{k}\uparrow}\right\rangle\text{ ,
}\Delta_{\boldsymbol{k}}=\sum_{\alpha}\gamma_{\alpha}(\boldsymbol{k})\Delta_{\alpha}$
(S10a) $\displaystyle\delta t_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{J}{2N}\sum_{\boldsymbol{k}}\gamma_{\alpha}(\boldsymbol{k})n_{\boldsymbol{k}}\text{
,
}\delta\xi_{\boldsymbol{k}}=\sum_{\alpha}\gamma_{\alpha}(\boldsymbol{k})\delta
t_{\alpha}\text{ ,
}\xi_{\boldsymbol{{k}}}=\xi^{0}_{\boldsymbol{{k}}}+\delta\xi_{\boldsymbol{k}},$
(S10b)

which allows to write $H_{tJ}^{MF}$ as:

$\displaystyle
H^{MF}_{tJ}=\sum_{\boldsymbol{k}}\left[\sum_{\sigma}\xi_{\boldsymbol{k}}c^{\dagger}_{\boldsymbol{k}\sigma}c_{\boldsymbol{k}\sigma}+\left(\Delta_{\boldsymbol{k}}c^{\dagger}_{\boldsymbol{k}\uparrow}c^{\dagger}_{-\boldsymbol{k}\downarrow}+h.c.\right)\right]+E_{0},$
(S11)

and:

$\displaystyle
E_{0}=\frac{N}{J}\sum_{\alpha}\left(\frac{\left|\Delta_{\alpha}\right|^{2}}{2}+\delta
t^{2}_{\alpha}\right).$ (S12)

$\Delta_{\alpha}$ and $\delta t_{\alpha}$ have to be obtained self-
consistently by using the Hamiltonian (S11) to compute the quantum averages.
This yields the self-consistent equations:

$\displaystyle\Delta_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{J}{N}\sum_{\boldsymbol{k}}\gamma_{\alpha}(\boldsymbol{k})\frac{\Delta_{\boldsymbol{k}}}{E_{\boldsymbol{k}}}\tanh\left(E_{\boldsymbol{k}}/2T\right),$
(S13a) $\displaystyle\delta t_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{J}{2N}\sum_{\boldsymbol{k}}\gamma_{\alpha}(\boldsymbol{k})\frac{\xi_{\boldsymbol{k}}}{E_{\boldsymbol{k}}}\tanh\left(E_{\boldsymbol{k}}/2T\right),$
(S13b)

that must be solved along with the equation for the electronic density:

$\displaystyle
n=1-\frac{1}{N}\sum_{\boldsymbol{k}}\frac{\xi_{\boldsymbol{k}}}{E_{\boldsymbol{k}}}\tanh\left(E_{\boldsymbol{k}}/2T\right),$
(S14)

which sets the value of $\mu$. In this work we consider: $n=0.9$,
corresponding to the hole doping: $\delta=10\%$. The Fig. S1 shows
$\Delta_{\alpha}$ as a function of the temperature, obtained for the chiral
$d_{x^{2}-y^{2}}+id_{xy}$,solution (a),(b) and for the pure $d$ one (c),(d).
The panels (a),(c) refer to the real part of $\Delta_{\alpha}$, while the
panels (b),(d) to the imaginary part. The different symmetries are color coded
as specified in the inset panel.

Figure S1:  $\Delta_{\alpha}$ as a function of the temperature, obtained for
the chiral $d_{x^{2}-y^{2}}+id_{xy}$,solution (a),(b) and for the pure $d$ one
(c),(d). The panels (a),(c) refer to the real part of $\Delta_{\alpha}$, while
the panels (b),(d) to the imaginary part. The different symmetries are color
coded as specified in the inset panel.

The meanfield energy per site is given by:

$\displaystyle\epsilon_{MF}=\frac{E_{0}}{N}+\frac{1}{N}\sum_{\boldsymbol{k}}\left[\xi_{\boldsymbol{k}}-E_{\boldsymbol{k}}\tanh\left(E_{\boldsymbol{k}}/2T\right)\right]+\mu
n,$ (S15)

where the first two terms come out from averaging $H^{MF}_{tJ}$, while the
last term has to be considered in $\epsilon_{MF}$ because the Hamiltonian
$H_{tJ}$ is defined in the grancanonical ensemble. The Fig. S2 shows the
difference, $\Delta\epsilon_{MF}$, between the meanfield energy per site of
the chiral solution and that of the non-chiral one, as a function of the
temperature.

Figure S2:  Difference between the meanfield energy per site of the chiral
solution and that of the non-chiral one, as a function of the temperature.

##  II Evaluation of the AFM coupling from first principles calculations

We evaluate the nearest-neighbor exchange constant ($J$) between different Sn
atoms through the Hubbard model described by the following Hamiltonian:

$\displaystyle H=J_{fp}\sum_{\langle
i,j\rangle}\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{j}$ (S16)

We adopted a super-cell approach considering a 2$\sqrt{3}\times 3$ cell, with
respect to the Si(111) surface periodicity, with different collinear magnetic
configurations: two Sn atoms per cell magnetized ferromagnetic or anti-
ferromagnetic each other.

Accordingly with literature[10, 6, 8, 9], we modeled the $\alpha$-Sn/Si(111)
surface by considering a layer of Sn atoms on top of three Si-bilayers; the
Si-dangling bonds at the opposite side are capped with hydrogen atoms fixed to
the relaxed positions obtained by capping the pristine Si(111) surface at one
side.

The atomic position of the first five atomic layers are optimized (Sn atoms
and the first two Si-bi-layers) whereas the remaining bi-layer is fixed to the
Si bulk positions. More than 15 Å of vacuum is included. Density functional
theory calculations are performed with the Quantum-Espresso[54, 55] code. We
used ultrasoft pseudopotentials with an energy cutoff up to 45 Ry. Integration
over the Brillouin zone was performed using uniform 12$\times 6\times 1$ grid.

The magnetic calculations were done using the semilocal approximation for the
exchange and correlation term in a GGA+U framework[10]. As showh in Fig.S3,
the obtained results appear to be ”slightly” affected by the choice of the $U$
parameter: in the range $U=$2÷5 eV, $J_{fp}$ is monotonically decreasing as a
function of $U$, without change of sign or order of magnitude.

The most favourable magnetic configuration is the anti-ferromagnetic one, with
an energy difference of the order of $\Delta E_{FM-AFM}\sim 35$÷15 meV with
respect to the the ferromagnetic phase. The calculated antiferromagnetic
exchange coupling results $J_{fp}\sim 10$ meV (for $U=4$ eV, $\Delta E_{FM-
AFM}=0.021$ eV)[10].

Figure S3:  Evolution of the antiferromagnetic exchange coupling as a function
of the Hubbard term $U$.


# Quantum Measurement Without Collapse

Pavel Bóna111email<EMAIL_ADDRESS>

###### Abstract

It is shown that the classical book by von Neumann proposing dynamics of
measured systems with “reduction (or collapse) of system’s wave packet”
contains also hints how to avoid this discontinuity in time evolution of the
measured system (hence it is in this point not quite selfconsistent). The
possibility of avoiding that collapse is a consequence of the observation that
any “human observer” can observe simultaneously just mutually compatible
“observables”. In the paper it is shown how to describe this fact and its
consequences. The proposed interpretation of quantum measurement leads also to
trivial solution of the “Schrödinger Cat Paradox”.

Department of Theoretical Physics, Comenius University, Bratislava

Let us sketch a simple model of any “measurement”:

We need to have a “system”, which is the object of our present interest
intended to be “measured”. The specification of this system should distinguish
it from other “systems”, i.e. some criteria of its identification should be
used. For repeated trials of the experiment, which are unavoidable in such a
statistical branch of science as quantum mechanics (QM), we have to be sure
that repeatedly used “systems” are equal to each other (according to the
experimenter’s criteria). Such a specification procedure is the decisive part
of “preparation” of the system in the first part of experiment. The prepared
system is called to exist in a “specific state”. Next the system is supposed
to move in known environment, so that its state can be considered to be known
from the known laws of nature and calculated from the corresponding known,
generally accepted “equations of motion”.

The remaining parts of the experiment with the prepared systems are
“measurement” and “reading of the results”. These parts of experiments in QM
will be our main interest in this paper. We shall describe a scheme of the
measuring process with observation of results in QM later in this paper,
starting with a brief analysis and criticism of some of its essential features
appearing in [1], which seem to be almost generally accepted in physical
community.

Before that, we shall sketch the key aspects of the general scheme of the
mathematical model of QM.

The formalism of (nonrelativistic) QM

QM describes ‘physical systems’, i.e. mainly microsystems like (stable)
elementary particles, atoms, molecules, and finite collections of them. Each
system is specified by an “algebra of observables” (a C*-algebra), usually
realized as a symmetric algebra of bounded operators on the Hilbert space
$\cal H$ ascribed to the system. Each system can be in any time characterized
by its state. States are described by normalized state-vectors
$\psi,\phi,...\in\cal H$ (equivalently by one-dimensional orthogonal
projectors ${\bf P}_{\psi},{\bf P}_{\phi}...$) or by the so called density
matrices, or density operators, being positive normalized operators
$\rho\in\cal{T(H)}$: $Tr\,\rho=1$, where $\cal{T(H)}$ is the linear space of
all trace-class operators on $\cal H$. Each state-vector $\psi\in\cal H$ is in
a physically equivalent way represented by the density matrix
$\rho_{\psi}:={\bf P}_{\psi}$, so that we can work with “states” and the
“density matrices” as with synonyms.

Observables in QM represent the physical quantities characterizing the
described system, they are identified with selfadjoint operators on $\cal H$.
It is postulated in QM that all selfadjoint operators (or at least the bounded
ones) represent observables, hence some measurable quantities. It is usually
unspecified which physical (measurable) meaning is to be assigned to each
selfadjoint operator. It is often useful to work with abstract setting if the
set of observables forms a C*-algebra (which can be always represented as an
algebra of bounded operators in some Hilbert space), cf. [2].

Each selfadjoint operator is characterized by its spectral projection measure
E, i.e. a projector-valued function of Borel sets on real line $\mathbb{R}$
with values in orthogonal projectors:

$\displaystyle E:B\ (\in{\cal{B}}(\mathbb{R}))\mapsto E(B),\ \
E(\mathbb{R})=I_{\cal H};$ $\displaystyle B_{j}\cap B_{k}=\emptyset\Rightarrow
E(B_{j})E(B_{k})=0,E(\cup_{k}B_{k})=\sum_{k}E(B_{k}).$

If an observable (i.e. a selfadjoint operator) $\mathfrak{A}$ has purely
discrete spectrum, i.e. there is a complete (i.e. such that it cannot be
extended) orthonormal system of “eigenfunctions” or “eigenvectors” in $\cal H$
$\supset$ $\\{\psi_{n},n=1,2,...\\}:\ (\psi_{k}|\psi_{n})=\delta_{k,n}$, and
$\mathfrak{A}\,\psi_{n}=\lambda_{n}\,\psi_{n},\forall n$, then the system can
attain only the values $\lambda_{n}$ as possible values of the quantity
$\mathfrak{A}$, i.e. by measuring $\mathfrak{A}$ only the results
$\\{\lambda_{n},n=1,2,...\\}$ can be obtained. If another observable
$\mathfrak{B}$ can have simultaneosly with $\mathfrak{A}$ (i.e. in the same
complete set of states) sharp values $\\{\mu_{n},n=1,2,...\\}$, the operators
mutually commute; and vice versa: If
$\mathfrak{A}\mathfrak{B}=\mathfrak{B}\mathfrak{A}$, then these operators have
a common complete orthonormal set of eigenvectors
$\\{\chi_{n},n=1,2,...\\},\mathfrak{A}\chi_{n}=\lambda_{n}\chi_{n};\mathfrak{B}\chi_{n}=\mu_{n}\chi_{n}.$

Each vector $\varphi\in\cal H$ can be expressed as a linear combination of any
such orthonormal system: $\varphi=\sum_{n}c_{n}\psi_{n}$, where the
coefficients $c_{n}=(\psi_{n}|\varphi)$ represent the probabilities
$|c_{n}|^{2}$ of obtaining the result $\lambda_{n}$ by measurement of the
observable $\mathfrak{A}$ in the state $\varphi$.

QM is an “intrinsically statistical theory”, which means that there is no
state of any system in which all the observables have specific values: the
state-vector or density matrix of a specific state contains just information
on “probability distributions” of all observables in that state; a sharp value
of some observable in a given state is a rather exceptional possibility. This
leads to serious problems of understanding and description of the measurement
process: Each time after measurement of some quantity of a single system (i.e.
after just a single “particle” was interacting with the measuring apparatus),
the apparatus finds one specific value also in the case if the measured
quantity of the considered system in the measured state has just probability
distribution of possible values of the quantity with nonzero dispersion. If
the observer wants to obtain the form of that probability distribution by the
measurement, he has to repeat the same measurement on many copies of “equally
prepared” system under consideration ($\approx$ particle). This leads to
questions we intend to discuss later in this paper.

The dynamics of QM-systems (similarly as in classical mechanics) is specified
by the specification of the operator $\bf H$, the “system’s Hamiltonian”
corresponding to the “energy”. If the considered system does not interact with
other systems, then the time evolution of its state-vectors is:

$\psi(t)=e^{-it\bf H}\,\psi(0),\ \text{or }\ {\bf P}_{\psi(t)}=e^{-it\bf
H}{\bf P}_{\psi(0)}e^{it\bf H}.$

If the considered system is not isolated, but interacts with another system,
then this pair of systems is considered as a new larger isolated system with
its own new Hamiltonian, consisting of the sum of the “free Hamiltonians” of
each system and, in addition, of some "interaction Hamiltonian" characterizing
the mutual interaction of the two systems.

The only problematic issue in this abstract general scheme is the question of
some satisfactory description of the evolution of the considered system in the
time of the physical act of measurement of some specific observable: Besides
the “measured system” also some macroscopic apparatus is acting (and
interacting), changing its state so that the human observer can read the
result of the measurement. The proposed description of this process leads also
to an answer to the question “how can the Schrödinger Cat be simultaneously
dead and alive”.

The essential points of the von Neumann scheme of measurement in QM

(i) The measured system S in the state $\rho$ interacts with the measuring
macroscopic apparatus M chosen for measurement of a specific observable
$\mathfrak{A}$. In general, according to von Neumann, a chain of other
similarly working apparatuses ${\bf M_{1}+M_{2}+...}$, all of them described
as quantum-mechanical systems, can be added to M, ending either by a human
observer (or even by his brain), or an apparatus in which the result can be
fixed by some “pointer position” (and prepared for observation by human
observers).

(ii) The states of above mentioned chain of apparatuses are changed due to
interaction with S and the following evolution by continuous unitary evolution
operator of the composed system (=process 2, cf. [1, Ch. V.1 p.230]). Hence
the measured state $\rho$ is transformed subsequently isomorphically in the
instant of the interaction-measurement by the measuring device M to the states
of the measuring systems ${\bf M+M_{1}+M_{2}+...}$ by unitary time evolution.
So, we shall call the subsequently arising states of the apparatuses ${\bf
M+M_{1}+}$ ${\bf M_{2}+...}$ “the measured state $\rho$ of S”. It can be seen
that the whole chain of apparatuses can be considered effectively as a single
apparatus M.

(iii) The last element of the chain modifies the observed state $\rho$ by an
abrupt change depending on the measured observable $\mathfrak{A}$. We shall
assume, for simplicity, that $\mathfrak{A}$ has purely discrete nondegenerate
spectrum with (complete orthonormal) set of eigenvectors
$\\{\psi_{n},n=1,2,...\\}$:
$\mathfrak{A}\,\psi_{n}=\lambda_{n}\,\psi_{n},\forall n$. The mentioned abrupt
change looks as follows [1, p.230]:

$\rho\mapsto\rho^{\prime}:=\sum_{n}(\psi_{n}|\rho|\psi_{n})\,{\bf
P}_{\psi_{n}}.$

This is description of the famous “collapse of the wave packet”; the change
$\rho\mapsto\rho^{\prime}$ is considered to be instantaneous. This change
cannot be reached by any unitary transformation, hence it is not accessible by
Schrödinger-like continuous evolution with any Hamiltonian (in nontrivial
cases $\rho^{\prime}\neq\rho$).

(iv) The von Neumann’s book [1] also contains, however, certain notes which
could (and will!) enable us to avoid the above mentioned collapse. Namely from
the considerations in [1, p.262] it clearly follows that any final human
observer (as the last element of the chain of M’s) can observe simultaneously
measurable, hence mutually commuting observables only.222This assertion was
even more explicitly expressed in the original German version of [1], in the
paragraph situated between the last two paragraphs of [1, p.261], which was
omitted in the English edition.

A proposal for alternative description of quantum measurement process

Let us note first that there is an ambiguity in von Neumann’s interpretation
of the reduced density matrix: Since the set of density matrices does not form
a simplex, the convex decomposition of nontrivial density matrices to extremal
elements is highly nonunique. Moreover, various decompositions of a given
density matrix may be incompatible: Different decompositions may contain
states corresponding to eigenvectors of mutually noncommuting observables. If
it is possible to describe the presently considered physical situation by
elements of a simplex which are unambiguously decomposable into extremal
elements having clear physical meaning, then these elements can be interpreted
as “proper mixtures”. We shall see that the state described by the reduced
density matrix can be naturally transformed into such a form.

Accepting (essentially) the point (ii) of our description of the von Neumann
scheme of the measurement on a system S occurring in a state
$\psi:=\sum_{j}c_{j}\psi_{j}$ by a measuring (quantum macroscopic) apparatus
M, we assume that the interaction of S+M leads to such a change of the state
of the combined system where the macroscopically observable and mutually
distinguishable state-vectors $\Phi_{j}$ corresponding to the values
$\lambda_{j}$ of the observable $\mathfrak{A}$ and to the states of $\psi_{j}$
of S (i.e. representing the corresponding “pointer positions”) are (via
unitary mapping by U) in one to one correspondence with the states $\psi_{j}$.
This is in agreement with [1, p.285-286]:

$U:\
\psi\otimes\Phi_{0}=\sum_{j}c_{j}\psi_{j}\otimes\Phi_{0}\mapsto\sum_{j}c_{j}\psi_{j}\otimes\Phi_{j},$

where $\Phi_{0}$ is the state of M just before the measurement interaction and
$\Phi:=\sum_{j}c_{j}\Phi_{j}$ is the state of M immediately after (or in the
instant of, if the measurement is instantaneous) the measurement, obtained by
partial trace of the above described “two-systems state-vector” w.r.t. the
measured system, cf. [3].

The following fate (i.e the subsequent dynamical evolution) of the state of
the system S after the interaction with macrosystem M might be very diverse,
c.f. [4, §7, esp. p.24]: S could be absorbed by the macrosystem M, or just
dispersed by M with various simultaneous changes of its state. For the result
of the measurement this is not important. However, such a simple (and even
general) scheme that the measured state of S after its interaction with the
macroscopic measuring device M will continue its dynamical evolution, just
“reduced” to the eigenstate of the measured quantity corresponding to the
obtained measured eigenvalue, is rather popular but nonrealistic
simplification and deformation of the process.

What is important, however, is the imprint left by S on M: Because the
resulting “pointer positions” after each single run of the experiment should
be repeatedly visible by different people, the states $\Phi_{j}$ should be
‘sufficiently stable’, or stationary w.r.t. the dynamics (of at least
macroscopic observables) of M. Then the observable of M with eigenvectors
$\Phi_{j}(j=1,2,\dots)$ can be denoted by $\tilde{\mathfrak{A}}$, because it
is ‘a copy‘ of the corresponding observable $\mathfrak{A}$ of the measured
system S. So, this $\tilde{\mathfrak{A}}$, or its eigenvalues, are what has to
be observed by any human observer taking part in the considered measuring
process. But according to (iv) above, all the observables of M that are
simultaneously observable by any human observer must, as the corresponding
operators, mutually commute.

Let us form from these mutually observable quantities of M, necessarily
including our macroscopic $\tilde{\mathfrak{A}}$, a C*-algebra $\cal A$. Here
there is some ambiguity in our construction – the choice of the C*-algebra
might depend on the ability of observers to observe certain observables from
$\cal A$.333This corresponds to the ambiguity of the definition of macroscopic
observables in finite systems; they cannot be unambiguously defined in an
abstract theory. We could simply choose $\cal A$ as the maximal commutative
subalgebra of all observables of M containing $\tilde{\mathfrak{A}}$; but this
choice would necessarily contain also microscopic variables of the macroscopic
system M. So, let us take some ‘convenient’ abelian C*-algebra $\cal A$
containing $\tilde{\mathfrak{A}}$, e.g. the C*-algebra with unit element
generated by $\tilde{\mathfrak{A}}$ (as the minimal possibility).

What are the states of M accessible to our human observers? These are the
states, i.e. positive normalized linear functionals, of the abelian C*-algebra
$\cal A$. Let us describe the state space of $\cal A$, cf. [2, 5]: For each
abelian C*-algebra $\cal A$ there exists its spectrum space $\cal M$, i.e. a
compact subspace in the w*-topology of the whole state space $\cal S(A)$ of
$\cal A$, consisting of its extremal (pure) states. The points of $\cal M$ are
in bijective correspondence with the pure (=extremal) states on $\cal A$. The
states $\cal S(A)$ consist of all probability Radon measures on $\cal M$,
which form a (Choquet) simplex (in the same sense as classical probability
distributions on the phase space of classical mechanics form a simplex: each
such probability distribution is a convex combination-integral of the
functions supported by single points ($\delta$ functions)). This is valid for
an arbitrary choice of the commutative C*-algebra. Note that the
characteristic property of any simplex is that it is a subset of a linear
space such that each of its elements has unequivocal convex decomposition into
its extremal elements.

Let us see how the state ${\bf P}_{\psi}$ of S corresponding to the measured
state-vector $\psi:=\sum_{j}c_{j}\psi_{j}$, or its unitary “macroscopic copy”
$\Phi$, as described above, looks from the point of view of a human observer.
In the basis $\Phi_{j}(j=1,2,\dots)$ – the unitary “copy” of
$\psi_{j}(j=1,2,\dots)$ – the measured state-vector
$\psi=\sum_{j}c_{j}\psi_{j}$ is transformed into:

$\Phi:=\sum_{j}c_{j}\Phi_{j}.$

Since we are interested in the states on $\cal A$ only, we have to restrict
the state ${\bf P}_{\Phi}$ as a linear functional on all the observables of M
to the observables from $\cal A$. The transformed macroscopic form
$\tilde{\mathfrak{A}}$ of $\mathfrak{A}$ has in the Hilbert space $\cal H_{\bf
M}$ of M the complete orthonormal set of eigenvectors $\Phi_{j}$. The
“measured” state $\Phi$ applied to any $a\in\cal A$ (as a linear functional on
$\cal A$) yields

$\Phi:a\mapsto
Tr(P_{\Phi}a)=(\Phi|a|\Phi)=(\sum_{j}c_{j}\Phi_{j}|a|\sum_{k}c_{k}\Phi_{k})=\sum_{j}|c_{j}|^{2}(\Phi_{j}|a|\Phi_{j}),\forall
a\in\cal A,$

because $\\{\Phi_{j}:j=1,2,\dots\\}$ is an orthonormal set of eigenvectors of
all the elements of $\cal A$.

We see that a “pure state-vector” on the whole algebra of observables
$L(H_{\bf M})$ decomposes into a genuine (=proper) mixture of the states
corresponding to single elements of the “eigenbasis” of the commutative
algebra $\cal A$. This is identical to the above described von Neumann’s
“collapse of the wave packet” for the case $\rho:={\bf P}_{\Phi}$.

Is the Schrödinger Cat simultaneously dead and alive ?

Accepting that any human observer of macroscopic (hence visible) system can
distinguish by his senses mutually commuting observables only, we have also
trivial resolution of the “Problem of the Cat”.

Let us have two macroscopic state-vectors $\Psi_{1},\ \Psi_{2}$ of the
observed system, which are mutually orthogonal (e.g. ‘dead and alive cat’, or
two orthogonal states of a very long spin chain). For any observable
$\mathfrak{A}$ for which these states are eigenstates it holds

$(\Psi_{1}|\mathfrak{A}|\Psi_{2})=0.$

This is valid for all $\mathfrak{A}$’s from any commutative algebra. For a
state of the form

$\Psi:=c_{1}\Psi_{1}+c_{2}\Psi_{2},\ c_{1}c_{2}\neq 0$

we have

$(\Psi|\mathfrak{A}|\Psi)=|c_{1}|^{2}(\Psi_{1}|\mathfrak{A}|\Psi_{1})+|c_{2}|^{2}(\Psi_{2}|\mathfrak{A}|\Psi_{2}),$

which describes a mixture applied to any given element of the commutative
algebra of “observable observables”. In the case of the Cat this expresses the
situation of looking at the Cat (statistically, in the sense of QM) and
observing the two possibilities: the Cat is either dead or alive, with
probabilities $|c_{1}|^{2}$ and $|c_{2}|^{2}$ respectively, but not both
simultaneously. The above expression corresponds to a genuine (=proper)
mixture of the states $\Psi_{1},\Psi_{2}$.

This also shows how to resolve the problem of possible interferences of states
of large but finite subsystems of an infinite system, e.g. a spin chain. We
can consider, at least “ideologically”, problems like those arising e.g. in
[6, Sec. 7.6] in the construction of “Models of Quantum Measurement” to be
solved.

## References

  * [1] J. von Neumann: Mathematical Foundations of Quantum Mechanics: New Edition, translated from German by R. T. Beyer (with the assistence of the author), 1949; edited by N. A. Wheeler, Princeton University Press, Princeton, 2018.
  * [2] S. Sakai: C*-algebras and $W^{*}$-algebras, Springer, Berlin, 1971.
  * [3] E. B. Davies: Quantum Theory of Open Systems, Academic Press, Massachussetts, 1976.
  * [4] L. D. Landau and E. M. Lifshitz: Quantum Mechanics – Non-relativistic Theory, vol.3 of the Course of Theoretical Physics, translated from Russian by J. B. Sykes and J. S. Bell, 1958; Pergamon Press, Princeton, 1991.
  * [5] M. A. Naimark: Normirovannyje kolca (Normed Rings), Nauka, Moscow, 1968.
  * [6] P. Bóna: Classical Systems in Quantum Mechanics, Springer, Berlin, 2020.