# Emergent quantum probability from full quantum dynamics and the role of
energy conservation

Chen Wang  Department of Physics, Zhejiang Normal University, Jinhua, Zhejiang
321004, China  Jincheng Lu  Jiangsu Key Laboratory of Micro and Nano Heat
Fluid Flow Technology and Energy Application, School of Physical Science and
Technology, Suzhou University of Science and Technology, Suzhou, 215009, China
Jian-Hua Jiang<EMAIL_ADDRESS>Suzhou Institute for Advanced Research,
University of Science and Technology of China, Suzhou, 215123, China

###### Abstract

We propose and study a toy model for the quantum measurements that yield the
Born’s rule of quantum probability. In this model, the electrons interact with
local photon modes and the photon modes are dissipatively coupled with local
photon reservoirs. We treat the interactions of the electrons and photons with
full quantum mechanical description, while the dissipative dynamics of the
photon modes are treated via the Lindblad master equation. By assigning double
quantum dot setup for the electrons coupling with local photons and photonic
reservoirs, we show that the Born’s rule of quantum probability can emerge
directly from microscopic quantum dynamics. We further discuss how the
microscopic quantities such as the electron-photon couplings, detuning, and
photon dissipation rate determine the quantum dynamics. Surprisingly, in the
infinite long time measurement limit, the energy conservation already dictates
the emergence of the Born’s rule of quantum probability. For finite-time
measurement, the local photon dissipation rate determines the characteristic
time-scale for the completion of the measurement, while other microscopic
quantities affect the measurement dynamics. Therefore, in genuine
measurements, the measured probability is determined by both the local devices
and the quantum mechanical wavefunction.

##  I Introduction

The emergence of probability is one of the key features in quantum mechanics
that has been puzzling since the birth of quantum mechanics sw1989ap ;
sw1989prl . In 1926 Max Born proposed his theory where quantum states are
interpreted as probability waves mborn1926zp ; mborn1954nl . This probability
wave theory connects the probability in quantum measurements with the well-
known quantum probability, i.e., the square of the absolute value of the
wavefunction. This interpretation, though gives ease in using quantum theory
to compute experimental observables, it never fully satisfied the questioning
minds in the past century usinha2010science ; abassi2013rmp ; pshadbolt2014np
; mop2021prl . In particular, how could a wave be probabilistic when the wave
equation (i.e., the Schrödinger equation) is deterministic? What if the
connection between the quantum wavefunction and the measured probability is an
emergent phenomenon, i.e., it emerges from the quantum dynamics of the
measurement process instead of a direct connection as interpreted by Born?

In the past decades, it has been accepted that probability emerges in quantum
mechanics due to the interaction between the quantum system and the
measurement instrument jaw1984book ; hmwiseman2011book . The instrument
generally contains very large degrees of freedom and serves as an environment
interacting with the quantum system. Such interaction causes quantum
decoherence that leads to the loss of certainty and the emergence of quantum
probability whz2003prl ; whz2005pra ; ms2005fop ; um2004ijqi . Several models
have been proposed to explore the underlying physics of quantum measurement
aea2013rmp , e.g., the Coleman-Hepp model bs2001book , the Gaveau-Schulman
model gb1990jsp , and the Curie-Weiss model aea2003epl . In particular, the
Curie-Weiss model consisting of a spin 1/2 measured by coupled-spins
interacting with one phonon reservoir, was used to elucidate the practical
measurement in spin systems aea2013rmp . However, a simple microscopic model
of quantum measurements which can demonstrate the emergence of Born’s quantum
probability from a full quantum mechanical description is still helpful in
understanding the underlying physics of quantum measurements.

Here, we propose such a model by considering the detection of electron’s
probability in a double quantum dot setup via the local electron-photon
interactions mrd2011prl ; yyliu2014prl ; xmi2017science ; trh2018prl . In this
model, electrons interacts with local photon modes and the photon modes
dissipate their energy into local photonic reservoirs. We treat the electron-
photon interaction in a fully quantum mechanical framework and describe the
photon dissipation via the Lindblad master equation uweiss2012book ;
hpb2002book . We propose to measure the quantum system by monitoring the
photonic energy dissipation. By solving the quantum dynamic equations, we find
that in the infinite long time limit, the photonic energy dissipation into the
reservoir is equal to the input electron energy. Since the latter is
proportional to the square of the absolute value of the wavefunction, the
energy conservation already dictates that the measured quantum probability is
equal to the Born’s rule of quantum probability. For finite time measurements,
the local photon dissipation rate determines the characteristic timescale for
the completion of the measurement, while other microscopic quantities affect
the measurement dynamics. We further discuss how electron-photon detuning,
strong electron-photon interaction, and finite reservoir temperature affect
the measurement dynamics. Our work reveals the underlying microscopic quantum
physics of quantum measurements and can serve as a prototype model to study
the origin of the Born’s rule on the quantum probability. Throughout this
work, we set Planck constant $\hbar\equiv 1$.

Figure 1: (Color online) (a) The schematic illustration of the electron-photon
model at Eq. (II.1). It is composed of two individual quantum dots, each
owning one electron and interacting with single-mode photonic resonator. (b)
The description of quantum probability readout from the quantum dots at Eq.
(7) via the electron-photon interaction $\lambda_{v}$ and the photon damping
process $\kappa_{v}$. (c) The microscopic energy conservation process during
the measurement process at Eq. (8), i.e., the energy loss from the electron-
photon hybrid system is identical with the gain into the reservoir, which is
characterized as the energy current $J_{v}$.

##  II Measurement setup

We begin by introducing a generic quantum system that consists of two
individual quantum dots interacting with one photonic resonator syzhu1988pla ;
vva2012prl ; pstegman2022acs , to describe the probability measurement of the
electron wavefunction through the electron-photon interactions, where the
photonic resonators may be treated as the probability detectors rl2020prx ;
ea2021nature . Then, we analyze the dissipative dynamics of the electron-
photon interacting model to practically read out information on the quantum
probability of the electron wavefunction in each quantum dot. We note that the
measurement description of the electron-photon interaction and photon
dissipation is fully quantum.

###  II.1 Electron-photon Model

The electron-photon system is modeled as two quantum dots each owning single
electron, separately coupled to two photonic resonators, shown in Fig. 1(a).
The Hamiltonian is described as $H_{s}=\sum_{v={l},{r}}H_{s,v}$. The $v$-th
system component is specified as

$\displaystyle~{}H_{s,v}$ $\displaystyle=$
$\displaystyle\sum_{j=e,g}E_{j,v}|j_{v}{\rangle}{\langle}j_{v}|+\omega_{0,v}a^{\dagger}_{v}a_{v}$
$\displaystyle+\lambda_{v}(a^{\dagger}_{v}|g_{v}{\rangle}{\langle}e_{v}|+a_{v}|e_{v}{\rangle}{\langle}g_{v}|),$

where $|e_{v}{\rangle}$ and $|g_{v}{\rangle}$ are the excited and ground
states with the corresponding occupation energies $E_{v,e}$ and $E_{v,g}$ in
the $v$-th quantum dot, $a^{\dagger}_{v}~{}(a_{v})$ is the creating
(annihilating) operator of the photonic resonator with the photon energy
$\omega_{0,v}$, $\lambda_{v}$ denotes the electron-photon coupling strength.
In the following, we set $E_{e,{l}}=E_{e,{r}}=0$ without losing any
generality.

Considering $H_{s,v}$ in the subspace
$\\{|n_{v}-1,e_{v}{\rangle},|n_{v},g_{v}{\rangle}\\}$ with $n_{v}$ being the
photon excitation number, it is straightforward to find that the $n$-th
eigenvalues of the electron-photon model are given by
$E_{n_{v},\pm}=\omega_{0,v}n_{v}+\frac{E_{g,v}-\omega_{0,v}}{2}\pm\sqrt{(\frac{\omega_{0,v}+E_{g,v}}{2})^{2}+\lambda^{2}_{v}n_{v}}$,
and the eigenvectors are given by

$\displaystyle|\psi_{n_{v},+}\rangle=\cos{\frac{\theta_{n_{v}}}{2}}|n_{v},g_{v}\rangle+\sin{\frac{\theta_{n_{v}}}{2}}|n_{v}-1,e_{v}\rangle,$
(2a)
$\displaystyle|\psi_{n_{v},-}\rangle=-\sin{\frac{\theta_{n_{v}}}{2}}|n_{v},g_{v}\rangle+\cos{\frac{\theta_{n_{v}}}{2}}|n_{v}-1,e_{v}\rangle,$
(2b)

with
$\tan\theta_{n_{v}}=2\lambda_{v}\sqrt{n_{v}}/(\omega_{0,v}+E_{g,v})~{}(n_{v}{\geq}1)$.
Moreover, we define the effective ground state
$|\psi_{0_{v}}{\rangle}=|0,g_{v}{\rangle}$ and the corresponding energy
$E_{0,v}=E_{g,v}$. It need note that the real bare ground state actually is
$|0,g_{v}{\rangle}$. However, this state will never take participate into the
dissipative dynamics of the dot-photon system.

###  II.2 Dynamics

We consider master dynamics of the electron-photon model with weak photon
dissipation uweiss2012book ; hpb2002book ; pstegman2022acs

$\displaystyle~{}\frac{d\rho_{s}}{dt}$ $\displaystyle=$
$\displaystyle-i[H_{s},\rho_{s}]$
$\displaystyle-\sum_{v={l},{r}}\frac{\kappa_{v}}{2}[a^{\dagger}_{v}a_{v}\rho_{s}+\rho_{s}a^{\dagger}_{v}a_{v}-2a_{v}\rho_{s}a^{\dagger}_{v}],$

where $\kappa_{v}$ means the dissipation strength of photons from the $v$-th
resonator into the corresponding zero-temperature environment (e.g., the
$v$-th photon counter). Here, the dissipation of quantum dots is assumed
negligible compared to the photon damping strength. Then, we select the
initial state of the electron-photon model. There exists one electron in each
quantum dot, i.e.,
$|\phi_{e}(0)=C_{l}|e_{{l}},g_{{r}}{\rangle}+C_{r}|g_{{l}},e_{{r}}{\rangle}$,
and resonators are initially both in vacuum before the procedures of the
probability readout. Hence, the initial wavefunction is generally described as

$\displaystyle~{}|\phi(0){\rangle}$ $\displaystyle=$ $\displaystyle
C_{l}|0_{l},e_{{l}}{\rangle}{\otimes}|0_{r},g_{{r}}{\rangle}+C_{r}|0_{l},g_{{l}}{\rangle}{\otimes}|0_{r},e_{{r}}{\rangle},$
(4)

where $C_{v}$ is the wavefunction coefficient.

We focus on the individually measuring the quantum probability in the local
dots man2002book ; jcorler2020prl . Consequently, the initial reduced density
matrix of the $v$-th system component is obtained as

$\displaystyle~{}\rho_{v}(0)=\mathbf{P}_{v}|0_{v},e_{v}{\rangle}{\langle}0_{v},e_{v}|+\mathbf{P}_{\overline{v}}|0_{v},g_{v}{\rangle}{\langle}0_{v},g_{v}|,$
(5)

where the Born’s quantum probabilities are expressed as

$\displaystyle~{}\mathbf{P}_{v}=|C_{v}|^{2},$ (6)

with ${\overline{v}}={r}$ for $v={l}$ and ${\overline{v}}={l}$ for $v={r}$.
Hence, the dissipative dynamics of the $v$-th component is expressed as

$\displaystyle~{}\frac{d\rho_{v}}{dt}=-i[H_{s,v},\rho_{v}]-\frac{\kappa_{v}}{2}[a^{\dagger}_{v}a_{v}\rho_{v}+\rho_{v}a^{\dagger}_{v}a_{v}-2a_{v}\rho_{v}a^{\dagger}_{v}],$
(7)

with $\rho_{{l}({r})}=\textrm{Tr}_{{r}({l})}\\{\rho_{s}\\}$, which is also
exhibited in Fig. 1(b). Moreover, the dynamics is constraint to the lowest
three eigenstates, i.e., $|\psi_{1_{v},+}\rangle$, $|\psi_{1_{v},-}\rangle$,
and $|\psi_{0_{v}}{\rangle}$. And it is straightforward to show the inverse
relations
$|0_{v},e_{v}{\rangle}=(\sin{\frac{\theta_{v}}{2}}|\psi_{1_{v},+}\rangle+\cos{\frac{\theta_{v}}{2}}|\psi_{1_{v},-}\rangle)$
and
$|1_{v},g_{v}{\rangle}=(\cos{\frac{\theta_{v}}{2}}|\psi_{1_{v},+}\rangle-\sin{\frac{\theta_{v}}{2}}|\psi_{1_{v},-}\rangle)$,
with $\tan{\theta_{v}}={2\lambda_{v}}/{(\omega_{0,v}+E_{g,v})}$.

##  III Quantum probability Measurements

In this section, we investigate the probability measurement at weak and strong
electron-photon couplings and finite temperature reservoir, respectively.

###  III.1 Measurement with weak electron-photon coupling

We first study the probability measurement based on the quantum dots and
photonic resonators in the long time evolution limit, i.e., complete
measurement. Then, we further analyze the finite time behavior of the emitted
photons, i.e., finite time measurement. Finally, we briefly discuss the
influence of finite temperature reservoir on measured probability.

####  III.1.1 Complete measurement

We investigate the probability measurement from the energy perspective in the
double quantum dot setup via the interaction with local photon resonators,
with the initial state at Eq. (4). Specifically, the energy flow from the
$v$-th system component into the dissipative reservoir (e.g., the $v$-th
photon counter) can be defined as

$\displaystyle~{}J_{v}=-d{\langle}H_{s,v}{\rangle}(t)/dt,$ (8)

which quantifies the energy loss of the $v$-th component, shown in Fig. 1(c).
Meanwhile, in the eigenspace of $H_{s,v}$, the expression of the energy flow
can be reexpressed as
$J_{v}=-[E_{+,v}(dP_{v,+}(t)/dt)+E_{-,v}(dP_{v,-}(t)/dt)]$, with the density
matrix elements denoting
$P_{v,\pm}(t)=\langle\psi_{1_{v},\pm}|\rho_{v}(t)|\psi_{1_{v},\pm}\rangle$ and
$P_{v,+-}(t)=\langle\psi_{1_{v},+}|\rho_{v}(t)|\psi_{1_{v},-}\rangle$, and
energies given by

$\displaystyle~{}E_{+(-),v}=\Big{[}(\omega_{0,v}-E_{g,v}){\pm}\sqrt{(\omega_{0,v}+E_{g,v})^{2}+4\lambda^{2}_{v}}\Big{]}/2,$
(9)

which characterizes the microscopic energy transport processes, e.g.,
$E_{+,v}[-dP_{v,+}(t)/dt]$ characterizing the relaxation of the system from
the channel $|\psi_{1_{v},+}{\rangle}$ to $|\psi_{0_{v}}{\rangle}$ by
releasing the energy $E_{+,v}$ and one excitation particle. After sufficient
long time evolution, it is straightforward to find that the total energy loss
of the quantum system is obtained as
$\int^{\infty}_{0}dt[-d{\langle}H_{v}{\rangle}(t)/dt]=-E_{g,v}|C_{v}|^{2}$.
Simultaneously, the microscopic transferred energy to the reservoir is given
by integrating the current

$\displaystyle-\int^{\infty}_{0}dt\Big{[}E_{+,v}\frac{dP_{v,+}(t)}{dt}+E_{-,v}\frac{dP_{v,-}(t)}{dt}\Big{]}=-E_{g,v}|C_{v}|^{2}.$
(10)

This clearly demonstrates the microscopic energy conservation, i.e., the
energy of the electron in the $v$-th dot is fully dissipated into the
corresponding reservoir via two eigen-channels.

Furthermore, it is interesting to find that Based on the master equation, the
energy flow at Eq. (8) can be reexpressed as

$\displaystyle~{}J_{v}$ $\displaystyle=$ $\displaystyle
E_{+,v}\kappa_{v}[\cos^{2}\frac{\theta_{v}}{2}P_{v,+}(t)-\cos\frac{\theta_{v}}{2}\sin\frac{\theta_{v}}{2}\textrm{Re}\\{P_{v,+-}\\}]$
$\displaystyle+E_{-,v}\kappa_{v}[\sin^{2}\frac{\theta_{v}}{2}P_{v,-}(t)-\cos\frac{\theta_{v}}{2}\sin\frac{\theta_{v}}{2}\textrm{Re}\\{P_{v,+-}\\}].$

Both the populations and coherence terms contribute to the energy flow.
Moreover, the dynamics of the photon number in the $v$-th resonator is
described as

$\displaystyle~{}\langle a^{\dagger}_{v}a_{v}\rangle(t)$ $\displaystyle=$
$\displaystyle\cos^{2}{\frac{\theta_{v}}{2}}P_{v,+}(t)+\sin^{2}{\frac{\theta_{v}}{2}}P_{v,-}(t)$
$\displaystyle-2\cos{\frac{\theta_{v}}{2}}\sin{\frac{\theta_{v}}{2}}\textbf{Re}[P_{v,+-}(t)].$

Then, we consider the dynamical equation at Eq. (7) and integrate the photonic
energy flow $\int^{\infty}_{0}J_{v}dt$ at Eq. (III.1.1), which is connected
with the countered photon numbers (see Appendix for the details). By defining
the measured quantum probability

$\displaystyle~{}\mathscr{P}_{v}(t){\equiv}\kappa_{v}\int^{\infty}_{0}dt\langle
a^{\dagger}_{v}a_{v}\rangle(t),$ (13)

the complete quantum probability measurement inherited from microscopic energy
conservation is established as

$\displaystyle~{}\mathscr{P}_{v}(\infty)=\mathbf{P}_{v},$ (14)

with $\mathbf{P}_{v}=|C_{v}|^{2}$. This relation denotes that by practically
counting the emitted photon number from the individual resonator to the
external sink (i.e., the measured probability), we are able to correctly
detect the quantum probability of the electron wavefunction in each dot, once
the dissipation rate $\kappa_{v}$ is available. And the measured probability
is irrelevant with the electron-photon system parameters, e.g.,
$\omega_{0,v}$, $E_{g,v}$, and $\lambda_{v}$. Therefore, we conclude that the
microscopic energy conservation is able to yield the complete probability
measurement.

Alternatively, from the particle conservation perspective, i.e., from the
excitation particles in the hybrid system to detected photon particles via the
photon dissipation processes, we find the dynamics of the ground state
population
($P_{v,0}(t)={\langle}\psi_{0_{v}}|\rho_{v}(t)|\psi_{0_{v}}{\rangle}$) is
tightly related with the photon flow, i.e.,
${dP_{v,0}(t)}/{dt}=\kappa_{v}\langle a^{\dagger}_{v}a_{v}\rangle(t)$. This
demonstrates that the probability of electrons transport into the ground state
can be read out via the emitted photon flows $\kappa_{v}\langle
a^{\dagger}_{v}a_{v}\rangle(t)$. After the long time evolution the complete
probability measurement at Eq. (14) can be fully recovered, considering
$P_{v,0}(0)=|C_{\overline{v}}|^{2}$ and $P_{v,0}(\infty)=1$.

####  III.1.2 Finite time measurement

Here, we study the probability measurement at finite time dynamics. Since the
complete probability measurement holds individually for each electron-photon
interacting component, we consider the initial state at Eq. (4). It should
note that even for such a concise reduced density matrix Eq. (5), the general
solution of dissipative dynamics is hard to obtain, due the coupling between
diagonal and off-diagonal elements of the electron-photon system. While if we
include the secular approximation, the equations of motion of system density
matrix are reduced to
${dP_{v,+}}/{dt}{\approx}{-\kappa_{v}}\cos^{2}\frac{\theta_{v}}{2}P_{v,+}$,
${dP_{v,-}}/{dt}{\approx}{-\kappa_{v}}\sin^{2}\frac{\theta_{v}}{2}P_{v,-}$,
and
${dP_{v,+-}}/{dt}\approx-(i\Lambda_{v}P_{v,+-}+\frac{\kappa_{v}}{2}P_{v,+-})$,
with the energy gap
$\Lambda_{v}=\sqrt{({\omega_{0,v}+E_{g,v}})^{2}+4\lambda^{2}_{v}}$ between
$|\psi_{1_{v},+}{\rangle}$ and $|\psi_{1_{v},-}{\rangle}$. The dynamics of
density matrix elements are given by
$P_{v,+}(t)\approx|C_{v}|^{2}\exp(-\kappa_{v}\cos^{2}\frac{\theta_{v}}{2}t)\sin^{2}\frac{\theta_{v}}{2}$,
$P_{v,-}(t)\approx|C_{v}|^{2}\exp(-\kappa_{v}\sin^{2}\frac{\theta_{v}}{2}t)\cos^{2}\frac{\theta_{v}}{2}$,
$P_{v,+-}(t)\approx|C_{v}|^{2}\exp[-(iE_{v,+-}+\kappa_{v}/2)t]\cos\frac{\theta_{v}}{2}\sin\frac{\theta_{v}}{2}$.
Then, the accumulated photonic energy base on Eq. (III.1.1) into the
dissipative reservoir is approximated as

$\displaystyle\int^{t}_{0}J_{v}(s)ds$ $\displaystyle\approx$
$\displaystyle|C_{v}|^{2}\Big{[}E_{+,v}\sin^{2}\frac{\theta_{v}}{2}(1-e^{-(\cos^{2}\frac{\theta_{v}}{2})\kappa_{v}t})$
(15)
$\displaystyle+E_{-,v}\cos^{2}\frac{\theta_{v}}{2}(1-e^{-(\sin^{2}\frac{\theta_{v}}{2})\kappa_{v}t})$
$\displaystyle-\kappa_{v}\frac{\sin^{2}\theta_{v}}{2}(E_{+,v}+E_{-,v})$
$\displaystyle{\times}\textrm{Re}\\{\frac{1-e^{-(i\Lambda_{v}+\frac{\kappa_{v}}{2})t}}{i\Lambda_{v}+\frac{\kappa_{v}}{2}}\\}\Big{]}.$

After long time evolution $t{\rightarrow}\infty$, the accumulated photonic
energy is given by
$\int^{\infty}_{0}J_{v}(s)ds{\approx}|C_{v}|^{2}(E_{+,v}\sin^{2}\frac{\theta_{v}}{2}+E_{-,v}\cos^{2}\frac{\theta_{v}}{2}{\approx}-E_{g,v}|C_{v}|^{2}$
under the condition $\lambda_{v}{\gg}\kappa_{v}$, which approaches the total
energy dissipated to the reservoir.

Similarly, the measured quantum probability, i.e., the detected photon numbers
at Eq. (13), is generally given by

$\displaystyle\mathscr{P}_{v}(t)$ $\displaystyle\approx$
$\displaystyle|C_{v}|^{2}\frac{\sin^{2}\theta_{v}}{4}\Big{[}\frac{1-e^{-(\cos^{2}\frac{\theta_{v}}{2})\kappa_{v}t}}{\cos^{2}\frac{\theta_{v}}{2}}$
$\displaystyle+\frac{1-e^{-(\sin^{2}\frac{\theta_{v}}{2})\kappa_{v}t}}{\sin^{2}\frac{\theta_{v}}{2}}-2\kappa_{v}\textrm{Re}\Big{\\{}\frac{1-e^{-(i\Lambda_{v}+\frac{\kappa}{2})t}}{\frac{\kappa}{2}+i\Lambda_{v}}\Big{\\}}\Big{]},$

In the limit $t{\rightarrow}\infty$, we find that
$\kappa_{v}\int^{\infty}_{0}\langle
a^{\dagger}_{v}a_{v}\rangle(s)ds\approx\Big{[}1-\frac{\cos^{2}\frac{\theta_{v}}{2}\sin^{2}\frac{\theta_{v}}{2}\kappa^{2}_{v}}{(\kappa_{v}/2)^{2}+E^{2}_{v,+-}}\Big{]}$.
In the coupling regime comparatively stronger than the dissipation strength
$\lambda_{v}{\gg}\kappa_{v}$, we obtain $\kappa_{v}\int^{\infty}_{0}\langle
a^{\dagger}_{v}a_{v}\rangle(s)ds\approx|C_{v}|^{2}$, which consistently
recovers the complete probability measurement. For convenience, we select
system parameters as $\omega_{0,v}=\omega_{0}$, $E_{g,v}=E_{g}$,
$\lambda_{v}=\lambda$, and the dissipative strength $\kappa_{v}=\kappa$.

Figure 2: (Color online) The finite time dynamics of (a) photon numbers and
(b) energy flow with various quantum dot energies. The initial state is
selected as
$|\phi(0){\rangle}=|0_{l},e_{l}{\rangle}{\otimes}|0_{r},g_{r}{\rangle}$. The
system parameters are given by $\omega_{0}=1$, $E_{e}=0$, $\lambda=0.01$, and
$\kappa=0.001$.

Then, we plot Fig. 2 to study the finite time dynamics of the dissipated
energy and the photon numbers with $C_{l}=1$ and $C_{r}=0$. Thus, only the
$l$-th part component contributes to the dissipative dynamics. In particular
at resonance ($E_{g}=-\omega_{0}$), it is found that the average photon number
and the photonic energy flow are reduced to

$\displaystyle{\langle}a^{\dagger}_{l}{a}_{l}{\rangle}(t)\,\approx$
$\displaystyle\,\exp(-{\kappa}t/2)(1-\cos 2\lambda{t})/2,~{}$ (16a)
$\displaystyle J_{l}(t)\,{\approx}\,$
$\displaystyle{\kappa}(-E_{g})\exp(-\kappa{t}/2)(1-\cos 2\lambda{t})/2,~{}$
(16b)

which both exhibits periodic damping oscillation behavior with the period
$T=\pi/\lambda$. Accordingly, the measured quantum probability and dissipated
energy show intermittent increase. In particular, the $n$-th “quasi-step”
note1  with the amplitude

$\displaystyle\mathscr{P}(nT)\,{\approx}\,$
$\displaystyle[1-e^{-n\pi\kappa/2\lambda}]~{},$ (17a)
$\displaystyle\frac{1}{-E_{g}}\int^{nT}_{0}J_{l}(s)ds\,{\approx}\,$
$\displaystyle[1-e^{-n\pi\kappa/2\lambda}],~{}$ (17b)

are shown around the time ${t}=nT$, where the photon counter detects the
negligible photon flow. From Eqs. (17a) and (17b) it is known that the
characteristic measurement number of times can be quantified as
$n_{c}{\approx}2\lambda/\kappa$ and the corresponding time is given by
$t_{c}=n_{c}T{\approx}2\pi/\kappa$. Then the measurement outcome of emitted
photons number and energy gradually becomes complete, which approaches the
complete measurement results. While at a given finite measurement time, the
effectiveness and systematic error of quantum probability measurement
generally rely on the specific properties of the measurement devices, which
are typically characterized as $\lambda$ and $\kappa$.

Moreover, we analyze the influence of the bias condition
($E_{g}{\neq}-\omega_{0}$) on the dissipative dynamics of photons. Both the
photon numbers and energy flow are monotonically suppressed with the increase
of energy bias $|\omega_{0}+E_{g}|$ (though not shown here, such monotonic
suppression also persists in the regime ($-E_{g}{>}\omega_{0}$). Hence, the
fast detection of the complete quantum probability and the underlying
conservation favors the resonant condition.

Figure 3: (Color online) The finite time dynamics of photon number and
accumulation with various photon dissipation strengths. The initial state is
the same as Fig. 2. The system parameters are given by $\omega_{0}=1$,
$E_{e}=0$, $E_{g}=-1$, and $\lambda=0.01$.

We also study the effect of the photon dissipation on the dynamics of photons
numbers at resonance in Fig. 3, considering the analogous behaviors of energy
flow as shown in Eqs. (16b) and (17b). For the average photon number, it is
shown that the profile is dramatically suppressed with the increase of the
dissipation strength $\kappa$, whereas the oscillating period is robust
against the dissipation process. This is consistent with Eq. (16a) and
quantified by the characteristic time $2\pi/\kappa$. Accordingly, the emitted
photon numbers exhibit fast accumulation in the photon counter. Therefore, we
conclude that enhancing the photon dissipation is helpful for the efficient
measurement of the emitted photons and energy.

Figure 4: (Color online) The finite time dynamics of photon number with strong
electron-photon interaction (a) at $\lambda=0.5\omega_{0}$ and (b) by tuning
coupling strength $\lambda$. The inset in (b) denotes the photon number as a
function of electron-photon coupling strength, where the circles and dashed
lines are calculated from full numerical calculation at the time
${\kappa}t=10$ and $(\lambda/\omega_{0})^{2}$, respectively. The initial state
is the same as Fig. 2. The system parameters are given by $\omega_{0}=1$,
$E_{e}=0$, and $E_{g}=-1$.

###  III.2 Measurement with strong electron-photon coupling

When the electron-photon coupling becomes strong and even comparable with the
bare frequency of the optical resonator, the counter-rotating terms of the
electron-photon interaction should be considered mk2014prb ; bka2016prb ;
jclu2019prb . The system then becomes analogous with the seminal quantum Rabi
model dbraak2011prl ; afk2019nrp ; pfd2019rmp , is modified to

$\displaystyle~{}H_{s,v}$ $\displaystyle=$
$\displaystyle\sum_{j=e,g}E_{j,v}|j_{v}{\rangle}{\langle}j_{v}|+\omega_{0,v}a^{\dagger}_{v}a_{v}$
$\displaystyle+\lambda_{v}(a^{\dagger}_{v}+a_{v})(|g_{v}{\rangle}{\langle}e_{v}|+|e_{v}{\rangle}{\langle}g_{v}|),$

Due to the inclusion of the counter-rotating terms
$(a^{\dagger}_{v}|e_{v}{\rangle}{\langle}g_{v}|+a_{v}|g_{v}{\rangle}{\langle}e_{v}|)$,
the particle conservation breaks down, whereas the energy conservation is
still valid. Accordingly, we find that the analytical eigensolution of
$H_{s,v}$ is hard to obtain until now, except the solution mapped to the
special function via Bargmann algebra dbraak2011prl  and photonic coherent
states qhchen2012pra . Here, we include the master equation Eq. (II.2) to
study the influence of strong electron-photon interaction on probability
measurement.

With the initial state
$|\phi(0)=|0_{l},e_{{l}}\rangle{\otimes}|0_{r},g_{{r}}\rangle$, we first plot
Fig. 4(a) to study the short-time dissipative dynamics of average photon
number ${\langle}a^{\dagger}_{l}a_{l}{\rangle}(t)$ at strong electron-photon
coupling, e.g., $\lambda=0.5\omega_{0}$, by comparing with the counterpart
based on the system Hamiltonian at Eq. (II.1). It is interesting to find that
the photon number shows the aperiodic oscillation due to the effect of
counter-rotating terms, which is significantly distinct from the one with the
electron-photon interaction
$(a^{\dagger}_{l}|g_{l}{\rangle}{\langle}e_{l}|+a_{l}|e_{l}{\rangle}{\langle}g_{l}|)$.
Furthermore, the peak of photon number breaches the unity, which is
unavailable in absence of counter-rotating terms.

Next, we study the photon number dynamics at long time evolution in Fig. 4(b).
It is shown that by increasing the electron-photon coupling strength, the
aperiodicity degree becomes increasingly apparent. Moreover, it is interesting
to find that after long time evolution the photon number becomes nonvanishing.
This directly demonstrates the breakdown of the complete measurement at Eq.
(14) with weak electron-photon interaction. We plot the inset of Fig. 4(b) to
quantify the robustness of the photon number.
${\langle}a^{\dagger}_{l}a_{l}{\rangle}(t{\rightarrow}\infty)$ becomes
monotonic enhancement with the increase of $\lambda$.

As is known that after long time evolution the system may reach the steady
state, which generally may not be the eigenstate, e.g., the ground state at
zero temperature. In particular for rather strong electron-photon coupling, by
considering the degenerate perturbation theory, the eigenvectors of $H_{s,l}$
at Eq. (III.2) are estimated as

$\displaystyle|\psi_{n,\pm}{\rangle}$ $\displaystyle{\approx}$
$\displaystyle\frac{1}{\sqrt{2}}\Big{(}\exp\Big{[}{-\frac{\lambda}{\omega_{0}}(a^{\dagger}_{l}-a_{l})}\Big{]}\frac{(a^{\dagger}_{l})^{n}}{\sqrt{n!}}|0{\rangle}_{a}{\otimes}|-{\rangle}$
(19)
$\displaystyle{\pm}\exp\Big{[}{\frac{\lambda}{\omega_{0}}(a^{\dagger}_{l}-a_{l})}\Big{]}\frac{(a^{\dagger}_{l})^{n}}{\sqrt{n!}}|0{\rangle}_{a}{\otimes}|+{\rangle}\Big{)},$

with $|\pm{\rangle}=(|e_{{l}}{\rangle}{\pm}|g_{{l}}{\rangle})/\sqrt{2}$ and
$a_{l}|0{\rangle}_{a}=0$. And the corresponding eigenvalues become
$E_{n,\pm}\approx\omega_{0}n{\mp}|D_{nn}(2\lambda/\omega_{0})|^{2}E_{g}/2-\lambda^{2}/\omega_{0}+E_{g}/2$,
with the coherent state overlap coefficient
$D_{nn}=e^{-2(\lambda/\omega_{0})^{2}}\sum^{n}_{r=0}(-1)^{r}n!(2\lambda/\omega_{0})^{2n-2r}/[(n-r)!(n-r)!r!]$
qhchen2008pra . Then, the transition rates between two eigenstate populations
via the master equation could be reestablished. Specifically, by including the
secular approximation the population dynamics is given by
$dP_{n,\pm}/dt{\approx}[\kappa{(n+1)}P_{n+1,\pm}-\kappa{n}P_{n,\pm}]+\kappa(\lambda/\omega_{0})^{2}(P_{n,\mp}-P_{n,\pm})$,
with the density matrix elements
$P_{n,\pm}={\langle}\psi_{n,\pm}|\rho_{l}|\psi_{n,\pm}{\rangle}$ and the
transition coefficients
${\langle}\psi_{n-1,\pm}|a_{l}|\psi_{n,\pm}{\rangle}=\sqrt{n}$ and
${\langle}\psi_{n-1,\pm}|a_{l}|\psi_{n,\mp}{\rangle}=0$. The dissipative
dynamics generally results in the steady state

$\displaystyle~{}\rho_{l}(\infty){\approx}\frac{1}{2}(|\psi_{0,+}{\rangle}{\langle}\psi_{0,+}|+|\psi_{0,-}{\rangle}{\langle}\psi_{0,-}|).$
(20)

Thus, the corresponding steady-state photon number is given by
${\langle}a^{\dagger}_{l}a_{l}{\rangle}(t{\rightarrow}\infty){\approx}({\lambda}/{\omega_{0}})^{2}$.
This partially explain the nonvanishing steady-state value of the average
photon number at strong electron-photon coupling in the inset of Fig. 4(b).
Therefore, we conclude that strong electron-photon interaction generally
breaks the microscopic particle conservation and complete measurement.

###  III.3 Finite temperature reservoir

We study the effect of finite-temperature reservoir on the quantum measurement
of probability. At weak electron-photon coupling
$\lambda_{v}/\omega_{0,v}{\ll}1$, the master equation for dissipative dynamics
of the electron-photon model can be described as

$\displaystyle\frac{d\rho_{s}}{dt}$ $\displaystyle=$
$\displaystyle-i\sum_{v={l},{r}}[H_{s,v},\rho_{s}]$
$\displaystyle+\sum_{v={l},{r}}\kappa_{v}(1+\overline{n}_{v})\mathcal{L}[a_{v}]\rho_{s}+\sum_{v={l},{r}}\kappa_{v}\overline{n}_{v}\mathcal{L}[a^{\dagger}_{v}]\rho_{s},$

where the dissipator denotes
$\mathcal{L}[a_{v}]\rho_{s}=(2a_{v}\rho_{s}a^{\dagger}_{v}-a^{\dagger}_{v}a_{v}\rho_{s}-\rho_{s}a^{\dagger}_{v}a_{v})/2$.
Here, we empirically describe the incoherent coupling between the photon
resonator and the bosonic reservoir at the temperature $T_{v}$ with the Bose-
Einstein distribution function
$\overline{n}_{v}=1/[\exp(\omega_{0,v}/k_{B}T_{v})-1]$ jmfink2010prl ;
tbosen2023prl . We select the initial state of the reduced $H_{s,v}$ as

$\displaystyle\rho_{v}(0)$ $\displaystyle=$
$\displaystyle(|C_{v}|^{2}|e_{v}{\rangle}{\langle}e_{v}|+|C_{\overline{v}}|^{2}|g_{v}{\rangle}{\langle}g_{v}|)$
$\displaystyle{\otimes}\frac{e^{-\beta_{v}\omega_{0,v}{a^{\dagger}}_{v}a_{v}}}{\mathcal{Z}_{v}},$

with the partition function $\mathcal{Z}_{v}=1/[1-\exp(-\beta\omega_{0,v})]$
and $\beta_{v}=1/k_{B}T_{v}$.

We investigate the energy flow into the $v$-th reservoir. At low temperature
regime, it is numerically found that the dissipative dynamics of the flow is
typically contributed by three lowest eigenstates, i.e.,
$|\psi_{1_{v},\pm}{\rangle}$ and $|\psi_{0_{v}}{\rangle}$. Accordingly,
relation between the energy flow and the photon number can be expressed as

$\displaystyle~{}J_{v}$ $\displaystyle\approx$
$\displaystyle\frac{(\omega_{0,v}-E_{g,v})}{2}[\kappa_{v}(1+\overline{n}_{v}){\langle}a^{\dagger}_{v}{a}_{v}{\rangle}-\kappa_{v}\overline{n}_{v}P_{0,v}]$
(23) $\displaystyle+\frac{\Lambda_{v}}{2}\frac{d(P_{-,v}-P_{+,v})}{dt},$

with $P_{\pm,v}={\langle}\psi_{1_{v},\pm}|\rho_{v}|\psi_{1_{v},\pm}\rangle$,
$P_{0,v}={\langle}\psi_{0_{v}}|\rho_{v}|\psi_{0_{v}}\rangle$, and
$\Lambda_{v}=\sqrt{(\omega_{0,v}+E_{g,v})^{2}+4\lambda^{2}_{v}}$. Moreover,
the total energy loss of the quantum system is obtained as
$\int^{\infty}_{0}J_{v}dt=[{\langle}H_{s,v}{\rangle}(0)-{\langle}H_{s,v}{\rangle}(\infty)]$,
which is approximated as
$\int^{\infty}_{0}J_{v}dt{\approx}E_{g,v}(1/[1+\exp(-\beta_{v}E_{g,v})]-|C_{v}|^{2})$.
Hence, based on the microscopic energy conservation law, we obtain the
effective probability measurement by integrating the current at Eq. (23) as

$\displaystyle~{}\kappa_{v}\int^{\infty}_{0}dt[{\langle}a^{\dagger}_{v}{a}_{v}{\rangle}(t)-\frac{\overline{n}_{v}P_{0,v}(t)}{1+\overline{n}_{v}}]{\approx}\frac{|C_{{v}}|^{2}-1/\mathcal{Z}^{\prime}_{v}}{(1+\overline{n}_{v})^{2}},$
(24)

with $\mathcal{Z}^{\prime}_{v}=(1+e^{-\beta_{v}E_{g,v}})$. If the temperature
approaches zero, i.e., $\overline{n}_{v}=0$, the effective relation is
naturally reduced to the complete probability measurement relation, i.e.,
$\kappa_{v}\int^{\infty}_{0}dt{\langle}a^{\dagger}_{v}{a}_{v}{\rangle}(t)=|C_{v}|^{2}$
at Eq. (14).

Alternatively, the dynamical equation of the ground state probability is
described as

$\displaystyle~{}\frac{d}{dt}P_{0,v}\approx\kappa(1+\overline{n}_{v}){\langle}a^{\dagger}_{v}{a}_{v}{\rangle}-\kappa_{v}\overline{n}_{v}P_{0,v},$
(25)

in which the change of the probability is contributed by the photon flow into
the reservoir and the backward flow from the reservoir thermal noise.
Straightforwardly, the effective probability measurement at Eq. (24) can be
recovered from Eq. (25) by considering
$P_{0,v}(0){\approx}|C_{\overline{v}}|^{2}/\mathcal{Z}_{v}$ and
$P_{0,v}(\infty){\approx}1/\mathcal{Z}_{v}\mathcal{Z}^{\prime}_{v}$.

##  IV Conclusion

In summary, we investigate a practical measurement of the quantum probability
and its underlying microscopic conservation laws through quantum electron-
photon interaction and quantum photon dissipation, connecting the quantum
probability with the probabilistic Born’s rule. We model the measurement setup
with two individual quantum dots, each with an electron occupation interacting
with a single-mode photonic resonator, which is ultimately dissipated to a
reservoir, e.g., a photon counter.

For the quantum measurement with weak electron-photon coupling, we first study
the complete probability measurement at Eq. (14), which exactly connects the
quantum probability of the electron wavefunction in the dot with the counted
photon numbers. The measurement rule is protected by the microscopic energy
conservation law with electron-photon energy exchange processes. Such rule can
be alternatively recovered under the microscopic particle conservation law.
This complete probability measurement relation is impervious to the quantum
system parameters, such as $E_{e(g),v}$, $\omega_{0,v}$, and $\lambda_{v}$.
Next, we study the finite time probability measurement at finite time
dynamics. Under the secular approximation, at resonance
($E_{e,v}-E_{g,v}=\omega_{0,v}$) the photon numbers at Eq. (16a) exhibit the
periodic oscillating damping behavior, which are determined by the the
electron-photon coupling and photon dissipation strengths. The emitted photon
number shows intermittent increase feature at Eq. (17a), with quasi-steps
appearing around the times $t_{v}=n\pi/\lambda_{v}$. The characteristic
measurement time is quantified as $t_{c,v}{\approx}2\pi/\kappa_{v}$, where the
measurement outcome of emitted photons numbers and energy becomes effectively
complete. Furthermore, it is found that the bias condition
($(E_{e,v}-E_{g,v}){\neq}\omega_{0,v}$) blocks the efficient measurement of
the quantum probability.

We further analyze the applicability of quantum measurement in the strongly-
coupled electron-photon model and consider the counter-rotating terms in
$H_{s,v}$ Eq. (III.2), which straightforwardly breaks the particle number
conservation. In short-time dynamics, by comparing the photon number dynamics
with the counterpart in absence of counter-rotating terms, we observe two
distinct features: (1) the photon number exhibits the aperiodic oscillations,
and (2) the peak of photon number breaches the unity. While in long time
dynamics, it is found that (1) the aperiodicity degree of photon number
oscillating behavior becomes apparent with the increase of electron-photon
coupling, and (2) the photon number still becomes nonvanishing, due to the
formation of the mixture of photonic coherent states, which demonstrates the
breakdown of the complete probability measurement relation.

Moreover, we investigate the influence of finite temperature reservoir on the
quantum measurement and explore the effective measurement relation connecting
the counted photon flux with quantum probability and thermal statistics.

We hope that the reported measurements of quantum probability and the
microscopic conservation laws may provide general avenues for practically
detecting quantum probability.

##  V ACKNOWLEDGEMENTS

This work was supported by the funding for the National Natural Science
Foundation of China under Grants No. 12125504, No. 12074281, and No. 11704093,
Jiangsu Key Disciplines of the Fourteenth Five-Year Plan (Grant No. 2021135),
and the Opening Project of Shanghai Key Laboratory of Special Artificial
Microstructure Materials and Technology.

*

##  Appendix A Derivation of complete quantum probability measurement relation

In the eigenbasis of $H_{s,v}$, the energy current
$J_{v}=-d{\langle}H_{s,v}{\rangle}/dt$ is expressed as

$\displaystyle
J_{v}=-E_{+,v}\frac{dP_{v,+}(t)}{dt}-E_{-,v}\frac{dP_{v,-}(t)}{dt},$ (26)

with the eigen-energies $E_{\pm,v}$ at Eq. (9). Including the dynamics of
density matrix operator in the eigenbasis of $H_{s,v}$

$\displaystyle\frac{d}{dt}P_{v,+}=$
$\displaystyle-\kappa\cos^{2}\frac{\theta}{2}P_{v,+}+\frac{\kappa}{2}\sin{{\theta}}\textrm{Re}[P_{v,+-}],$
(27a) $\displaystyle\frac{d}{dt}P_{v,-}=$
$\displaystyle-\kappa\sin^{2}\frac{\theta}{2}P_{v,-}+\frac{\kappa}{2}\sin{{\theta}}\textrm{Re}[P_{v,+-}],$
(27b) $\displaystyle\frac{d}{dt}P_{v,+-}=$
$\displaystyle-(i\Lambda_{v}+\frac{\kappa}{2})P_{v,+-}+\frac{\kappa}{2}{\sin{\theta}}\textrm{Re}[P_{v,+-}],$
(27c)

with $\Lambda_{v}=E_{+,v}-E_{-,v}$ and
$\textrm{Re}[P_{v,+-}]=(P_{v,+-}+P_{v,-+})/2$, the current is specified as

$\displaystyle J_{v}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2}(2\cos^{2}\frac{\theta}{2}P_{v,+}-\sin{{\theta}}\textrm{Re}[P_{v,+-}])$
(28)
$\displaystyle+\frac{\kappa}{2}(2\sin^{2}\frac{\theta}{2}P_{v,-}-\sin{{\theta}}\textrm{Re}[P_{v,+-}]).$

Considering the expression of photon number
${\langle}a^{\dagger}_{l}a_{l}{\rangle}$ at Eq. (III.1.1), the relation
between the current and the photon number can be established as

$\displaystyle~{}J_{v}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2}(\omega_{0,v}-E_{g,v}){\langle}a^{\dagger}_{l}a_{l}{\rangle}$
$\displaystyle-\frac{\Lambda_{v}}{2}\Big{(}\frac{d}{dt}P_{v,+}-\frac{d}{dt}P_{v,-}\Big{)}.$

By integrating Eq. (A) at both sides, we find
$\int^{\infty}_{0}J_{v}dt=-E_{g,v}|C_{v}|^{2}$,
$P_{v,-}(0)-P_{v,+}(0)=|C_{v}|^{2}\cos\theta$, and
$P_{v,-}(\infty)-P_{v,+}(\infty)=0$. Finally, we obtain the complete quantum
measurement relation at Eq. (14).

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$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle
S_{u}^{d}(s)=\text{diag}(X_{1},...,X_{n}),\quad
X_{i}\sim_{iid}\Gamma^{-1}(\alpha_{i},\beta_{i}),$

where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Gamma^{-1}(\alpha_{i},\beta_{i})$
denotes an Inverse-Gamma distribution such that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{i}>1$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{E}[X_{i}]=\frac{\beta_{i}}{\alpha_{i}-1}=[\Psi_{u}]_{i}i$
for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i$.
This ensures that the means of diagonal elements are the same across the two
samples.

  3. 3.

Solve equation (49) using off-the-shelf constrained quasi-Newton algorithms in
both cases and construct
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S$
realizations of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f_{s}^{*}=f_{min}(S_{u}^{*})$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f_{s}^{d}=f_{min}(S_{u}^{d})$
for
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s=1,...,S$.

  4. 4.

Construct the numerical p-value using

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p=\frac{1}{S}\sum_{s=1}^{S}\mathbbm{1}\\{f_{s}^{*}\geq
f_{s}^{d}\\},$

and repeat Step 3 for different values of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Phi}_{u}$.
Reported results in Table 12.

  5. 5.

For optimal
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W_{min}^{spec}=\operatorname*{arg\,min}f(W,S_{u}^{spec})$,
consider the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\times
n$ elementwise difference
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{\Delta}_{s}$
for each iteration
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s$:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\tilde{\Delta}_{s}=f(W_{min}^{d},S_{u}^{d})-f({W}_{min}^{*},S_{u}^{*})$

Then I test the marginal hypotheses
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H_{0}:[\tilde{\Delta}_{s}]_{ij}=0$
for all
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i,j$.
This test corresponds to the numerical t-statistic
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t_{ij}=m_{ij}/SE_{ij}$,
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m_{ij}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}SE_{ij}$
are the mean and standard error of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}[\tilde{\Delta}_{s}]_{ij}$,
respectively. Reported results in Table 13.

Table 12: Simulation p-values Specification | n=2 | n=3 | n=4 | n=5
---|---|---|---|---
(1) | 0.000 | 0.002 | 0.036 | 0.004
(2) | 0.000 | 0.000 | 0.041 | 0.005
(3) | 0.000 | 0.003 | 0.046 | 0.006
(4) | 0.000 | 0.001 | 0.048 | 0.009
(5) | 0.000 | 0.000 | 0.053 | 0.008
(6) | 0.000 | 0.003 | 0.060 | 0.008
(7) | 0.000 | 0.001 | 0.061 | 0.006
(8) | 0.000 | 0.001 | 0.061 | 0.010
(9) | 0.000 | 0.004 | 0.068 | 0.011
(10) | 0.000 | 0.001 | 0.072 | 0.011
Note: this table reports numerical p-values from the procedure described in Appendix F for different values of $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$ and $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Phi}$. Number of iterations is $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S=1000$. Table 13: Simulation t-statistics n=2 | n=3 | n=4 | n=5
---|---|---|---
121 | -27 | -54 | -118
863 | 48 | 133 | 201
861 | 60 | 163 | 152
-504 | 48 | 2 | -502
| -68 | 133 | -152
| 80 | -247 | 201
| 60 | -26 | -583
| 80 | 190 | 172
| -91 | 163 | 205
|  | -26 | 61
|  | -255 | 152
|  | 204 | 172
|  | 2 | -177
|  | 190 | 174
|  | 204 | 138
|  | -60 | -502
|  |  | 205
|  |  | 174
|  |  | -89
|  |  | -334
|  |  | -152
|  |  | 61
|  |  | 138
|  |  | -334
|  |  | -75
Note: this table reports elementwise t-statistcs from the procedure described
in Appendix F for different values of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n$
and a fixed value
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\Phi}$.
Number of iterations is
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S=1000$.

##  Appendix G Comovement in Industry Volatility

Recent research documents significant common variation in both market and
fundamental volatility at the granular level.343434Herskovic et al. (2016)
show that a single common factor explains around 30% of variation in log
variance for the panel of CRSP stocks. Other work also documents common
variation in option-implied volatilities (Engel and Figlewski (2015)), intra-
daily returns (Barigozzi et al. (2014)), and dispersion in firm sales growth
(Bloom et al. (2018)). Importantly, the factor structure in volatility is
significant even after removing all common variation in returns and cash flow
growth, which suggests that this is being driven by underlying sources of
systematic risk and rather than an omitted set of returns or sales growth
factors.353535Herskovic et al. (2016) verify that there is a factor structure
even when the pairwise correlation between idiosyncratic return or sales
growth residuals is statistically indistinguishable from zero. I also verify
that the factor structure holds in residual returns after a non-parametric
regression using deep feed-forward neural networks, which have favorable
universal approximation properties (see e.g., Hornik et al. (1989)).  When
economic units are connected via input-output networks, a shock to any given
unit can generate systematic effects. Moreover, networks mechanically generate
volatility comovement regardless of whether shocks to individual units are
uncorrelated. When shocks are correlated, this comovement is even more
pronounced.

My results so far establish a significant relationship between network
concentration across and between customers and suppliers and realized
variance. Consequently, comovement in supply chain concentration should also
generate comovement in realized variance. Table 14 reports average loadings
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{2}$
values of univariate factor regressions for panel of industry network
concentration and realized volatility measures. The main takeaway is that
20-30% of dynamic variation in input-output concentration and return and sales
growth volatilities across industries of these variables can be explained by a
single factor. Moreover, there is a significant degree of comovement between
these common factors. That is, network concentration factors can explain up to
40% of time-series variation in both market and sales growth volatility
factors. These results are robust to a variety of specifications.

Table 14: Comovement in Industry Variance Panel A: Loadings
---
Factor / Outcome | Across (d) | Between (d) | Across (s) | Between (s) | Var (mkt) | Var (cf)
Across (d) | 0.788 | 1.576 | 0.364 | 0.507 | 0.410 | 0.574
Between (d) | 0.584 | 0.735 | 0.394 | 0.567 | 0.206 | 0.383
Across (s) | 0.701 | 1.185 | 0.700 | 0.518 | 0.156 | 0.497
Between (s) | 0.455 | 0.422 | 0.401 | 0.736 | 0.157 | 0.310
Var (mkt) | 0.091 | 0.124 | 0.016 | 0.035 | 0.880 | 0.185
Var (cf) | 0.109 | 0.181 | 0.156 | 0.218 | 0.283 | 0.721
Panel B:
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{2}$
(avg univariate)
Factor / Outcome | Across (d) | Between (d) | Across (s) | Between (s) | Var (mkt) | Var (cf)
Across (d) | 0.201 | 0.258 | 0.217 | 0.318 | 0.170 | 0.079
Between (d) | 0.189 | 0.318 | 0.257 | 0.400 | 0.174 | 0.087
Across (s) | 0.164 | 0.274 | 0.229 | 0.344 | 0.168 | 0.082
Between (s) | 0.189 | 0.322 | 0.265 | 0.347 | 0.165 | 0.086
Var (mkt) | 0.042 | 0.046 | 0.041 | 0.042 | 0.343 | 0.068
Var (cf) | 0.063 | 0.097 | 0.085 | 0.115 | 0.147 | 0.104
Panel C:
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{2}$
(aggregate)
Factor / Outcome | Across (d) | Between (d) | Across (s) | Between (s) | Var (mkt) | Var (cf)
Across (d) | 1 | 0.620 | 0.575 | 0.566 | 0.373 | 0.352
Between (d) |  | 1 | 0.628 | 0.670 | 0.255 | 0.422
Across (s) |  |  | 1 | 0.668 | 0.196 | 0.469
Between (s) |  |  |  | 1 | 0.229 | 0.428
Var (mkt) |  |  |  |  | 1 | 0.165
Var (cf) |  |  |  |  |  | 1
Notes: This table reports the results of factor regressions for industry
variance components. I calculate factors as the first principal component of
an industry panel of the variable of interest. Panels A and B report the
average loading and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{2}$
from the regressions
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}outcome_{it}=\alpha_{i}+\beta_{i}\cdot
f_{t}+u_{it}$, respectively. Each column, row pair denotes a different
outcome, factor pair. Panel C reports the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{2}$
of aggregate time-series regressions of factor pairs (i.e.,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y_{t}=\alpha+\beta\cdot
x_{t}+u_{t}$, where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y_{t}$
is the column factor and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{t}$
is the row factor). All variables are log-transformed. Sample is at an annual
frequency from 1997 to 2019 for 66 non-government BEA industries.

##  Appendix H Additional Firm-Level Results

Table 15: Predicting Firm Sales Growth  Variable | (1) | (2) | (3)
---|---|---|---
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t-1}$ | 0.031** | 0.029** | 0.007**
| (0.002) | (0.001) | (0.000)
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t-1}$ | -0.016** | -0.018** | 0.015
| (0.003) | (0.003) | (0.011)
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}ROA_{i,t-1}$ | 0.080** | 0.063** | 0.064**
| (0.003) | (0.002) | (0.002)
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}size_{i,t-1}$ | -0.008** | -0.008** | -0.008**
| (0.000) | (0.000) | (0.000)
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}age_{i}$ | 0.037** | 0.050** | 0.046**
| (0.004) | (0.004) | (0.003)
Constant | 0.44 | 0.372 | 0.39
Obs | 259,976 | 259,976 | 259,976
Adj R2 | 0.265 | 0.259 | 0.258
Notes: This table reports the regression results based on the model in (21).
The dependent variable is year-on-year quarterly sales growth and the
covariates are aggregate TFP growth from Fernald (2012), procurement proxy
from Briganti and Sellemi (2022), log return on assets, size (log market
value), and age as year appears on the database. Each column reports results
for different industry aggregations: (1) is 66 BEA non-government industries,
(2) is 405 BEA non-government industries, and (3) is 15 major BEA non-
government industries. Standard errors are clustered at the same granularity.
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}**$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}*$
indicates significance at the 1% and 5% levels, resp. Sample is quarterly
between 1997-2019 for 10,700 firms.

(a)

(b)

(c)

Figure 7: Distribution of Calibrated Propensities with No Network Connections
Notes: This figure plots the kernel density of calibrated parameters
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k_{iq}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{iq}$
as described in Section 5. Note that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{p}_{iq}=1/(1+\exp(-k_{iq}x_{iq}))$.

(a)

(b)

(c)

Figure 8: Network Propagation Risk Factors Notes: This figure plots the time
series of network propagation risk factors (Panel A), the cross-sectional
average industry substitutability (Panel B), and aggregate TFP and procurement
demand growth (Panel C). Shaded regions indicate NBER-dated recession periods.
Table 16: Descriptive Statistics Calibrated Firm Parameters

Table 17: One-Way Sorted Portfolios on Network Propagation Factors
(controlling for volatility factors) Panel A: One-way sorts on upstream
propagation beta (controlling for
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}$,
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma_{t}^{civ}$)
---
| 1 (Low) | 2 | 3 | 4 | 5 (High) | H-L | t(H-L) | MR p-val
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{E}[r]-r_{f}$ | 12.97 | 12.12 | 11.34 | 10.44 | 10.11 | -2.86 | -5.83 | 0.22
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{capm}$ | -0.04 | -0.21 | -0.21 | -0.28 | -0.68 | -0.64 | -4.06 | 0.00
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{ff3}$ | -0.10 | -0.14 | -0.17 | -0.25 | -0.39 | -0.29 | -0.83 | 0.01
Volatility (%) | 13.37 | 14.81 | 14.71 | 19.1 | 12.64 | - | - | -
Book-to-market | 0.52 | 0.53 | 0.58 | 0.54 | 0.50 | - | - | -
Market value ($bn) | 12.18 | 6.21 | 18.07 | 6.14 | 15.75 | - | - | -
Panel B: One-way sorts on downstream propagation beta (controlling for
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}$,
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma_{t}^{civ}$)
| 1 (Low) | 2 | 3 | 4 | 5 (High) | H-L | t(H-L) | MR p-val
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{E}[r]-r_{f}$ | 14.97 | 12.94 | 12.15 | 9.78 | 9.35 | -5.61 | -9.36 | 0.09
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{capm}$ | 0.03 | -0.24 | -0.29 | -0.32 | -0.39 | -0.43 | -1.88 | 0.00
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{ff3}$ | -0.02 | -0.10 | -0.26 | -0.29 | -0.30 | -0.29 | -3.40 | 0.01
Volatility (%) | 18.69 | 16.42 | 12.11 | 14.20 | 13.02 | - | - | -
Book-to-market | 0.59 | 0.55 | 0.51 | 0.51 | 0.52 | - | - | -
Market value ($bn) | 5.11 | 12.13 | 8.96 | 16.43 | 15.64 | - | - | -
Panel C: One-way sorts on upstream propagation beta (controlling for
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}$,
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma_{t}^{mkt}$)
| 1 (Low) | 2 | 3 | 4 | 5 (High) | H-L | t(H-L) | MR p-val
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{E}[r]-r_{f}$ | 13.83 | 13.07 | 11.33 | 10.10 | 9.73 | -4.09 | -8.10 | 0.21
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{capm}$ | 0.01 | -0.26 | -0.28 | -0.35 | -0.45 | -0.47 | -6.84 | 0.04
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{ff3}$ | -0.07 | -0.17 | -0.20 | -0.29 | -0.30 | -0.23 | -2.06 | 0.03
Volatility (%) | 13.68 | 16.44 | 17.09 | 14.12 | 13.03 | - | - | -
Book-to-market | 0.51 | 0.54 | 0.55 | 0.54 | 0.53 | - | - | -
Market value ($bn) | 11.43 | 5.87 | 7.24 | 19.44 | 14.40 | - | - | -
Panel D: One-way sorts on downstream propagation beta (controlling for
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{t}$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g_{t}$,
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma_{t}^{mkt}$)
| 1 (Low) | 2 | 3 | 4 | 5 (High) | H-L | t(H-L) | MR p-val
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{E}[r]-r_{f}$ | 13.66 | 13.5 | 12.67 | 9.77 | 9.24 | -4.42 | -7.17 | 0.37
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{capm}$ | 0.07 | -0.26 | -0.32 | -0.39 | -0.41 | -0.48 | -1.81 | 0.00
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{capm}$ | 0.02 | -0.17 | -0.24 | -0.25 | -0.33 | -0.35 | -1.52 | 0.01
Volatility (%) | 19.47 | 12.94 | 15.7 | 14.27 | 13.03 | - | - | -
Book-to-market | 0.63 | 0.54 | 0.44 | 0.53 | 0.52 | - | - | -
Market value ($bn) | 4.60 | 10.24 | 13.33 | 15.77 | 14.30 | - | - | -
Notes: This table reports average excess returns and post-sample alphas in
annual percentages for value-weighted portfolios sorted into quintiles on
annual upstream and downstream propagation factors. Sample is between
1997-2021 for more than 10,000 stocks belonging to the BEA 66 non-government
industry classifications. Panels A and B control for productivity growth and
federal procurement demand growth, while Panels C and D have no controls. I
also report average return volatility, book-to-market ratio and market value
for each portfolio. To test for significant return spreads, I report
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t$-statistics
for the null hypothesis
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H_{0}:xr_{5}=xr_{1}$,
where
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}xr_{q}$
is the average return of the
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q^{th}$
quintile single sorted portfolio. Moreover, I report p-values for the test
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H_{0}:xr_{q+1}<xr_{q}\forall
q\leq 4$, calculated via bootstrap following Patton and Timmermann (2010).


# Cryogenic spin Seebeck effect

Mehrdad Elyasi  Institute for Materials Research, Tohoku University, 2-1-1
Katahira, 980-8577 Sendai, Japan  Gerrit E. W. Bauer  Institute for Materials
Research & AIMR & CSRN, Tohoku University, 2-1-1 Katahira, 980-8577 Sendai,
Japan  Zernike Institute for Advanced Materials, University of Groningen, The
Netherlands

###### Abstract

We present a theory of the non-linearities of the spin Seebeck effect (SSE) in
a ferromagnetic nanowire at cryogenic temperatures. We adopt a microscopic
quantum noise model based on a collection of two-level systems. At certain
positions of Pt detectors to the wire, the transverse SSE changes sign as a
function of temperature and/or temperature gradient. On the other hand, the
longitudinal SSE does not show significant non-linearities even far outside
the regime of validity of linear response theory.

We address the spin Seebeck effect (SSE) in electrically insulating magnets,
i.e. the spin current caused by a temperature gradient as detected by the
inverse spin Hall effect voltage in heavy metal contacts [2, 4, 5, 6, 7, 8,
9]. The longitudinal SSE (LSSE) is observed in a planar configuration in which
the heat and spin currents flow in parallel and normal to the interfaces [10].
The transverse or non-local [11] SSE (TSSE) refers to more complicated
configurations, usually two contacts on the surface of a magnetic slab or
film. The spin current is injected into the metal contact by spin pumping [4]
but the signal is usually dominated by the currents that are generated by
temperature gradients in the bulk of the magnet [12]. The reported signals are
in general proportional to the applied temperature differences $\triangle T.$
However, several recent studies of the SSE at low temperatures [13, 14, 16,
15, 17, 18, 19, 20] do not address a fundamental issue of thermal transport at
ultralow temperatures. Linear response is valid when the perturbation is
sufficiently small, but the properly normalized driving force is not
$\triangle T$ but $\triangle T/T$ (or $\partial T/T),$ i.e. the temperature
difference divided by the average one [21]. This condition is increasingly
difficult to fulfill at low temperatures, or positively formulated, it should
become easier to access non-linear thermomagnonic transport phenomena.

Existing theoretical treatments of the spin Seebeck effect are not suitable to
address the low-temperature and non-linear regimes. The low frequency magnons
that dominate at cryogenic temperatures are strongly affected by dipolar
interactions, so exchange-only magnon models fail. The assumption of a
semiclassical magnon accumulation in terms of a local chemical potential and
magnon temperature [22] breaks down because thermalization becomes weak. With
a classical magnetization noise model and in linear response, the non-thermal
distribution functions governing the SSE can be described by mode- (rather
than position-) dependent magnon temperatures and chemical potentials [24].
Treatments of the stochastic magnetization dynamics in terms of classical
white noise sources [26, 27, 25, 28] do not work at low temperatures. This can
be repaired by a noise spectrum that obeys the quantum fluctuation dissipation
theorem [29], but at the cost of introducing phenomenological damping
constants. A recent linear response study of the LSSE at low temperatures [31]
focuses on the magnon-polaron hybrid state at large magnetic fields [30].

The broadening of the ferromagnetic resonance of a YIG sphere increases
$\propto T$ for $T>1\,$K. The minimum in the damping followed by an increase
and saturation with decreasing temperatures $<1\,$K [32] is caused by
impurities and disorder, presumably two-level systems (TLS) [34, 35, 33]. The
spin and heat transport in this regime has to our knowledge not been addressed
in the literature and is the focus of this Letter. We study the cryogenic SSE
of a ferromagnetic (FM) nanowire with a microscopic TLS model for the thermal
noise at weak magnetic fields. In this regime magnon-magnon and magnon-phonon
interactions may be safely disregarded. We predict that the antisymmetry of
the TSSE signal as a function of position of a Pt detector [4, 5, 26] is
broken in the non-linear regime and a non-monotonous temperature dependence at
certain contact positions emerges. These effects are caused by the non-uniform
gradient of the spin distribution functions in spite of a constant temperature
gradient. The LSSE signal is on the other hand surprisingly robust, with a
linear dependence on a global temperature difference $\Delta T$ much larger
than $T_{M}$.

Model. We consider YIG nanostructures with high quality surfaces [36] in which
scattering at low temperatures is dominated by rare earth (RE) substitutional
impurities, e.g. Tb or Yb, on the Y sites [32, 34, 35, 33]. Two degenerate
atomic levels of a RE atom form a two-level system (TLS) with pseudo-spin
$\vec{\Omega}$ that interacts with the local iron magnetic moments of spin
$\vec{S}_{Fe}$ by an exchange interaction
$H_{TLS}=\vec{S}_{Fe}\cdot\bar{K}\vec{\Omega}$, where $\bar{K}$ is an
anisotropic exchange interaction tensor, which splits the pseudo-spin levels
by $\omega_{01}$. Since the RE angular momentum strongly couples to the
lattice, spin waves can be efficiently dissipated via $H_{TLS}$. The isotropic
Heisenberg exchange contribution $S_{x(y)}\Omega_{x(y)}$ couples the
precessional dynamics and leads to a “transverse” relaxation that preserves
the total magnetization. The anisotropy introduces “longitudinal” terms like
$S_{x}\Omega_{z}$ and $S_{y}\Omega_{z}$ by which the splitting $\omega_{01}$
depends on the magnetization direction. Van Vleck [35, 33] computed the life
time broadenings due to $H_{TLS}$ as a function of the ratio of rare earth to
Fe concentration $c$. For the longitudinal process he reports

Figure 1: Model. (a) The dissipation caused by two TLS ensembles that fit the
experiments of Tabuchi et al. [32].
$\Gamma_{tot}=\sum_{j=1,2}\Gamma_{L(T),j}$, where $j$ indicates the ensembles
parametrized by $c_{1}=3\times 10^{-7}$, $c_{2}=1\times 10^{-4}$,
$\tau_{1}=10\,$ns, $\tau_{2}=0.1\,$ps, $\omega_{1,01}=2\pi\times 10\,$GHz,
$\omega_{2,01}=2\pi\times 150\,$GHz and $\omega_{U}=2\pi\times 10\,$GHz. (b) A
spin Seebeck current $\mathcal{J}^{(s)}$ polarized along $\hat{z}$ flows from
the magnet into the metal contact. The color indicates the temperature
profile, where white (black) is hottest (coldest). (c) Left: Array of spins
$\vec{S}_{i}$, coordinate system, lattice spacing $d$, external magnetic field
$\vec{h}$, and local reservoirs at temperature $T_{i}$. Right: mesoreservoir
of $N_{TLS}$ two-level systems (TLS) with frequency splittings
$\omega_{j,01}$. The TLS ensemble is in contact with a thermal bath
(relaxation rate $\xi_{j}=2\pi/\tau_{j}$) at temperature $T_{i}$ and interacts
with a spin (green lines) by $V_{i,j}$.

$\displaystyle\Gamma_{L}$
$\displaystyle=\sum_{j}\Gamma_{L,j}=\sum_{j}\frac{c_{j}\hbar}{6k_{B}T}\sum_{n=1}^{6}$
$\displaystyle\omega_{j,n,01}^{2}f_{j,n}(\theta,\phi)\frac{\tau_{j,n}\omega_{U}}{1+\tau_{j,n}^{2}\omega_{U}^{2}}\left[1-\tanh^{2}\frac{\hbar\omega_{j,n,01}}{2k_{B}T}\right],$
(1)

where $j$ indicates a certain type of impurity or a certain corresponding TLS,
while $n$ indicates a yttrium site in the YIG unit cell. $\tau_{j,n}$ is the
relaxation time of an excited TLS, $f_{j,n}(\theta,\phi)\in\left[0,1\right]$
depends on the polar magnetization direction angles $\theta,\phi,$ and
$\omega_{U}$ is the FMR frequency. The transverse relaxation is dominated by
the isotropic exchange and reads [35, 33]

$\displaystyle\Gamma_{T}$
$\displaystyle=\sum_{j}\Gamma_{T,j}=-\mathrm{Im}\sum_{j}\sum_{n=1}^{6}\frac{c_{j}\omega_{j,n,01}}{12}\tanh\frac{\hbar\omega_{j,n,01}}{2k_{B}T}\times$
$\displaystyle\left[\frac{\omega_{U}}{\omega_{U}-\omega_{j,n,01}+i/\tau_{j,n}}+\frac{\omega_{U}}{\omega_{U}+\omega_{j,n,01}+i/\tau_{j,n}}\right].$
(2)

For $\omega_{j,n,01}=\omega_{j,01}\forall n$ and $\tau_{j,n}=\tau_{j}\forall
n$,

$\displaystyle\Gamma_{T,j}$
$\displaystyle=\frac{1}{2}\omega_{U}c_{j}\tanh\frac{\hbar\omega_{j,01}}{2k_{B}T}\times$
$\displaystyle\left\\{\begin{array}[]{c}\omega_{j,01}\tau_{j}/\left(1+\tau_{j}^{2}\omega_{j,01}^{2}\right)\\\
\omega_{j,01}\tau_{j}\end{array}\ \mathrm{for}\
\begin{array}[]{c}\omega_{j,01}\gg\omega_{U}\\\
\omega_{j,01}\sim\omega_{U}\end{array}.\right.$ (7)

$\Gamma_{L,j}$ is a non-monotonous function of temperature, increasing from
zero at $T=0\ $up to a maximum at $\sim\hbar\omega_{01}/k_{B}$. $\Gamma_{T,j}$
monotonically increases from zero as $T$ decreases and saturates to a finite
value at $T=0$ since the transverse relaxation is proportional to polarization
of the TLS, i.e.
$\tanh\left[\hbar\omega_{j,01}/\left(2k_{B}T\right)\right]/2$. The
proportionality of $\Gamma_{T,j}$ with $\tau_{j}$ when
$\omega_{j,01}\sim\omega_{U}$, only holds for $1/\tau_{j}\ll\omega_{U}.$
$\Gamma_{T,j}$ vanishes with $\tau_{j}$ because of the associated lifetime
broadening of the TLS density of states. Tabuchi et al. [32] found an
excellent agreement for the temperature dependent broadening at $T<1\,$K
assuming $\omega_{01}/2\pi\sim\omega_{U}/2\pi\sim 10\,\mathrm{GHz}$, and a
temperature independent bias. The increase in damping for $T\geq 1\,$K might
be phonon induced, but could indicate also the existence of a second family of
levels with larger exchange splitting. Figure 1(a) shows that the total
dissipation due to combination of two distinct TLS,
$\Gamma_{tot}=\sum_{j=1,2}\left(\Gamma_{L,j}+\Gamma_{T,j}\right)$ explains the
observed damping very well for up to $T\sim 5\,$K, where we used
$c_{1}=3\times 10^{-7}$, $c_{2}=1\times 10^{-4}$, $\tau_{1}=10\,$ns,
$\tau_{2}=10\,$ps, $f_{1}=1$, $f_{2}=1$, $\omega_{1,01}/2\pi=10\,$GHz, and
$\omega_{2,01}/2\pi=150\,$GHz. Figure 1(a) shows that at $T<1\,$K,
$\Gamma_{T,1}\gg\Gamma_{L,1(2)},\Gamma_{T,2}$. In the following we therefore
consider only a single TLS type with
$\Gamma_{tot}\approx\alpha_{\mathrm{TLS}}\omega_{\text{U}}$, where
$\alpha_{\mathrm{TLS}}=c_{1}\tau_{1}\omega_{1,01}\tanh\left(\hbar\omega_{1,01}/2k_{B}T\right)\approx
10^{-4}$ is the (Gilbert) damping coefficient. We proceed to predict the
consequence of TLS dominated dissipation for the spin Seebeck effect.

Figure 1(b) shows the schematics of the physical system, a nanowire magnetized
along its length, while Fig. 1(c) shows the schematics of the corresponding
spin lattice-reservoir model. The dipolar interactions affect the magnon
dispersion only for wave lengths that are much larger than the unit cell. We
therefore adopt a micromagnetic approach in which the local magnetization
represents an average over slices of typically 50 nm that contain many local
moments. The macrospin site $i$ then interacts with a “mesoreservoir” composed
of several TLS as described earlier. Since the latter are local impurities
with short-range exchange interactions, we may disregard their cross-
correlation. The mesoreservoir in turn interacts with a large reservoir with
well-defined temperature $T_{i}$ that is allowed to vary slowly in space. The
Hamiltonian for the model in Fig. 1(c) now reads

$H=H_{S}+H_{R}+H_{SR},$ (8)

where $H_{S}$ describes the magnet, $H_{R}$ the mesoreservoirs, and $H_{SR}$
the interaction between them. We expand the Heisenberg Hamiltonian for the
spin chain of Fig. 1(c) to the second order of the Holstein-Primakoff
transformation, for a spin $S$ on site $i$, i.e.
$S_{i}^{+}=\sqrt{2S}a_{i}^{{\dagger}}{[1-a_{i}^{{\dagger}}a_{i}/}\left({2S}\right){]}^{1/2}$,
$S_{i}^{-}=\sqrt{2S}{[1-a_{i}^{{\dagger}}a_{i}/}\left({2S}\right){]}^{1/2}a_{i}$,
$S_{i}^{z}=S-a_{i}^{{\dagger}}a_{i}$, in terms of magnons
$a_{i}^{{\dagger}}$($a_{i}$) created (annihilated) at site $i$. This leads to

$\displaystyle H_{S}$
$\displaystyle=\sum_{i}\left(\mathcal{A}_{i}-S\sum_{j}F_{zz}^{i,j}\right)a_{i}^{{\dagger}}a_{i}+$
$\displaystyle\sum_{i,j}\left\\{\left[SJ\delta\left(i\pm
1,j\right)+\mathcal{B}_{i,j})\right]a_{i}^{{\dagger}}a_{j}+\right.$
$\displaystyle\left.\mathcal{C}_{i,j}a_{i}a_{j}+\mathrm{H.C.}\right\\},$ (9)

where $\mathcal{A}_{i}=-2SJ+\gamma_{e}h^{z}$.
$\mathcal{A}_{1(N_{L})}=-2SJ+\gamma_{e}h^{z}$ indicates that the edges are in
contact with a pinned spin, otherwise
$\mathcal{A}_{1(N_{L})}=-SJ+\gamma_{e}h^{z}$. $h^{z}$ is the magnetic field in
the $\hat{z}$ direction, $\gamma_{e}$ is the gyromagnetic ratio, $\delta$ is
the Kronecker delta,
$\mathcal{B}_{i,j}=S\left(F_{xx}^{i,j}+F_{yy}^{i,j}\right)/2$,
$\mathcal{C}_{i,j}=S\left(F_{xx}^{i,j}-F_{yy}^{i,j}\right)/2$, where
$F_{xx(yy)}^{i,j}$ is the dipolar field of $S_{i}^{x}$ ($S_{i}^{y}$) exerted
on $S_{j}^{x}$ ($S_{j}^{y}$), and $J$ is the exchange interaction. We compute
the dipolar interactions assuming uniform dynamics along the thickness of the
nanowire ($\|\hat{x}$) and a nodeless cosine function amplitude with an
effective width along $\hat{y}$ [37].

We parametrize $J$ by its value in the continuum limit. For long wave lengths
$J=\gamma_{e}\mu_{0}M_{s}\lambda^{2}/\left(d^{2}S\right),$ where $S=NS_{0}$
and $N=w_{x}w_{y}d/l^{3}$ is the number of unit cells in each 1D segment,
$w_{x}$ ($w_{y}$) is the thickness (width) of the nanowire, $l$ is the unit
cell dimension. $S_{0}=l^{3}M_{s}/\left(2\pi\gamma_{e}\right)\approx 14$ is
the net number of spins in the YIG unit cell, with magnetization
$M_{s}=1.46\times 10^{5}\,$A/m, $\gamma_{e}=26\,$GHz/T and $l=1.2\,$nm. The
exchange length for YIG $\lambda=\sqrt{3}\times 10^{-8}\,\text{m}$ [38].
Figure 2(a) shows the dipolar-exchange magnon dispersion for three values of
$w_{y}$, where $d=50\,$nm, number of segments $N_{L}=200$, i.e. a nanowire of
length $L=10\,\mathrm{\mu}$m. Figure 2(a) shows that the dispersion minimum
becomes deeper for larger $w_{y}$. When the wire is not too narrow (e.g.
$w_{y}>100\,$nm for $w_{x}=100\,$nm [37]), the dispersion relation is non-
monotonous or “backward moving” for small wave vectors along the magnetization
[40, 41, 39].

Next, we Bosonize the Hamiltonian
$H_{R}=\sum_{i}\sum_{j}\omega_{j,01}r_{i,j}^{{\dagger}}r_{i,j}$ and its
interaction with the system
$H_{SR}=\sum_{i}\sum_{j}\left(V_{i,j}r_{i,j}^{{\dagger}}a_{i}+\mathrm{H.c.}\right)$,
where $i$ labels the magnetic segments and $j$ the TLS for weak excitations,
i.e. $\hbar\omega_{j,01}\gg k_{B}T$. The TLS pseudo-spin $\Omega$ Hamiltonian
can be simplified by another Holstein-Primakoff transformation
$\mathcal{L}_{i,j}^{+}=\sqrt{2\mathcal{L}_{j}}r_{i,j}^{{\dagger}}$ and
$\mathcal{L}_{i,j}^{-}=\sqrt{2\mathcal{L}_{j}}r_{i,j}$, where
$r_{i,j}^{{\dagger}}$ ($r_{i,j}$) creates (annihilates) a boson with frequency
$\omega_{j,01}$. The polarization of a TLS with index $j$ in the collection,
$\mathcal{L}_{j}=\langle\Omega_{z}\rangle=\tanh\left[\hbar\omega_{j,01}/\left(2k_{B}T\right)\right]/2$.
$V_{i,j}=\omega_{j,01}\sqrt{c_{j}\mathcal{L}_{j}}/\sqrt{S_{0}}$ is the
interaction between a magnon on site $i$ with pseudo-spin $j$ at relative
concentration $c_{j}$. Each TLS collection is in contact with a large
reservoir at a (slowly varying) temperature $T_{i}$ and dissipation
$\xi_{j}=2\pi/\tau_{j}$. This dissipation is accompanied by the fluctuating
field acting on TLS collection $g_{i,j}=\sqrt{\xi_{j}}\mathcal{F}_{i,j}$,
where
$\langle\mathcal{F}_{i,j}(t)\mathcal{F}_{i,j}^{{\dagger}}(t^{\prime})\rangle=\left(n_{i,j}^{th}+1\right)\delta(t-t^{\prime})$,
$\langle\mathcal{F}_{i,j}^{{\dagger}}\mathcal{F}_{i,j}\rangle=n_{i,j}^{th}\delta(t-t^{\prime})$,
and $n_{i,j}^{th}=\left(e^{\hbar\omega_{j,01}/k_{B}T_{i}}-1\right)^{-1}$. The
white noise correlation functions hold as long as $\hbar\xi_{j}\ll k_{B}T_{i}$
for each $i$, which is a safe assumption for $T_{i}>10\,$mK and
$\tau_{1}=10\,$ns. Here, we focus on low temperatures $T<1\,$K and a single
TLS parametrized by $c_{1}=3\times 10^{-7}$, $\tau_{1}=10\,$ns, and
$\omega_{1,01}/2\pi=10\,$GHz, leading to $V_{i,1}/2\pi\approx 1.5\,$MHz at
$T=0$. [see Fig. 1(a)]. We are safely in the regime $\langle
r_{i,1}^{{\dagger}}r_{i,1}\rangle\ll Nc_{1}\mathcal{L}_{1}$, where $N$ is
number of unit cells, i.e. far from the saturation of TLS excitations.

We now address the steady state for the model defined above, i.e. a closed
system of a magnetic nanowire with a large temperature gradient and at low
temperatures. The Pt side contacts non-invasively detect the non-thermal
component of site-dependent magnon distributions, i.e. the TSSE, which we
compute numerically without additional approximations. The objective is the
matrix $\Lambda_{\infty}$ of the equal time correlation function of the phase
space variables $m_{x,i}=a_{i}+a_{i}^{{\dagger}}$,
$m_{y,i}=-i(a_{i}-a_{i}^{{\dagger}})$, $X_{i}=r_{i}+r_{i}^{{\dagger}}$, and
$Y_{i}=-i(r_{i}-r_{i}^{{\dagger}})$ (or symmetric covariance matrix) in the
steady state that governs the spatially dependent magnon population and spin
currents [see e.g. Eq. (11)] [42]. This is the long-time limit of the time-
dependent covariance matrix ${\Lambda}$ that obeys the equation of motion
$\dot{\Lambda}=\mathcal{O}\Lambda+\Lambda\mathcal{O}+\Upsilon$ [42], where
$\dot{\mathbf{v}}=\mathcal{O}\mathbf{v}+\mathbf{\mathfrak{c}}$,
$\mathbf{v}=\left[m_{x,1},m_{y,1},X_{1},Y_{1},\cdots,m_{x,L},m_{y,L},X_{L},Y_{L}\right]$,
and $\mathcal{O}$ is determined by the Heisenberg equation
$\mathbf{v}(p)=-i\left[H,\mathbf{v}(p)\right]-\mathbf{\zeta}(p)\mathbf{v}(p)/2$.
$\mathbf{\zeta}(p)=\xi_{1}$ for $p\in\\{4(i-1)+3,4(i-1)+4\\}$ $\forall i$,
while $\mathbf{\zeta}(p)=0$ for $p\in\\{4(i-1)+1,4(i-1)+2\\}$ $\forall i$.
$\mathbf{c}$ is the vector of fluctuating fields and determines
$\Upsilon=\langle\left(\mathbf{c}^{T}\mathbf{c}+\mathbf{c}\mathbf{c}^{T}\right)/2\rangle$.
$\Upsilon$ is diagonal with elements
$\Upsilon(p,p)=\zeta(p)(2n_{i}^{th}+1)=\zeta(p)\left[2\left(e^{\hbar\omega_{1,01}/k_{B}T_{i}}-1\right)^{-1}+1\right]$
($n_{i}^{th}$ is the Planck distribution). We obtain $\Lambda_{\infty}$ by
solving $\mathcal{O}\Lambda_{\infty}+\Lambda_{\infty}\mathcal{O}=-\Upsilon$.
The latter equation can be cast into a linear system of equations in the phase
space variables that we solve numerically by inverting a (non-sparse)
$(4\times N_{L})^{2}\times(4\times N_{L})^{2}$ matrix, which in practice
limits the system size to $N_{L}<100$.

The spin Seebeck spin current can be detected by the inverse spin-Hall voltage
in Pt contacts generated by the spin current pumped by a non-equilibrium
magnetization at the YIG$|$Pt interface [43, 44, 4, 5]. The spin pumping at
site $i$

$\tilde{\mathcal{J}}_{i}^{(SP)}\approx\frac{\hbar
g_{r}}{4\pi}\langle\vec{S}_{i}\times\dot{\vec{S}}_{i}\rangle=\frac{\hbar
g_{r}}{4\pi}\langle S_{i}^{x}\dot{S}_{i}^{y}-S_{i}^{y}\dot{S}_{i}^{x}\rangle,$
(10)

where $g_{r}$ is the real part of the complex spin mixing conductance. We
adress ${\mathcal{J}}_{i}^{(SP)}=4\pi\tilde{\mathcal{J}}_{i}^{(SP)}/\hbar
g_{r}$ in the rest of the paper.
$\dot{S}^{x,(y,z)}=-i\left[S^{x(y,z)},H_{S}\right]$ leads to

$\displaystyle\mathcal{J}_{i}^{(SP)}$
$\displaystyle=\frac{1}{4}\sum_{j}\left\\{\left[SJ\delta\left(i\pm
1,j\right)+\mathcal{B}_{i,j}\right]\times\right.$
$\displaystyle\left.\left(\langle m_{x}^{i}m_{x}^{j}\rangle+\langle
m_{y}^{i}m_{y}^{j}\rangle\right)+\mathcal{C}_{ij}\left(-\langle
m_{x}^{i}m_{x}^{j}\rangle+\langle m_{y}^{i}m_{y}^{j}\rangle\right)\right\\},$
(11)

which at equilibrium is canceled exactly by the torque induced by the thermal
spin current noise emitted by the metal contact [4]. Therefore, for a certain
temperature profile $T_{i}$, the net spin current pumped from site $i$ into
NM,
$\mathcal{J}_{i}^{(s)}=\mathcal{J}_{i}^{(SP,NEQ)}-\mathcal{J}_{i}^{(SP,EQ)}$,
where $\mathcal{J}_{i}^{(SP,NEQ)}$ ($\mathcal{J}_{i}^{(SP,EQ)})$ are the non-
equilibrium (equilibrium) currents at a contact $i$ with temperature $T_{i}$.
Disregarding any spin accumulation in the metal contacts, the SSE spin current
is pumped by non-equilibrium magnons. Indeed, the dominant term (confirmed by
calculations) in $J_{i}^{(SP)}$ is proportional to $\langle
m_{x}^{i}m_{x}^{i}\rangle+\langle m_{y}^{i}m_{y}^{i}\rangle=4\langle
a_{i}^{{\dagger}}a_{i}\rangle+2$. Therefore, the local magnon accumulation at
each site, i.e. the difference of $\langle a_{i}^{{\dagger}}a_{i}\rangle$ at
equilibrium and non-equilibrium, drives the spin current
$\mathcal{J}_{i}^{(s)}$. The thermalization is weak so the local distribution
functions cannot be parametrized by magnon temperatures or chemical
potentials. We disregard the effect of the pumping on the magnon system for
simplicity, which is allowed when the mixing conductance is small, e.g. for
sufficiently small contacts. The distributed spin pumping currents in Eq. (11)
may then be expressed in terms of the steady-state covariance matrix
$\Lambda_{\infty}$, i.e. $\langle
m_{x}^{i}m_{x}^{j}\rangle=\Lambda_{\infty}\left[4(i-1)+1,4(j-1)+1\right]$ and
$\langle
m_{y}^{i}m_{y}^{j}\rangle=\Lambda_{\infty}\left[4(i-1)+2,4(j-1)+2\right]$.

Figure 2: (a) The magnon dispersion of a wire with different widths $w_{y}$.
The wavenumber $k$ corresponds to the peak of the Fourier transform of the
spatial wave function. (b) and (c) Spin Seebeck current as a function of
average temperature $T_{M}=(T_{L}+T_{R})/2$ and temperature difference $\Delta
T=T_{R}-T_{L}$, respectively. Left: $|\mathcal{J}^{(s)}/\Delta T|$. Right:
$\mathcal{J}^{(s)}/|\mathcal{J}^{(s)}|$. In (b), $\Delta T=10\,$mK. In (c),
$T_{L}=20\,$mK. In (b) and (c), $w_{y}=500\,$nm [red curve in (a)]. In
(a)-(c), $h^{z}=20\,$mT. Figure 3: (a) Site dependence of
$\mathcal{J}_{i}^{(s)}$ for the $\Delta T$ $=107,135,894\,$mK from Fig. 2(c).
(b) Dependence of spin current $\mathcal{J}_{i}^{(s)}\left(\Delta T\right)$ of
four sites $i$ on $\Delta T$, from Fig. 2(c). The dashed lines correspond to a
finer mesh $d=25\,$nm (rather than 50 nm), illustrating convergence. In (a)
and (b), $T_{L}=20\,$mK.

Temperature dependence - We apply a linear temperature gradient with $\Delta
T=T_{R}-T_{L}$, $T_{L}$ ($T_{R}$) is the temperature at the left (right) edge
of the nanowire. In linear response, the TSSE signal is antisymmetric,
changing sign in the middle of the wire [4, 5, 26, 25, 24]. Figures 2(b) and
(c) show amplitudes (left panels) and signs (right panels) of
$\mathcal{J}_{i}^{s}/\Delta T$ as a function of
$T_{M}=\left(T_{R}+T_{L}\right)/2$ and temperature difference $\Delta T$,
respectively, for $w_{y}=500\,$nm and $h^{z}=20\,$mT [with dispersion in Fig.
2(a)]. In Fig. 2(b), we show the dependence on the mean temperature $T_{M}$
for fixed $\Delta T=10\,$mK. In Fig. 2(c), $T_{L}=20\,$mK is fixed and the
gradient $\Delta T$ is varied. According to Figure 2(b), the signal increases
with increasing $T_{M}$. The early saturation is an artifact by the frequency
cut-off, $\omega_{\mathrm{max}}$, introduced by the finite mesh size $d$, so
results are valid for $T<\hbar\omega_{\mathrm{max}}/k_{B}$. The qualitative
features nevertheless remain intact for half the mesh size $d$, i.e. a $\sim
4$ times larger cutoff frequency, as illustrated in Fig. 3(b). The site index
$i_{\pm}$ at which $\mathcal{J}_{i}^{s}$ changes sign in Fig. 2(b) shifts to
the right edge with decreasing $T_{M}$ below $\hbar\omega_{c}/k_{B}=0.1\,$K,
where $\omega_{c}$ is the uniform (Kittel) mode frequency. According to Fig.
2(c), the asymmetry survives at higher temperatures with increasing
temperature difference $\Delta T$ and fixed $T_{L}\ll\hbar\omega_{c}/k_{B}$.
Figures 3(a)-(b) emphasize the essence of the results in Fig. 2(c) (see also
Fig. S1 [50]). Figure 3(a) shows the deviation of the signal from an
antisymmetric profile. In Fig. 3(b), we observe that for a contact on the
right half of the nanowire and fixed small $T_{L}=20\,$mK, the TSSE signal
changes sign and a maximum appears at relatively large $\Delta T$ and $T_{M}$.
The dashed curves in Fig. 3(b) show that for smaller mesh size $d$, i.e.
higher cutoff frequency, the peak and sign change features remain intact. In
the blue curves of Fig. 3(b), the $\Delta T^{\prime}$s that cause the sign
change and peak are $\sim 0.4\,$K and $\sim 0.2\,$K, respectively.

The deviation from an antisymmetric signal can partly be understood in a
semiclassical picture. The additional occupation of a magnon mode with
frequency $\omega_{0}$ scales like $\delta
n_{BE}\sim\tau_{r}v(\omega_{0})\nabla n_{BE}(\omega_{0})$ [12], where
$\tau_{r}$ is a relaxation time, $v({\omega_{0}})$ is the group velocity, and
$n_{BE}(\omega_{0})=\left(e^{\hbar\omega_{0}/k_{B}T}-1\right)^{-1}$.
$\int_{\omega}\delta n_{BE}\neq 0$ because the magnons pile up or get drained
at the edges. In linear response and a long spin-diffusion length the
dependence is linear with a zero in the center. An expansion in
$\hbar\omega_{0}/\left(k_{B}T\right)$ can only indicate that $\nabla
n_{BE}(\omega_{0})$ is uniform for $T_{M}>\hbar\omega_{0}/k_{B}$ when $\Delta
T\ll T_{M}$ [see Fig. 2(b)], and for $\Delta T\gg\hbar\omega_{0}/k_{B}$ when
$T_{M}\sim\Delta T/2$ [see Fig. 2(c)]. A detailed calculation is indeed
necessary to determine the spatial dependence of magnon accumulation in the
nonlinear regime of temperature, where we observe a substantial non-
monotonicity of signal at certain contact positions [see Fig. 3(b)].

The TSSE voltage
$V_{i}^{(TSSE)}=2\rho\theta_{H}ew_{y}g_{r}\mathcal{J}_{i}^{s}/\left(4\pi
w_{y}d\right)$, where $-e$ is the electron charge, and for Pt, the
conductivity $\rho=0.9\,\mathrm{\mu{\Omega}\,}$m [46], spin Hall angle
$\theta_{H}=0.07$ [45], while for the YIG/Pt interface
$g_{r}/\left(w_{y}d\right)=10^{16}\,1/\text{m}^{2}$ [46]. This leads to
$V_{i}^{(TSSE)}\approx 8\times 10^{-18}\mathcal{J}_{i}^{s}\,$V. The low
temperature maximum of $\mathcal{J}_{i}^{s}/\Delta T\sim 10^{10}$ for $\Delta
T=0.2\,$K [see Fig. 3(b)] leads to a substantial $V_{i}^{(TSSE)}\sim 16\,$nV.

Figure 4: Longitudinal spin Seebeck effect. The temperature dependence of the
spin current at the right endpoint flowing into a spin sink (a) $\Delta
T=10\,$mK. (b) $T_{L}=20\,$mK. Here we plot
$(\mathcal{J}_{L}^{(LSSE)}-\mathcal{J}_{R}^{(LSSE)})/2$ (red line labeled Sym)
and $\mathcal{J}_{R}^{(LSSE)}+\mathcal{J}_{L}^{(LSSE)}$ (black line labeled
Asym.)

LSSE - The LSSE records the total spin current generated in the magnet within
the spin relaxation length and not just the magnon accumulation at the contact
as in the TSSE. We can access the LSSE by modifying the boundary conditions at
the terminals of the wire in order to allow the spin currents to flow
unimpeded into the contacts that act as spin and energy sinks. To this end, we
introduce two reservoirs to the left and right of the nanowire. We assume the
reservoirs to be non-magnetic metals (NM) with a large spin mixing conductance
and interfacial damping at both ends as detailed in [50]. This is in contrast
to the contacts in the TSSE, which we assumed to be non-invasive. Figures 4(a)
and (b) show temperature dependence of the average current through the wire
$\mathcal{J}^{(LSSE)}=\left(\mathcal{J}_{R}^{(LSSE)}-\mathcal{J}_{L}^{(LSSE)}\right)/2$.
The temperature combinations of Figs. 4(a) and (b) are the same as in Figs.
2(a) and (b), i.e. fixed $\Delta T=10\,$mK but varying $T_{M}$ or fixed
$T_{L}=20\,$mK but varying $\Delta T$, respectively. Figures 4(a) and (b) are
featureless and illustrate that even for low $T_{M}\ll\Delta T$,
$\mathcal{J}^{(LSSE)}$ depends (quasi-)linearly on $\Delta T$ [see Fig. 4(b)].
In Fig. 4(b), we also show difference of the spin currents into the left and
out of the right contacts $\mathcal{J}_{R}^{(LSSE)}+\mathcal{J}_{L}^{(LSSE)}$,
which turns out to be relatively very small because LSSE is dominated by the
bulk spin current which is the same for both contacts. It should be emphasized
that our TLS model is strictly valid only for $T<1\,$K.

Conclusion - We investigate SSE at cryogenic temperatures $T\lesssim 1\,$K
with dissipation by two-level systems. In the nonlinear temperature regime,
i.e. large $\Delta T/T$, we predict a non-monotonic TSSE signal at certain
position of the detector contacts. For a linear temperature gradient, and a
contact position in the hot region, the sign changes and a substantially large
$\sim 10\,$nV voltage peak emerges at $2T_{M}\approx\Delta T\sim 0.2\,$K. On
the other hand, the LSSE signal follows a (quasi-)linear dependence on $\Delta
T$, even when much larger than the average temperature.

Acknowledgments -We acknowledge support by JSPS KAKENHI Grants with Nos.
20H02609 and 19H006450.

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