# Intrinsic decoherence for the displaced harmonic oscillator

Alejandro R. Urzúa111email<EMAIL_ADDRESS>and Héctor M. Moya-
<EMAIL_ADDRESS>
Coordinación de Óptica
Instituto Nacional de Astrofísica, Óptica y Electrónica
Luis Enrique Erro #1, 72840, Tonantzintla, Puebla

###### Abstract

By using the complete solution of the Milburn equation (beyond the Lindblad
form that it is generally used) that describes intrinsic decoherence, we study
the decaying dynamics of a displaced harmonic oscillator. We calculate the
expectation values of position quadrature, and the number operator in initial
coherent and squeezed states.

##  1 Introduction

Being decoherence the main enemy of the nonclassical properties that may be
generated in quantum mechanical systems, it is of great interest its study to
learn how the interesting properties that arise in such systems may be
maintained. Some years ago Milburn [1] proposed a modification of the
Schrödinger equation that accounts for (intrinsic) decoherence. By assuming
that the system evolves by a random sequence of unitary phase changes on
sufficiently small time scales Milburn produced a Lindblad equation with the
Hamiltonian as the relevant operator. Since then, several authors have studied
the decay of coherences for different systems, for instance it has been shown
that in the atom-field interaction such loss of coherences prevents the
revivals to occur for the atomic population inversion [2]. Yang et al. [3]
determined the performance of quantum Fisher information of the two-qutrit
isotropic Heisenberg $XY$ chain subject to decoherence; Mohamed et al. [4]
analyzed the robustness of quantum correlations of the nearest neighbour and
the next-to-neighbour qubits in an intrinsic noise model describing the
dynamics of the decoherence for a system formed by three-qubit Heisenberg $XY$
chain. Zheng and Zhang [5] applied Milburn’s scheme to study the entanglement
in the Jaynes Cummings model, where a pair of atoms undergo Heisenberg type
interactions; He and Chao [6] studied the coherence dynamics of two atoms in a
Kerr-like medium; Muthuganesan and Chandrasekar [7] applied intrinsic
decoherence when studying an exactly solvable model of two interacting
spin-$\frac{1}{2}$ qubits described by the Heisenberg anisotropic interaction.
Chlih et al. [8] used the intrinsic decoherence to study a variety of initial
states, where they obtain the temporal evolution of quantum correlations in a
two-qubit XXZ Heisenberg spin chain model subject to a Dzyaloshinskii–Moriya
(DM) interaction and to an external uniform magnetic field. Intrinsic
decoherence scheme offers and alternative to the study of coherence phenomena
under symmetry breaking, just as Gong et. al [9] showed that in a ring
arrangement of coupled harmonic oscillators. Guo-Hui and Bing Bing [10]
estimated the quantum discord of two qubits that loose coherence through
intrinsic mechanisms. Furthermore, Mohamed et. al [11] have used the intrinsic
decoherence effect for two qubits interacting with a coherent field, with the
purpose to protect the entropy and entanglement from the dipole-dipole
interaction. León-Montiel et al. [12], have shown that noise may be helpful to
assist energy transfer in coupled oscillators. Bayen and Mandal [13] reported
the analytical solution of quartic anharmonic oscillator with driven force
that show squeezing effects on the coherent states. some time ago, Mandal
[14], shows a way to calculate the quasidistribution of photons in the
squeezed states of light when the parameters are taken as purely real. Germain
et al. [15] explicitly shows the effects of a non-commutative bath of
oscillators on a magneto-oscillator, they gives the solution for decoherence
without dissipation using the master’s equation formulation. Recently, Lu
[16], reported the evolution of orthogonal coherent states given the solution
of the master’s equation, showing how the decoherence affects the negativity
of the associate Wigner’s function. Mohamed [17] and Abhignan [18] have
studied the role of intrinsic decoherence in the subsystems correlations by
using the quantum discord, finding how the decoherence parameter leads to
sudden death and revivals of some geometrical and dynamical quantities, like
entanglement.

The present paper is organized as follow: In Section 2 we give a review of the
Milburn’s intrinsic decoherence method. Section 3 introduce the displaced
harmonic oscillator in terms of annihilation and creation operators; there, we
calculate the expectation values for the position quadrature and the number
operators for a general initial condition. Subsection 3.1 details the
expectation values when we have an initial coherent state. Subsection 3.2
details the expectation values when we have an initial squeezed state for
complex squeezing parameter; here, we hold the real part constant and vary the
phase to look for changes in the dynamics. Section 4 is left to discuss
results and conclusions.

##  2 Intrinsic decoherence equation and solution

Decoherence is a central research topic in quantum mechanics, since it shows
how a quantum system starts to lose it quantumness because the system under
study is usually surrounded by an environment that affects it. However, there
may be other causes of decoherence that are intrinsic to the system [1]. We
may say that there exist two main causes of decoherence [19]: First, those
that are driven by environmental interactions, oscillators or spins bath, for
example, where the Linblad master equations plays the role of the describing
mathematical framework; and the other is intrinsic decoherence that is driven
by the mere existence of the system dynamics in the equations that described
their nature. We will study here the second,by using the equation that Milburn
[1] introduced to modifiy the Schrödinger equation and that describes
(intrinsic) decoherence via the equation

$\dot{\rho}=\gamma\left(e^{-\mathrm{i}\frac{\hat{H}}{\gamma}}\rho
e^{\mathrm{i}\frac{\hat{H}}{\gamma}}-\rho\right),$ (1)

where $\hat{H}$ is the system’s Hamiltonian, and $\gamma$ is the intrinsic
decoherence parameter that states the rate of decaying in the dynamics. By
developing the exponentials functions above in Taylor series, and keeping
terms up to second order, we obtain

$\dot{\rho}\approx\gamma\left(\left[1-\mathrm{i}\frac{\hat{H}}{\gamma}-\frac{\hat{H}^{2}}{\gamma^{2}}\right]\rho\left[1+\mathrm{i}\frac{\hat{H}}{\gamma}-\frac{\hat{H}^{2}}{\gamma^{2}}\right]-\rho\right),$
(2)

that can be rewritten in the Lindblad form

$\dot{\rho}=-\mathrm{i}\left[\hat{H},\rho\right]-\frac{1}{\gamma}\left[\hat{H},\left[\hat{H},\rho\right]\right],$

where the Schrödinger equation is recovered when $\gamma\rightarrow\infty$.
This last equation shows us that a Linbladian may be obtained from the
complete intrinsic decoherence evolution, describing the system with different
levels of accuracy, since the parameter expansion leads to drop highers order
of decoherence.

However, a complete solution of (1) may be obtained

$\rho(t)=e^{-\gamma t}e^{\hat{S}t}\rho(0),$ (3)

where the superoperator

$\hat{S}\rho=\gamma e^{-\mathrm{i}\frac{\hat{H}}{\gamma}}\rho
e^{\mathrm{i}\frac{\hat{H}}{\gamma}},$

has been used, such that

$e^{\hat{S}t}\rho(0)=\sum_{k=0}^{\infty}\frac{(\gamma t)^{k}}{k!}\,\rho_{k},$

with the $k$-th element of the density matrix $\rho$ defined by

$\rho_{k}=\ket{\psi_{k}}\bra{\psi_{k}},\qquad\ket{\psi_{k}}=e^{-\mathrm{i}k\frac{\hat{H}}{\gamma}}\ket{\psi(0)},$
(4)

with $\ket{\psi(0)}$ the initial wavefunction.

##  3 Displaced harmonic oscillator

We start with the Hamiltonian for a displaced harmonic oscillator (we set
$\hbar=1$)

$\hat{H}=\omega\hat{a}^{\dagger}\hat{a}+\lambda\left(\hat{a}+a^{\dagger}\right),$
(5)

where $\omega$ is the natural frequency of the standard harmonic oscillator,
and $\lambda$ is the amplitude of the displacement, meaning that the potential
function is moved and scaled.

It is not difficult to show that the Hamiltonian (5) may be rewritten as

$\hat{H}=\omega\hat{D}^{\dagger}\left(\frac{\lambda}{\omega}\right)\hat{a}^{\dagger}\hat{a}\hat{D}^{\dagger}\left(\frac{\lambda}{\omega}\right)-\frac{\lambda^{2}}{\omega},$
(6)

where $\hat{D}\left(\tfrac{\lambda}{\omega}\right)$ is the usual displacement
operator. With this last equation, we obtain the $k$-th element of the
wavefunction (4) as

$\ket{\psi_{k}}=e^{\mathrm{i}\frac{k\lambda^{2}}{\gamma\omega}}\hat{D}^{\dagger}\left(\frac{\lambda}{\omega}\right)e^{-\mathrm{i}\hat{a}^{\dagger}\hat{a}\frac{k\omega}{\gamma}}\hat{D}\left(\frac{\lambda}{\omega}\right)\ket{\psi(0)}.$
(7)

We now study two properties of quantum mechanical system, namely the position
quadrature and the average number of photons. It is interesting first, to
denote how the decoherence parameter affects the position quadrature, since it
represent the real part of the complex amplitude of the annihilation operator
$\hat{a}$; next, the effect on the photon number is explicit since we can
observe how the population tends to decay in time depending on the strength of
the decaying parameter $\gamma$.

#### Average of the position quadrature operator

We start by calculating the matrix elements of (twice) the position quadrature
operator over the $k$-th components of the wavefunction, we have then

$\displaystyle\bra{\psi_{k}}\hat{a}^{\dagger}+\hat{a}\ket{\psi_{k}}=$ (8)
$\displaystyle\quad\left[\bra{\psi(0)}\hat{D}^{\dagger}\left(\frac{\lambda}{\omega}\right)e^{\frac{\mathrm{i}k\omega}{\gamma}\hat{a}^{\dagger}\hat{a}}\hat{D}\left(\frac{\lambda}{\omega}\right)\right]\left(\hat{a}^{\dagger}+\hat{a}\right)$
$\displaystyle\qquad\qquad\times\left[\hat{D}^{\dagger}\left(\frac{\lambda}{\omega}\right)e^{-\frac{\mathrm{i}k\omega}{\gamma}\hat{a}^{\dagger}\hat{a}}\hat{D}\left(\frac{\lambda}{\omega}\right)\ket{\psi(0)}\right]$
$\displaystyle=\bra{\psi(0)}\hat{D}^{\dagger}\left(\frac{\lambda}{\omega}\right)e^{\frac{\mathrm{i}k\omega}{\gamma}\hat{a}^{\dagger}\hat{a}}\left[\hat{a}^{\dagger}+\hat{a}-2\frac{\lambda}{\omega}\right]$
$\displaystyle\qquad\qquad\times
e^{-\frac{\mathrm{i}k\omega}{\gamma}\hat{a}^{\dagger}\hat{a}}\hat{D}\left(\frac{\lambda}{\omega}\right)\ket{\psi(0)}$
$\displaystyle=\bra{\psi(0)}\hat{D}^{\dagger}\left(\frac{\lambda}{\omega}\right)\left[\hat{a}^{\dagger}e^{\frac{\mathrm{i}k\omega}{\gamma}}+\hat{a}e^{-\frac{\mathrm{i}k\omega}{\gamma}}-2\frac{\lambda}{\omega}\right]\hat{D}\left(\frac{\lambda}{\omega}\right)\ket{\psi(0)}$
$\displaystyle=\bra{\psi(0)}\left[\left(\hat{a}^{\dagger}+\frac{\lambda}{\omega}\right)e^{\frac{\mathrm{i}k\omega}{\gamma}}+\left(\hat{a}+\frac{\lambda}{\omega}\right)e^{-\frac{\mathrm{i}k\omega}{\gamma}}-2\frac{\lambda}{\omega}\right]\ket{\psi(0)},$

which gives finally

$\displaystyle\braket{(\hat{a}^{\dagger}+\hat{a})}=e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\psi_{k}}\hat{a}\ket{\psi_{k}}$ (9)
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\psi(0)}\left[\left(\hat{a}^{\dagger}+\frac{\lambda}{\omega}\right)e^{\frac{\mathrm{i}k\omega}{\gamma}}\right.$
$\displaystyle\hskip
160.43727pt\left.+\left(\hat{a}+\frac{\lambda}{\omega}\right)e^{-\frac{\mathrm{i}k\omega}{\gamma}}-2\frac{\lambda}{\omega}\right]\ket{\psi(0)}.$

Given a suitable initial wavefunction $\ket{\psi(0)}$, we can sum up the terms
and find analytical expressions for the average dynamics.

#### Average of the number of photons operator

For the average of the number of photons, we follow the procedure depicted in
equations (8) and (9), first calculating the matrix elements of the operator
$\hat{a}^{\dagger}\hat{a}$

$\displaystyle\bra{\psi_{k}}\hat{a}^{\dagger}\hat{a}\ket{\psi_{k}}=$ (10)
$\displaystyle\quad\left[\bra{\psi(0)}\hat{D}^{\dagger}\left(\frac{\lambda}{\omega}\right)e^{\frac{\mathrm{i}k\omega}{\gamma}\hat{a}^{\dagger}\hat{a}}\hat{D}\left(\frac{\lambda}{\omega}\right)\right]\hat{a}^{\dagger}\hat{a}$
$\displaystyle\hskip
56.9055pt\times\left[\hat{D}^{\dagger}\left(\frac{\lambda}{\omega}\right)e^{-\frac{\mathrm{i}k\omega}{\gamma}\hat{a}^{\dagger}\hat{a}}\hat{D}\left(\frac{\lambda}{\omega}\right)\ket{\psi(0)}\right]$
$\displaystyle=\bra{\psi(0)}\hat{D}^{\dagger}\left(\frac{\lambda}{\omega}\right)e^{\frac{\mathrm{i}k\omega}{\gamma}\hat{a}^{\dagger}\hat{a}}$
$\displaystyle\hskip
56.9055pt\times\left[\hat{a}^{\dagger}\hat{a}-\frac{\lambda}{\omega}\left(\hat{a}^{\dagger}+\hat{a}\right)+\frac{\lambda^{2}}{\omega^{2}}\right]$
$\displaystyle\hskip 113.81102pt\times
e^{-\frac{\mathrm{i}k\omega}{\gamma}\hat{a}^{\dagger}\hat{a}}\hat{D}\left(\frac{\lambda}{\omega}\right)\ket{\psi(0)}$
$\displaystyle=\bra{\psi(0)}\hat{D}^{\dagger}\left(\frac{\lambda}{\omega}\right)$
$\displaystyle\hskip
56.9055pt\times\left[\hat{a}^{\dagger}\hat{a}-\frac{\lambda}{\omega}\left(\hat{a}^{\dagger}e^{\frac{\mathrm{i}k\omega}{\gamma}}+\hat{a}e^{-\frac{\mathrm{i}k\omega}{\gamma}}\right)+\frac{\lambda^{2}}{\omega^{2}}\right]$
$\displaystyle\hskip
170.71652pt\times\hat{D}\left(\frac{\lambda}{\omega}\right)\ket{\psi(0)}$
$\displaystyle=\bra{\psi(0)}\left[\left(\hat{a}^{\dagger}\hat{a}+\frac{\lambda}{\omega}\left\\{\hat{a}^{\dagger}+\hat{a}\right\\}+2\frac{\lambda^{2}}{\omega^{2}}\right)-\right.$
$\displaystyle\hskip
42.67912pt\left.\frac{\lambda}{\omega}\left(\left\\{\hat{a}^{\dagger}+\frac{\lambda}{\omega}\right\\}e^{\frac{\mathrm{i}k\omega}{\gamma}}+\left\\{\hat{a}+\frac{\lambda}{\omega}\right\\}e^{-\frac{\mathrm{i}k\omega}{\gamma}}\right)\right]\ket{\psi(0)},$

that in the same fashion of equation (9), we only need to give a suitable
initial condition $\ket{\psi(0)}$ to obtain explicit expressions.

###  3.1 Initial coherent state

Let $\ket{\psi(0)}=\ket{\alpha}$ be a coherent state, we can use this initial
condition to evaluate the position quadrature and number operators averages in
the way that equation (9) dictates.

First, we start by calculating the dynamics of (twice) the position quadrature
operator, since $\bra{\alpha}\hat{a}^{\dagger}\ket{\alpha}=\alpha^{*}$ and
$\bra{\alpha}\hat{a}\ket{\alpha}=\alpha$, we obtain

$\displaystyle\braket{\hat{a}^{\dagger}+\hat{a}}=e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\psi_{k}}\hat{a}^{\dagger}+\hat{a}\ket{\psi_{k}}$ (11)
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\alpha}$ $\displaystyle\hskip
42.67912pt\times\left[\left(\hat{a}^{\dagger}+\frac{\lambda}{\omega}\right)e^{\frac{\mathrm{i}k\omega}{\gamma}}+\left(\hat{a}+\frac{\lambda}{\omega}\right)e^{-\frac{\mathrm{i}k\omega}{\gamma}}-2\frac{\lambda}{\omega}\right]\ket{\alpha}$
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\left[\left(\alpha^{*}+\frac{\lambda}{\omega}\right)e^{\frac{\mathrm{i}k\omega}{\gamma}}+\left(\alpha+\frac{\lambda}{\omega}\right)e^{-\frac{\mathrm{i}k\omega}{\gamma}}-2\frac{\lambda}{\omega}\right]$
$\displaystyle=e^{-\gamma
t}\left[\left(\alpha^{*}+\frac{\lambda}{\omega}\right)e^{\gamma
te^{\frac{\mathrm{i}\omega}{\gamma}}}+\left(\alpha+\frac{\lambda}{\omega}\right)e^{\gamma
te^{-\frac{\mathrm{i}\omega}{\gamma}}}-2\frac{\lambda}{\omega}e^{\gamma
t}\right]$
$\displaystyle=\left(\alpha^{*}+\frac{\lambda}{\omega}\right)e^{-\gamma
t\left(1-e^{\frac{\mathrm{i}\omega}{\gamma}}\right)}+\left(\alpha+\frac{\lambda}{\omega}\right)e^{-\gamma
t\left(1-e^{-\frac{\mathrm{i}\omega}{\gamma}}\right)}-2\frac{\lambda}{\omega},$

that checks $\braket{(\hat{a}^{\dagger}+\hat{a})}=\alpha^{*}+\alpha\equiv
2\mathrm{Re}\\{\alpha\\}$, when $t=0$. We plot the expectation value for
$\braket{(\hat{a}^{\dagger}+\hat{a})}$ for several displacemet strengths in
Fig. 1, showing the same behaviour, i.e., a damping in the oscillations. Such
damping is not amplified for larger displacement amplitudes.

Figure 1: Expectation value of $\braket{(\hat{a}^{\dagger}+\hat{a})}$ given by
(11). The parameters are $\alpha=4,\omega=4,\gamma=10$. We vary the
displacement strength $\lambda\in\\{0.1,0.7,1.5\\}$.

For the average dynamics of the number of photons we have

$\displaystyle\braket{\hat{a}^{\dagger}\hat{a}}=e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\psi_{k}}\hat{a}^{\dagger}\hat{a}\ket{\psi_{k}}$ (12)
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\alpha}\left[\left(\hat{a}^{\dagger}\hat{a}+\frac{\lambda}{\omega}\left\\{\hat{a}^{\dagger}+\hat{a}\right\\}+2\frac{\lambda^{2}}{\omega^{2}}\right)-\right.$
$\displaystyle\hskip
56.9055pt\left.\frac{\lambda}{\omega}\left(\left\\{\hat{a}^{\dagger}+\frac{\lambda}{\omega}\right\\}e^{\frac{\mathrm{i}k\omega}{\gamma}}+\left\\{\hat{a}+\frac{\lambda}{\omega}\right\\}e^{-\frac{\mathrm{i}k\omega}{\gamma}}\right)\right]\ket{\alpha}$
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\left[\left(\left|\alpha\right|^{2}+\frac{\lambda}{\omega}\left\\{\alpha+\alpha^{*}\right\\}+2\frac{\lambda^{2}}{\omega^{2}}\right)-\right.$
$\displaystyle\hskip
56.9055pt\left.\frac{\lambda}{\omega}\left(\left\\{\alpha+\frac{\lambda}{\omega}\right\\}e^{\frac{\mathrm{i}k\omega}{\gamma}}+\left\\{\alpha^{*}+\frac{\lambda}{\omega}\right\\}e^{-\frac{\mathrm{i}k\omega}{\gamma}}\right)\right]$
$\displaystyle=e^{-\gamma
t}\left[\left(\left|\alpha\right|^{2}+\frac{\lambda}{\omega}\left\\{\alpha+\alpha^{*}\right\\}+2\frac{\lambda^{2}}{\omega^{2}}\right)e^{\gamma
t}-\right.$ $\displaystyle\hskip
56.9055pt\left.\frac{\lambda}{\omega}\left(\left\\{\alpha+\frac{\lambda}{\omega}\right\\}e^{\gamma
te^{\frac{\mathrm{i}\omega}{\gamma}}}+\left\\{\alpha^{*}+\frac{\lambda}{\omega}\right\\}e^{\gamma
te^{-\frac{\mathrm{i}\omega}{\gamma}}}\right)\right]$
$\displaystyle=\left[\left|\alpha\right|^{2}+\frac{\lambda}{\omega}\left\\{\alpha+\alpha^{*}\right\\}+2\frac{\lambda^{2}}{\omega^{2}}\right]-$
$\displaystyle\hskip
28.45274pt\frac{\lambda}{\omega}\left[\left\\{\alpha+\frac{\lambda}{\omega}\right\\}e^{-\gamma
t\left(1-e^{\frac{\mathrm{i}\omega}{\gamma}}\right)}+\left\\{\alpha^{*}+\frac{\lambda}{\omega}\right\\}e^{-\gamma
t\left(1-e^{-\frac{\mathrm{i}\omega}{\gamma}}\right)}\right],$

that also checks $\braket{\hat{a}^{\dagger}\hat{a}}=\left|\alpha\right|^{2}$,
for $t=0$. In Fig. 2 we plot the average number of photons which shows a
similar behaviour as the quadrature, this is, a damping of the oscillations.

Figure 2: Expectation value of $\braket{\hat{a}^{\dagger}\hat{a}}$ given by
(12). The parameters are $\alpha=4,\omega=4,\gamma=10$. We variate the
displacement strength $\lambda\in\\{0.1,0.7,1.5\\}$.

###  3.2 Initial squeezed state

For an initial squeezed state $\ket{\alpha,z}=\hat{S}(z)\ket{\alpha}$ we use
that $\hat{S}^{\dagger}(z)\hat{a}\hat{S}(z)=\mu\hat{a}-\nu\hat{a}^{\dagger}$
and
$\hat{S}^{\dagger}(z)\hat{a}^{\dagger}\hat{S}(z)=\mu\hat{a}^{\dagger}-\nu^{*}\hat{a}$,
and because $z=re^{\mathrm{i}\theta}$, we have $\mu=\cosh r$ and
$\nu=e^{\mathrm{i}\theta}\sinh r$.

For the average of (twice) the position quadrature operator it is easy to show
that the evolution gives

$\displaystyle\braket{\hat{a}^{\dagger}+\hat{a}}=e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\psi_{k}}\hat{a}^{\dagger}+\hat{a}\ket{\psi_{k}}$ (13)
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\alpha,r}\left[\left(\hat{a}^{\dagger}+\frac{\lambda}{\omega}\right)e^{\frac{\mathrm{i}k\omega}{\gamma}}+\right.$
$\displaystyle\hskip
85.35826pt\left.\left(\hat{a}+\frac{\lambda}{\omega}\right)e^{-\frac{\mathrm{i}k\omega}{\gamma}}-2\frac{\lambda}{\omega}\right]\ket{\alpha,r}$
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\alpha}\hat{S}^{\dagger}(r)\left[\left(\hat{a}^{\dagger}+\frac{\lambda}{\omega}\right)e^{\frac{\mathrm{i}k\omega}{\gamma}}+\right.$
$\displaystyle\hskip
85.35826pt\left.\left(\hat{a}+\frac{\lambda}{\omega}\right)e^{-\frac{\mathrm{i}k\omega}{\gamma}}-2\frac{\lambda}{\omega}\right]\hat{S}(r)\ket{\alpha}$
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\left[\left(\mu\alpha^{*}-\nu^{*}\alpha+\frac{\lambda}{\omega}\right)e^{\frac{\mathrm{i}k\omega}{\gamma}}+\right.$
$\displaystyle\hskip
85.35826pt\left.\left(\mu\alpha-\nu\alpha^{*}+\frac{\lambda}{\omega}\right)e^{-\frac{\mathrm{i}k\omega}{\gamma}}-2\frac{\lambda}{\omega}\right]$
$\displaystyle=\left(\mu\alpha^{*}-\nu^{*}\alpha+\frac{\lambda}{\omega}\right)e^{-\gamma
t\left(1-e^{\frac{\mathrm{i}\omega}{\gamma}}\right)}+$ $\displaystyle\hskip
85.35826pt\left(\mu\alpha-\nu\alpha^{*}+\frac{\lambda}{\omega}\right)e^{-\gamma
t\left(1-e^{-\frac{\mathrm{i}\omega}{\gamma}}\right)}-2\frac{\lambda}{\omega}.$

Figure 3 shows that, as in the case of the coherent states, the oscillations
decay in a fast fashion, independently of the phase of the squeezed states.

Figure 3: Expectation value of $\braket{(\hat{a}^{\dagger}+\hat{a})}$ given by
(13). The parameters are $\alpha=4,\omega=4,\gamma=10,\lambda=0.7,r=0.3$,
where we choose to maintain the real part $r$ of the squeeze coefficient and
variate their complex phase $\theta$.

For the average of the number operator, because the action of the squeezing
operator results in

$\displaystyle\hat{S}^{\dagger}(z)\hat{a}^{\dagger}\hat{a}S(z)$
$\displaystyle=\left(\mu\hat{a}^{\dagger}-\nu^{*}\hat{a}\right)\left(\mu\hat{a}-\nu\hat{a}^{\dagger}\right)$
(14)
$\displaystyle=\mu^{2}\hat{a}^{\dagger}\hat{a}+|\nu|^{2}\hat{a}\hat{a}^{\dagger}-\mu\left(\nu\hat{a}^{\dagger^{2}}+\nu^{*}\hat{a}^{2}\right)$
$\displaystyle=\left(\mu^{2}+|\nu|^{2}\right)\hat{a}^{\dagger}\hat{a}-\mu\left(\nu\hat{a}^{\dagger^{2}}+\nu^{*}\hat{a}^{2}\right)+|\nu|^{2},$

we can then evaluate the average as

$\displaystyle\braket{\hat{a}^{\dagger}\hat{a}}=e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\psi_{k}}\hat{a}^{\dagger}\hat{a}\ket{\psi_{k}}$ (15)
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\alpha,r}\left[\left(\hat{a}^{\dagger}\hat{a}+\frac{\lambda}{\omega}\left\\{\hat{a}^{\dagger}+\hat{a}\right\\}+2\frac{\lambda^{2}}{\omega^{2}}\right)-\right.$
$\displaystyle\hskip
28.45274pt\left.\frac{\lambda}{\omega}\left(\left\\{\hat{a}^{\dagger}+\frac{\lambda}{\omega}\right\\}e^{\frac{\mathrm{i}k\omega}{\gamma}}+\left\\{\hat{a}+\frac{\lambda}{\omega}\right\\}e^{-\frac{\mathrm{i}k\omega}{\gamma}}\right)\right]\ket{\alpha,r}$
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\alpha}\hat{S}^{\dagger}(r)\left[\left(\hat{a}^{\dagger}\hat{a}+\frac{\lambda}{\omega}\left\\{\hat{a}^{\dagger}+\hat{a}\right\\}+2\frac{\lambda^{2}}{\omega^{2}}\right)\right.$
$\displaystyle\hskip
14.22636pt-\left.\frac{\lambda}{\omega}\left(\left\\{\hat{a}^{\dagger}+\frac{\lambda}{\omega}\right\\}e^{\frac{\mathrm{i}k\omega}{\gamma}}+\left\\{\hat{a}+\frac{\lambda}{\omega}\right\\}e^{-\frac{\mathrm{i}k\omega}{\gamma}}\right)\right]\hat{S}(r)\ket{\alpha}$
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\left[\left\\{\mu^{2}+|\nu|^{2}\right\\}\left|\alpha\right|^{2}-\mu\left\\{\nu\alpha^{*^{2}}+\nu^{*}\alpha^{2}\right\\}+\right.$
$\displaystyle\hskip
42.67912pt\left.\frac{\lambda}{\omega}\left\\{\left\\{\mu-\nu^{*}\right\\}\alpha+\left\\{\mu-\nu\right\\}\alpha^{*}\right\\}+2\frac{\lambda^{2}}{\omega^{2}}+|\nu|^{2}-\right.$
$\displaystyle\left.\frac{\lambda}{\omega}\left(\left\\{\mu\alpha^{*}-\nu^{*}\alpha+\frac{\lambda}{\omega}\right\\}e^{\frac{\mathrm{i}k\omega}{\gamma}}+\left\\{\mu\alpha-\nu\alpha^{*}+\frac{\lambda}{\omega}\right\\}e^{-\frac{\mathrm{i}k\omega}{\gamma}}\right)\right]$
$\displaystyle=\left[\left\\{\mu^{2}+|\nu|^{2}\right\\}\left|\alpha\right|^{2}-\mu\left\\{\nu\alpha^{*^{2}}+\nu^{*}\alpha^{2}\right\\}+\right.$
$\displaystyle\hskip
28.45274pt\left.\frac{\lambda}{\omega}\left\\{\left\\{\mu-\nu^{*}\right\\}\alpha+\left\\{\mu-\nu\right\\}\alpha^{*}\right\\}+2\frac{\lambda^{2}}{\omega^{2}}+|\nu|^{2}\right]-$
$\displaystyle\hskip
56.9055pt\frac{\lambda}{\omega}\left[\left\\{\mu\alpha^{*}-\nu^{*}\alpha+\frac{\lambda}{\omega}\right\\}e^{-\gamma
t\left(1-e^{\frac{\mathrm{i}\omega}{\gamma}}\right)}+\right.$
$\displaystyle\hskip
99.58464pt\left.\left\\{\mu\alpha-\nu\alpha^{*}+\frac{\lambda}{\omega}\right\\}e^{-\gamma
t\left(1-e^{-\frac{\mathrm{i}\omega}{\gamma}}\right)}\right],$

Figure 4: Expectation value of $\braket{\hat{a}^{\dagger}\hat{a}}$ given by
(15). The parameters are $\alpha=4,\omega=4,\gamma=10,\lambda=0.7,r=0.3$,
where we choose to maintain the real part $r$ of the squeeze coefficient and
variate their complex phase $\theta$.

In Figure 4 we plot the average number of photons which also shows, as former
examples, the rapid decay of oscillations due to intrinsic decoherence.

##  4 Conclusions

We have given a solution of the Milburn equation (beyond the Lindblad form
that it is generally used). This equation describes a modification of the
Scrhödinger equation that accounts for (intrinsic) decoherence. We have
studied the decaying dynamics of a displaced harmonic oscillator for initially
coherent and squeezed states.

## References

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# Supplemental Material for “Dipole-Obstructed Cooper Pairing: Theory and
Application to $j=3/2$ Superconductors”

Penghao Zhu<EMAIL_ADDRESS>Department of Physics, The Ohio State
University, Columbus, OH 43210, USA  Rui-Xing Zhang<EMAIL_ADDRESS>Department of Physics and Astronomy, The University of Tennessee, Knoxville,
TN 37996, USA  Department of Materials Science and Engineering, The University
of Tennessee, Knoxville, TN 37996, USA

###### Contents

  1. S1 Proof of Chern-vorticity theorem
  2. S2 Mirror symmetry and quantized Berry-dipole flux in two-band tight-binding models
  3. S3 Cooper pairings of Berry-dipole Fermi surface
    1. S3.1 Projection onto FS
    2. S3.2 Symmetry analysis and robust nodal points under certain channels
  4. S4 Annihilation of nodal points upon varying Fermi level in normal Weyl semimetals
  5. S5 Projection of inter-sector pairing onto FS and symmetry analysis
    1. S5.1 Projection onto FS
    2. S5.2 Symmetry analysis
  6. S6 Perturbations from asymmetric spin-orbit coupling
  7. S7 Berry-dipole superconductor

##  S1 Proof of Chern-vorticity theorem

We prove the Chern-vorticity theorem for

${\cal O}({\bf
k}_{1})=\langle\psi_{1}(\mathbf{k}_{1})|\hat{O}|\psi_{2}(\mathbf{k}_{2})\rangle=|{\cal
O}({\bf k}_{1})|e^{i\varphi({\bf k}_{1})}.$ (S1)

by constructing a vector:

$\mathbf{S}=\partial_{\mathbf{k}_{1}}\varphi(\mathbf{k}_{1})-\mathbf{A}_{1}(\mathbf{k}_{1})+g^{T}\mathbf{A}_{2}(\mathbf{k}_{2}),$
(S2)

where

$\mathbf{A}_{a}(\mathbf{k}_{a})=i\langle\psi_{a}(\mathbf{k}_{a})|\partial_{\mathbf{k}_{a}}\psi_{a}(\mathbf{k}_{a})\rangle,\
a=1,2,$ (S3)

is the Berry connection. Importantly, vector $\mathbf{S}$ is invariant under
the $U(1)$ gauge transformation $|\psi_{a}(\mathbf{k}_{a})\rangle\rightarrow
e^{i\alpha_{a}(\mathbf{k}_{a})}|\psi_{a}(\mathbf{k}_{a})\rangle$ with $a=1,2$.
If there is a nonzero Chern-number of $|\psi_{1}(\bf k_{1})\rangle$
($|\psi_{2}(\bf k_{2})\rangle$) on effectively closed $\mathcal{M}_{1}$
($\mathcal{M}_{2}$), then $\mathbf{A}_{1}(\mathbf{k}_{1})$
($\mathbf{A}_{2}(\mathbf{k}_{2})$) cannot be globally smooth over the entire
manifold. Without loss of generality, one can have two locally smooth gauge
patches over the manifold as illustrated for an annulus geometry in Fig.
S1(a). Then, across the boundary (i.e., $\mathcal{L}_{0}$ in Fig. S1(a)) of
the two patches, there are singularities of $\mathbf{A}_{1}(\mathbf{k}_{1})$
and $\mathbf{A}_{2}(\mathbf{k}_{2})$ such that

$\displaystyle\lim_{\delta\rightarrow
0}\left(\int_{\mathcal{L}_{0}+\delta}-\int_{\mathcal{L}_{0}-\delta}\right)\mathbf{A}_{1}(\mathbf{k}_{1})\cdot
d\mathbf{k}_{1}=2\pi\mathcal{C}_{1},$ (S4)
$\displaystyle\lim_{\delta\rightarrow
0}\left(\int_{g(\mathcal{L}_{0}+\delta)+\mathbf{t}}-\int_{g(\mathcal{L}_{0}-\delta)+\mathbf{t}}\right)\mathbf{A}_{2}(\mathbf{k}_{2})\cdot
d\mathbf{k}_{2}=2\pi(\det g)\mathcal{C}_{2}.$

Here, $\det g$ shows up in the second equation above because we have defined
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ over $\mathcal{M}_{1}$ and
$\mathcal{M}_{2}$ under the same orientation-choice, e.g., for a 2D closed
surface in 3D, we choose the normal vectors of $\mathcal{M}_{1}$ and
$\mathcal{M}_{2}$ both to point outward from their enclosed region. If $g$ is
an improper transformation such as inversion, $\det g=-1$ and the path
integral $\lim_{\delta\rightarrow
0}\left(\int_{g(\mathcal{L}_{0}+\delta)+\mathbf{t}}-\int_{g(\mathcal{L}_{0}-\delta)+\mathbf{t}}\right)\mathbf{A}_{2}(\mathbf{k}_{2})\cdot
d\mathbf{k}_{2}$ will correspond to the Chern number defined with respect to
normal vectors pointing inwards the enclosed region as illustrated in Fig.
S1(b), which is opposite to $\mathcal{C}_{2}$.

Figure S1: (a) Illustration of the two gauge patches over an annulus geometry,
and the paths for the line integrals. The purple crosses represent vortex of
$\mathcal{O}(\mathbf{k}_{1})$. The inner (outer) blue dashed circle represents
$\mathcal{L}_{0}+\delta$ ($\mathcal{L}_{0}-\delta$). (b) Illustration of the
orientation of the Berry flux over $\mathcal{M}_{2}$ associated with the path
integral in Eq. (S4) when $g$ is an spatial inversion symmetry. The red and
blue arrows denote the positive direction of the Berry flux.

In contrast to the Berry connection, $\mathbf{S}$ is gauge invariant, and thus
is continuous across the gauge patch boundary, i.e., $\lim_{\delta\rightarrow
0}\left(\int_{\mathcal{L}_{0}+\delta}-\int_{\mathcal{L}_{0}-\delta}\right)\mathbf{S}\cdot
d\mathbf{k}_{1}=0$. Consequently,
$\partial_{\mathbf{k}_{1}}\varphi(\mathbf{k}_{1})$ must have its own
singularities (i.e., vortices) to cancel or balance those from the Berry
connection, which leads to

$\displaystyle\lim_{\delta\rightarrow
0}\left(\int_{\mathcal{L}_{0}+\delta}-\int_{\mathcal{L}_{0}-\delta}\right)\partial_{\mathbf{k}_{1}}\varphi\cdot
d\mathbf{k}_{1}=\lim_{\delta\rightarrow
0}\left(\int_{\mathcal{L}_{0}+\delta}-\int_{\mathcal{L}_{0}-\delta}\right)[\mathbf{A}_{1}(\mathbf{k}_{1})-g^{T}\mathbf{A}_{2}(\mathbf{k}_{2})]\cdot
d\mathbf{k}_{1}$ (S5) $\displaystyle=\lim_{\delta\rightarrow
0}\left(\int_{\mathcal{L}_{0}+\delta}-\int_{\mathcal{L}_{0}-\delta}\right)\mathbf{A}_{1}(\mathbf{k}_{1})\cdot
d\mathbf{k}_{1}-\lim_{\delta\rightarrow
0}\left(\int_{g(\mathcal{L}_{0}+\delta)+\mathbf{t}}-\int_{g(\mathcal{L}_{0}-\delta)+\mathbf{t}}\right)\mathbf{A}_{2}(\mathbf{k}_{2})\cdot
d\mathbf{k}_{2}$
$\displaystyle=2\pi(\mathcal{C}_{1}-(\det{g})\mathcal{C}_{2}).$

As illustrated in Fig. S1(a), the net vorticity of
$\mathcal{O}(\mathbf{k}_{1})$, defined as
$\nu=\sum_{\text{vortex}}\oint_{\partial\text{vortex}}\partial_{\mathbf{k}_{1}}\varphi\cdot
d\mathbf{k}_{1}$, can be calculated through line integrals over
$\mathcal{L}_{0}\pm\delta$ and $b_{1,2}$:

$\displaystyle\nu$
$\displaystyle=\frac{1}{2\pi}\left(\oint_{b_{2}}-\oint_{b_{1}}\right)\partial_{\mathbf{k}_{1}}\varphi\cdot
d\mathbf{k}_{1}+\frac{1}{2\pi}\lim_{\delta\rightarrow
0}\left(\int_{\mathcal{L}_{0}+\delta}-\int_{\mathcal{L}_{0}-\delta}\right)\partial_{\mathbf{k}_{1}}\varphi\cdot
d\mathbf{k}_{1}$ (S6) $\displaystyle=\mathcal{C}_{1}-(\det
g)\mathcal{C}_{2}+\frac{1}{2\pi}\left(\oint_{b_{2}}-\oint_{b_{1}}\right)\partial_{\mathbf{k}_{1}}\varphi\cdot
d\mathbf{k}_{1}$

Rewriting
$1/(2\pi)\left(\oint_{b_{2}}+\oint_{b_{1}}\right)\partial_{\mathbf{k}_{1}}\varphi\cdot
d\mathbf{k}_{1}$ formally as
$1/(2\pi)\sum_{b}\oint_{b}\partial_{\mathbf{k}_{1}}\varphi\cdot
d\mathbf{k}_{1}$ results in the Chern-vorticity theorem depicted in Eq. (2) in
the main text.

For SC systems where
$|\psi_{2}(\mathbf{k}_{2})\rangle=|\xi_{2}^{(e)\star}(\mathbf{k}_{2})\rangle$,

$\mathbf{A}_{2}(\mathbf{k}_{2})=i\langle\xi^{(e)\star}_{2}(\mathbf{k}_{2})|\partial_{\mathbf{k}_{2}}\xi^{(e)\star}_{2}(\mathbf{k}_{2})\rangle=-i\langle\xi^{(e)}_{2}(\mathbf{k}_{2})|\partial_{\mathbf{k}_{2}}\xi^{(e)}_{2}(\mathbf{k}_{2})\rangle,$

and thus

$\mathcal{C}_{2}\equiv\lim_{\delta\rightarrow
0}\left(\int_{-\mathcal{L}_{0}+\delta}-\int_{-\mathcal{L}_{0}-\delta}\right)\left[-i\langle\xi^{(e)}_{2}(\mathbf{k}_{2})|\partial_{\mathbf{k}_{2}}\xi^{(e)}_{2}(\mathbf{k}_{2})\rangle\right]\cdot
d\mathbf{k}_{2},$

which is opposite to the Chern number of electronic Bloch states
$|\xi^{(e)}_{2}(\mathbf{k}_{2})\rangle$.

##  S2 Mirror symmetry and quantized Berry-dipole flux in two-band tight-
binding models

In this part, we prove that the mirror symmetry can quantize Berry-dipole flux
over the FS [1, 2] for general two-band models. Let us consider a general two-
band Hamiltonian with $H({\bf k})=\mathbf{h}(\mathbf{k})\cdot\bm{\sigma}$ that
respects a $\hat{z}$-directional mirror symmetry $M_{z}=i\sigma_{z}$. Here
$\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are Pauli matrices.
Crucially, $M_{z}$ requires $h_{x,y}(-k_{z})=-h_{x,y}(k_{z})$ and
$h_{z}(-k_{z})=h_{z}(k_{z})$. For a spherical-like FS ${\cal M}$ enclosing
$\Gamma=(0,0,0)$, we denote ${\cal M}_{0}$ as the 1D “equator” at $k_{z}=0$
[i.e., the black loop in Fig. (1))(b) in the main text] and ${\cal M}_{\pm}$
as the FS patches with $k_{z}>0$ and $k_{z}<0$, respectively. Since $M_{z}$
requires $H(k_{z}=0)$ to be diagonal, either valence or conduction states
along ${\cal M}_{0}$ shall be identical, up to an unimportant $U(1)$ phase. In
other words, by choosing a proper gauge to eliminate this $U(1)$ phase, the 1D
equator ${\cal M}_{0}$ is topologically equivalent to a 0D point, and both
${\cal M}_{\pm}$ are then effectively closed 2D manifolds, over which the
Berry flux must be quantized. Please note that this quantization condition
does not generally apply to $N$-band models with $N>2$, which may require
additional symmetry protection.

##  S3 Cooper pairings of Berry-dipole Fermi surface

The general pairing matrix for a Berry-dipole system is given by

$\Delta(\mathbf{k})=d_{x}(\mathbf{k})\sigma_{x}+d_{y}(\mathbf{k})\sigma_{y}+d_{+}\sigma_{+}+d_{-}\sigma_{-},$
(S7)

where $\sigma_{\pm}=(\sigma_{z}\pm\sigma_{0})/2$. The Fermi statistics
requires $\Delta(\mathbf{k})=-\Delta^{T}(-\mathbf{k})$, and thus
$d_{x,\pm}(\mathbf{k})$ ($d_{y}(\mathbf{k})$) are (is) odd (even) in
$\mathbf{k}$.

Note that when $\Sigma=0$ in Eq. (3) of the main text, $\mathcal{M}_{\pm}$ are
hemispheres and the only boundary for both of them is the equator. This
corresponds to the case where $b_{1}$ in Fig. S1(a) shrinks into a point.

###  S3.1 Projection onto FS

Utilizing the gauge patterns and the wavefunction specified in the main text,
we can derive a simple formula for
$\Delta_{\text{eff}}=\langle\xi^{(e)\star}_{-}(-\mathbf{k})|\Delta(\mathbf{k})|\xi^{(e)}_{+}(\mathbf{k})\rangle$
in the two gauge patches:

$\displaystyle\Delta_{\text{eff},I}=d_{x}e^{i\phi}\sin
2\theta-(d_{+}e^{2i\phi}\cos^{2}\theta+d_{-}\sin^{2}\theta),$ (S8)
$\displaystyle\Delta_{\text{eff},II}=d_{x}e^{-i\phi}\sin
2\theta-(d_{+}\cos^{2}\theta+d_{-}e^{-2i\phi}\sin^{2}\theta).$

The consequence of each pairing channel is summarized as follows:

  * •

$d_{x}$ channel has zeros at $\theta=0,\pi/2,\pi$. If we count the vorticity
on $\mathcal{M}_{+}$ following Eq. (S6), we will find a net vorticity $-2$:
Phase winding $2\pi$ of $\Delta_{\text{eff},I}$ indicates a vorticity $-1$ in
patch I, and the $-2\pi$ of $\Delta_{\text{eff},II}$ indicates a vorticity
$-1$ in patch II. This is because there is a sign difference between the
integrals over $\mathcal{L}_{0}+\delta$ and $\mathcal{L}_{0}-\delta$ in the
definition of the vorticity [c.f. Eq. (S6)]. Please note that the above
counting of $\nu=2$ has NOT taken into account the explicit expression of
$d_{x}({\bf k})$, which can update $\nu$ via the boundary integral ${\cal
I}_{\varphi}$.

  * •

$d_{y}$ channel vanishes uniformly when projecting onto the Fermi surface.

  * •

$d_{+}$ and $d_{-}$ channels both manifest zeros at $\theta=\pi/2$ (equator)
and $\theta=0,\pi$ (poles), of which the vorticity can be counted in the same
way as that of $d_{x}$ pairing.

For a general $\Delta({\bf k})$ that consists of non-zero $d_{x}$ and
$d_{\pm}$ components, the pairing zeros can appear at arbitrary positions. As
an example, we study $d_{x}=k_{z}$ and $d_{\pm}=k_{\mp}$, where
$k_{\pm}=k_{x}\pm ik_{y}$. Substituting these into Eq. (S8), we solve

$\Delta_{\text{eff},I}=0=e^{i\phi}\cos\theta\sin
2\theta-e^{i\phi}\sin\theta\cos^{2}\theta-e^{i\phi}\sin^{3}\theta,$ (S9)

which gives us zeros at $\theta=0,\pi/4,3\pi/4,\pi$. This can be understood in
the following way: (1) When $d_{x}$ is the only nonzero term, there are zeros
at the poles and the equator; (2) When $d_{\pm}$ are turned on, the zeros at
the equator are separated into two loops at general latitude. In this case,
the boundary term in Eq.(2) is determined by the phase winding of $d_{-}$ over
the equator, which contributes $+1$ to $\nu$ according to Eq. (S6). Since the
$\mathcal{C}_{1}-(\det g)\mathcal{C}_{2}=-2$ for $\mu<0$, the net vorticity in
this case is $-1$. Importantly, the nodal points observed at poles in this
example are the origin of those observed in LSM, as discussed in the main
text.

###  S3.2 Symmetry analysis and robust nodal points under certain channels

Here, we first examine the $C_{4h}$ point symmetry group as specified in the
main text and discuss the constraints imposed on pairings (i.e. $d_{x,y,\pm}$
in Eq. (S7)). The character table for $C_{4h}$ is shown in Table. 1. In SC
systems, we call pairing $\Delta(\mathbf{k})$ is an irreducible representation
(irrep)/symmetric channel of $C_{4h}$ group if

$\mathring{\mathcal{G}}\Delta(\mathbf{k})\mathring{\mathcal{G}}^{T}=\alpha_{\mathcal{G}}\Delta(\check{\mathcal{G}}\mathbf{k}),\
\forall\mathcal{G}\in C_{4h},$ (S10)

where $\mathring{\mathcal{G}}$ ($\check{\mathcal{G}}$) is the representation
of $\mathcal{G}$ in the Hilbert space of electrons (momentum space), and
$\alpha_{\mathcal{G}}$ is the character of $\mathcal{G}$ given by the body of
Table. 1. Different irreps/symmetric channels correspond to different
representations of symmetry operators (denoted as $\acute{\mathcal{G}}$) for
the BdG Hamiltonian, i.e.,

$\acute{\mathcal{G}}=\begin{pmatrix}\mathring{\mathcal{G}}&\\\
&\alpha_{\mathcal{G}}\mathring{\mathcal{G}}^{\star}\end{pmatrix}$ (S11)

First, for irreps/symmetric channels $A_{g},B_{g},^{1}E_{g},^{2}E_{g}$
($A_{u},B_{u},^{1}E_{u},^{2}E_{u}$), $d_{x,y,\pm}$ are enforced to be even
(odd) in $\mathbf{k}$ because $P\sigma_{x,y,\pm}P^{T}=\sigma_{x,y,\pm}$.
Combining with the requirements of Fermi statistics, we conclude that
$d_{x,\pm}$ ($d_{y}$) vanish(es) for irreps/symmetric channels
$A_{g},B_{g},^{1}E_{g},^{2}E_{g}$ ($A_{u},B_{u},^{1}E_{u},^{2}E_{u}$). Since
the $d_{y}$ term vanishes uniformly when projecting onto the FS, we should
next focus on irreps/symmetric channels $A_{u},B_{u},^{1}E_{u},^{2}E_{u}$ and
consider the constraints from $C_{4}$. In our case, $C_{4}$ acts on Pauli
matrices as

$C_{4}\sigma_{\pm}C_{4}^{T}=\pm i\sigma_{\pm},\
C_{4}\sigma_{x,y}C_{4}^{T}=-\sigma_{x,y}.$ (S12)

Consequently, $d_{x/\pm}(\mathbf{k})=\beta_{x/\pm}d_{x/\pm}(C_{4}\mathbf{k})$,
here $\beta_{x/\pm}$ is determined by the character of $C_{4}$:
$\beta_{x}=-\alpha_{C_{4}},\beta_{\pm}=\mp i\alpha_{C_{4}}$. Importantly,
(i)in irreps/symmetric channels where $\beta_{x,\pm}\neq 1$, $d_{x,\pm}$ must
vanish at the poles of the FS which are $C_{4}$ fixed points; and (ii) in
irreps/symmetric channels where $\beta_{x,\pm}=\pm 1$, $d_{x,\pm}$ are even in
$(k_{x},k_{y})$ and thus must be odd in $k_{z}$. Consequently, $d_{x,\pm}$
vanish at the equator where $k_{z}$=0. Following points (i-ii), we can
conclude that for irreps $A_{u}$ and $B_{u}$, $d_{x}$ vanishes when $k_{z}=0$,
and $d_{\pm}$ vanish at the poles ($k_{x}=k_{y}=0$) of the FS. In contrast,
for irreps ${}^{1,2}E_{u}$, $d_{x}$ vanishes at poles, and $d_{\pm}$ vanish
when $k_{z}=0$ plane.

For $|\mu|>|\Sigma|$ where the FS of $h_{0}$ is always singly connected and
sphere-like, the normal states are identically $(1,0)^{T}$ ($(0,1)^{T}$) for
$\mu>0$ ($\mu<0$) over the equator of the FS at $k_{z}=0$ plane. Hence,
$\Delta_{\text{eff}}=d_{+}(d_{-})$ for $\mu>0$ ($\mu<0$). Consequently, the
irreps ${}^{1,2}E_{u}$ with vanishing $d_{\pm}$ corresponds guarantees the
boundary term in Eq. (2) of the main text to be zero. However, the irrep
$A_{u}$ ($B_{u}$) can have nonzero $d_{\pm}$ with $\beta_{\pm}=\mp i(\pm i)$.
Considering the constraints imposed by $C_{4}$ rotation symmetry, a smooth
pairing factor, $d_{\pm}$, in irrep $A_{u}$ ($B_{u}$) will generally takes the
form $d_{\pm}=\sum_{m,n}f_{m,n}k_{+}^{m}k_{-}^{n}$ with $m-n=4p\pm 1$
($m-n=4p\mp 1$) and $m,n,p$ to be integers. Assuming $k_{x}$ and $k_{y}$ are
small, which is automatically valid for our discussion on
$\mathbf{k}\cdot\mathbf{p}$ models, the phase winding of $d_{\pm}$ will be
determined by the leading term with smallest $m+n$, and generally takes the
value $4p\pm 1$ ( $4p\mp 1$ ) times of $2\pi$ where $p$ is an integer.
Therefore, the boundary term in Eq. (2) of the main text either has no
contribution or contributes an odd number to $\nu$.

For $|\mu|<|\Sigma|$, the WSM case ($\Sigma>0$) has two spherical FSs with no
boundaries. Therefore, the inter-FS pairings are always nodal due to the
Chern-vorticity theorem. However,the NLSM case ($\Sigma<0$) has its FS to be a
torus, where each of $\mathcal{M}_{+}$ and $\mathcal{M}_{-}$ has two
boundaries. Following the discussion in last paragraph, in $A_{u}$ and $B_{u}$
irreps, each boundary of $\mathcal{M}_{+}$ ($\mathcal{M}_{-}$) can contribute
an odd number to the net vorticity $\nu$. Therefore, the net vorticity can be
zero and the pairings are not necessarily nodal. Indeed, as shown in Fig. S2,
when we have pairing $k_{z}\sigma_{x}+k_{-}\sigma_{+}+k_{+}\sigma_{-}$, the
BdG Hamiltonian is totally gapped when $\Sigma<0$ and $|\mu|<|\Sigma|$. In
conclusion, the $\Sigma<0$ and $|\mu|<|\Sigma|$ case with pairing channels
$A_{u}$ and/or $B_{u}$ irreps of $C_{4h}$ is the only case where the SC
pairing over a singly-connected Berry-dipole FS is not topologically
obstructed.

| $A_{g}$ | $B_{g}$ | ${}^{1}E_{g}$ | ${}^{2}E_{g}$ | $A_{u}$ | $B_{u}$ | ${}^{1}E_{u}$ | ${}^{2}E_{u}$
---|---|---|---|---|---|---|---|---
$C_{4}$ | $1$ | $-1$ | $-i$ | $+i$ | $1$ | $-1$ | $-i$ | $+i$
$P$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$
Table 1: Character Table for $C_{4h}$. Figure S2: Superconducting gap over the
FS for $\Sigma=-0.1$ and pairing
$k_{z}\sigma_{x}+k_{-}\sigma_{+}+k_{+}\sigma_{-}$ belongs to the $B_{u}$
irrep, with various chemical potentials $\mu=-0.25,-0.15,-0.1,-0.05$.

##  S4 Annihilation of nodal points upon varying Fermi level in normal Weyl
semimetals

Here, we study a minimal, inversion symmetric model for the Weyl semimetal:

$h_{\text{wsm}}(\mathbf{k})=k_{x}\sigma_{x}+k_{y}\sigma_{y}+(m-1/2(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}))\sigma_{z},$
(S13)

and the corresponding BdG Hamiltonian:

$H_{\text{wsm}}(\mathbf{k})=\begin{pmatrix}h_{\text{wsm}}(\mathbf{k})&\Delta(\mathbf{k})\\\
\Delta^{{\dagger}}(\mathbf{k})&-h^{T}_{\text{wsm}}(-\mathbf{k})\end{pmatrix}.$
(S14)

To make a direct comparison with the Berry-dipole WSM discussed in the main
text, we focus on the same pairing
$\Delta(\mathbf{k})=(k_{x}+ik_{y})\sigma_{x}$, and calculate the SC gap over
the FS for various chemical potentials. As shown in Fig. S3, the two nodal
points merge and annihilate each other upon decreasing $\mu$ ($\mu<0$). This
is fundamentally distinct from the node-loop conversion found for the Berry-
dipole FS in the main text.

Figure S3: Superconducting gap over the FS for $H_{\text{wsm}}(\mathbf{k})$
with $m=0.1$ and various chemical potentials $\mu=-0.2,-0.1,-0.05$. In this
case, the annihilation of BdG Weyl nodes by decreasing $\mu$ leads to a full
energy gap.

##  S5 Projection of inter-sector pairing onto FS and symmetry analysis

In this section, we analyze the nodal behaviors of inter-sector pairings in
Dirac dipole semimetals, consisting of two copies of Berry-dipole semimetals
($h_{0}$ in the main text) in opposite orbital sectors ($l=\pm 1$):

$h_{DD}(\mathbf{k})=\begin{pmatrix}h_{0}(\mathbf{k})&0\\\
0&-h_{0}(\mathbf{k})\end{pmatrix}.$ (S15)

$h_{DD}$ has the $C_{4h}\times\mathcal{T}$ symmetry as mentioned in the main
text. To simplify the analysis, we focus on $\Sigma=0$ and $\mu<0$ in the
following.

Now the pairing matrix is a $4\times 4$ matrix, which can be formally
described by

$\displaystyle\Delta=\Delta_{\text{intra}}+\Delta_{\text{inter}}=\begin{pmatrix}\Delta_{++}&0\\\
0&\Delta_{--}\\\ \end{pmatrix}+\begin{pmatrix}0&\Delta_{+-}\\\
\Delta_{-+}&0\\\ \end{pmatrix}.$ (S16)

The nodal structures of $\Delta_{\text{intra}}$ trivially duplicate those of
Berry-dipole semimetals, and here we focus on $\Delta_{\text{inter}}$ that
generally takes the form

$\Delta_{\text{inter}}(\mathbf{k})=d_{i,j}(\mathbf{k})\tau_{i}\sigma_{j},\
i=x,y\ \operatorname{and}\ j=0,x,y,z,$ (S17)

where $\sigma_{x,y,z}$ ($\tau_{x,y,z}$) represent Pauli matrices for spin
(orbital).

###  S5.1 Projection onto FS

Under the same gauge used for Berry-dipole semimetals, the eigenstate
wavefunctions of $h_{DD}$ in two gauge patches are

$\displaystyle|\xi^{(e)}_{l=+1,I}\rangle=(-\cos\theta
e^{-i\phi},\sin\theta,0,0)^{T},\
|\xi^{(e)}_{l=-1,I}\rangle=-(0,0,\sin\theta,\cos\theta e^{i\phi})^{T},$
$\displaystyle|\xi^{(e)}_{l=+1,II}\rangle=(-\cos\theta,\sin\theta
e^{i\phi},0,0)^{T},\ |\xi^{(e)}_{l=-1,II}\rangle=(0,0,\sin\theta
e^{-i\phi},\cos\theta)^{T}.$ (S18)

A direct calculation of
$\Delta_{\text{eff}}^{\text{inter}}=\langle\xi^{(e)}_{l=+1}(\mathbf{k})|\Delta_{\text{inter}}|\xi^{(e)\star}_{l=-1}(-\mathbf{k})\rangle$
in the two gauge patches gives

$\displaystyle\Delta^{\text{inter}}_{\text{eff},I}=\Delta^{\text{inter}}_{\text{eff},II}$
$\displaystyle=[-d_{y,y}+\cos 2\theta d_{x,x}+\sin
2\theta(d_{x,z}\cos\phi+d_{y,0}\sin\phi)]\sigma_{x}$ (S19)
$\displaystyle+[d_{x,y}+\cos 2\theta d_{y,x}+\sin 2\theta(d_{y,z}\cos\phi-
d_{x,0}\sin\phi)]\sigma_{y}.$

Let us first consider each channel independently, and reveal the nodal
structures:

  * •

$d_{x,0}$ and/or $d_{y,0}$: The pairing vanishes at $\sin 2\theta=0$ or
$\sin\phi=0$. This leads to two intersected nodal loops, one at $k_{z}=0$
($\theta=\frac{\pi}{2}$) and the other at $k_{x}=0$ ($\phi=0,\pi$).

  * •

$d_{x,z}$ and/or $d_{y,z}$: Two intersected nodal loops, one at $k_{z}=0$ and
the other at $k_{y}=0$.

  * •

$d_{x,x}$ and/or $d_{y,x}$: Two parallel nodal loops at $\theta=\frac{\pi}{4}$
and $\theta=\frac{3\pi}{4}$.

  * •

$d_{x,y}$ and $d_{y,y}$: No apparent nodal structure.

Since these nodal structures are completely determined by the information of
wavefunctions of normal states and independent of the detials of $d_{i,j}$, we
generally refer to them as geometry obstructed nodal pairing. The
topologically obstructed nodal pairing is a special case of the geometry
obstructed nodal pairing, where the existence of nodal points can be
attributed to some well-defined topological invariants of the normal states.
Importantly, for Dirac semimetals, $d_{x/y,0},d_{x/y,z}$ channels are always
geometrically-obstructed nodal at poles ($\theta=0,\pi$) and the equator
($\theta=\pi/2$).

###  S5.2 Symmetry analysis

Next, let us analyze constraints imposed by symmetries on $d_{ij}$. Following
Eq. (S10), we identify different pairing channels as different irreps of
$C_{4h}\times\mathcal{T}$ group. For simplicity, we label each irrep by
$(X_{g/u},\alpha_{\mathcal{T}})$, where $X_{g/u}$ specify the irrep of
$C_{4h}$. The symmetry operators act on $\tau_{i}\sigma_{j}$ as

$\displaystyle C_{4}\tau_{x/y}\sigma_{x,y}C_{4}^{T}=s_{x/y}\sigma_{x,y},\
C_{4}\tau_{x/y}\sigma_{\pm}C_{4}^{T}=\mp is_{x/y}\sigma_{\pm}$ (S20)
$\displaystyle
U_{\mathcal{T}}(\tau_{x/y}\sigma_{0})^{\star}U_{\mathcal{T}}^{T}=s_{x/y}\sigma_{0},\
U_{\mathcal{T}}(\tau_{x/y}\sigma_{x,y,z})^{\star}U_{\mathcal{T}}^{T}=-\tau_{x/y}\sigma_{x,y,z},$

where $U_{\mathcal{T}}=i\tau_{x}\sigma_{y}$ is the unitary part of the
$\mathcal{T}$. Let us now concentrate on the nodal structures at the poles and
the equator of the FS, which are most relevant to the discussion in the main
text. Since $d_{x/y,0/z}$ are already guaranteed to be nodal at poles and
equators by the geometry of normal states, here we list the symmetry
constraints for $d_{x/y,x/y}$ in different irreps:

  * •

($X_{g}$,+1) with $X=A,B,^{1}E,^{2}E$: First, $d_{x,x}$ and $d_{y,y}$ are
ruled out, because Fermi statistics requires them to be odd in $\mathbf{k}$
but spatial inversion symmetry requires them to be even in $\mathbf{k}$. Next,
$d_{x,y}$ and $d_{y,x}$ are purely imaginary because of $\mathcal{T}$.
Finally, if $X=^{1}E,^{2}E$, $d_{x,y}$ and $d_{y,x}$ are nontrivial irrep of
$C_{4}$ with characters $\pm i$, and thus are odd in both $(k_{x},k_{y})$ and
$k_{z}$. This indicates that $d_{x,y}$ and $d_{y,x}$ vanish both at poles and
the equator. For $X=B$, $d_{x,y}$ and $d_{y,x}$ are nontrivial irrep of
$C_{4}$ with characters $\pm i$, and thus must vanish at the poles that are
$C_{4}$ fixed point. For $X=A$, there is no further constraints, and the
system can be gapped out by a $s$-wave pairing.

  * •

($X_{g}$,-1) with $X=A,B,^{1}E,^{2}E$: Similar to the first case, except that
$d_{x,y}$ and $d_{y,x}$ are enforced to be purely real by $\mathcal{T}$ in
this case.

  * •

($X_{u}$,+1) with $X=A,B,^{1}E,^{2}E$: All anti-symmetric
$\tau_{i}\sigma_{j}$s are ruled out by the spatial inversion symmetry, and the
only available terms are $d_{x,x}$ and $d_{y,y}$, which are enforced to be
purely real by $\mathcal{T}$. For $X=^{1}E,^{2}E$, $d_{x,x}$ and $d_{y,y}$ are
odd in $(k_{x},k_{y})$ but even in $k_{z}$, and thus will vanish at poles. For
$X=B$, $d_{x,x}$ and $d_{y,y}$ are odd in $k_{z}$ and form nontrivial irrep of
$C_{4}$, and thus they vanish at both the poles and the equator. For $X=A$,
$d_{x,x}$ and $d_{y,y}$ are odd in $k_{z}$ and thus vanish at the equator.

  * •

($X_{u}$,-1) with $X=A,B,^{1}E,^{2}E$: Similar to the third case, except that
$d_{x,y}$ and $d_{y,x}$ are enforced to be purely imaginary by $\mathcal{T}$.

The s-wave and p-wave pairing in Eq. (9) of the main text correspond to
$(A_{g},+1)$ and $(B_{u},+1)$, but they fall in the same $A_{1}$ irrep of
$T_{d}$ group.

Figure S4: (a-b) Superconducting gap over the FS for a Dirac-dipole semimetal
with the p-wave pairing and a Luttinger semimetal with both s-wave and p-wave
pairings, respectively. (c-d) Energy spectra along the green-arrow paths in
(a-b). The calculations are done for $\mu=-0.2$, $t=0.2$, $\Delta_{s}=0.01$,
and $\Delta_{p}=0.05$. The red circles highlight the degeneracy rooted in the
dipole-obstructed intra-sector pairing. Note that the surfaces in (a-b) are
FSs with an approximately (but not exactly) constant energy, which are
specifically chosen to better illustrate the nodal structures of projected
pairings.

##  S6 Perturbations from asymmetric spin-orbit coupling

In real materials, the spatial inversion symmetry breaking spin-3/2 pairing
happens because of the asymmetric spin-orbit coupling (ASOC) [3, 4, 5]. Here,
we consider the perturbative effect of the ASOC on the dipole-obstructed nodal
points. Specifically, we consider the linear Dresselhaus term

$h_{SOC}=t(k_{x}(J_{y}J_{x}J_{y}-J_{z}J_{x}J_{z})+k_{y}(J_{z}J_{y}J_{z}-J_{x}J_{y}J_{z})+k_{z}(J_{x}J_{z}J_{x}-J_{y}J_{z}J_{x}),$
(S21)

where $J_{x,y,z}$ are the rotation generators for $j=3/2$:

$J_{x}=\frac{1}{2}\left(\begin{array}[]{cccc}0&\sqrt{3}&0&0\\\
\sqrt{3}&0&2&0\\\ 0&2&0&\sqrt{3}\\\
0&0&\sqrt{3}&0\end{array}\right),J_{y}=\frac{1}{2}\left(\begin{array}[]{cccc}0&-\sqrt{3}i&0&0\\\
\sqrt{3}i&0&-2i&0\\\ 0&2i&0&-\sqrt{3}i\\\
0&0&\sqrt{3}i&0\end{array}\right),J_{z}=\frac{1}{2}\left(\begin{array}[]{cccc}3&0&0&0\\\
0&1&0&0\\\ 0&0&-1&0\\\ 0&0&0&-3\end{array}\right).$ (S22)

We add $h_{SOC}$ to the Luttinger-Khon model and study its effect on the
dipole-obstructed intra-sector pairing, which manifests as degeneracy
highlighted by the purple points in Fig. 3(f) of the main text. We first look
at the Dirac-dipole semimetal with $p$-wave pairing. As shown in Fig. S4(a)
and (c), the nodal points due to the dipole-obstructed intra-FS pairing at two
poles are deformed into nodal loops. Therefore, the effect of the ASOC
resembles that of the s-wave pairing term, which couples the nodal points in
different FSs and partially lifts the four-fold degeneracy guaranteed by the
dipole-obstructed intra-pairing. By turning on both $\alpha$ and $\Delta_{s}$,
we find that the nodal loop is still robust with only a slight change to its
location on the FS, as shown in Fig. S4(b) and (d).

Figure S5: (a)-(c) Superconducting gap over FS for Berry-dipole WSM
($\Sigma=0.1$) at various $\mu=-0.2,-0.1,-0.05$. The numbers in (c) label the
nodal points. (d) Illustration of Gaussian surfaces enclosing nodal points 2
(blue) and 3 (red). (e) Berry phase versus polar angle over the green spheres
in (c), where the blue (red) circles are calculated for the blue (red) nodal
point in (c). (f) Berry phase versus polar angle over the gray ellipsoid in
(c).

##  S7 Berry-dipole superconductor

In this part, we discuss a case where the dipole-obstructed BdG nodal points
themselves carry a Berry-dipole charge. This case is a BdG analog of the
Berry-dipole semimetals [1, 2], and thus is referred to as a Berry-dipole
superconductor.

Consider the BdG Hamiltonain in Eq. (4) in the main text, and choose
$\Sigma=0.1$ and $\Delta(\mathbf{k})=k_{z}\sigma_{x}$. The normal states
exhibit Weyl points, and thus when $|\mu|<|\Sigma|$, the FSs are two spheres
as shown in Fig. S5(c). Focusing on nodal points $2$ and $3$, numerical
calculations of Berry phases along the azimuthal direction over Green and Gray
Gaussian surfaces shown in Fig. S5(d) reveal that (i) points $2$ and $3$ are
BdG Weyl nodes [c.f. Fig, S5(e)]; and (ii) they carry a Berry-dipole charge
[c.f. Fig, S5(f)], i.e., on a Gaussian surface surrounding both of the nodes,
there is a $2\pi$ ($-2\pi$) quantized Berry flux over the upper (lower) hemi-
sphere. This Berry-dipole charge clearly puts constraints on pair
annihilation, and thus when nodal points $2$ and $3$ meet each as we vary the
chemical potential, they will become a nodal loop in $k_{z}=0$ plane instead
of being gapped out. The Berry dipole charge presents here because
$\Delta(\mathbf{k})=0$ at $k_{z}=0$, and hence all eigenstates of the BdG
Hamiltonian are identical on the $k_{z}=0$ plane and thus over the equator of
the Gaussian surface surrounding both nodes. Then, the equator of this
Gaussian surface can always be viewed as a point, and each half of this
Gaussian surface is effectively closed. Consequently, the Berry flux will be
quantized over each half of the Gaussian surface.

We note that this Berry-dipole superconductor is not a symmetry-protected
crystalline superconducting phase, because we can include symmetric
$d_{\pm}\sigma_{\pm}$ pairings to make the eigenstates over the equator
momentum-dependent. The $C_{4h}$ symmetry considered in our work cannot
isolate $k_{z}\sigma_{x}$ pairing and rule out all others. Thus, it could be
an interesting direction to search for symmetry-protected Berry-dipole
superconductors and conduct relevant classification studies.

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