#  Current-driven mechanical motion of double stranded DNA results in
structural instabilities and chiral-induced-spin-selectivity of electron
transport

Nicholas S. Davis  College of Science and Engineering, James Cook University,
Townsville, QLD, 4811, Australia  Julian A. Lawn  College of Science and
Engineering, James Cook University, Townsville, QLD, 4811, Australia  Riley J.
Preston  Institute of Physics, University of Freiburg, Hermann-Herder-Strasse
3, 79104 Freiburg, Germany  Daniel S. Kosov  College of Science and
Engineering, James Cook University, Townsville, QLD, 4811, Australia

###### Abstract

Chiral-induced-spin-selectivity of electron transport and its interplay with
DNA’s mechanical motion is explored in a double stranded DNA helix with spin-
orbit-coupling. The mechanical degree of freedom is treated as a stochastic
classical variable experiencing fluctuations and dissipation induced by the
environment as well as force exerted by nonequilibrium, current-carrying
electrons. Electronic degrees of freedom are described quantum mechanically
using nonequilibrium Green’s functions. Nonequilibrium Green’s functions are
computed along the trajectory for the classical variable taking into account
dynamical, velocity dependent corrections. This mixed quantum-classical
approach enables calculations of time-dependent spin-resolved currents. We
showed that the electronic force may significantly modify the classical
potential which, at sufficient voltage, creates a bistable potential with
considerable effect on electronic transport. The DNA’s mechanical motion has a
profound effect on spin transport; it results in chiral-induced spin
selectivity increasing spin polarisation of the current by 9% and also
resulting in temperature-dependent current voltage characteristics. We
demonstrate that the current noise measurement provides an accessible
experimental means to monitor the emergence of mechanical instability in DNA
motion. The spin resolved current noise also provides important dynamical
information about the interplay between vibrational and spin degrees of
freedom in DNA.

##  I Introduction

Chiral-induced-spin-selectivity (CISS) is an intriguing phenomenon observed in
a wide range of molecules and conditions[1, 2, 3, 4]. Primarily this effect
pertains to the preferential spin current flowing in each direction across the
molecule, attracting interest in biochemistry due to the link between spin
currents and enantioselectivity [5] and controlling reactions [6, 7] as well
as spintronics[8, 9, 10, 11, 12, 13] due to the magnitude of spin-polarisation
(SP) [14]. This significant SP is surprising given the weak spin-orbit-
coupling (SOC) in organic molecules[15], and many avenues have been explored
to reconcile the difference between experimental observation and theoretical
prediction. Consequently, the emergence of a large SP has been attributed to
many different effects, yet a definitive description of CISS remains elusive.

Measurements of the CISS effect have continued since its discovery in
attemptsto elucidate its underlying mechanisms. This has lead to a few key
observations that warrant consideration when designing effective models. The
first is that the magnitude of SP increases with the length of the
molecule[16, 17, 18], indicating a compounding mechanism as electrons traverse
the molecule. Also apparent is the sensitivity of SP to temperature[19];
sparking a strong interest in vibrational effects.

It has been well established that there is no spin separability in a system
involving only a single transport channel. Time-reversibility of SOC allows
its removal via a gauge transformation; thus spin cannot separate[11, 20, 21].
In junctions with single-point contact to only two leads, it is necessary for
the molecule itself to possess multiple spin-transport channels, which is the
case for double-stranded DNA with inter-strand exchange.

Beyond attempts to magnify the SOC within the molecule [22, 23, 24], several
aspects of molecular transport have featured prominently as candidates for
core functions of CISS. Many models neglect electron-electron interactions,
yet their inclusion has proved to greatly enhance SP [25, 26]. A similar case
holds for electron-phonon interactions used to include molecular vibration[27,
16]. Furthermore, these effects are necessary to include as the single-
electron treatment cannot fully describe CISS [28, 29].

The vibration mechanism is expanded upon here; considering DNA’s mechanical
motion coupled to electronic degrees of freedom. The mechanical motion is
considered classically in which it evolves in time under the influence of the
force produced by nonequilibrium quantum electrons. Inherently advantageous to
this approach is the versatility of the mechanical coordinate, which can
experience large amplitude motion in complex potential energy landscapes. The
dynamics of DNA’s mechanical motion is modelled using a Langevin equation with
nonequilibrium electronic forces computed via nonequilibrium Green’s functions
(NEGF). These calculations of time-dependent current include non-adiabatic
dynamical corrections arising from the time-dependence of the electronic
Hamiltonian which depends parametrically on the mechanical variable. Aside
from the SP, which is the topic of most studies, this approach facilitates the
calculation of spin-resolved current noise where dynamical corrections plays
an important role.

The paper is organised as follows. Section II describes the theory; the model
Hamiltonian, NEGF calculations of current and forces, and the numerical scheme
to solve the Langevin equation with NEGF forces. Results are presented and
discussed in Section III, which discusses the average nonequilibrium energy
landscape for DNA’s mechanical motion, along with current and current noise
induced by DNA’s mechanical motion. Section IV presents the conclusions.

We use atomic units in our equations: $\hbar=e=1$. Most values of physical
quantities will also be stated in atomic units (a.u.); however, we will
present the values for all energy related quantities in eV and voltages in V,
for clarity.

##  II Theory

###  II.1 Model

The system consists of a double-stranded DNA molecule with the ends linked to
two macroscopic electrodes. Each nucleotide in the DNA is represented as a
single electronic level; the electrons are allowed to hop within a single
strand between neighbouring nucleotides as well between nucleotides within the
base pair. The electronic Hamiltonian depends on DNA conformation, for which
the dynamics are described by a single mechanical degree of freedom.

The system is described by the Hamiltonian

$H(t)=H_{M}+H_{L}+H_{R}+H_{ML}+H_{MR}\\\
+H_{\text{e-mech}}(x)+H_{\text{mech}}(p,x),$ (1)

which is partitioned into the DNA’s $H_{M}$, left and right electrodes
$H_{L}+H_{R}$ and their coupling with DNA $H_{ML}+H_{MR}$, the coupling
between tunneling electrons and mechanical degree of freedom
$H_{\text{e-mech}}$, and, finally, the mechanical degree of freedom
$H_{\text{mech}}(p,x)$. The electrodes’ Hamiltonians take the form of non-
interacting electronic reservoirs:

$H_{L}+H_{R}=\sum_{\alpha k\sigma}\epsilon_{\alpha k\sigma}d_{\alpha
k\sigma}^{\dagger}d_{\alpha k\sigma},$ (2)

where $d^{\dagger}$ and $d$ are the standard creation and annihilation
operators and the subscript $\alpha k\sigma$ denotes the action in electrode
$\alpha\in\left\\{L,R\right\\}$ on single-particle state $k$ with spin
$\sigma\in\left\\{\uparrow,\downarrow\right\\}$.

The molecular Hamiltonian describes a double-stranded DNA helix with spin-
orbit-coupling, as in [30, 31]:

$\displaystyle H_{M}$ $\displaystyle=\sum_{\beta
j\sigma}\epsilon_{\beta}d_{\beta j\sigma}^{\dagger}d_{\beta
j\sigma}+\sum_{\beta j\sigma}t_{\beta}(d_{\beta
j\sigma}^{\dagger}d_{\beta,j+1,\sigma}+\text{h.c.})$
$\displaystyle+\sum_{j\sigma}v(d_{Aj\sigma}^{\dagger}d_{Bj\sigma}+\text{h.c.})+V_{\text{SOC}}$
(3)

Here, subscript $j$ is the base-pair index for strand
$\beta\in\left\\{A,B\right\\}$, $\epsilon_{\beta}$ is the on-site energy,
$t_{\beta}$ and $v$ are the intra-chain and inter-chain hopping matrix
elements, respectively. These hopping integrals are treated as equal for all
sites within each strand, however, the strands are assumed to be different. As
such [31]: $\epsilon_{A}=-0.2$ eV, $\epsilon_{B}=0.1$ eV, $t_{A}=0.1$eV,
$t_{B}=-0.14$ eV and $v=-0.08$ eV.

The spin-orbit coupling (SOC) is

$V_{\text{SOC}}=\sum_{\beta
j\sigma\sigma^{\prime}}i\gamma_{\beta}\Lambda_{\sigma\sigma^{\prime}}^{\beta
j}d_{\beta j\sigma}^{\dagger}d_{\beta,j+1,\sigma^{\prime}}+\text{h.c.}$ (4)

The SOC matrix $\Lambda_{\sigma\sigma^{\prime}}^{\beta j}$ contains constants
determined by the geometry of the helix, defined by pitch $\theta$ and
rotation per base pair $\phi$,

$\displaystyle\Lambda_{\uparrow\uparrow}^{\beta j}$
$\displaystyle=2\cos\theta,\;\;\;\Lambda_{\downarrow\downarrow}^{\beta
j}=-2\cos\theta,$ (5) $\displaystyle\Lambda_{\uparrow\downarrow}^{\beta j}$
$\displaystyle=i(-)_{\beta}(e^{-i\phi(j-1)}+e^{-i\phi
j})\sin\theta,\;\;\;\Lambda_{\downarrow\uparrow}^{\beta
j}=\Big{(}\Lambda_{\uparrow\downarrow}^{\beta j}\Big{)}^{*},$ (6)

where $(-)_{\beta}$ is (+1) if $\beta=A$ and (-1) if $\beta=B$. The SOC
strength constants are taken as $\gamma_{A}=-0.014$ eV and $\gamma_{B}=0.01$
eV, similar to[31].

The mechanical degree of freedom $(x,p)$ is described by Hamiltonian for a
classical particle of mass $m$

$H_{\text{mech}}(x)=\frac{p^{2}}{2m}+U(x),$ (7)

with harmonic potential $U(x)=\frac{1}{2}m\omega_{0}x^{2}$. This frequency is
chosen within a range to reflect the vibrational dynamics of DNA, with
standard value $\omega_{0}=0.005$ eV [32]. Finally, the mass, $m=5.9244\times
10^{5}$ a.u., is taken as the approximate mass of a single nucleotide.

The interaction between electrons and mechanical degree of freedom is assumed
to be described by linear response to the classical coordinate $x$, and is
given by

$H_{\text{e-mech}}(x)=-\chi_{0}\sqrt{2m\omega_{0}}x\sum_{\beta
j\sigma}d_{\beta j\sigma}^{{\dagger}}d_{\beta j\sigma}\\\
-\chi_{1}\sqrt{2m\omega_{0}}x\sum_{\beta j\sigma}(d_{\beta
j\sigma}^{{\dagger}}d_{\beta,j+1,\sigma}+\text{h.c.})\\\
-\chi_{2}\sqrt{2m\omega_{0}}x\sum_{j\sigma}(d_{Aj\sigma}^{{\dagger}}d_{Bj\sigma}+\text{h.c}).$
(8)

Here, the first term describes the on-site electron-mechanical motion
interaction, the second term takes into account the influence of the
mechanical motion on intra-strand electronic hopping, and the last term deals
with inter-strand electron transfer assisted by mechanical motion of DNA. The
corresponding coupling strengths are described by $\chi_{0}$, $\chi_{1}$ and
$\chi_{2}$. Following [31, 33, 34], we assume that $\chi_{1}=0.2\chi_{0}$,
$\chi_{2}=\chi_{1}$. The value of $\chi_{0}$ will be treated as a parameter to
study the role of the strength of electron-vibration coupling on the results.

The system-electrode coupling is described by

$H_{ML}=\sum_{k\beta\sigma}\left(t_{Lk,\beta 1}d_{Lk\sigma}^{\dagger}d_{\beta
1\sigma}+\text{h.c}\right),$ (9) $H_{MR}=\sum_{k\beta\sigma}\left(t_{Rk,\beta
N}d_{Rk\sigma}^{\dagger}d_{\beta N\sigma}+\text{h.c}\right),$ (10)

where $t_{\alpha k,\beta 1}$ and $t_{\alpha k,\beta N}$ are the tunnelling
amplitudes between electrodes’ states $\alpha k\sigma$ and corresponding DNA
base-pair 1 or N for left and right couplings, respectively. Thus, it is
assumed that both strands of DNA are attached to the electrodes.

###  II.2 Green’s Functions with nonadiabatic corrections

The self-energies and Green’s functions are matrices in molecular spin-orbital
space with dimension $(2\times 2\times N)$, where $N$ is the number of base-
pairs in the DNA, 2 is for the number of DNA strands and another 2 for the
spin degrees of freedom.

We treat self-energies in wide-band approximation, within which the matrix
elements of the retarded component are

$\displaystyle{\Sigma}_{L,\beta
j\sigma,\beta^{\prime}j^{\prime}\sigma^{\prime}}^{R}$
$\displaystyle=-\frac{i}{2}\Gamma_{L}\delta_{\beta\beta^{\prime}}\delta_{j1}\delta_{j^{\prime}1}\delta_{\sigma\sigma^{\prime}},$
(11) $\displaystyle{\Sigma}_{R,\beta
j\sigma,\beta^{\prime}j^{\prime}\sigma^{\prime}}^{R}$
$\displaystyle=-\frac{i}{2}\Gamma_{R}\delta_{\beta\beta^{\prime}}\delta_{jN}\delta_{j^{\prime}N}\delta_{\sigma\sigma^{\prime}},$
(12)

where $\Gamma_{\alpha}$ is the level broadening function due to the coupling
to electrode $\alpha$. The advanced self energy matrices are obtained via the
Hermitian conjugate of the retarded components

$\mathbf{\Sigma}_{\alpha}^{A}=(\mathbf{\Sigma}_{\alpha}^{R})^{\dagger}.$ (13)

Finally, the lesser self-energies are computed via fluctuation-dissipation
relation as

$\mathbf{\Sigma}_{\alpha}^{<}(\omega)=f_{\alpha}(\omega)(\mathbf{\Sigma}_{\alpha}^{A}-\mathbf{\Sigma}_{\alpha}^{R}),$
(14)

where

$f_{\alpha}(\omega)=\frac{1}{1+e^{(\omega-\mu_{\alpha})/k_{B}T}}$ (15)

is the Fermi-Dirac occupation number for electrode $\alpha$ with chemical
potential $\mu_{\alpha}$ and temperature $T$. In all calculations we assume
the the voltage $V$ is applied symmetrically: $\mu_{L}=V/2$ and
$\mu_{R}=-V/2$.

Suppose we know a trajectory $(x(t),p(t))$ for the mechanical degree of
freedom. The electronic part of the Hamiltonian (1) becomes explicitly time-
dependent through the parametric dependence on $x(t)$, which means that the
Green’s functions should be obtained from the solution of the full Keldysh-
Kadanoff-Baym equations. Following [35, 36, 37], we use a time-separation
technique to solve the Keldysh-Kadanoff-Baym equations in the Wigner space,
resulting in the expression

$\mathbf{G}(t,\omega)=\mathbf{G}_{(0)}(t,\omega)+\mathbf{G}_{(1)}(t,\omega).$
(16)

Here

$\mathbf{G}(t,\omega)=\int
d(t_{1}-t_{2})e^{i\omega(t_{1}-t_{2})}\mathbf{G}(t_{1},t_{2})$ (17)

is the Wigner space Green’s function and $t=(t_{1}+t_{2})/2$ is the central
time; $\mathbf{G}_{(0)}(t,\omega)$ is the adiabatic Green’s function (Green’s
function which instantaneously follows the changes in the reaction coordinate)
and $\mathbf{G}_{(1)}(t,\omega)$ contains non-adiabatic corrections
(accounting for the dynamics of mechanical motion). The adiabatic Green’s
functions are standard Green’s functions computed with static Hamiltonian. The
advanced and retarded Green’s functions are computed by matrix inversion

$\mathbf{G}_{(0)}^{A/R}(t,\omega)=\left(\omega-\mathbf{h}\left(x(t)\right)-\mathbf{\Sigma}^{A/R}\right)^{-1},$
(18)

where matrix $\mathbf{h}$ is formed from the electronic Hamiltonian including
the coupling to mechanical motion term:

$\displaystyle h_{\beta
j\sigma,\beta^{\prime}j^{\prime}\sigma^{\prime}}=(\epsilon_{\beta}-\chi_{0}\sqrt{2m\omega_{0}}x)\delta_{\beta\beta^{\prime}}\delta_{jj^{\prime}}\delta_{\sigma\sigma^{\prime}}$
$\displaystyle+(t_{\beta}-\chi_{1}\sqrt{2m\omega_{0}}x)\delta_{\beta\beta^{\prime}}\delta_{j\pm
1,j^{\prime}}\delta_{\sigma\sigma^{\prime}}$
$\displaystyle+(v-\chi_{2}\sqrt{2m\omega_{0}}x)\left(1-\delta_{\beta\beta^{\prime}}\right)\delta_{jj^{\prime}}\delta_{\sigma\sigma^{\prime}}$
$\displaystyle+i\Lambda_{\sigma\sigma^{\prime}}^{\beta
j}\gamma_{\beta}\delta_{\beta\beta^{\prime}}\delta_{j\pm 1,j^{\prime}}.$ (19)

The lesser adiabatic Green’s function is computed using advanced and retarded
components

$\mathbf{G}_{(0)}^{<}(t,\omega)=\mathbf{G}_{(0)}^{R}(t,\omega)\mathbf{\Sigma}^{<}(\omega)\mathbf{G}_{(0)}^{A}(t,\omega).$
(20)

The first order nonadiabatic corrections to advanced and retarded Green’s
functions are (functional dependence on $t$ and $\omega$ is suppressed for
brevity)

$\mathbf{G}_{(1)}^{A/R}=\frac{1}{2i}\mathbf{G}_{(0)}^{A/R}\left[\mathbf{G}_{(0)}^{A/R},\dot{\mathbf{h}}\right]_{-}\mathbf{G}_{(0)}^{A/R},$
(21)

where $\dot{\mathbf{h}}$ is the time derivative of matrix (19)

$\displaystyle\dot{h}_{\beta
j\sigma,\beta^{\prime}j^{\prime}\sigma^{\prime}}=-\dot{x}\sqrt{2m\omega_{0}}\delta_{\sigma\sigma^{\prime}}\Big{(}\chi_{0}\delta_{\beta\beta^{\prime}}\delta_{jj^{\prime}}$
$\displaystyle+\chi_{1}\delta_{\beta\beta^{\prime}}\delta_{j\pm
1,j^{\prime}}+\chi_{2}\left(1-\delta_{\beta\beta^{\prime}}\right)\delta_{jj^{\prime}}\Big{)}.$
(22)

The nonadiabatic correction to the lesser component is

$\mathbf{G}_{(1)}^{<}=\mathbf{G}_{(0)}^{A/R}\mathbf{\Sigma}^{<}\mathbf{G}_{(1)}^{A}+\mathbf{G}_{(1)}^{R}\mathbf{\Sigma}^{<}\mathbf{G}_{(0)}^{A}\\\
+\frac{1}{2i}\mathbf{G}_{(0)}^{R}\left(\dot{\mathbf{h}}\
\mathbf{G}_{(0)}^{R}\partial_{\omega}\mathbf{\Sigma}^{<}+\mathbf{G}_{(0)}^{<}\dot{\mathbf{h}}+\text{h.c}\right)\mathbf{G}_{(0)}^{A}.$
(23)

###  II.3 Spin-resolved current with dynamical corrections and force exerted
by electrons on mechanical degree of freedom

Spin resolved electronic current, which includes dynamical corrections due to
the mechanical motion, is obtained as in [35, 36, 37, 38]. Adiabatic current
(depends on instantaneous position $x(t)$) is given by the standard expression

$J_{\alpha\sigma}^{(0)}(t)=\intop_{-\infty}^{\infty}\frac{d\omega}{2\pi}\text{Tr}\Big{[}\mathbf{G}_{(0)}^{<}\mathbf{\Sigma}_{\alpha\sigma}^{A}+\mathbf{G}_{(0)}^{R}\mathbf{\Sigma}_{\alpha\sigma}^{<}\\\
-\mathbf{\Sigma}_{\alpha\sigma}^{<}\mathbf{G}_{(0)}^{A}-\mathbf{\Sigma}_{\alpha\sigma}^{R}\mathbf{G}_{(0)}^{<}\Big{]},$
(24)

while the first order correction is given by

$J_{\alpha\sigma}^{(1)}(t)=\intop_{-\infty}^{\infty}\frac{d\omega}{2\pi}\
\text{Tr}\Big{[}\mathbf{G}_{(1)}^{<}\mathbf{\Sigma}_{\alpha\sigma}^{A}\\\
+\mathbf{G}_{(1)}^{R}\mathbf{\Sigma}_{\alpha\sigma}^{<}-\mathbf{\Sigma}_{\alpha\sigma}^{<}\mathbf{G}_{(1)}^{A}-\mathbf{\Sigma}_{\alpha\sigma}^{R}\mathbf{G}_{(1)}^{<}\\\
+\frac{1}{2i}\left(\mathbf{\dot{G}}_{(0)}^{R}\partial_{\omega}\mathbf{\Sigma}_{\alpha\sigma}^{<}+\partial_{\omega}\mathbf{\Sigma}_{\alpha\sigma}^{<}\mathbf{\dot{G}}_{(0)}^{A}\right)\Big{]},$
(25)

where

$\mathbf{\dot{G}}_{(0)}^{A}=\mathbf{G}_{(0)}^{A}\mathbf{\dot{h}}\mathbf{G}_{(0)}^{A},$
(26)
$\mathbf{\dot{G}}_{(0)}^{R}=\Big{(}\mathbf{\dot{G}}_{(0)}^{A}\Big{)}^{{\dagger}},$
(27)

and the derivative $\partial_{\omega}\mathbf{\Sigma}_{\alpha\sigma}^{<}$ can
be easily computed using the wide-band approximation expression for
$\mathbf{\Sigma}_{\alpha\sigma}^{<}$.

We turn our attention now to calculations of the force exerted by
nonequilibrium electrons on the mechanical degree of freedom. The force
operator is computed by differentiating the corresponding part of the
electronic Hamiltonian

$\displaystyle\hat{F}_{\text{e}}(t)=-\partial_{x}H_{\text{e-mech}}(t)$
$\displaystyle=$ $\displaystyle-\sum_{\beta
j\sigma,\beta^{\prime}j^{\prime}\sigma^{\prime}}(\partial_{x}h)_{\beta
j\sigma,\beta^{\prime}j^{\prime}\sigma^{\prime}}d_{\beta
j\sigma}^{{\dagger}}d_{\beta^{\prime}j^{\prime}\sigma^{\prime}},$ (28)

where

$(\partial_{x}h)_{\beta
j\sigma,\beta^{\prime}j^{\prime}\sigma^{\prime}}=-\sqrt{2m\omega_{0}}\delta_{\sigma\sigma^{\prime}}\Big{(}\chi_{0}\delta_{\beta\beta^{\prime}}\delta_{jj^{\prime}}\\\
+\chi_{1}\delta_{\beta\beta^{\prime}}\delta_{j\pm
1,j^{\prime}}+\chi_{2}\left(1-\delta_{\beta\beta^{\prime}}\right)\delta_{jj^{\prime}}\Big{)}.$
(29)

Taking a quantum average along with truncation due to time separation yields

$F_{\text{e}}(t)=i\int\frac{d\omega}{2\pi}\text{Tr}\left[\partial_{x}\mathbf{h}\mathbf{G}_{(0)}^{<}(t,\omega)\right].$
(30)

This force can then be integrated over $x$ to produce effective potentials for
the classical coordinates produced by nonequilibrium current-carrying
electrons.

###  II.4 Dynamics of mechanical degree freedom

The mechanical degree of freedom $x$ is treated as a stochastic classical
variable experiencing fluctuations and dissipation induced by the environment
as well as force exerted by nonequilibrium, current carrying electrons. To
simulate the effect of the environment, the equation of motion of the
mechanical coordinate is defined by a Langevin equation,

$p=m\dot{x},$ (31) $\dot{p}(t)=f(t)-\zeta(t)p+\eta(t),$ (32)

where force

$f(t)=-m\omega_{0}^{2}x+F_{\text{e,neq}}(t)$ (33)

contains classical oscillator term with spring constant $m\omega_{0}^{2}$ and
nonequilibrium electronic force $F_{\text{e,neq}}$. The nonequilibrium
electronic force is defined as

$F_{\text{e,neq}}=F_{\text{e}}(x)-F_{\text{e,eq}}(x),$ (34)

where the electronic force $F_{\text{e}}$ is given by (30) and
$F_{\text{e,eq}}(x)$ is the equilibrium electronic force computed at zero
voltage bias. This equilibrium force removal is justified based on the
assumption that the equilibrium potential energy surface is completely
represented by harmonic potential term $\frac{1}{2}m\omega_{0}^{2}x^{2}$. The
environment terms, viscosity $\zeta$ and white noise random force $\eta(t)$
are related to each other by the fluctuation dissipation theorem
$\langle\eta(t)\eta(t^{\prime})\rangle=m\zeta kT\delta(t-t^{\prime})$.

To integrate the Langevin equation (31,32) we use the BAOAB splitting
algorithm[39]. The algorithm breaks up the Langevin equation into three
separate finite difference equations: Inertial momentum update (B), position
update (A) and a momentum update due to the stochastic force (O); it involves
NEGF calculations of force on each internal momentum update step (B). This
algorithm has been shown to be more accurate in comparison to other Langevin
integrators, particularly being very robust to changes in time-step; it also
provides accurate barrier crossing times for 1D double-welled potentials[39]
which is important for bistable DNA’s mechanical motion considered in our
paper. We use integration time-step 1 a.u. and viscosity $\zeta=2.228\times
10^{-4}$ a.u. in all our simulations.

##  III Results

###  III.1 Electronic current induces mechanical instability

Figure 1: Left-side axis: Spin-resolved current profile for adiabatic current
$J_{R\sigma}^{(0)}(x)$. Dashed horizontal lines indicate mean current values
computed via averaging over values of the current over Langevin trajectory.
Right-side axis: Total potential $U(x)$. Dashed horizontal line indicates
thermal energy above minimum. Voltages shown: 0.05 V (a-b), 0.14 V (c-d), 0.29
V (e-f). $T=300$ K (left column) and $T=77$ K (right column),
$\chi_{0}=0.0361$ eV.

We begin with examining the effective potential energy experienced by
mechanical degree of freedom

$U(x)=\frac{1}{2}m\omega_{0}^{2}x^{2}-\int_{x_{0}}^{x}dxF_{\text{e,neq}}(x),$
(35)

where the electronic force $F_{\text{e,neq}}$ is given by (30) and choice of
$x_{0}$ is arbitrary.

The inclusion of the nonequilibrium electronically-induced force on the $x$
coordinate is observed to introduce a double-well potential feature to the
ordinarily harmonic oscillator for some parameters. Firstly, considering high
temperature (300 K), it is observed that for low voltages $V<0.11$ V the
effective potential is roughly parabolic with a single potential minimum. As
the voltage is increased the effective potential widens and a centralised
energy barrier $U_{B}$ emerges separating two minima. As can be seen in
Figures 1(a),(b),(c), this change is gradual and initially the barrier is
below the thermal energy with respect to the lowest minima, $(U_{B}<k_{B}T)$,
but for higher voltages $V>0.17$ V it is above $(U_{B}>k_{B}T)$. This
behaviour continues up to $V=0.34$ V, after which the barrier begins to
subside. The qualitatively similar behaviour is observed at $T=77$ K, although
the low temperature ensures that the barrier, once present, always exceeds
thermal energy; shown in Figures 1(b),(d),(f). The influence of the
nonequilibrium electronic force is more pronounced at low temperature –
additional minima start to develop at smaller voltage and the minima are
separated by higher potential barriers.

The parameter space is further explored at $T=300$ K in Figure 2 where the
shaded regions indicate the qualitative nature of the potential resulting from
the strength of electron-mechanical-motion coupling $\chi_{0}$ and voltage
$V$. At the relatively low voltages considered, the molecular electronic
population tends to increase with voltage, and so does the electronic force
(30). This force is due to the electron-mechanical motion interaction, so
$\chi_{0}$ also scales the magnitude and thus both parameters contribute to
the development of double-minima potential. In addition to the distinction
between single well and double well potentials, Figure 2 also emphasises the
distinction between the double-potential when the barrier is above and below
the thermal energy. Results presented in 1(a),(b), and (c) corresponds to
points A, B, C in Figure 2. We will use value $\chi_{0}=0.0316$ eV in all our
calculations for the rest of the paper.

Figure 2: System parameter map indicating the nature of the effective
potential $U(x)$ (35) for the combination of electron-mechanical motion
strength $\chi_{0}$ and voltage $V$. The potential is defined by the number of
local minima: either a single or double potential well and, for the double
potential, whether the barrier between minima is greater than the thermal
energy. Point A, B, and C correspond to voltages 0.05 V, 0.14 V, and 0.29 V,
respectively. Dashed line represents $\chi_{0}=0.0316$ eV. $T=300$ K. Figure
3: Occupation probability for each left/right region of double potential in
blue/red, respectively, computed at temperatures $T=300$ K and $T=77$ K.

To understand the dynamics of DNA’s mechanical motion in bistable potential it
is instructive to view occupation probabilities for left and right minima. For
this we perform Langevin simulations by numerically solving equations (31, 32)
at $T=300$ K and $T=77$ K. These occupation probabilities are defined as
$P_{\alpha}=\tau_{\alpha}/\tau_{\text{total}}$, where $\tau_{\alpha}$ is the
time which the mechanical coordinate spends in potential side $\alpha=L,R$
during total trajectory $\tau_{\text{total}}$. As can be seen from the
occupation of the left and right minima in Figure 3, in the room temperature
regime ($T=300$ K), the occupation probability of the left minima is not
substantially suppressed even when the barrier exceeds the thermal energy.
Thermal fluctuations allow the coordinate to overcome the barrier. In the
lower temperature regime ($T=77$ K), the leftmost minima is “frozen out” of
the dynamics leading to an effectively zero occupation when the barrier energy
is sufficiently above the temperature.

###  III.2 Mechanical motion induces spin-polarisation of electronic current

Figure 4: Time-averaged spin currents computed at $T=300$ K and $T=77$ K as a
function of voltage.  Figure 5: Spin polarisation,
$\text{SP}=(J_{\uparrow}-J_{\downarrow})/(J_{\uparrow}+J_{\downarrow})$,
computed at $T=300$ K and $T=77$ K as a function of voltage.

Figure 1(a)-(f) shows a nontrivial dependence of the spin-resolved electric
currents on the mechanical coordinate $x$. The metastable left minima, which
is populated more at higher temperatures, corresponds to more conductive DNA
junctions – the electronic current is larger for both spin components here
comparing to the stable right minima. These differences in electronic currents
for stable and metastable DNA configurations are reflected in current-voltage
characteristics shown in Figure 4. As one can see in Figure 4, electronic
current computed at $T=300$ K is larger than the current at $T=77$ K; this
observation is qualitatively consistent with the experimental data [26]. Note
that there is no noticeable temperature dependence of electronic current in
the absence of DNA’s mechanical motion coupling to electronic degrees of
freedom.

Figure 1(a)-(f) also shows that the DNA’s mechanical motion in the stable
(right) potential induces considerable majority-spin polarisation of the
current whereas metastable DNA configuration favours minority-spin to a lesser
extent. Given that high temperature enables greater access to the left region,
the time-averaged current will receive greater contribution to minority-spin
current. As a result, the spin polarisation of electronic current decreases
with increasing temperature as one can see from Figure 5. For $T=300$ K, spin-
polarisation increases from roughly 2-5% and is maximised at 0.3 V. Beyond
this the spin polarisation reduces with voltage. Meanwhile for $T=77$ K, spin-
polarisation increases from 3-9% up to 0.2 V. SP maintains up to 0.4 V then
also decays at high voltage. This temperature dependence is contrary to the
aforementioned experiment [26] – we delegate this problem to future study.

###  III.3 Spin-resolved current noise assisted by DNA’s mechanical motion

Being a time-averaged quantity, the current-voltage characteristic (Figure 4)
does not provide information about the DNA dynamics; it does not reveal the
important time-scales related to the effective potential shapes. To reveal
experimentally accessible information about DNA’s mechanical motion and its
interplay with spin polarisation we turn our attention to spin-resolved
current noise calculations. Note also that the dynamical (velocity dependent)
corrections to the electric current do not play any role so far, since their
contributions disappear upon averaging over trajectory due to being linear in
the velocity. However, the corrections play a role in current noise.

Noise spectroscopy in molecular junctions has proven useful in identifying
transport mechanisms not discernible from the current-voltage characteristics
alone. Such measurements can reveal atomistic details of the local environment
and metal-molecule interfaces[40, 41], coupling between electronic and
vibrational degrees of freedom[42, 43, 44, 45, 46], individual conduction
transport channels[47, 48, 49, 50], and mechanical stability of the
junction[51]. Naturally, extending these ideas to consider spin noise provides
insight into spin transport. Indeed, shot noise has been used to identify
spin-polarised transport within a specific spin channel[52]. Spin fluctuations
can reveal Rabi splittings, Zeeman shifts, and the formation of doublet and
triplet states [53]; also enabling disturbance-free probing of spin
dynamics[54, 55]. Shot noise calculations[56, 57] have explored contributions
to quantum noise and its correlations; however the role of molecular,
mechanical motion is largely unexplored. In what follows, we quantify the
noise as a result of this motion.

The spin-resolved current noise is formally defined as

$S_{\alpha,\sigma\sigma^{\prime}}(\tau)=\lim_{T\rightarrow+\infty}\frac{1}{T}\int_{0}^{T}dt\langle\Big{[}\delta\hat{J}_{\alpha\sigma}(t),\delta\hat{J}_{\alpha\sigma}(t+\tau)\Big{]}_{+}\rangle,$
(36)

where $\delta\hat{J}_{\alpha\sigma}(t)$ describes the instantaneous deviation
of the $\sigma$-spin current at time $t$ from its mean value and
$[...,...]_{+}$ is the anti-commutator. The quantum expectation value
$\langle...\rangle$ averages over electronic degrees of freedom while the time
average over the mechanical motion of DNA is described by
$\lim_{T\rightarrow+\infty}\frac{1}{T}\int_{0}^{T}dt...$; equivalent to an
ensemble average of DNA geometries. This total noise is a complex amalgamation
of various sources of noise. Quantum noise is inherent to electrons due to
discrete charge, Pauli exclusion principle, shot noise, finite temperature and
the correlations arising from electron-electron interactions. The latter is
excluded from our model of non-interacting electrons. Additionally, a
mechanical noise results from the current-induced motion of the DNA. Due to
the time-scale separation of electrons and this molecular motion, this
contribution from the noise can be isolated [58]. The characteristic time
scale of shot noise decay is $1/\Gamma$, whereas the noise due to nuclear
motion appears on much longer times, $1/\zeta$. For the viscosity considered,
$\zeta/\Gamma\approx 0.006$; hence the noise induced by geometrical
fluctuations dominates the noise power spectrum at low frequencies, and can
exceed the shot noise contribution by orders of magnitude. In what follows we
focus on the “mechanical” noise as

$S_{\alpha,\sigma\sigma^{\prime}}(\tau)=2\lim_{T\rightarrow+\infty}\frac{1}{T}\int_{0}^{T}dt\delta
J_{\alpha\sigma}(t+\tau)\delta J_{\alpha\sigma^{\prime}}(t),$ (37)

where the current fluctuation at time $t$ is

$\delta
J_{\alpha\sigma}(t)=J_{\alpha\sigma}(t)-\lim_{T\rightarrow+\infty}\frac{1}{T}\int_{0}^{T}dtJ_{\alpha\sigma}(t).$
(38)

The adiabatic mechanical noise

$S_{\alpha,\sigma\sigma^{\prime}}^{(0)}(\tau)=2\lim_{T\rightarrow+\infty}\frac{1}{T}\int_{0}^{T}dt\delta
J_{\alpha\sigma}^{(0)}(t+\tau)\delta J_{\alpha\sigma^{\prime}}^{(0)}(t),$ (39)

and dynamical corrections, which not only depend on the instantaneous position
but also on the velocity,

$S_{\alpha,\sigma\sigma^{\prime}}^{(1)}(\tau)=2\lim_{T\rightarrow+\infty}\frac{1}{T}\int_{0}^{T}dt\\\
\Big{[}\dot{x}(t+\tau)B_{\alpha\sigma}(t+\tau)\delta
J_{\alpha\sigma^{\prime}}^{(0)}(t)+\delta
J_{\alpha\sigma}^{(0)}(t+\tau)\dot{x}(t)B_{\alpha\sigma^{\prime}}(t)\\\
+\dot{x}(t+\tau)B_{\alpha\sigma}(t+\tau)\dot{x}(t)B_{\alpha\sigma^{\prime}}(t)\Big{]},$
(40)

are included in total noise

$S_{\alpha,\sigma\sigma^{\prime}}(\tau)=S_{\alpha,\sigma\sigma^{\prime}}^{(0)}(\tau)+S_{\alpha,\sigma\sigma^{\prime}}^{(1)}(\tau).$
(41)

Here we factorised the dynamical correction to electric current

$J_{\alpha\sigma}^{(1)}(t)=\dot{x}B_{\alpha\sigma}(x).$ (42)

As we will demonstrate, the terms (39), (40) are responsible for different
features of noise correlation functions. The adiabatic mechanical noise
$S_{\alpha,\sigma\sigma^{\prime}}^{(0)}(\tau)$ is generally featureless and
shows overall exponential decay. The first two cross correlation terms in
$S_{\alpha,\sigma\sigma^{\prime}}^{(1)}(\tau)$ (the terms which are linear in
velocity of mechanical motion) are responsible for deviation of
$S_{\alpha,\uparrow\downarrow}(\tau)$ from
$S_{\alpha,\downarrow\uparrow}(\tau)$ for bistable mechanical motion. The last
term in $S_{\alpha,\sigma\sigma^{\prime}}^{(1)}(\tau)$ involves velocity-
velocity correlations containing the underlying dip and peak features at half-
period and full-period of motion in the right region.

Figure 6: Spin-resolved noise $S_{R,\sigma\sigma^{\prime}}(\tau)$ for single-
minima effective potential (corresponds to point A in Figure 2). (a) adiabatic
noise (39), (b) total noise computed with dynamical corrections (41).

Figures 6 (a,b) show spin resolved components of current noise for DNA’s
mechanical motion in single-potential shown in Figure 1-a. Adiabatic
mechanical noise $S^{(0)}(\tau)$ demonstrates simple decay of correlations in
time with no distinct features. Once dynamical corrections are included, noise
reveals negative correlation at time comparable to the half-period \- this dip
is a characteristic feature of classical velocity-velocity correlation
function and, unsurprisingly, it can be shown that it originates predominantly
from the last term in (40) which contains $\dot{x}(t+\tau)\dot{x}(t)$.

Figure 7: Spin-resolved noise $S_{R,\sigma\sigma^{\prime}}(\tau)$ for bistable
mechanical motion (corresponds to point C in Figure 2). a) adiabatic noise
(39), (b) total noise computed with dynamical corrections (41).

Figures 7 (a,b) show spin resolved components of current noise for DNA
bistable mechanical motion shown in Figure 1-e. Adiabatic noise
$S^{(0)}(\tau)$ remains always positive and begins to develop small positive-
correlation features, due to the consistently large deviation of the current
from its average value for stable (right) potential and at the same time
consistently smaller current dispersion in metastable (left) potential \- see
1-e. Dynamically corrected noise reveals a distinct negative contribution to
correlations (dip) at approximately half-period of mechanical motion in the
stable potential and positive-correlation (peak) appears at approximately of
full period of above motion - this feature again takes it origin from the
velocity-velocity correlation term in (40). It is interesting to note that
although the average current is dominated by $\sigma=\uparrow$ electrons - see
Figure 5, the noise shows the opposite spin selectivity:
$S_{\alpha,\uparrow\uparrow}(\tau)$ is aways smaller than
$S_{\alpha,\downarrow\downarrow}(\tau)$ due to smaller deviation of
$J_{\alpha\uparrow}(x)$ from the mean compared to $J_{\alpha\downarrow}(x)$.

The interesting new feature associated with mechanical bistability is the
different temporal correlations between fluctuations of spin-up and spin-down
currents depending upon which fluctuations occurred first – as it is apparent
from Figure 7 (b), cross-spin noise $S_{\alpha,\uparrow\downarrow}(\tau)$
deviates from $S_{\alpha,\downarrow\uparrow}(\tau)$. Analysis of the sum of
two terms in current noise dynamical corrections,
$\dot{x}(t+\tau)B_{\alpha\sigma}(t+\tau)\delta
J_{\alpha\sigma^{\prime}}^{(0)}(t)$ and $\delta
J_{\alpha\sigma}^{(0)}(t+\tau)\dot{x}(t)B_{\alpha\sigma^{\prime}}(t)$ shows
that the up-down component is negative whereas the down-up component is
positive (at the timescale of noise correlation decay).

##  IV Conclusions

We have explored mechanical motion due to current induced forces in a double
stranded DNA helix with spin-orbit-coupling and its role in the spin
polarisation of current and noise induced by DNA chirality. We represent the
mechanical motion as dynamics of a stochastic classical variable experiencing
fluctuations and dissipation induced by the environment as well as force
exerted by nonequilibrium, current carrying electrons. Electronic degrees of
freedom are described quantum mechanically using NEGF. NEGF are computed along
the trajectory for the classical variable taking into account dynamical,
velocity dependent corrections.

We observe that

  * •

DNA’s mechanical instability is induced by tunneling electrons. The
instability emerges at moderate applied voltage bias given that the strength
of electron-mechanical motion coupling exceeds certain critical value. The
physical regimes for stable and bistable DNA motion are identified.

  * •

Mechanical motion results in moderate increase of SP, achieving 3-9% depending
on the temperature and voltage bias.This is compared to the absence of any SP
without mechanical motion. The comparatively small SP is expected given the
short helix.

  * •

The temperature-dependent energy landscape together with temperature-dependent
dynamics of the mechanical motion lead to considerable temperature dependence
of the electronic current. The temperature dependence of the total current
agrees with experimental observation, however, SP does not show the expected
response to increased temperature. This raises questions about the importance
of additional interactions such as allowing the mechanical motion to flip the
spin via SOC mechanism which are not included in the present study.

  * •

The spin resolved noise induced by DNA’s mechanical motion can be a useful
experimental tool to extract information about DNA mehanical motion and detect
the emergence of mechanical instabilities.

## References

  * Aiello  _et al._ [2022] C. D. Aiello, J. M. Abendroth, M. Abbas, A. Afanasev, S. Agarwal, A. S. Banerjee, D. N. Beratan, J. N. Belling, B. Berche, A. Botana, J. R. Caram, G. L. Celardo, G. Cuniberti, A. Garcia-Etxarri, A. Dianat, I. Diez-Perez, Y. Guo, R. Gutierrez, C. Herrmann, J. Hihath, S. Kale, P. Kurian, Y.-C. Lai, T. Liu, A. Lopez, E. Medina, V. Mujica, R. Naaman, M. Noormandipour, J. L. Palma, Y. Paltiel, W. Petuskey, J. C. Ribeiro-Silva, J. J. Saenz, E. J. G. Santos, M. Solyanik-Gorgone, V. J. Sorger, D. M. Stemer, J. M. Ugalde, A. Valdes-Curiel, S. Varela, D. H. Waldeck, M. R. Wasielewski, P. S. Weiss, H. Zacharias, and Q. H. Wang, A chirality-based quantum leap, ACS Nano 16, 4989 (2022) .
  * Naaman  _et al._ [2019] R. Naaman, Y. Paltiel, and D. Waldeck, Chiral molecules and the electron spin, Nat Rev Chem 3, 250 (2019).
  * Naaman  _et al._ [2020a] R. Naaman, Y. Paltiel, and D. H. Waldeck, Chiral induced spin selectivity gives a new twist on spin-control in chemistry, Accounts of Chemical Research 53, 2659 (2020a).
  * Naaman and Waldeck [2015] R. Naaman and D. H. Waldeck, Spintronics and chirality: Spin selectivity in electron transport through chiral molecules, Annual Review of Physical Chemistry 66, 263 (2015).
  * Michaeli  _et al._ [2016] K. Michaeli, N. Kantor-Uriel, R. Naaman, and D. H. Waldeck, The electron’s spin and molecular chirality - how are they related and how do they affect life processes?, Chem. Soc. Rev. 45, 6478 (2016).
  * Naaman  _et al._ [2020b] R. Naaman, Y. Paltiel, and D. H. Waldeck, Chiral molecules and the spin selectivity effect, The Journal of Physical Chemistry Letters 11, 3660 (2020b) .
  * Evers  _et al._ [2022] F. Evers, A. Aharony, N. Bar-Gill, O. Entin-Wohlman, P. Hedegard, O. Hod, P. Jelinek, G. Kamieniarz, M. Lemeshko, K. Michaeli, V. Mujica, R. Naaman, Y. Paltiel, S. Refaely-Abramson, O. Tal, J. Thijssen, M. Thoss, J. M. van Ruitenbeek, L. Venkataraman, D. H. Waldeck, B. Yan, and L. Kronik, Theory of chirality induced spin selectivity: Progress and challenges, Advanced Materials 34, 2106629 (2022).
  * Naaman and Waldeck [2012] R. Naaman and D. H. Waldeck, Chiral-induced spin selectivity effect, The Journal of Physical Chemistry Letters 3, 2178 (2012) .
  * Bloom  _et al._ [2017] B. P. Bloom, B. M. Graff, S. Ghosh, D. N. Beratan, and D. H. Waldeck, Chirality control of electron transfer in quantum dot assemblies, Journal of the American Chemical Society 139, 9038 (2017) .
  * Mondal  _et al._ [2016] P. C. Mondal, C. Fontanesi, D. H. Waldeck, and R. Naaman, Spin-dependent transport through chiral molecules studied by spin-dependent electrochemistry, Accounts of Chemical Research 49, 2560 (2016) .
  * Yang  _et al._ [2019] X. Yang, C. H. van der Wal, and B. J. van Wees, Spin-dependent electron transmission model for chiral molecules in mesoscopic devices, Phys. Rev. B 99, 024418 (2019).
  * Shang  _et al._ [2022] Z. Shang, T. Liu, Q. Yang, S. Cui, K. Xu, Y. Zhang, J. Deng, T. Zhai, and X. Wang, Chiral-molecule-based spintronic devices, Small 18, 2203015 (2022).
  * Al-Bustami  _et al._ [2018] H. Al-Bustami, G. Koplovitz, D. Primc, S. Yochelis, E. Capua, D. Porath, R. Naaman, and Y. Paltiel, Single nanoparticle magnetic spin memristor, Small 14 (2018).
  * Kulkarni  _et al._ [2020] C. Kulkarni, A. K. Mondal, T. K. Das, G. Grinbom, F. Tassinari, M. F. J. Mabesoone, E. W. Meijer, and R. Naaman, Highly efficient and tunable filtering of electrons’ spin by supramolecular chirality of nanofiber-based materials, Advanced Materials 32, https://doi.org/10.1002/adma.201904965 (2020).
  * Min  _et al._ [2006] H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, and A. H. MacDonald, Intrinsic and rashba spin-orbit interactions in graphene sheets, Phys. Rev. B 74, 165310 (2006).
  * Fransson [2021] J. Fransson, Charge redistribution and spin polarization driven by correlation induced electron exchange in chiral molecules, Nano Letters 21, 3026 (2021).
  * Mishra  _et al._ [2020] S. Mishra, A. K. Mondal, S. Pal, T. K. Das, E. Z. B. Smolinsky, G. Siligardi, and R. Naaman, Length-dependent electron spin polarization in oligopeptides and dna, The Journal of Physical Chemistry C 124, 10776 (2020).
  * Kettner  _et al._ [2018] M. Kettner, V. V. Maslyuk, D. Nurenberg, J. Seibel, R. Gutierrez, G. Cuniberti, K.-H. Ernst, and H. Zacharias, Chirality-dependent electron spin filtering by molecular monolayers of helicenes, The Journal of Physical Chemistry Letters 9, 2025 (2018).
  * Fransson [2023] J. Fransson, Temperature activated chiral induced spin selectivity, The Journal of Chemical Physics 159, 084115 (2023).
  * Bardarson [2008] J. H. Bardarson, A proof of the kramers degeneracy of transmission eigenvalues from antisymmetry of the scattering matrix, Journal of Physics A: Mathematical and Theoretical 41, 405203 (2008).
  * Vittmann  _et al._ [2022] C. Vittmann, R. K. Kessing, J. Lim, S. F. Huelga, and M. B. Plenio, Interface-induced conservation of momentum leads to chiral-induced spin selectivity, The Journal of Physical Chemistry Letters 13, 1791 (2022) .
  * Dalum and Hedegard [2019] S. Dalum and P. Hedegard, Theory of chiral induced spin selectivity, Nano Letters 19, 5253 (2019) .
  * Gutierrez  _et al._ [2012] R. Gutierrez, E. Diaz, R. Naaman, and G. Cuniberti, Spin-selective transport through helical molecular systems, Phys. Rev. B 85, 081404 (2012).
  * Yeganeh  _et al._ [2009] S. Yeganeh, M. A. Ratner, E. Medina, and V. Mujica, Chiral electron transport: Scattering through helical potentials, The Journal of Chemical Physics 131, 014707 (2009) .
  * Fransson [2019] J. Fransson, Chirality-induced spin selectivity: The role of electron correlations, The Journal of Physical Chemistry Letters 10, 7126 (2019) .
  * Das  _et al._ [2022] T. K. Das, F. Tassinari, R. Naaman, and J. Fransson, Temperature-dependent chiral-induced spin selectivity effect: Experiments and theory, The Journal of Physical Chemistry C 126, 3257 (2022) .
  * Diaz  _et al._ [2018] E. Diaz, P. Albares, P. G. Estevez, J. M. Cervero, C. Gaul, E. Diez, and F. Dominguez-Adame, Spin dynamics in helical molecules with nonlinear interactions, New Journal of Physics 20, 043055 (2018).
  * Fransson [2022a] J. Fransson, Charge and spin dynamics and enantioselectivity in chiral molecules, The Journal of Physical Chemistry Letters 13, 808 (2022a) .
  * Fransson [2022b] J. Fransson, The chiral induced spin selectivity effect what it is, what it is not, and why it matters, Israel Journal of Chemistry 62, e202200046 (2022b) .
  * Guo and Sun [2012] A.-M. Guo and Q.-f. Sun, Spin-selective transport of electrons in dna double helix, Phys. Rev. Lett. 108, 218102 (2012).
  * Du  _et al._ [2020] G.-F. Du, H.-H. Fu, and R. Wu, Vibration-enhanced spin-selective transport of electrons in the dna double helix, Phys. Rev. B 102, 035431 (2020).
  * Lee  _et al._ [2006] C. Lee, K.-H. Park, and M. Cho, Vibrational dynamics of dna. i. vibrational basis modes and couplings, The Journal of Chemical Physics 125, 114508 (2006).
  * Voityuk  _et al._ [2001] A. A. Voityuk, J. Jortner, M. Bixon, and N. Rosch, Electronic coupling between watson-crick pairs for hole transfer and transport in desoxyribonucleic acid, The Journal of Chemical Physics 114, 5614 (2001).
  * Senthilkumar  _et al._ [2005] K. Senthilkumar, F. C. Grozema, C. F. Guerra, F. M. Bickelhaupt, F. D. Lewis, Y. A. Berlin, M. A. Ratner, and L. D. A. Siebbeles, Absolute rates of hole transfer in dna, Journal of the American Chemical Society 127, 14894 (2005) .
  * Kershaw and Kosov [2017] V. F. Kershaw and D. S. Kosov, Nonequilibrium Green’s function theory for nonadiabatic effects in quantum electron transport, Journal of Chemical Physics 147, 224109 (2017).
  * Kershaw and Kosov [2018] V. F. Kershaw and D. S. Kosov, Non-adiabatic corrections to electric current in molecular junctions due to nuclear motion at the molecule-electrode interfaces, The Journal of Chemical Physics 149, 044121 (2018).
  * Kershaw and Kosov [2019] V. F. Kershaw and D. S. Kosov, Non-equilibrium Green’s function theory for non-adiabatic effects in quantum transport: Inclusion of electron-electron interactions, Journal of Chemical Physics 150, 074101 (2019) .
  * Kershaw and Kosov [2020] V. F. Kershaw and D. S. Kosov, Non-adiabatic effects of nuclear motion in quantum transport of electrons: A self-consistent Keldysh–Langevin study, The Journal of Chemical Physics 153, 154101 (2020) .
  * Leimkuhler and Matthews [2013] B. Leimkuhler and C. Matthews, Robust and efficient configurational molecular sampling via Langevin dynamics, The Journal of Chemical Physics 138, 174102 (2013) .
  * Kim  _et al._ [2010] Y. Kim, H. Song, D. Kim, T. Lee, and H. Jeong, Noise characteristics of charge tunneling via localized states in metal-molecule-metal junctions,  _ACS Nano_ , ACS Nano 4, 4426 (2010).
  * Adak  _et al._ [2015] O. Adak, E. Rosenthal, J. Meisner, E. F. Andrade, A. N. Pasupathy, C. Nuckolls, M. S. Hybertsen, and L. Venkataraman, Flicker noise as a probe of electronic interaction at metal–single molecule interfaces, Nano Letters 15, 4143 (2015) .
  * Galperin  _et al._ [2006] M. Galperin, A. Nitzan, and M. A. Ratner, Inelastic tunneling effects on noise properties of molecular junctions, Phys. Rev. B 74, 075326 (2006).
  * Secker  _et al._ [2011] D. Secker, S. Wagner, S. Ballmann, R. Härtle, M. Thoss, and H. B. Weber, Resonant vibrations, peak broadening, and noise in single molecule contacts: The nature of the first conductance peak, Phys. Rev. Lett. 106, 136807 (2011).
  * Kumar  _et al._ [2012] M. Kumar, R. Avriller, A. L. Yeyati, and J. M. van Ruitenbeek, Detection of vibration-mode scattering in electronic shot noise, Phys. Rev. Lett. 108, 146602 (2012).
  * Schinabeck  _et al._ [2014] C. Schinabeck, R. Härtle, H. B. Weber, and M. Thoss, Current noise in single-molecule junctions induced by electronic-vibrational coupling, Phys. Rev. B 90, 075409 (2014).
  * Kosov [2017] D. S. Kosov, Non-renewal statistics for electron transport in a molecular junction with electron-vibration interaction, J. Chem. Phys. 147, 104109 (2017).
  * Djukic and van Ruitenbeek [2006] D. Djukic and J. M. van Ruitenbeek, Shot noise measurements on a single molecule, Nano Letters 6, 789 (2006) .
  * Wheeler  _et al._ [2010] P. J. Wheeler, J. N. Russom, K. Evans, N. S. King, and D. Natelson, Shot noise suppression at room temperature in atomic-scale au junctions, Nano Letters 10, 1287 (2010) .
  * Tsutsui  _et al._ [2010] M. Tsutsui, M. Taniguchi, and T. Kawai, Single-molecule identification via electric current noise, Nat. Commun. 1, 138 (2010).
  * Cho  _et al._ [2017] D. Cho, H. Lee, S. Shekhar, M. Yang, J. Y. Park, and S. Hong, Direct mapping of electrical noise sources in molecular wire-based devices, Scientific Reports 7, 43411 (2017).
  * Xiang  _et al._ [2014] D. Xiang, T. Lee, Y. Kim, T. Mei, and Q. Wang, Origin of discrete current fluctuations in a single molecule junction, Nanoscale 6, 13396 (2014).
  * Burtzlaff  _et al._ [2015] A. Burtzlaff, A. Weismann, M. Brandbyge, and R. Berndt, Shot noise as a probe of spin-polarized transport through single atoms, Phys. Rev. Lett. 114, 016602 (2015).
  * Glasenapp  _et al._ [2014] P. Glasenapp, N. A. Sinitsyn, L. Yang, D. G. Rickel, D. Roy, A. Greilich, M. Bayer, and S. A. Crooker, Spin noise spectroscopy beyond thermal equilibrium and linear response, Phys. Rev. Lett. 113, 156601 (2014).
  * Oestreich  _et al._ [2005] M. Oestreich, M. Römer, R. J. Haug, and D. Hägele, Spin noise spectroscopy in gaas, Phys. Rev. Lett. 95, 216603 (2005).
  * Crooker  _et al._ [2004] S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D. L. Smith, Spectroscopy of spontaneous spin noise as a probe of spin dynamics and magnetic resonance, Nature 431, 49–52 (2004).
  * Mirzanian and Shokri [2013] S. Mirzanian and A. Shokri, Angular dependence of shot noise in the presence of rashba spin–orbit coupling in semiconductor spintronics junctions, Physica E: Low-dimensional Systems and Nanostructures 54, 59 (2013).
  * Pradhan and Fransson [2018] S. Pradhan and J. Fransson, Shot noise as a probe of spin-correlated transport through single atoms, Phys. Rev. B 97, 115409 (2018).
  * Preston  _et al._ [2020] R. J. Preston, V. F. Kershaw, and D. S. Kosov, Current-induced atomic motion, structural instabilities, and negative temperatures on molecule-electrode interfaces in electronic junctions, Phys. Rev. B 101, 155415 (2020).


# Lensing reconstruction from the cosmic microwave background polarization
with machine learning

Ye-Peng Yan  Institute for Frontiers in Astronomy and Astrophysics, Beijing
Normal University, Beijing 100875, China<EMAIL_ADDRESS>Department of
Astronomy, Beijing Normal University, Beijing 100875, China  Guo-Jian Wang
School of Chemistry and Physics, University of KwaZulu-Natal, Westville
Campus, Private Bag X54001 Durban, 4000, South Africa  NAOC-UKZN Computational
Astrophysics Centre (NUCAC), University of KwaZulu-Natal, Durban, 4000, South
Africa  Si-Yu Li  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Science, P. O. Box 918-3 Beijing 100049,
People’s Republic of China  Yang-Jie Yan  Institute for Frontiers in Astronomy
and Astrophysics, Beijing Normal University, Beijing 100875, China;
<EMAIL_ADDRESS>Department of Astronomy, Beijing Normal University, Beijing
100875, China  Jun-Qing Xia  Institute for Frontiers in Astronomy and
Astrophysics, Beijing Normal University, Beijing 100875, China;
<EMAIL_ADDRESS>Department of Astronomy, Beijing Normal University, Beijing
100875, China

###### Abstract

The lensing effect of the cosmic microwave background (CMB) is a powerful tool
for our study of the distribution of matter in the universe. Currently, the
quadratic estimator (EQ) method, which is widely used to reconstruct lensing
potential, has been known to be sub-optimal for the low-noise levels
polarization data from next-generation CMB experiments. To improve the
performance of the reconstruction, other methods, such as the maximum
likelihood estimator and machine learning algorithms are developed. In this
work, we present a deep convolutional neural network model named the Residual
Dense Local Feature U-net (RDLFUnet) for reconstructing the CMB lensing
convergence field. By simulating lensed CMB data with different noise levels
to train and test network models, we find that for noise levels less than
$5\mu$K-arcmin, RDLFUnet can recover the input gravitational potential with a
higher signal-to-noise ratio than the previous deep learning and the
traditional QE methods at almost the entire observation scales.

Cosmic microwave background radiation (322); Observational cosmology (1146);
Convolutional neural networks (1938)

##  1 Introduction

In observations of the cosmic microwave background (CMB) over the past
decades, its anisotropy has been established as a robust and powerful
cosmological probe, as shown through many experiments over the past two
decades, such as COBE (Mather et al., 1994), Boomerang (Lange et al., 2001),
WMAP (Bennett et al., 2013), Planck (Planck Collaboration et al., 2020a), SPT
(Louis et al., 2017). An important goal for current and next-generation CMB
experiments is the precise measurement of anisotropies in its polarization
states, particularly the search for the faint primordial polarization B-modes
induced by primordial gravitational waves. A lot of upcoming and proposed CMB
surveys will be devoted to this direction, such as SPT-3G (Benson et al.,
2014), CMB-S4 (Abazajian et al., 2016), the Simons Observatory (Ade et al.,
2019), LiteBIRD (Hazumi et al., 2019), AliCPT (Li et al., 2017). Gravitational
lensing effect of the large-scale structure has a significant impact on CMB
observations. Gravitational lensing of the CMB arises from the deflection of
CMB photons as they pass through the matter distribution between the surface
of the last scattering and us, which will lead to a subtle remapping of the
temperature and polarization anisotropies (Lewis & Challinor, 2006). More
specifically, lens smoothes the acoustic peaks of the CMB power spectra and
generates non-stationary statistics of CMB fluctuations, and converts E-mode
polarization into B-mode polarization (Zaldarriaga & Seljak, 1998; Lewis et
al., 2001; Hotinli et al., 2022). It is therefore essential that we remove the
lensing effect (delensing) from the observed CMB to reveal the unaltered
primordial signal. On the other hand, CMB lensing is a powerful probe for
observing the growth of cosmic structures because it encodes information about
the matter fluctuations in the CMB maps observed by our telescopes.
Reconstructing and analysing a lensing map from CMB data can help constrain
the $\Lambda$CDM cosmological model, the neutrino mass scale, modified
gravity, and a wealth of other cosmological physics (Smith et al., 2006;
Planck Collaboration et al., 2020b). All in all, reconstruction of the lensing
potential and delensing from observable CMB data are key for decoding early
universe physics.

CMB photons deflection angle is proportional to the gradient of the lensing
potential $\phi$, which is the projected weighted gravitational potential
along the line-of-sight between us and the CMB. Quadratic estimator
(Zaldarriaga & Seljak, 1999; Hu & Okamoto, 2002; Maniyar et al., 2021) has
been used with great success to reconstruct the CMB lensing potential from CMB
temperature and polarization maps and to detect the effects of gravitational
lensing at high significance with existing CMB surveys (Planck Collaboration
et al., 2020b; Wu et al., 2019; Darwish et al., 2021). QE is nearly optimal at
current noise level of CMB experiments. However, it will become significantly
sub-optimal once the noise level falls below the $\sim 5$ $\mu$K-arcmin
(Hirata & Seljak, 2003a; Carron & Lewis, 2017; Horowitz et al., 2019), which
could be achieved in the next-generation CMB surveys, such as CMB-S4. Previous
works (Hirata & Seljak, 2003a, b; Carron & Lewis, 2017; Carron, 2019; Millea
et al., 2019, 2020; Legrand & Carron, 2022) demonstrated that the likelihood-
based approaches might significantly enhance the lensing reconstruction by
efficiently utilizing higher order statistics of the CMB fields at low noise
levels.

Machine learning has demonstrated exceptional capabilities in the field of
image processing, such as image recognition and denoising, which has been
widely used in astrophysical research (Mehta et al., 2019; Fluke & Jacobs,
2020; Petroff et al., 2020; Wang et al., 2022; Yan et al., 2023) as a result
of the impressive advancements in computer science in recent years. Machine
learning recently demonstrated its potential for CMB lensing reconstruction.
Specifically, Caldeira et al. (2019) and Heinrich et al. (2022) applied the
Residual-Unet (ResUnet) model (Kayalibay et al., 2017; Zhang et al., 2018a) to
the CMB convergence $\kappa$ reconstruction. In addition, Guzman & Meyers
(2021) and Guzman & Meyers (2022) used the ResUnet model to reconstruct maps
of patchy Reionization and cosmic polarization rotation. In their
investigation, this model successfully reconstructs $\kappa$ or rotation maps,
and has lower lensing reconstruction noise than the QE technique at low noise
levels. Their results, however, are noise-dependent, and the power spectrum of
$\kappa$ field is difficult to reconstruct accurately in the presence of non-
negligible noise. Li et al. (2022) employs a generative adversarial network
(GAN) to precisely reconstruct $\kappa$ power spectrum despite the presence of
noise. However, the GAN model’s reconstruction of the $\kappa$ map contains
more noise than the ResUnet model. As an improvement of the ResUnet model, we
introduce a convolutional neural network to reconstruct the CMB lensing
convergence field.

This paper is organized as follows. In Section 2, we briefly review a basic
background for the CMB lensing and the quadratic estimator. In Section 3, We
describe the simulated data sets as well as our network model. In Section 4,
we present the results of our method and its comparison with the ResUnet model
and quadratic estimator approach. In section 5 discussions about using the CNN
method are presented. Finally, we conclude in Section 6.

##  2 WEAK LENSING OF THE CMB

In this section, the physical underpinnings of CMB lensing, as well as its
construction using the quadratic estimator, will be briefly reviewed. We work
in the flat-sky approximation and denote two-dimensional Fourier wave numbers
by $\boldsymbol{\ell}$ for CMB fields and $\boldsymbol{L}$ for the lensing
potential. Gravitational lensing deflects the path of CMB photons resulting in
a remapping of the primary CMB in the sky (Hu & Okamoto, 2002; Lewis &
Challinor, 2006),

$\displaystyle\tilde{T}(\hat{\boldsymbol{n}})=T(\hat{\boldsymbol{n}}+\boldsymbol{\alpha}(\hat{\boldsymbol{n}})),$
(1) $\displaystyle(\tilde{Q}(\hat{\boldsymbol{n}})\pm
i\tilde{U}(\hat{\boldsymbol{n}}))(\hat{\boldsymbol{n}})=(Q\pm
iU)(\hat{\boldsymbol{n}}+\boldsymbol{\alpha}(\hat{\boldsymbol{n}})),$ (2)

where $\hat{\boldsymbol{n}}$ denotes the line-of-sight direction,
($\tilde{T}$, $\tilde{Q}$, $\tilde{U}$) are the lensed CMB fields, ($T$, $Q$,
$U$) are the primordial CMB fields, and $\boldsymbol{\alpha}$ is the
deflection angle that describes the remapping. The deflection angle is the
gradient of the lensing potential ($\phi$) that is a weighted integral of the
gravitational potential ($\psi$) along the line of sight, which can be written
as:

$\displaystyle\boldsymbol{\alpha}=\nabla\phi,$ (3) $\displaystyle\phi=-2\int
dD\frac{(\chi_{s}-\chi)}{\chi\chi_{s}}\psi(\chi\hat{\boldsymbol{n}},\chi),$
(4)

where $\chi$ is the comoving distance along the line-of-sight and $\chi_{s}$
is the distance to the last-scattering surface. The lensing potential can be
also represented by the convergence field $\kappa$ because one can move
between the two fields using Poisson equation. In Fourier space, using the
flat-sky approximation, this relationship can be written as:

$\displaystyle\kappa(\boldsymbol{L})$
$\displaystyle=-\frac{\boldsymbol{L}^{2}}{2}\phi(\boldsymbol{L}).$ (5)

Gravitational lensing will affect auto- and cross-spectra of CMB fields and
produces correlations between different modes $\ell$, proportional to the
lensing potential. In the flat-sky approximation, the off-diagonal mode
coupling has the following form

$\displaystyle\langle\tilde{X}(\boldsymbol{\ell_{1}})\tilde{Y}(\boldsymbol{\ell_{2}})\rangle$
$\displaystyle\propto
f^{\phi}_{XY}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})\phi(\boldsymbol{L}),$
(6)

where $\boldsymbol{L}=\boldsymbol{\ell_{1}}+\boldsymbol{\ell_{2}}$, ($X$,$Y$)
are CMB temperature or polarization fluctuations ($T$,$E$,$B$), and
$\langle\rangle$ denotes the average over CMB realization. In this work, we
focus on the quadratic estimator (Hu & Okamoto, 2002), in which the minimum-
variance quadratic estimator for the CMB lensing potential is written as

$\displaystyle\hat{\phi}_{\beta}$
$\displaystyle=A_{\beta}(\boldsymbol{\ell})\int\frac{d^{2}\boldsymbol{\ell_{1}}}{(2\pi)^{2}}X^{\rm
obs}(\boldsymbol{\ell_{1}})Y^{\rm
obs}(\boldsymbol{\ell_{2}})F_{\beta}^{\phi}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}}),$
(7)

where the superscript ‘obs’ denotes the observed map including instrumental
noise, and $F_{\beta}^{\phi}$ is the variance filter. $\beta=XY$, where
$X,Y\in\\{T,E,B\\}$. The normalization factor
$A^{\phi}_{\beta}(\boldsymbol{\ell})$ is chosen to make the estimator unbiased
$\langle\hat{\phi}_{\beta}(\boldsymbol{\ell})\rangle=\phi(\boldsymbol{\ell})$
and has the following form for the least variance estimator

$\displaystyle A_{\beta}(\boldsymbol{\ell})$
$\displaystyle=\left[\int\frac{d^{2}\boldsymbol{\ell}_{1}}{(2\pi)^{2}}f^{\phi}_{\beta}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})F_{\beta}^{\phi}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})\right]^{-1},$
(8)

For the EB estimator, we have

$\displaystyle F_{EB}^{\phi}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})$
$\displaystyle=\frac{f^{\phi}_{EB}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})}{C_{\boldsymbol{\ell_{1}}}^{EE,\text{obs}}C_{\boldsymbol{\ell_{1}}}^{BB,\text{obs}}},$
(9)

and

$\displaystyle f^{\phi}_{EB}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})=$
$\displaystyle\left[(\boldsymbol{\ell_{1}}\cdot\boldsymbol{L})C_{\boldsymbol{\ell_{1}}}^{EE}-(\boldsymbol{\ell_{2}}\cdot\boldsymbol{L})C_{\boldsymbol{\ell_{1}}}^{BB}\right]$
(10) $\displaystyle\times\sin
2(\varphi(\boldsymbol{\ell_{1}})-\varphi(\boldsymbol{\ell_{2}})).$

For the EE estimator, we have

$\displaystyle F_{EE}^{\phi}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})$
$\displaystyle=\frac{f^{\phi}_{EE}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})}{2C_{\boldsymbol{\ell_{1}}}^{EE,\text{obs}}C_{\boldsymbol{\ell_{2}}}^{EE,\text{obs}}},$
(11)

and

$\displaystyle f^{\phi}_{EE}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})=$
$\displaystyle\left[(\boldsymbol{\ell_{1}}\cdot\boldsymbol{L})C_{\boldsymbol{\ell_{1}}}^{EE}+(\boldsymbol{\ell_{2}}\cdot\boldsymbol{L})C_{\boldsymbol{\ell_{2}}}^{EE}\right]$
(12) $\displaystyle\times\cos
2(\varphi(\boldsymbol{\ell_{1}})-\varphi(\boldsymbol{\ell_{2}})).$

The noise properties of these estimators follow from

$\displaystyle\langle\hat{\phi}_{\beta}(\boldsymbol{L})\hat{\phi}_{\gamma}(\boldsymbol{L^{\prime}})\rangle$
$\displaystyle=(2\pi)^{2}\delta(\boldsymbol{L}-\boldsymbol{L^{\prime}})\left[C_{\boldsymbol{L}}^{\phi\phi}+N^{\phi}_{\beta\gamma}(\boldsymbol{L})\right],$
(13)

where

$\displaystyle N^{\phi}_{\beta\gamma}(L)=$ $\displaystyle
A_{\beta}(L)A_{\gamma}(L)\int\frac{d^{2}\boldsymbol{\ell}_{1}}{(2\pi)^{2}}F^{\phi}_{\beta}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})[F_{\gamma}^{\phi}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})$
(14) $\displaystyle\times
C_{\ell_{1}}^{X_{\beta}X_{\gamma}}C_{\ell_{2}}^{Y_{\beta}Y_{\gamma}}+F_{\gamma}^{\phi}(\boldsymbol{\ell_{1}},\boldsymbol{\ell_{2}})C_{\ell_{1}}^{X_{\beta}Y_{\gamma}}C_{\ell_{2}}^{Y_{\beta}X_{\gamma}}].$

Figure 1: (a) RDLFUnet: The architecture of residual dense local feature
U-net. Conv and $\bigoplus$ represent the convolution layer and summation. The
cat denotes concatenation operation along the channel dimension. C, K, S, and
P represent the number of output channels, kernel (filter) size, stride, and
padding of the convolutional layer, respectively. Bottom panels: BN, SELU, and
$\bigotimes$ represent batch normalization, Scaled Exponential Linear Unit
(SELU) function, and multiplication, respectively. (b) RLFB: residual local
feature block from Kong et al. (2022). (c) RDLFB: the residual dense local
feature block. (d) ESA: Enhanced Spatial Attention from Kong et al. (2022).
(e) ResUnet. The shapes (channel, height, width) of feature maps are shown in
the panel (e).

Similarly to Heinrich et al. (2022), we use in this paper the minimum variance
combination of all the polarization pairs,

$\displaystyle\hat{\phi}^{QE}(\boldsymbol{L})$
$\displaystyle=\sum_{\beta}\omega_{\beta}(L)\hat{\phi}^{QE}(\boldsymbol{L}),$
(15)

instead of the EB-only estimator used in Caldeira et al. (2019). Here, the
minimum variance weighting is

$\displaystyle\omega_{\beta}$
$\displaystyle=\frac{\sum_{\gamma}(N^{-1})_{\beta\gamma}}{\sum_{\gamma\eta}(N^{-1})_{\gamma\eta}}.$
(16)

The reconstruction noise of the minimum-variance estimator is then

$\displaystyle N_{L}$
$\displaystyle=\frac{1}{\sum_{\beta\gamma}(N^{-1})_{\beta\gamma}}.$ (17)

##  3 METHODOLOGY

In this section, we will introduce the deep learning architecture we use in
this work and the details of the data pipeline and network training.

###  3.1 Network architecture

The convolutional neural network (CNN) has been widely applied to image
processing tasks. Its power relies on the network architecture that consists
of a stack of non-linear parametric models. By training, the network
architecture can convert complex problems into parameter optimization. The
convolutional layer (Dumoulin & Visin, 2016) is the core of the network
architecture. Each convolutional layer accepts the feature images from the
former layer as an input, then convolves them with a bank of local spatial
filters (or kernels) whose values are parameters to be learned, and finally
gives output to the next layer after a nonlinear activation function. The
output size of a convolutional layer is controlled by three hyper-parameters:
the number of output channels (the number of convolutional filters), stride,
and amount of zero padding. The stride is defined as the distance in pixels
between the centers of adjacent filters. Residual connection (He et al.,
2015), which widely used in image processing tasks, is designed to learn the
residual mapping by adding inputs of a given layer to the outputs of the layer
that one. The U-Net (Ronneberger et al., 2015) is a successful architecture
that takes an encoder–decoder with convolutional layers and adds extra
shortcuts (skip connections) between the encoding and decoding layers to allow
for propagation of small-scale information that might be lost when the size of
the images decreases. In this work, we provide Residual Dense Local Feature
U-Net (RDLFUnet), a deep learning architecture designed to reconstruct the CMB
lensing potential. This network architecture combines Residual-Unet (ResUnet)
(Kayalibay et al., 2017; Zhang et al., 2018a) and the modified Residual Local
Feature Network (RLFN) (Kong et al., 2022). The ResUnet is created in U-Net
using the residual connections. The RLFN is an excellent architecture for
image super resolution reconstruction.

Figure 2: Example of the observed Q (top) and U (bottom) maps with different
noise levels.

Our network architecture is divided into two branches as shown in Figure 1.
The first branch is a modified RLFN. The residual local feature block (RLFB),
which employs three convolutional layers for residual local feature learning,
is the fundamental building block of the RLFN as depicted in the bottom panel
(b). The RLFB finally employs an Enhanced Spatial Attention (ESA) block (Liu
et al., 2022) to produce the end output. The ESA block can make the residual
features to be more focused on critical spatial contents. In this work, the
residual dense local feature block (RDLFB), which is depicted in the bottom
panel (c), replaces the RLFB. In contrast to the RLFN with residual
connection, the RDLFB uses a residual dense connection (Huang et al., 2016;
Zhang et al., 2018b). In our test, such a change might enhance the ability to
reconstruct CMB lensing potential. The second branch, known as ResUnet, is
shown in the bottom panel (e) of Figure 1. ResUnet encodes relevant
information from the input maps into smaller maps, and then decodes that
information to generate the output maps. The output of these two branches is
concatenated along the channel dimension, then pass to three residual blocks,
and the gravitational convergence $\kappa$ map will be finally constructed.
Our network model is built in PyTorch111https://pytorch.org/ environment,
which is an open-source optimized tensor library for deep learning.

Figure 3: Example of the reconstructed $\hat{\kappa}$ for four noise levels
(0, 1, 2, 5 $\mu$K-arcmin; left to right). The top-left map is the ground
truth $\kappa$. The ResUnet (Caldeira et al., 2019) and RDLFUnet predictions
of $\hat{\kappa}$ are shown in the top and second rows. The related residual
maps $\kappa-\hat{\kappa}$ are shown in the third and fourth rows.

The network model is optimized by minimizing a loss function. The training set
consist of $S$ pairs of samples $\\{x_{i},y_{i}\\}_{i=1}^{S}$, where $x$
represents the observed Q/U maps, and $y$ is the corresponding ground truth of
$\kappa$ map. Our loss function defines as

$\displaystyle\mathcal{L}=\mathcal{L}_{\rm LAD}+\beta\mathcal{L}_{\rm FFT},$
(18)

where $\mathcal{L}_{\rm LAD}$ is the least absolute deviation (LAD, also
called L1 loss), $\mathcal{L}_{\rm FFT}$ is the Fourier space loss, and
$\beta$ is a coefficient representing the contribution of $\mathcal{L}_{\rm
FFT}$ to the total loss and we set $\beta=1$ throughout the paper. The L1 loss
has form

$\displaystyle\mathcal{L}_{\rm LAD}$
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\left[\frac{1}{WH}\sum_{w=1}^{W}\sum_{h=1}^{H}(|I^{n}_{w,h}-y^{n}_{w,h}|)\right],$
(19)

where $N$ is the batch size, $H$ ($W$) is the height (width) of the images in
pixels, and $I=f(x)$ ($f(\cdot)$ is the network model) is the predicted image.
The form of $\mathcal{L}_{\rm FFT}$ is defined as

$\displaystyle\mathcal{L}_{\rm FFT}$
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\left[\frac{1}{WH}\sum_{w=1}^{W}\sum_{h=1}^{H}(|A_{\rm
F}(I^{n}_{w,h})-A_{\rm F}(y^{n}_{w,h})|)\right],$ (20)

where $A_{\rm F}$ is the amplitude of FFT, which has the form of

$\displaystyle A_{\rm F}(I)$ $\displaystyle=\sqrt{Re[{\rm FFT}(I)]^{2}+Im[{\rm
FFT}(I)]^{2}},$ (21)

where $Re[\cdot]$ and $Im[\cdot]$ denote the real and imaginary part,
respectively.

###  3.2 Data pipeline and network training

As a supervised machine learning technique, the CNN method needs a training
dataset with values that are already known to be true. Our data set is based
on a simulation of a flat sky. First, the publicly available
CAMB222https://github.com/cmbant/CAMB package is used to calculate the
primordial CMB and lensing power spectra (Lewis et al., 2000). Here, we use a
standard $\Lambda$ cold dark matter model with parameters
($H_{0},\Omega_{b}h^{2},\Omega_{c}h^{2},\tau,A_{s},n_{s}$), and their best-fit
value and standard deviation can be obtained from the Planck 2015 data (Planck
Collaboration et al., 2016). Then, using the theoretical power spectra from
CAMB, we simulate two-dimensional maps that cover a $5^{\circ}\times
5^{\circ}$ patch of sky with $128\times 128$ pixels by using a modified
version of Orphics333https://github.com/msyriac/orphics and resunet-
cmb444https://github.com/EEmGuzman/resunet-cmb.

Previous works (Caldeira et al., 2019; Guzman & Meyers, 2021) employed a fixed
lensing power spectrum to generate observed CMB map, however they indicated
that the trained network model is sensitive to the variations in cosmological
parameters. Li et al. (2022) used a varying lensing power spectrum $\alpha
C_{\ell}^{\kappa\kappa}$ ($\alpha$ is random values within range $0.75-1.25$),
while the cosmological parameters remained constant. In this work, in order to
be more realistic and eliminate the dependence of the cosmological parameters,
we treat the values of cosmological parameters as independent Gaussian random
variables, where the mean values and standard deviations are taken from the
best-fit values and standard deviation of Planck-2015 results (Planck
Collaboration et al., 2016).

Finally, similar to Caldeira et al. (2019), Guzman & Meyers (2021) and Li et
al. (2022), we smooth the maps with a Gaussian beam size of FWHM= 1 acrmin and
choose 4 levels of noise with $0.0$ $\mu$K-arcmin, $1.0$ $\mu$K-arcmin, $2.0$
$\mu$K-arcmin, $5.0$ $\mu$K-arcmin. Each generated map applies a cosine taper
of $0.5^{\circ}$ to reduce edge effects. In total five different types of maps
are generated, ($Q^{\rm prim}$, $U^{\rm prim}$, $Q^{\rm obs}$, $U^{\rm obs}$,
$\kappa$). Here, the superscript ‘prim’ represents the primordial CMB map,
while ‘obs’ denotes the observed CMB map including the lensing and
instrumental information. To generate the observed maps ($Q^{\rm obs}$,
$U^{\rm obs}$), the primordial maps ($Q^{\rm prim}$, $U^{\rm prim}$) are first
lensed with the convergence map, then a noise map is added after these lensed
maps have been smoothed with a Gaussian beam. The example observed CMB maps
with varying noise levels are shown in Figure 2. The variations between noise
levels of 0 and $>1\mu$K-arcmin are readily visible on the maps.

For each noise level, 25000 sets of the five maps described above are
generated and split into training, validation, and test sets with a ratio of
8:1:1. Each noise level is trained using a separate network. To train the
network, mock observed CMB maps ($Q^{\rm obs}$, $U^{\rm obs}$) will be fed
into the network model, going through each level, and outputting a
$\hat{\kappa}$ map. We use a batch size of 32, adopt the Adam optimizer
(Kingma & Ba, 2014), and initially set the learning rate to 0.005, which
gradually drops to $10^{-6}$ during the iteration. Iterations totaling 60,000
are used to train the network model. We used two NVIDIA Quadro GV100 GPUs to
train network, and one network model requires $\sim$12 hr to train.

##  4 Results

Results for the reconstructed $\kappa$ map using our network and its power
spectra are presented in this section. We will compare our results to those of
the ResUnet model (Caldeira et al., 2019; Guzman & Meyers, 2021). The
reconstructed map is represented by $\hat{\kappa}$, whereas the true map is
represented by $\kappa$.

###  4.1 Reconstructed map

First, we apply our network to the task of $\kappa$ map reconstruction. We
take the observed $Q^{\rm obs}$ and $U^{\rm obs}$ maps as input, and the
desired output is the ${\kappa}$ map. The test set is fed into the network
after the network training is finished. The reconstructed $\hat{\kappa}$ maps
are shown in Figure 3 for the four noise experiments. In the case of noiseless
input, the residual map contains minimal information, implying that the
network can recover the $\kappa$ map reasonably effectively. However, as noise
levels increase, all residual map amplitudes increase, and some small-scale
structure details are missed by the reconstructed $\kappa$ maps. Despite the
presence of noise, our network can also recover the most large-scale structure
of the $\kappa$ map.

To compare with the ResUnet model, we train the ResUnet model separately for
each noise level. For the ResUnet training, we use the same training pipeline
and network architecture as Caldeira et al. (2019). In Figure 3, we compare
the results of the ResUnet with our network model at the level of the
reconstructed map. We can see that, for noiseless input, the residual map
$\kappa-\hat{\kappa}$ has a lower amplitude than ResUnet’s. We also calculate
the structural similarity index measure (SSIM) (Zhou, 2004) of the
reconstructed and true $\kappa$ map. The SSIM is a method used for measuring
the similarity between two images, as follows

$\displaystyle{\rm
SSIM}(x,y)=\frac{(2\mu_{x}\mu_{y}+c_{1})(2\sigma_{xy}+c_{2})}{(\mu_{x}^{2}+\mu_{y}^{2}+c_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+c_{2})},$
(22)

here, $x$ and $y$ represent two images, $\mu_{x}$ and $\mu_{y}$ are the pixel
sample mean of $x$ and $y$, $\sigma_{x}$ and $\sigma_{y}$ are the variance of
$x$ and $y$, $\sigma_{xy}$ is the covariance of $x$ and $y$, and $c_{1}$ and
$c_{2}$ are two constants to stabilize the division with weak denominator. The
two images are more similar the closer the value of SSIM is to 1. The two
images are precisely the same if its value is 1. For the noiseless input case,
the SSIMs are $0.98\pm 0.003$ for our model and $0.89\pm 0.015$ for the
ResUnet model, indicating that our reconstructed $\hat{\kappa}$ map is closer
to the true $\kappa$ than result from ResUnet model.

When considering the non-zero noise inputs, from visual inspection, our
reconstructed $\hat{\kappa}$ maps can also have more small-scale structure
features than the ResUnet model. Compared to ResUnet model, our network can
capture more details on small-scale structure. We also calculate the SSIM of
the reconstructed and true $\hat{\kappa}$ map. The SSIMs of our network for
the $1,\,2,\,5\,\mu$K-arcmin noise levels experiments are $0.68\pm 0.010$,
$0.56\pm 0.012$ and $0.41\pm 0.015$, while the ResUnet model are $0.62\pm
0.015$, $0.52\pm 0.014$ and $0.39\pm 0.017$, respectively. For experiments of
three noise levels, our model performs better than the ResUnet model.

Figure 4: Power spectra of predicted $\hat{\kappa}$ map from the ResUnet
(dashed line) and RDLFUnet (solid line) for four noise levels (0, 1, 2, 5
$\mu$K-arcmin) averaged over all realizations in the test set, and the length
of each $L$ bin is set to be 50 here. The ratio of the predicted power
spectrum to the true power spectrum is also shown.

###  4.2 Reconstructed power spectrum

Next, we present the power spectrum of the reconstructed $\hat{\kappa}$ map at
each noise level and compare them to the results of the ResUnet and QE
approaches.

Figure 4 shows the power spectra of reconstructed $\hat{\kappa}$ for four
input noise levels. The power spectra are the mean result over the testing
set. For the noiseless input, the constructed $\kappa$ power spectra from our
network is quite consistent with the target truth power spectrum. However, as
noise levels rise, the reconstructed power spectrum visibly degrades at small
scales, which implies the small-scale fluctuations are suppressed. The ResUnet
model’s reconstructed power spectra for each noise level are also shown in
figure 4 for comparison’s sake. At all noise levels, we can clearly observe
that $C_{L}^{\hat{\kappa}\hat{\kappa}}$ reconstructed by our model contains
more small-scale information than the ResUnet model, which is consistent with
the reported results of the reconstructed $\hat{\kappa}$ map in Figure 3.

Figure 5: Reconstruction noise power spectra of the ResUnet model, the
quadratic estimator, and our model RDLFUnet for noise levels of $1\
\mu$K-arcmin (top panel) and $5\ \mu$K-arcmin (bottom panel). ‘X’ denotes
ResUnet or RDLFUnet.

Finally, we compare the performance of the quadratic estimator in
reconstructing $\hat{\kappa}$ power spectrum to that of our network model.
First we need to define the reconstructed noise. As shown in Figure 4 and
Refs. (Caldeira et al., 2019; Guzman & Meyers, 2021), the reconstructed
$\kappa$ power spectrum $C^{\hat{\kappa}\hat{\kappa}}$ by network model is a
bias result. We use a method from Caldeira et al. (2019) to treat this bias.
We normalize the biased $\hat{\kappa}$ field by a factor $R_{L}$ defined as

$\displaystyle R_{L}$ $\displaystyle=\left[\frac{\langle
C_{L}^{\kappa\hat{\kappa}}\rangle}{\langle
C_{L}^{\kappa\kappa}\rangle}\right]^{-1},$ (23)

where $\langle...\rangle$ represents the average over the test set. It should
be noted that $R_{L}$ is a function of scales $L$. Finally, the equivalent of
the unbiased $\kappa$ field is given by

$\displaystyle\hat{\hat{\kappa}}_{L}$ $\displaystyle=R_{L}\hat{\kappa}_{L}.$
(24)

We assume that $\hat{\hat{\kappa}}$ can be modelled as
$\hat{\hat{\kappa}}=\kappa+n$, where $n$ is an uncorrelated noise term. The
equivalent of the reconstruction noise spectrum can be defined as

$\displaystyle N^{\kappa\kappa}_{L}$ $\displaystyle=\langle
C^{\hat{\hat{\kappa}}\hat{\hat{\kappa}}}_{L}\rangle-\langle
C^{\kappa\kappa}_{L}\rangle,$ (25)

where $C^{\kappa\kappa}_{L}$ is true power spectrum, and we have

$\displaystyle\langle C^{\hat{\hat{\kappa}}\hat{\hat{\kappa}}}_{L}\rangle$
$\displaystyle=R_{L}^{2}\langle C^{\hat{\kappa}\hat{\kappa}}_{L}\rangle.$ (26)

Figure 6: Null test shows the average power spectra
($C^{\hat{\hat{\kappa}}\hat{\hat{\kappa}}}$) of rescaled lensing maps
predicted by the RDLFUnet when we feed unlensed versions of the test set to
fully trained network. We compare these rescaled power spectra to the
reconstruction noise power spectra ($N_{L}^{\hat{\kappa}\hat{\kappa}}$) of the
ResUnet model shown in Figure 5. Figure 7: Reconstructed $\kappa$ map using
the ResUnet model with FFT loss for the noise level of $1\mu$K-arcmin. As a
comparison, reconstructed $\hat{\kappa}$ from the original ResUnet and
RDLFUnet also are shown in panels. Here, original ResUnet represents the
ResUnet model without FFT loss function. Power spectra of prediction map
$\hat{\kappa}$ averaged over all realizations in the test set are also shown
in the bottom panel.

Figure 5 shows the reconstruction noise power spectrum for the experiments
with $1\,\mu$K-arcmin (upper panel) and $5\,\mu$K-arcmin (lower panel) noise
levels using the ResUnet model, the standard quadratic estimator, and our
network model. The noise power spectra (see Eq. (17)) of the quadratic
estimator are obtained using publicly available
Symlens555https://github.com/simonsobs/symlens package.

We can see that the ResUnet model predictions have a lower reconstruction
noise than the quadratic estimator at all scales for the noise level of $1\
\mu$K-arcmin and closely matches the performance of the quadratic estimator
for the noise level of $5\ \mu$K-arcmin. The ResUnet model performance is also
obviously better than the quadratic estimator for the noise level of $2\
\mu$K-arcmin (the result is not shown in the figure). These imply that
ResUnets outperform the QE method for lower noise levels. These conclusions
are consistent with those made by Caldeira et al. (2019).

For experiments with noise levels below $5\ \mu$K-arcmin, the reconstruction
noise levels from our network model are lower than those from the ResUNet
model and QE at almost all angular scales. These results suggest that our
approach outperforms both the QE and the ResUnet model in terms of signal-to-
noise ratio on the reconstructed $\kappa$ map.

Finally, we use the Fisher matrix method to compute the signal-to-noise ratio
(SNR) (Liu et al., 2022)

$\displaystyle{\rm
SNR}=\sqrt{\sum_{L,L^{\prime}}C_{L}\mathbb{C}_{L,L^{\prime}}^{-1}C_{L^{\prime}}},$
(27)

where the numerator $C_{L}$ is the theoretical $\kappa$ spectrum, and
$\mathbb{C}_{L,L^{\prime}}$ is the covariance matrix obtained from our test
set via

$\displaystyle\mathbb{C}_{L,L^{\prime}}=\frac{1}{N-1}\sum_{n=1}^{N=2500}[(C^{\hat{\kappa}\hat{\kappa}}_{L}-\bar{C}^{\hat{\kappa}\hat{\kappa}}_{L})\times(C^{\hat{\kappa}\hat{\kappa}}_{L^{\prime}}-\bar{C}^{\hat{\kappa}\hat{\kappa}}_{L^{\prime}})],$
(28)

where $C^{\hat{\kappa}\hat{\kappa}}_{L}$ is reconstructed $\kappa$ power
spectrum and $\bar{C}^{\hat{\kappa}\hat{\kappa}}_{L}$ is the averaged
reconstructed $\kappa$ power spectrum across the test set. For experiments
with the noise level of $1\ \mu$K-arcmin, the SNRs for our network model,
ResUNet, and QE estimator are 14.7, 12.4, 9.1, respectively. For experiments
with the noise level of $5\ \mu$K-arcmin, the SNRs for our network model,
ResUNet, and QE estimator are 6.1, 5.5, 5.2, respectively. Our method can get
SNR higher than ResUNet and QE methods.

###  4.3 Null test

Finally, a null test is performed to ensure that the network learns a sensible
mapping of the input lensed ($Q^{\rm obs}$, $U^{\rm obs}$) maps to the
predicted $\hat{\kappa}$ map. In other words, we need to check that the
predicted $\hat{\kappa}$ maps by our network model are due to the presence of
the true $\kappa$ signal rather than an artifact of the network (i.e., non-
physical signal). In order to check the output results of our network model,
we feed unlensed versions of ($Q$, $U$) maps to the network trained on
observed ($Q^{\rm obs}$, $U^{\rm obs}$) maps.

We add noise to unlensed ($Q^{\rm prim}$, $U^{\rm prim}$) maps introduced in
the section 3.2 and feet it to network trained on observed ($Q^{\rm obs}$,
$U^{\rm obs}$) maps. The cross-correlation spectrum
($C^{\hat{\kappa}\kappa}_{L}$) of reconstructed $\hat{\kappa}$ derived from
the null test and true $\kappa$ is first calculated, and it is discovered that
they are uncorrelated at all noise levels. This suggests that when we input
the unlensed ($Q$, $U$) maps into network trained on observed maps, the output
reconstructed $\kappa$ map of the network is merely random noise.

The power spectrum of the output $\hat{\kappa}$ map from the null test is then
calculated and compared to the reconstructed noise. To address bias, we first
use Eq.(24) to normalize the biased $\hat{\kappa}$ map from the null test. We
can calculate the rescaled power spectrum
$C^{\hat{\hat{\kappa}}\hat{\hat{\kappa}}}_{L}$ in Eq.(26) from the output of
null test, and compare it to the reconstructed noise shown in Figure 5. Figure
6 shows the rescaled power spectra $\langle
C^{\hat{\hat{\kappa}}\hat{\hat{\kappa}}}_{L}\rangle$ for each noise level and
the reconstructed noise. We can see that the rescaled power spectra from the
null test are basically consistent with the reconstructed noise at all noise
levels, which is also consistent with the result of Caldeira et al. (2019).
All these null tests illustrate that our network is capable of successfully
learning the reasonable mapping from the input lensed ($Q^{\rm obs}$, $U^{\rm
obs}$) maps to the predicted $\hat{\kappa}$ map correctly.

We also notice that there is the difference within a few percent between
$C^{\hat{\hat{\kappa}}\hat{\hat{\kappa}}}_{L}$ and $N^{\kappa\kappa}_{L}$ at
large scales, and this difference becomes more obvious for the input maps with
lower noise levels. We think that this could be caused by the normalization
factor $R$ in Eq.(23), which may be not very accurate and will bias the
$\hat{\hat{\kappa}}$ field in computation because the $R$ computed from the
training set fluctuates high/low.

##  5 Discussions

###  5.1 Comparing with ResUnet Methods

In this work, we compared the performances of the ResUnet and RDLFUnet in
reconstructing $\kappa$ map. To be consistent with Caldeira et al. (2019), the
ResUnet model uses mean squared error in image space as the loss function for
training, whereas the RDLFUnet uses Eq.(18). The two models should utilize the
same loss function for an accurate comparison.

We use the loss function contained FFT loss, Eq.(18), to train the ResUnet
model for the experiment of $1\mu$K-arcmin noise (other experiments have
similar results), and results of reconstructing $\kappa$ are presented in
Figure 7. We can see that the reconstructed $\hat{\kappa}$ map by the ResUnet
with FFT loss function has more small-scale structures than the results of the
original ResUnet model. This implies that the FFT loss function can help the
network capture more small-scale structures at the map level, implying that
using the FFT loss function could enhance the model’s ability to reconstruct
$\kappa$ map.

From the aspect of the reconstructed power spectrum, the reconstructed
$C^{\hat{\kappa}\hat{\kappa}}$ power spectrum from the ResUnet with FFT loss
obviously outperforms the result from the original ResUnet model, but it is
still worse than the result of the RDLFUnet. These demonstrate that the
$\kappa$ reconstruction process is aided by our loss function and the RDLFUnet
model outperforms the ResUnet model. The RDLFUnet model actually has a more
complicated structure than the ResUnet model, which suggests that it may have
a stronger capacity for fitting than the ResUnet model. As a result, compared
with the ResUnet method, both our network structure and loss function can
improve the performance of reconstructing $\kappa$ map.

###  5.2 Variability of noise level

In Section 4, each noise level experiment is trained using a separate network.
Here, we alter the noise level for a trained network to see how sensitive our
method is to changes in noise level. We train the RDLFUnet model on a training
set with noise level of $1\ \mu$K-arcmin and then feed it three test sets with
noise levels of $0.8\ \mu$K-arcmin, $1\ \mu$K-arcmin, and $1.2\ \mu$K-arcmin.
We calculate the predicted power spectrum by RDLFUnet model as shown in Figure
8. The predicted $\hat{\kappa}$ power spectrum for test sets with noise levels
of $0.8\ \mu$K-arcmin and $1.2\ \mu$K-arcmin are consistent with the test sets
with noise level of $1\ \mu$K-arcmin. This implies that our network is
insensitive to small changes in noise level.

Figure 8: Power spectra of the predicted $\hat{\kappa}$ map from RDLFUnet
model. RDLFUnet model is trained on a training set with noise level of $1\
\mu$K-arcmin. ‘$1\ \mu$K-arcmin’ represents the prediction of RDLFUnet on test
sets with noise level of $1\ \mu$K-arcmin. ‘$0.8\ \mu$K-arcmin’ and ‘$1.2\
\mu$K-arcmin’ are the prediction of RDLFUnet on test set with noise level of
$0.8\ \mu$K-arcmin and $1.2\ \mu$K-arcmin, respectively. The ratio of the
predicted power spectrum to the true power spectrum is also shown.

##  6 CONCLUSIONS

In this work, we present a machine-learning method for reconstructing the CMB
lensing potential. The network was trained using simulated CMB maps with
$5\times 5$ ${\rm deg}^{2}$ size. We compared RDLFUnet’s performance with that
of ResUnet model from previous work. We demonstrated that the RDLFUnet can
capture more small-scale structure features than ResUnet at the reconstructed
$\kappa$ map level based on the reconstructed results of four tests with
varying noise levels (0, 1, 2, 5 $\mu$K-arcmin).

At the reconstructed power spectrum level, we demonstrated that our method
outperforms the ResUnet in reconstructing power spectra of lensing convergence
and has a higher signal-to-noise ratio than the ResUnet and quadratic
estimator methods at almost the entire observation scales.

All results in this paper are based on simulated data. To extend this network
usage for actual data, we need to include foreground residual in the input
maps. We plan to reconstruct CMB lensing from the foreground cleaned CMB
polarization data in future work. In addition, since actual data are often
taken from larger regions of the sky, we also need to use simulations from
larger sky patches. As a result, the flat sky approximation will no longer be
valid, necessitating the modeling of a spherical sky map. The spherical
HEALPix format map needs to be converted into 2D data since the neural network
needs 2D data for its inputs. This issue has been addressed in Wang et al.
(2022) and Yan et al. (2023). We also investigate the impact of different
foreground removal methods on the results of reconstructing CMB lensing in
future work.

We will further investigate the ability of our method to other aspects, such
as the reconstruction of foregrounds and CMB delensing, as well as the
component separation of future radio surveys, in future works.

## Acknowledgements

J.-Q. Xia is supported by the National Science Foundation of China under
grants No. U1931202 and 12021003; the National Key R&D Program of China No.
2017YFA0402600 and 2020YFC2201603.

## References

  * Abazajian et al. (2016) Abazajian, K. N., Adshead, P., Ahmed, Z., et al. 2016, arXiv:1610.02743
  * Ade et al. (2019) Ade, P., Aguirre, J., Ahmed, Z., et al. 2019, J. Cosmology Astropart. Phys, 2019, 056.
  * Bennett et al. (2013) Bennett, C. L., Larson, D., Weiland, J. L., et al. 2013, ApJS, 208, 20.
  * Benson et al. (2014) Benson, B. A., Ade, P. A. R., Ahmed, Z., et al. 2014, Proc. SPIE, 9153, 91531P.
  * Caldeira et al. (2019) Caldeira, J., Wu, W. L. K., Nord, B., et al. 2019, Astronomy and Computing, 28, 100307.
  * Carron & Lewis (2017) Carron, J. & Lewis, A. 2017, Phys. Rev. D, 96, 063510.
  * Carron (2019) Carron, J. 2019, Phys. Rev. D, 99, 043518.
  * Darwish et al. (2021) Darwish, O., Madhavacheril, M. S., Sherwin, B. D., et al. 2021, MNRAS, 500, 2250.
  * Dumoulin & Visin (2016) Dumoulin, V. & Visin, F. 2016, arXiv:1603.07285
  * Fluke & Jacobs (2020) Fluke, C. J. & Jacobs, C. 2020, WIREs Data Mining and Knowledge Discovery, 10, e1349.
  * Guzman & Meyers (2021) Guzman, E. & Meyers, J. 2021, Phys. Rev. D, 104, 043529.
  * Guzman & Meyers (2022) Guzman, E. & Meyers, J. 2022, J. Cosmology Astropart. Phys, 2022, 030.
  * Hazumi et al. (2019) Hazumi, M., Ade, P.A.R., Akiba, Y. et al. 2019, J Low Temp Phys, 194, 443
  * He et al. (2015) He, K., Zhang, X., Ren, S., et al. 2015, arXiv:1512.03385
  * Heinrich et al. (2022) Heinrich, C., Driskell, T., & Heinrich, C. 2022, arXiv:2210.07391
  * Hirata & Seljak (2003a) Hirata, C. M. & Seljak, U. 2003a, Phys. Rev. D, 67, 043001.
  * Hirata & Seljak (2003b) Hirata, C. M. & Seljak, U. 2003b, Phys. Rev. D, 68, 083002.
  * Horowitz et al. (2019) Horowitz, B., Ferraro, S., & Sherwin, B. D. 2019, MNRAS, 485, 3919.
  * Hotinli et al. (2022) Hotinli, S. C., Meyers, J., Trendafilova, C., et al. 2022, J. Cosmology Astropart. Phys, 2022, 020.
  * Hu & Okamoto (2002) Hu, W. & Okamoto, T. 2002, ApJ, 574, 566.
  * Huang et al. (2016) Huang, G., Liu, Z., van der Maaten, L., et al. 2016, arXiv:1608.06993
  * Kayalibay et al. (2017) Kayalibay, B., Jensen, G., & van der Smagt, P. 2017, arXiv:1701.03056
  * Kingma & Ba (2014) Kingma, D. P. & Ba, J. 2014, arXiv:1412.6980
  * Kong et al. (2022) Kong, F., Li, M., Liu, S., et al. 2022, arXiv:2205.07514
  * Lange et al. (2001) Lange, A. E., Ade, P. A., Bock, J. J., et al. 2001, Phys. Rev. D, 63, 042001.
  * Legrand & Carron (2022) Legrand, L. & Carron, J. 2022, Phys. Rev. D, 105, 123519.
  * Lewis & Challinor (2006) Lewis, A. & Challinor, A. 2006, Phys. Rep., 429, 1.
  * Lewis et al. (2001) Lewis, A., Challinor, A., & Turok, N. 2001, Phys. Rev. D, 65, 023505
  * Lewis et al. (2000) Lewis, A., Challinor, A., & Lasenby, A. 2000, ApJ, 538, 473.
  * Li et al. (2017) Li, H., Li, S.-Y., Liu, Y., et al. 2017, arXiv:1710.03047
  * Li et al. (2022) Li, P., Ilayda Onur, I., Dodelson, S., et al. 2022, arXiv:2205.07368
  * Liu et al. (2022) Liu, J., Zhang, W., Tang, Y., et al. 2020, In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 2359–2368.
  * Liu et al. (2022) Liu, J., Sun, Z., Han, J., et al. 2022, Science China Physics, Mechanics, and Astronomy, 65, 109511. doi:10.1007/s11433-022-1966-4
  * Louis et al. (2017) Louis, T., Grace, E., Hasselfield, M., et al. 2017, J. Cosmology Astropart. Phys, 2017, 031.
  * Maniyar et al. (2021) Maniyar, A. S., Ali-Haïmoud, Y., Carron, J., et al. 2021, Phys. Rev. D, 103, 083524.
  * Mather et al. (1994) Mather, J. C., Cheng, E. S., Cottingham, D. A., et al. 1994, ApJ, 420, 439.
  * Mehta et al. (2019) Mehta, P., Bukov, M., Wang, C.-H., et al. 2019, Phys. Rep., 810, 1.
  * Millea et al. (2019) Millea, M., Anderes, E., & Wandelt, B. D. 2019, Phys. Rev. D, 100, 023509.
  * Millea et al. (2020) Millea, M., Anderes, E., & Wandelt, B. D. 2020, Phys. Rev. D, 102, 123542.
  * Nah et al. (2016) Nah, S., Kim, T. H., & Lee, K. M. 2016, arXiv:1612.02177
  * Petroff et al. (2020) Petroff, M. A., Addison, G. E., Bennett, C. L., et al. 2020, ApJ, 903, 104.
  * Planck Collaboration et al. (2020a) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020a, A&A, 641, A6.
  * Planck Collaboration et al. (2020b) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020b, A&A, 641, A8.
  * Planck Collaboration et al. (2016) Planck Collaboration, Adam, R., Ade, P. A. R., et al. 2016, A&A, 594, A1.
  * Ronneberger et al. (2015) Ronneberger, O., Fischer, P., & Brox, T. 2015, arXiv:1505.04597
  * Smith et al. (2012) Smith, K. M., Hanson, D., LoVerde, M., et al. 2012, J. Cosmology Astropart. Physoi:10.1088/1475-7516/2012/06/014, 2012, 014.
  * Smith et al. (2006) Smith, K. M., Hu, W., & Kaplinghat, M. 2006, Phys. Rev. D, 74, 123002.
  * Tian et al. (2020) Tian, C., Xu, Y., & Zuo, W. 2020, Neural Networks, 121, 461.
  * Wang et al. (2022) Wang, G.-J., Shi, H.-L., Yan, Y.-P., et al. 2022, ApJS, 260, 13.
  * Wu et al. (2019) Wu, W. L. K., Mocanu, L. M., Ade, P. A. R., et al. 2019, ApJ, 884, 70.
  * Waqas Zamir et al. (2021) Waqas Zamir, S., Arora, A., Khan, S., et al. 2021, arXiv:2102.02808
  * Yan et al. (2023) Yan, Y.-P., Wang, G.-J., Li, S.-Y., et al. 2023, ApJ, 947, 29.
  * Zaldarriaga & Seljak (1998) Zaldarriaga, M. & Seljak, U. 1998, Phys. Rev. D, 58, 023003.
  * Zaldarriaga & Seljak (1999) Zaldarriaga, M. & Seljak, U. 1999, Phys. Rev. D, 59, 123507.
  * Zhang et al. (2018a) Zhang, Z., Liu, Q., & Wang, Y. 2018a, IEEE Geoscience and Remote Sensing Letters, 15, 749.
  * Zhang et al. (2018b) Zhang, Y., Tian, Y., Kong, Y., et al. 2018b, arXiv:1802.08797
  * Zhou (2004) Zhou Wang and Bovik, A.C. and Sheikh, H.R. & Simoncelli, E.P. 2004, IEEE Transactions on Image Processing, 13, 600-612.