<EMAIL_ADDRESS>
# On the amplification of acoustic phonons in carbon nanotube

K. A. Dompreh11footnotemark: 1 N. G. Mensah  D. Sakyi-Arthur  S. Y. Mensah
Department of Physics, College of Agriculture and Natural Sciences, U.C.C,
Ghana.  Department of Mathematics, College of Agriculture and Natural
Sciences, U.C.C, Ghana  [

###### Abstract

We present a theoretical study of acoustic phonons amplification in Carbon
Nanotubes (CNT). The phenomenon is via Cerenkov emission (CE) of acoustic
phonons using intraband transitions proposed by Mensah et. al., [1] in
Semiconductor Superlattices (SSL) and confirmed in [2]. From this, an
asymmetric graph of $\Gamma^{CNT}$ on $\frac{V_{d}}{V_{s}}$ and $\Omega\tau$
were obtained where amplification ($\Gamma_{amp}^{CNT}$) $>>$ absorption
($\Gamma_{abs}^{CNT}$). The ratio,
$\frac{|\Gamma_{amp}^{CNT}|}{|\Gamma_{abs}^{CNT}|}\approx 3.5$, at
$V_{d}=1.02V_{s}$, $\omega_{q}=3.0\ \mathrm{THz}$ and $T=85\ K$ for scattering
angle $\theta>0$ . A threshold field at which $\Gamma_{abs}^{CNT}$ switches
over to $\Gamma_{amp}^{CNT}$ was calculated to be $E_{z}^{dc}=6.2\times
10^{3}\ \mathrm{V/m}$. This field is far less than that deduced using Bloch-
Type Oscillation (BTO) [3] which is $E_{BTO}^{dc}=3.0\times 10^{5}\
\mathrm{V/m}$. The obtained $\Gamma_{amp}^{CNT}$ would enable the use of CNT
for the production of SASER.

Keywords: Amplification, Carbon Nanotubes, acoustic phonon, hypersound

## Introduction

The study of amplification of acoustic phonons propagated along the axes of
low-dimensional and bulk materials such as Semiconductor Superlattices (SL)
[4, 5, 6, 7], $\mathrm{2D}$-Graphene sheet [8, 9], Quantum Wells (QW) [10,
11], and Carbon Nanotubes (CNT) [12, 13] are actively persued recently using
microscopic theory of electron - phonon interactions. This is due to the
electronic and optoelectronic applications including the production of SASER
(Sound Amplification by Stimulated Emission of Radiation), for dynamic storage
of light in quantum wells and acoustic wave induced carrier transport [9].
When a non-quantizing electric field is applied to these material, and the
drift velocity $V_{d}>V_{s}$ ($V_{s}$ is sound velocity) an amplification of
acoustic phonons occurs but in the reverse when $V_{d}<V_{s}$ it lead to
absorption of acoustic phonons. The nature of this phenomenon is related to
Cerenkov’s emission of phonons by moving carriers [14, 15]. In SSL,
amplification of acoustic phonons via Cerenkov’s emission was proposed
theoretically by Mensah et. al., [1]. The Mensah formalism of intra-miniband
transitions via deformation potential (dp) in SSL has been confirmed
experimentally by Shinokita [2] to amplify over $200\%$ of acoustic phonons.
Recently, a series of related studies in Graphene and CNT with degenerate
energy dispersion has been conducted (theoretically) in the hypersound regime
$ql>>1$ ($q$ is the acoustic wavenumber, $l$ is the mean free path). The
results obtained qualitatively agreed with an experimentally obtained results
[12, 19]. In multilayer-graphene-based system, Yurchenko et. al., [16]
obtained amplification in the hydrodynamic regime $ql<<1$ in a collisionless
system with condition $2V_{D}>V_{S}$. Dagher, et. al., [4] utilised the
Boltzmann transport equation (BTE) to investigate the amplification of
travelling waves in metallic CNT biased by a dc field. The amplification
attained was as a results of Bloch-Type Oscillations (BTO). In CNT, the
$\pi$-bonding and anti-bonding ($\pi^{*}$) energy band crosses at the Fermi
level in a linear manner [17] where, intra-band scattering process depends on
the phonon modes. These modes are the Longitudinal acoustci (LA), Transverse
acoustic (TA) and the Radial Breathing Mode (RBM) which is the weakest
scattering mode. For each phonon branch, an electron can be scattered either
by zone center phonon, or by a zone boundary phonon. The energy dispersion
$\varepsilon(p_{z})$ is near the Fermi level therefore, at low temperatures,
conduction occurs through well seperated discrete electron states. This leads
to the emission of large number of coherent acoustic phonons [14, 15, 18]. The
extreme electron mobilities makes CNT, a good candidate for the amplification
of acoustic phonons. Till date, few studies [3, 18] has been conducted to
understand amplification in CNT. In this paper, we utilised the theory
proposed by [1] and verified experimentally by [2] to study amplification of
acoustic phonons in CNT. The paper is organised as follows: In section $2$,
the kinetic theory based on the linear approximation for the phonon
distribution function is setup, where, the rate of growth of the phonon
distribution is deduced and the amplification is obtained. In section $3$, the
final equation is analysed numerically in a graphical form at the harmonic.
Lastly the conclusion is presented in section $4$.

## Theory

We will proceed following the works of [18, 19, 20] where the kinetic equation
for the phonon distribution is given as

$\displaystyle\frac{\partial N_{{q}}}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{\hbar}\sum_{p}|{C_{{q}}}|^{2}\\{[N_{{q}}(t)+1]f_{{p}}(1-f_{{p}^{\prime}})\delta(\varepsilon_{{p}^{\prime}}-\varepsilon_{{p}}+\hbar\omega_{{p}})$
(1) $\displaystyle-$ $\displaystyle
N_{{q}}(t)f_{{p}^{\prime}}(1-f_{{p}})\delta(\varepsilon_{{p}^{\prime}}-\varepsilon_{{p}}+\hbar\omega_{{q}})\\}-\gamma
N_{q}(t)$

where $N_{{q}}(t)$ represent the number of phonons with wave vector ${q}$ at
time $t$. The factor $N_{{q}}+1$ accounts for the presence of $N_{{q}}$
phonons in the system when the additional phonon is emitted. The
$f_{{p}}(1-f_{{p}})$ represent the probability that the initial ${p}$ state is
occupied and the final electron state ${p}^{\prime}$ is empty whilst the
factor $N_{{q}}f_{{p}^{\prime}}(1-f_{{p}})$ is that of the boson and fermion
statistics. $\gamma$ denotes phonon losses which includes phonon scattering or
phonon absorption due to non-electronic mechanisms, phonon decay due to
anharmonicity of the lattice. In a more convenient form, Eqn. ($1$) can be
written as

$\displaystyle\frac{\partial N_{{q}}(t)}{\partial
t}=2\pi|{C_{{-q}}}|^{2}[{\frac{N_{{q}}(t)+1}{1-exp(\beta(\hbar\omega_{{q}}-\hbar\vec{q}\cdot
V_{D}))}+{\frac{N_{{q}}}{1-exp(-\beta(\hbar\omega_{{q}}-\hbar\vec{q}\cdot
V_{D}))}}}]$
$\displaystyle\times\sum_{{p}}{(f_{{p}}-f_{{p}^{\prime}})\delta(\varepsilon_{{p}^{\prime}}-\varepsilon_{{p}}+\hbar\omega_{{q}})}$
(2)

$\beta=1/k_{B}T$, $k_{B}$ is the Boltzmann constant and $T$ is the absolute
temperature. Here, phonon loses were ignored and the lowest order in the
electron-phonon coupling is approximated by $f_{p}$ and

$N_{{q}}=[exp(-\beta(\hbar\omega_{{q}}-\hbar\vec{q}\cdot V_{D})-1)]^{-1}$

Eqn.($2$) can further be expressed as

$\displaystyle\frac{\partial N_{{q}}(t)}{\partial
t}=2\pi|{C_{{q}}}|^{2}[N_{{q}}(t)-\frac{1}{1-exp(\hbar\omega_{{q}}-\hbar
q\cdot V_{D})-1}]$ $\displaystyle\times ImQ(q,\hbar\omega_{q}-q\cdot V_{D})$
(3)

where

$Q=\sum_{{p}}{\frac{f_{{p}}-f_{{p}^{\prime}}}{\varepsilon_{{p}}-\varepsilon_{p^{\prime}}-\hbar\omega_{q}-i\delta}}$
(4)

and

$f_{{p}}=[exp(-\beta(\varepsilon_{{p}}-\mu))+1]^{-1}$ (5)

From Eqn. ($3$) the phonon generation rate is given as

$\Gamma_{{q}}=-2|{C_{{q}}}|^{2}ImQ(\hbar\vec{q},\hbar\omega_{{q}}-\hbar\vec{q}\cdot
V_{D})$ (6)

this simplifies to the phonon transition as

$\Gamma_{{q}}=2\pi|{C_{{q}}}|^{2}\sum_{{p}}{(f_{{p}}-f_{{p}^{\prime}})\delta(\varepsilon_{{p}}-\varepsilon_{{p}^{\prime}}-(\hbar\omega_{{q}}-{\hbar\vec{q}}\cdot
V_{D}))}$ (7)

the total rate of absorption and emission of phonon is obtained by the
summation over all the initial and final electrons states. In Eqn.($7$),
$f_{{p}}>f_{{p}\prime}$ if $\varepsilon_{{p}}<\varepsilon_{{p}^{\prime}}$.
When $\hbar\omega_{{q}}-\hbar\vec{q}\cdot V_{D}>0$, the system would return to
its equilibrium configuration when perturbed but
$\hbar\omega_{{q}}-\hbar\vec{q}\cdot V_{D}<0$ leads to the Cerenkov condition
of phonon instability (amplification). From perturbation theory, the
transition probability per unit time from the initial state $|p\rangle$,
consisting of electron having momentum $p_{z}$, to the final state
$|p^{\prime}\rangle$, which consists of an electron with momentum
$p_{z}^{\prime}$ and a phonon with wave vector $q$. The phonon and the
electric field are directed along the CNT axis therefore
${p_{z}}^{\prime}=({p}+\hbar{q})cos(\theta)$ where $\theta$ is the scattering
angle. In CNT, the linear energy dispersion $\varepsilon({p})$ relation is
given as [13]

$\varepsilon({p_{z}})=\varepsilon_{0}\pm\frac{\sqrt{3}}{2\hbar}\gamma_{0}b({p_{z}}-{p}_{0})$
(8)

The $\varepsilon_{0}$ is the electron energy in the Brillouin zone at momentum
$p_{0}$, $b$ is the lattice constant , $\gamma_{0}$ is the tight binding
overlap integral ($\gamma_{0}=2.54$eV). The $\pm$ sign indicates that in the
vicinity of the tangent point, the bands exhibit mirror symmetry with respect
to each point. At low temperature, when, $k_{B}T<<1$, Eqn.($5$) reduces to

$f_{{p}}=exp(-\beta(\varepsilon(p_{z})-\mu))$ (9)

Inserting Eqn.($8$ and $9$) into Eqn.(7), and after some cumbersome
calculations yield

$\Gamma^{CNT}=\frac{4\hbar\pi|C_{{q}}|^{2}exp(-\beta(\varepsilon_{0}-\chi{p}_{0}))}{\gamma_{0}b\sqrt{3}(1-cos(\theta))}\\{exp(-\beta\chi(\eta+\hbar{q})cos(\theta))-exp(-\beta\chi\eta)\\}$
(10)

where $\chi={\sqrt{3}\gamma_{0}b}/{2\hbar}$, and

$\eta=\frac{2{\hbar}^{2}\omega_{{q}}(1-\frac{V_{d}}{V_{s}})+\gamma_{0}b\sqrt{3}\hbar{q}cos(\theta)}{\gamma_{0}b\sqrt{3}(1-cos(\theta))}$
(11)

## Analysis

In the formulation, we utilise a perturbation theory of electron transition
where, electrons are assumed to drift relative to the lattice ions. The
wavelength of the phonon is short compared with the screening length for the
electrons. Electron-electron interactions and phonon loses are ignored but the
electron-phonon interaction $C_{q}$ is asumed to be weak and treated as
perturbation. In Eqn. ($1$), the quantum-mechnical matrix element describing
the electron-phonon coupling for a highly excited (intense acoustic wave)
phonon is $|C_{-q}|^{2}\approx|C_{q}|^{2}$. Considering the finite electron
concentration, the matrix element can be modified as

$|C_{{q}}|^{2}\rightarrow\frac{|C_{q}|^{2}}{|{\aleph^{(el)}({q})|^{2}}}$ (12)

where $\aleph^{(el)}({q})$ is the electron permitivity. However, for acoustic
phonons, $|{C_{{q}}}|=\sqrt{{\Lambda^{2}\hbar{q}}/{2\rho V_{s}}}$, where
$\Lambda$ is the deformation potential constant and $\rho$ is the density of
the material. From Eq.($10$), taking $\varepsilon_{0}={p}_{0}=\mu=0$, the
Eqn.($10$) finally reduces to

$\Gamma^{CNT}=\frac{|{\Lambda}|^{2}\hbar^{3}q^{2}exp(-\beta\chi\eta)}{2\pi{\hbar\omega_{q}}\gamma_{0}b\sqrt{3}(1-cos(\theta))}\\{\sum_{n=-\infty}^{\infty}{\frac{exp(-n(\theta)+\beta\chi\eta)}{I_{n}(\beta\chi(\eta+\hbar{q}))}-1}\\}$
(13)

where $I_{n}(x)$ is the modified Bessel function. When the scattering angle
$\theta=0$, $\Gamma=\infty$ whereas for $\theta>0$ and $V_{d}>V_{s}$,
$\Gamma^{CNT}$ changes sign from positive ($+$) to ($-$) and amplification is
obtained. To analyse Eqn. ($15$), the following parameters were used:
$|{\Lambda}|=9$eV, $b=1.42\times 10^{-9}$m, $q=10^{4}$ m-1,
$\omega_{q}=10^{12}$s-1, $V_{s}=4.7\times 10^{3}$ m s-1, $T=85\ K$, and
$\theta>0$. The choice of these parameters espercially that of the acoustic
wavenumber is based on our previous studies [18]. The dependence of
$\Gamma^{CNT}$ on $\frac{V_{d}}{V_{s}}$ at $n=1$ is presented below (see
Figure $1a$).

Figure 1: Dependance of $\Gamma^{CNT}$ on: (a) $\frac{V_{d}}{V_{s}}$ at
various $\omega_{q}$ (b) $\Omega\tau$ at $T=85\ K$, $\omega_{q}=3\ THz$ and
$\theta=15^{o}$

From the graph, there is absorption ($\Gamma_{abs}^{CNT}$) when $V_{d}<V_{s}$
but when $V_{d}>V_{s}$ it swiches over to amplifcation ($\Gamma_{amp}^{CNT}$).
This satisfy the Cerenkov condition for acoustic phonon emission. The maximum
$\Gamma_{amp}^{CNT}$ occurred at $V_{d}=1.02V_{S}$. The amplification obtained
far exceed the absorption. The ratio of the amplification to absorption

$\frac{|\Gamma_{amp}^{CNT}|}{|\Gamma_{abs}^{CNT}|}\approx 3.5$ (14)

To determine the threshold field at which $\Gamma_{abs}^{CNT}$ switches to
$\Gamma_{amp}^{CNT}$, we calculated and found the $V_{d}$ to be

$V_{d}=\frac{8\gamma_{0}}{\sqrt{3}\hbar
bm}\sum_{r=1}^{\infty}\frac{r^{2}\Omega\tau}{1+(r\Omega\tau)^{2}}\sum_{s=1}^{m}F_{rs}E_{rs}$
(15)

Here,

$F_{rs}=\frac{a}{2\pi}\int_{0}^{\frac{2\pi}{a}}{\frac{exp(-iarp_{z})}{1+exp(\varepsilon(p_{z})/k_{\beta}T)}dp_{z}}$
(16)

and

$E_{rs}=\frac{a}{2\pi}\int_{0}^{\frac{2\pi}{a}}\varepsilon(p_{z})exp(-irp_{z})dp_{z}$
(17)

where $\Omega=eaE$ ($E$ is the electric field, r the radius of the CNT, and
$a=3b/2\hbar$). The $V_{d}$ is solved from the Boltzmann kinetic equation in
the $\tau$-approximation [21, 22, 23]. The justification for the
$\tau$-approximation can be found in [17]. By substituting Eqn.($15$) into
Eqn.($11$). A graph of $\Gamma^{CNT}$ against $\Omega\tau$ is plotted in
Figure ($1b$). It can be observed that the threshold field for which
$\Gamma_{abs}^{CNT}$ changes over to $\Gamma_{amp}^{CNT}$ occurs at
$\Omega\tau=0.04$ which gives $E_{z}^{dc}=6.2\times 10^{3}\ \mathrm{V/m}$.
This value is far lower than that calculated by Dagher et. al. [4] to be
$E_{z}^{dc}=3\times 10^{5}\ \mathrm{V/m}$ using Bloch-type oscillations (BTO).
This is of the order of $2$ which is quite high.

## Conclusion

We studied the amplification of acoustic phonon in CNT theoretically in the
hypersound regime. The method used involves the Cerenkov emission of acoustic
phonons where, when $V_{d}<V_{s}$, gave an absorption ($\Gamma_{abs}^{CNT}$),
but when $V_{d}>V_{s}$, an amplification ($\Gamma_{amp}^{CNT}$) was obtained.
The ratio $\frac{|\Gamma_{amp}^{CNT}|}{|\Gamma_{abs}^{CNT}|}\approx 3.5$, at
$V_{d}=1.02V_{s}$, and $\omega=3.0\ \mathrm{THz}$. The threshold field at
which $\Gamma_{abs}^{CNT}$ switched over to $\Gamma_{amp}^{CNT}$ was
calculated as $E_{z}^{dc}=6.2\times 10^{3}\ \mathrm{V/m}$. This is far lower
than that calculated via the BTO to be $E_{z}^{dc}=3\times 10^{5}\
\mathrm{V/m}$. We therefore propose the use of Carbon Nanotube as a material
for the production of SASER.

## Bibliography

## Bibliography

  * [1] Mensah, S. Y., Allotey, F. K. A., Mensah, N. G., & Elloh, V. W. (2003). Amplification of acoustic phonons in a degenerate semiconductor superlattice. Physica E: Low-dimensional Systems and Nanostructures, 19(3), 257-262.
  * [2] Shinokita, K., Reimann, K., Woerner, M., Elsaesser, T., Hey, R., & Flytzanis, C. (2016). Strong Amplification of Coherent Acoustic Phonons by Intraminiband Currents in a Semiconductor Superlattice. Physical review letters, 116(7), 075504.
  * [3] Dagher, M., Chamanara, N., Sounas, D., Martel, R., & Caloz, C. (2012). Theoretical investigation of traveling-wave amplification in metallic carbon nanotubes biased by a DC field. IEEE Transactions on Nanotechnology, 11(3), 463-471.
  * [4] Mensah, S. Y., & Kangah, G. K. (1991). Amplification of acoustic waves due to an external temperature gradient in superlattices. Journal of Physics: Condensed Matter, 3(22), 4105.
  * [5] Glavin, B. A., Kochelap, V. A., & Linnik, T. L. (1999). Generation of high-frequency coherent acoustic phonons in a weakly coupled superlattice. Applied physics letters, 74(23), 3525-3527.
  * [6] Mensah, S. Y., Allotey, F. K., Adjepong, S. K., & Mensah, N. G. (1997). Photostimulated attenuation of hypersound in superlattices. Superlattices and microstructures, 22(4), 453-457.
  * [7] Dompreh, K. A., Mensah, S. Y., Mensah, N. G., Abukari, S. S., Allotey, F. K., & Nkrumah-Buandoh, G. K. (2011). Amplification of acoustic phonons in superlattice. arXiv preprint arXiv:1101.1854.
  * [8] Dompreh, K. A., Mensah, N. G., & Mensah, S. Y. (2015). Amplification of Hypersound in Graphene with degenerate energy dispersion. arXiv preprint arXiv:
  * [9] Nunes, O. A. C., & Fonseca, A. L. A. (2012). Amplification of hippersound in graphene under external direct current electric field. Journal of Applied Physics, 112(4), 043707.
  * [10] Komirenko, S. M., Kim, K. W., Demidenko, A. A., Kochelap, V. A., & Stroscio, M. A. (2001). Amplification of transverse acoustic phonons in quantum well heterostructures with piezoelectric interaction. Journal of Applied Physics, 90(8), 3934-3941.
  * [11] Sekyi-Arthur, D., Mensah, S. Y., Mensah, N. G., Dompreh, K. A., & Edziah, R. (2016). Absorption of acoustic phonons in Fluorinated Carbon Nanotubes with non-parabolic, double periodic band. arXiv preprint arXiv:1604.05699.
  * [12] Dompreh, K. A., Mensah, N. G., Mensah, S. Y., Sam, F., & Twum, A. K. (2015). Acoustoelectric Effect in degenerate Carbon Nanotube. arXiv preprint arXiv:1504.05484.
  * [13] Mavrinsky, A. V., Baitinger, E. M., “Semiconductors”, vol. 43, issue 4, pp 480-484
  * [14] Kubakaddi, S. S. (2016). Cerenkov emission of acoustic phonons electrically generated from three-dimensional Dirac semimetals. Journal of Applied Physics, 119(19), 195701.
  * [15] Zhao, C. X., Xu, W., & Peeters, F. M. (2013). Cerenkov emission of terahertz acoustic-phonons from graphene. Applied Physics Letters, 102(22), 222101.
  * [16] Yurchenko, S. O., Komarov, K. A., & Pustovoit, V. I. (2015). Multilayer-graphene-based amplifier of surface acoustic waves. AIP Advances, 5(5), 057144.
  * [17] Maksimenko, A. S., & Slepyan, G. Y. (2000). Negative differential conductivity in carbon nanotubes. Physical review letters, 84(2), 362.
  * [18] Dompreh, K. A., Mensah, N. G., Mensah, S. Y., Abukari, S. S., Sam, F., & Edziah, R. (2015). Hypersound Absorption of Acoustic Phonons in a Degenerate Carbon Nanotube. Graphene, 4(03), 62.
  * [19] Mensah, S. Y., Allotey, F. K. A., & Adjepong, S. K. (1994). Acoustoelectric effect in a semiconductor superlattice. Journal of Physics: Condensed Matter, 6(34), 6783.
  * [20] Mensah, S. Y., Allotey, F. K. A., & Mensah, N. G. (2000). Nonlinear acoustoelectric effect in a semiconductor superlattice. Journal of Physics: Condensed Matter, 12(24), 5225.
  * [21] Maksimenko, S. A., & Slepyan, G. Y. (2003). Electromagnetics of carbon nanotubes. Introducton to complex medium for optics and electromagnetics, 507-546.
  * [22] Abukari, S. S., Mensah, S. Y., Mensah, N. G., Adu, K. A., Rabiu, M., & Twum, A. (2012). High frequency conductivity in carbon nanotubes. AIP Advances, 2(4), 042178.
  * [23] Abukari, S. S., Mensah, S. Y., Adu, K. W., Dompreh, K. A., & Twum, A. K. (2011). Negative Differential Conductivity in Carbon Nanotubes in the Presence of an External Electric Field. arXiv preprint arXiv:1101.5498.


∎

11institutetext: D. M. Janseitov 22institutetext: Ibragimova St. 1
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22email<EMAIL_ADDRESS>
# Investigation of spectral properties of 11Be in breakup reactions

Dinara Valiolda1,2,3 Daniyar Janseitov1,2,3 Vladimir Melezhik3,4

1 Al-Farabi Kazakh National University, Almaty, Republic of Kazakhstan
2 Institute of Nuclear Physics, Almaty, Republic of Kazakhstan
3 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, Dubna, Russian Federation
4 Dubna State University, Dubna, Russian Federation

(Received: date / Accepted: date)

###### Abstract

We investigate the breakup of the 11Be halo nuclei on a light target (12C)
within quantum-quasiclassical approach in a wide range of beam energy (5–67
MeV/nucleon) including bound states and low-lying resonances in different
partial and spin states of 11Be. The obtained results are in good agreement
with existing experimental data at 67 MeV/nucleon. We also demonstrate that
the developed computational scheme can be used for investigation of nuclei
spectral properties in low-energy breakup experiments on different targets.

###### Keywords:

halo nucleus, breakup cross section, low-lying resonances, computational
scheme

##  1 Introduction

Breakup reactions, in which the valence neutron is removed from the projectile
in its interaction with a target nucleus, have played a valuable role in
probing the exotic nuclear structures, such as halo nuclei tan1996 ;
hansen1987 ; nakamura1994 . At breakup reactions of these nuclei, the angular
distributions of neutrons measured in coincidence with the core nuclei
anne1990  are strongly forward peaked and the parallel momentum distributions
of the core fragment have very narrow widths orr1995 ; bertul1992 ; banerj1995
, which provide a pure confirmation of their halo structure. 11Be is a famous
example of one-neutron halo nuclei, where the loosely bound valence neutron
has a large spatial extension with respect to the corresponding core.

The breakup of the halo nuclei has been investigated theoretically by several
authors using a number of different approaches: the time-dependent (TD) models
kido1994 ; melezhik1999 ; melezhik2003 ; kido1996 ; esbensen1995 , the
coupled-channel technique with a discretized continuum (CDCC) tostevin2001 ;
yahiro1982 , models based on the eikonal approximation glaub1959 ,
particularly the dynamical eikonal approximation (DEA) baye2005 ; gold2006 .
Each model has its own peculiarities. In particular, an advantage of the non-
perturbative time-dependent model melezhik1999 ; melezhik2003  is the
inclusion in the calculation of all higher-order effects in the relative
motion of the breakup fragments, which provides a fully dynamical description
of the projectile excitation caused by both the Coulomb and the nuclear
interactions between the projectile and the target.

In our previous works valiolda2022 ; valipepan2022 , we have carried out with
the extended quantum-quasiclassical model based on the non-perturbative time-
dependent approach melezhik1999 ; melezhik2003  a comprehensive analysis of
the influence of resonant states on the breakup of a 11Be halo nucleus on a
heavy target (208Pb) in a wide energy region (5 - 70 MeV/nucleon). It was
shown that including of the low-lying resonances ($5/2^{+}$, $3/2^{-}$ and
$3/2^{+}$) of 10Be+n system made a significant contribution to the breakup
cross section, which gives a better agreement with existing experimental data
fukuda2004 .

Here, we perform calculations of the breakup cross section of 11Be on a light
carbon target at beam energies 5 - 67 MeV/nucleon with the curvilinear
trajectory of the projectile and including low-lying resonances ($5/2^{+}$,
$3/2^{-}$, $3/2^{+}$) of 10Be+n system. The obtained results are in good
agreement with existing experimental data at 67 MeV/nucleon fukuda2004 . In
the low energy region there are only a few theoretical results hebborn2019 ;
hebborn2018 , with which we compare our calculations. The performed analysis
demonstrates the possibility of studying spectral properties of halo nuclei in
their breakup reactions.

The second section is devoted to a description of the non-perturbative time-
dependent model with linear trajectories and a quantum-quasiclassical approach
with “real” trajectories of 11Be in breakup reactions. In the third section
the convergence of the computational scheme is demonstrated and the obtained
results are presented. In the fourth section the investigation of spectral
structure of 11Be in breakup reactions is given. The last section is devoted
to concluding remarks.

##  2 Description of the model

###  2.1 Non-perturbative time-dependent approach with linear trajectories of
11Be

This approach is based on the integration of the time-dependent three-
dimensional Schrödinger equation for halo neutron for describing its dynamics
during collision of 11Be with a target melezhik1999 ; melezhik2003 .

The halo neutron is treated as a structureless particle weakly bound by the
potential U(r) to the 10Be core nucleus, where r is the the relative variable
between the neutron and the core. The time-dynamics of the halo neutron
relative the 10Be core in the breakup reaction 11Be+12C
$\rightarrow^{10}$Be+n+12C is depicted by the time-dependent Schrödinger
equation

$i\hslash\frac{\partial}{\partial\textit{t}}\Psi(\textbf{r},t)=H(\textbf{r},t)\Psi(\textbf{r},t)=[H_{0}(r)+V(\textbf{r},\textbf{R},t)]\Psi(\textbf{r},t)$
(1)

in the projectile rest frame, where $\Psi(\textbf{r},t)$ is the wave packet of
the neutron relative the 10Be core. In this expression

$H_{0}(\textbf{r})=-\frac{\hslash^{2}}{2\mu}\Delta_{r}+U(\textbf{r})$ (2)

is the Hamiltonian describing the halo nucleon-core system with reduced mass
$\mu=m_{n}m_{c}/M$, where $m_{n},m_{c}$ and M=$m_{n}+m_{c}$ are the neutron,
10Be-core, and 11Be masses, respectively. The potential $U$(r) consists of the
sum of the l-dependent Woods-Saxon potential $V_{l}(r)$ and the spin-orbit
interaction $V_{l}^{s}(r)(\textbf{l s})$.

The parameters of the spherical Woods-Saxon potentials
$V_{l}(r)=V_{l}/(1+\exp(r-R_{0})/a)$, describing the energy spectrum of 11Be
nucleus, had been determined as $V_{l}$= 62.52 MeV (l-even) and $V_{l}$= 39.74
MeV ($l$-odd) baye2005  in order to reproduce the $1/2^{+}$ ground state of
11Be at -0.503 MeV, the $1/2^{-}$ excited state at -0.183 MeV and two
resonance states $5/2^{+}$ and $3/2^{+}$ with the position of peaks at
$E_{5/2^{+}}$= 1.232 MeV and $E_{3/2^{+}}$= 3.367 MeV NNDC ; ershov2014 . For
these states, the radius is $R_{0}$= 2.585 fm and the diffuseness is a= 0.6
fm. To fix the position of the $3/2^{-}$ resonance ($l$=1) close to the
experimental NNDC  and theoretical ershov2014  value $E_{3/2^{+}}$= 2.789 MeV
we tuned the set of parameters as $V_{l}$= 6.8 MeV, $R_{0}$= 2.5 fm and a=
0.35 fm in our recent works valiolda2022 ; valipepan2022 . For $l\geq 3$, the
spherical potential $V_{l}(r)$ was set to zero. More details of
parameterization of potentials between the neutron and 10Be core and how the
resonant states were included in the analysis of the breakup reaction were
discussed at valiolda2022 ; valejpfm2022 .

The time-dependent potential $V(\textbf{r},\textbf{R},t)$ in Eq.(1) simulates
the interaction of the target with the projectile. It was assumed to be purely
Coulombic for breakup reactions with a heavy target (208Pb) at collision
energies around 70 MeV/nucleon melezhik1999 ; melezhik2003 ; valiolda2022 ,
which is defined as
$V_{C}(\textbf{r},\textbf{R},t)=\frac{Z_{c}Z_{t}e^{2}}{|m_{n}\textbf{r}/M+\textbf{R}(t)|}-\frac{Z_{c}Z_{t}e^{2}}{R(t)}$,
where $Z_{c}$ and $Z_{t}$ are charge numbers of the core and target,
respectively, and $\textbf{R}(t)$ is the relative coordinate between the
projectile and the target. As it has been shown in previous studies with time-
dependent non-perturbative approach valiolda2022 ; valipepan2022 , the
contribution of the nuclear part of the projectile-target interaction in the
breakup cross sections on a heavy target is significant for lower beam
energies (30 - 5 MeV/nucleon). It is known melezhik2003  that as the impact
parameter of 11Be and a target approaches zero, nuclear interaction
predominates in the breakup. In this work we evaluate this effect in the case
of light target (12C) using the approach of optical potential $\Delta
V_{N}(\textbf{r},\textbf{R},t)=V_{cT}(\overrightarrow{\textbf{r}}_{cT}(t))+V_{nT}(\overrightarrow{\textbf{r}}_{nT}(t))$
for the nuclear interaction between the target and projectile:

$V(\textbf{r},\textbf{R},t)=V_{C}(\textbf{r},\textbf{R},t)+\Delta
V_{N}(\textbf{r},\textbf{R},t)\,.$ (3)

Here, $\overrightarrow{\textbf{r}}_{cT}(t)$ and
$\overrightarrow{\textbf{r}}_{nT}(t)$ are the core-target
$\overrightarrow{\textbf{r}}_{cT}(t)=\textbf{R}(t)+m_{n}\textbf{r}/M$ and
neutron-target
$\overrightarrow{\textbf{r}}_{nT}(t)=\textbf{R}(t)-m_{c}\textbf{r}/M$ relative
variables and optical potentials $V_{cT}$ and $V_{nT}$ have the form:

$V_{xT}(r_{xT})=-V_{x}f(r_{xT},R_{R},a_{R})-iW_{x}f(r_{xT},R_{I},a_{I})\,$ (4)

with Woods-Saxon form-factors $f(r_{xT},R_{R},a_{R})=1/(1+\exp(r_{xT}-R)/a)$,
where $x$ stands for either core or neutron.

The analytical expressions of such potentials were obtained by selecting the
parameters of general form-factors so as to fit the calculated scattering
cross sections onto experimental data. A compilation of optical potentials for
different projectiles and targets can be found in Ref. hebborn2019 ; parey1976
; bechetti1969 . We use here the parameters of the optical potentials (4) from
the work bonin1985 , which are given in Table I. The potential of the core-
target interaction is proposed by Al-Khalili, Tostevin, and Brooke khalili1997
consistent with the elastic scattering of 10Be on 12C (denoted as ATB in the
following). For the n- 12C interaction, commonly used parametrization of
Becchetti and Greenlees bechetti1969  (BG) is considered. The expediency of
using this parameterization of the nuclear part of the interaction was
analyzed in the work bechetti1969 .

Table 1: Parameters of the core-target khalili1997  and neutron-target bechetti1969  optical potentials at 67 MeV/nucleon. c or n | $V_{x}(MeV)$ | $W_{x}(MeV)$ | $R_{R}(fm)$ | $R_{I}(fm)$ | $\textit{a}_{R}(fm)$ | $\textit{a}_{I}(fm)$
---|---|---|---|---|---|---
10Be | 123.0 | 65.0 | 3.33 | 3.47 | 0.80 | 0.80
n | 34.54 | 13.40 | 2.68 | 2.88 | 0.75 | 0.58

In early works on non-perturbative time-dependent approach melezhik1999 ;
melezhik2003  the straight-line trajectories
$\textbf{R}(t)=\textbf{b}+\textbf{v}_{0}t$ were used, where b is the impact
parameter orthogonal to the initial velocity of the projectile
$\textbf{v}_{0}$. However, advancement into the region of low collision
energies, where the deviation of the projectile trajectory from a straight
line increases, required go in beyond this approximation valiolda2022 .

###  2.2 Quantum-quasiclassical approach with “real” trajectories of 11Be

In our previous work valiolda2022 , we have extended the description of the
11Be breakup on 208Pb to low energies using the quantum-quasiclassical
approach developed in melezhik2004 ; melezhik2021 ; melezhik2001 ;
melezhik2023j  and successfully applied previously in various problems of
atomic physics. This made it possible to calculate with satisfactory accuracy
the cross section of this reaction at low collision energies of up to 5
MeV/nucleon. Here we apply this approach to describe the breakup of 11Be on a
light target 12C.

In this approach, simultaneously with the time-dependent Schrödinger equation
(1) for the halo-nucleon wave function $\Psi(\textbf{r},t)$ we integrate the
set of Hamilton equations

$\frac{d}{dt}\textbf{P}=-\frac{\partial}{\partial\textbf{R}}H_{BP}(\textbf{P},\textbf{R},t)\,,\,\frac{d}{dt}\textbf{R}=\frac{\partial}{\partial\textbf{P}}H_{BP}(\textbf{P},\textbf{R},t)$
(5)

describing relative projectile-target dynamics. Here, the classical
Hamiltonian $H_{BP}(\textbf{P},\textbf{R},t)$ is given by

$H_{BP}(\textbf{P},\textbf{R},t)=\frac{\textbf{P}^{2}}{2M}+\langle\Psi(\textbf{r},t)|V(\textbf{r},\textbf{R},t)|\Psi(\textbf{r},t)\rangle\,\,,$
(6)

where the term
$\langle\Psi(\textbf{r},t)|V(\textbf{r},\textbf{R},t)|\Psi(\textbf{r},t)\rangle$
represents the quantum-mechanical average of the projectile-target interaction
over the halo-nucleon density instantaneous distribution
$|\Psi(\textbf{r},t)|^{2}$ during the collision. Thus, the Hamiltonian (6)
defined in such a way has a parametric dependence on the halo-neutron position
$\textbf{R}(t)$ at every time moment. This quantum-quasiclassical model
permits to include in the computational scheme the deformation of the
projectile trajectories and energy transfer between the target and projectile
and vice versa during collisions. The required stability and accuracy of the
integration of Eq. (5) simultaneously with the time-dependent Schrödinger
equation (1) with the same step of integration over time was ensured by using
a computational finite-difference scheme developed in melezhik2021  based on
the Störmer–Verlet method.

The total breakup cross section is calculated as a function of the relative
energy $E$ between the emitted neutron and the core nucleus including neutron
interaction with the core in the final state of the breakup process by the
formula melezhik1999 ; melezhik2003 ; valiolda2022 ; melezhik2001 :

$\displaystyle\frac{d\sigma_{bu}(E)}{dE}=\frac{4\mu\textit{k}}{\hslash^{2}}\int_{b_{min}}^{b_{max}}\sum_{j=l+s}\sum_{lm}|$
(7)
$\displaystyle\int\phi_{ljm}(k,r)Y_{lm}(\hat{r})\Psi(\mathbf{r},T_{out})d\mathbf{r}|^{2}bdb\,.$

Here $\Psi(\mathbf{r},T_{out})$ is the neutron wave packet at the end of the
collision process 11Be+12C $\rightarrow^{10}$Be+n+12C (at $t=T_{out}$), which
is found by numerical integration of the hybrid system of equations consisting
of the time-dependent Schrödinger equation (1) and the classical Hamilton
equations (5). $\phi_{ljm}(k,r)$ is the radial part of the eigenfunction of
the Hamiltonian $H_{0}(\textbf{r})$ (2) in the continuum spectrum (
$E=k^{2}\hbar^{2}/(2\mu)>0$), normalized to a spherical Bessel function
$\textit{j}_{l}(kr)$ as $kr\rightarrow\infty$ if $U$(r)=0, $Y_{lm}(\hat{r})$
are the spherical harmonics. Summation over ($l,m$) in (7) includes all 16
partial waves up to $l_{max}$= 3 inclusive, as in valiolda2022 .

The time evolution starts at initial time $T_{in}$ and stops at final time
$T_{out}$ by iteration over $N_{T}$ time steps $\bigtriangleup\textit{t}$ as
explained in melezhik1999 ; melezhik2003 . The initial (final) time
$T_{in}(T_{out})$ has to be sufficiently big
$|\textit{T}_{in}|,T_{out}\rightarrow+\infty$, fixed from the demand for the
time-dependent potential $V(\textbf{r},\textbf{R},t)$ to be negligible at the
beginning (end) of the time evolution at $t$=$T_{in}$($T_{out}$). Following
the investigation performed in valejpfm2023 , the time interval is fixed as
$T_{in}$= -10 $\hslash$/MeV and $T_{out}$=10 $\hslash$/MeV, the time step
$\bigtriangleup\textit{t}$ equals to 0.02 $\hslash$/MeV. Technically, the most
difficult part when numerically integrating the system of equations (1), (5)
is finding $\Psi(\mathbf{r},t)$ \- solving the three-dimensional time-
dependent Schrödinger equation (1). To solve it, a two-dimensional discrete-
variable representation (2D DVR) is used to approximate the desired wave
function $\Psi(r,\Omega,t)$ in terms of angular variables
$\Omega=(\theta,\varphi)$ melezhik1999 ; melezhik2003 ; melezhik1997 .
Moreover, as shown in our previous work valiolda2022 , as the collision energy
of 11Be with a heavy target decreases, the convergence of this approach in the
number of basis functions $N$ of the 2D DVR (which is equal to the number of
grid points in angular variables $N=N_{\theta}\times N_{\varphi}$) slows down.
In the next section, we investigate the convergence of the 2D DVR in $N$ for
our problem of the breakup of 11Be on a light target in the intire energy
range under study.

For discretizing with respect to the radial variable r, a sixth-order (seven
point) finite-difference approximation on a quasiuniform grid has been used on
the interval $r\in[0,r_{m}]$ with $r_{m}$= 600 fm. The grid has been realized
by the mapping $r\rightarrow$_x_ of the initial interval onto $x\in[0,1]$ by
the formula $r=r_{m}(e^{8x}-1)/((e^{8}-1))$ valiolda2022 ; melezhik1997 . The
choice of the edges of integration over radial variables and convergence of
the method with respect to radial meshes was discussed at valejpfm2023 .

In the calculation of the breakup cross section the choice of edges of
integration over impact parameters $b_{min}$ and $b_{max}$ must be carefully
tested. For breakup reaction 11Be+12C $\rightarrow^{10}$Be+n+12C the evolution
is computed from impact parameters $b_{min}$= 0 fm up to $b_{max}$. The step
$\vartriangle{b}$ is chosen in order to ensure the convergence of the integral
in Eq. (7). It varies from $\bigtriangleup\textit{b}$= 0.25 fm at small b up
to $\bigtriangleup\textit{b}$= 2 fm at large b as in capel2004 . The inclusion
of a strong interaction between the target and the projectile using the
optical potential leads to a faster convergence of the integral (7) along the
upper limit $b_{max}$ (as it was shown in valejpfm2023  at $b_{max}$= 150 fm).
More details of numerical integration can be found in works melezhik1999 ;
melezhik2003 ; valiolda2022 .

##  3 Results and discussions

In our recent work valejpfm2023 , the convergence with respect to $N$ of the
time-dependent approach with linear trajectories of projectile for the breakup
of the 11Be nucleus on a light target (12C) was investigated at 67 MeV/nucleon
and also accuracy of the numerical technique was discussed there. Here, we
investigate the convergence in $N$ of the quantum-quasiclassical approach
including the curvature of the trajectory of the 11Be in collision with
12C-target (effect important at low collision energies of 11Be on 208Pb-target
valiolda2022 . The main task of our investigation is to extend a quantum-
quasiclassical approach for calculation of the breakup cross sections at low
beam energies.

Therefore, the breakup cross sections $d\sigma_{bu}(E)/dE$ were computed with
this approach on the 2D DVR basis functions extended from $N$= 49
($N_{\theta}=N_{\varphi}=$7) to $N$= 225 ($N_{\theta}=N_{\varphi}=$15) with
including two bound states (ground $1/2^{+}$ and first excited $1/2^{-}$
states) and three low-lying resonances ($5/2^{+}$, $3/2^{-}$, $3/2^{+}$) of
11Be at beam energy of E= 20 MeV/nucleon. Fig. 1 shows that it is sufficient
to use ($N_{\theta}=N_{\varphi}=$15) angular grid points $N$= 225 (2D DVR
basis functions) for computing the breakup cross section of 11Be on a 12C with
demanded accuracy of the order of a few percents. This number of 2D DVR basis
functions was used in the subsequent calculations.

Figure 1: The convergence of the calculated with quantum-quasiclassical
approach (1), (5) breakup cross section $d\sigma_{bu}(E)/dE$ (7) of 11Be at 20
MeV/nucleon as a function of the number
$N=N_{\theta}\times\textit{N}_{\varphi}$ of angular grid points (2D DVR basis
functions).

Figure 2: The comparison of breakup cross sections calculated with the time-
dependent approach with linear trajectories of the projectile and with
quantum-quasiclassical approach with realistic trajectories of the projectile
for beam energies of 5 and 67 MeV/nucleon.

In Fig. 2, we demonstrate the influence of the projectile curvature on the
breakup cross sections. Here, the breakup cross sections calculated with the
hybrid quantum-quasiclassical approach (realistic trajectories) and with time-
dependent approach with linear trajectories are presented for beam energies 5
MeV/nucleon (graph a) and 67 MeV/nucleon (graph b). In both calculations two
bound states ($1/2^{+}$, $1/2^{-}$) and three low-lying resonances ($5/2^{+}$,
$3/2^{-}$, $3/2^{+}$) of 10Be+n system were included. The Coulomb and nuclear
interactions between the projectile and the target (3) were also included. The
results given in Fig.2 show that in the energy range 5 - 67 MeV/nucleon the
contribution of the nonlinearity of the projectile curve does not exceed
several percent. Nevertheless, at low beam energy (5 MeV/nucleon), a decrease
in the ($d\sigma_{bu}(E)/dE$) cross section calculated with real trajectories
in the region 0.3 MeV $\leq E\leq$ 1.1 MeV and above 1.7 MeV (in the region of
low-lying resonances $3/2^{-}$ and $3/2^{+}$ with resonant energies
$E_{3/2^{-}}$= 2.789 MeV and $E_{3/2^{+}}$= 3.367 MeV, graph (a)) is
considerable. Including into consideration the deformation of projectile
trajectory also slightly decreases the maximum of the resonance $5/2^{+}$ at
$E_{5/2^{+}}$= 1.232 MeV. At our previous studies of the influence of the low-
lying resonances into the breakup of 11Be on 208Pb, it was found that for the
incident beam energies at 5 - 30 MeV/nucleon, the contribution of the
$5/2^{+}$ resonance state of 11Be to the breakup cross sections is clearly
visible, while at energies $\sim$70 MeV/nucleon, resonant states $3/2^{-}$ and
$3/2^{+}$ make the largest contribution valiolda2022 . It should be noted that
in the case of a heavy target (208Pb), the differences between the cross
sections calculated with the linear and realistic trajectories of the
projectile were about several percent in the energy range 30 - 20 MeV/nucleon,
for 10 MeV/nucleon the discrepancy was 10$\%$ and reached a value of more than
20$\%$ at 5 MeV/nucleon, which exceeded the effect of nuclear interaction
valiolda2022 . Here, for a breakup on the light target (12C), the influence of
projectile trajectory curvature on the breakup cross section with decreasing
collision energy is significantly less than in case of a heavy target (208Pb)
due to weaker Coulomb interaction between 11Be and 12C than between 11Be and
208Pb, as we assume.

Fig. 3 illustrates the comparison of calculated breakup cross section of 11Be
on 12C with the experimental data measured by Fukuda et al. at 67 MeV/nucleon
fukuda2004  and with the results of calculations of earlier work capel2004
performed with non-perturbative time-dependent approach melezhik1999 ;
melezhik2003 . Both the theoretical breakup cross sections at energy $E_{0}$
are convoluted with the instrumental energy resolution
$(E_{0}^{-1/2}/0.48exp[-(E-E_{0})^{2}/0.073E_{0}]$, with $E_{0}$ and $E$ in
MeV) capel2004  to obtain a value at energy $E$ comparable to experiment. The
major effect of this convolution is to resize the peaks, that are broadened
and slightly shifted towards lower energies. In both computations the
parameters of optical potentials and the edge parameters of integral (7) are
the same except the angular basis functions as we use an angular grid with the
equal numbers $N_{\theta}=N_{\varphi}=$15 of grid return $N=N_{\theta}\times
N_{\varphi}$ over $\theta$ and $\varphi$, whereas in capel2004  the authors
applied angular basis with $N_{\theta}=$12, $N_{\varphi}=$23\. Our results
(red full line) were obtained with including of two bound ($1/2^{+}$,
$1/2^{-}$) and three resonance ($5/2^{+}$, $3/2^{-}$ and $3/2^{+}$) states of
11Be into hybrid quantum-quasiclassical approach accounting the curvilinear
trajectories of the projectile. Otherwise, in calculations of capel2004 , two
bound and only one resonance $5/2^{+}$ were included in the time-dependent
approach with linear trajectories of the projectile. As it is shown at Fig.3,
the computation of the breakup cross section, adjusted for curvature of the
projectile trajectory due to the interaction with the target and including
low-lying resonances ($5/2^{+}$, $3/2^{-}$ and $3/2^{+}$), provides slightly
better description of the experimental data at the region of resonance
$5/2^{+}$.

Figure 3:  The comparison of theoretical capel2004  and experimental
fukuda2004  breakup cross sections with our results, obtained with quantum-
quasiclassical approach, including two bound ($1/2^{+}$, $1/2^{-}$) and three
resonance ($5/2^{+}$, $3/2^{-}$, $3/2^{+}$) states of 11Be at 67 MeV/nucleon.

Figure 4: Breakup cross sections of 11Be on 12C at 20 MeV/nucleon calculated
within quantum-quasiclassical approach in comparison with another theoretical
approaches from hebborn2018 .

To date, there are no experimental data for low beam energies for the breakup
reaction of 11Be on carbon and also on lead targets. However, attempts of
calculations at lower energies have already been carried out within the
framework of various approaches hebborn2019 ; hebborn2018 . In Fig.4, we
present breakup cross sections of 11Be on 12C calculated at 20 MeV/nucleon as
a function of the 10Be+n relative energy. It is plotted in comparison with
existing data, obtained with alternative theoretical approaches: the coupled-
channel technique with a discretized continuum (CDCC) (a purple dotted-
dashes), the eikonal model (a red dashes line) and dynamical eikonal
approximation (DEA) (a blue dotted line) hebborn2018 . Our results,
highlighted with a solid line, are obtained with including two bound
($1/2^{+}$, $1/2^{-}$) and three resonant ($5/2^{+}$, $3/2^{-}$, $3/2^{+}$)
states of 10Be+n system in the frame of the quantum-quasiclassical approach
with curvilinear trajectories of the projectile. Theoretical calculations from
hebborn2018  were performed with straight-line trajectories of the projectile
and with including only one resonance $5/2^{+}$ and two bound ($1/2^{+}$,
$1/2^{-}$) states of the core-neutron system. As it is shown in Fig.4, the
CDCC result hebborn2018  underestimated our cross sections by approximately
30$\%$ over the whole energy range. Otherwise DEA results exceeded essentially
the value of breakup cross sections in the peak, which correspond to the
resonance position of $5/2^{+}$, in comparison with eikonal approximation
hebborn2018  and our quantum-quasiclassical approach. Our results agreed well
with the eikonal approximation hebborn2018  at 10Be+n relative energies up to
1.3 MeV and with DEA at energies above 1.4 MeV.

In Fig. 5 we present the breakup cross sections of 11Be on the 12C target
calculated for two bound ($1/2^{+}$, $1/2^{-}$) states broad range 5 - 67
MeV/nucleon of beam energies. We took into account the influence of the
resonant ($5/2^{+}$, $3/2^{-}$, $3/2^{+}$) states and the effect of the
deformation and “vibration” of the projectile trajectory, as well as the
transfer of energy from the target to the projectile and vice versa
valiolda2022  during the breakup process.

Figure 5: The breakup cross section $d\sigma_{bu}(E)/dE$ calculated as a
function of beam energy and the energy between 10Be and neutron, including two
bound ($1/2^{+}$, $1/2^{-}$) states and three resonances ($5/2^{+}$,
$3/2^{-}$, $3/2^{+}$) of 11Be.

##  4 Investigation of spectral structure of 11Be in breakup reactions

Let us note two features in the dynamics of the breakup cross section of 11Be
with decreasing the colliding energy (see Fig.5): a dramatic increase of the
$5/2^{+}$ resonance at 1.25 MeV and the transformation of the plateau in the
0.3 - 1.0 MeV region into a broad peak at 0.3 MeV as the colliding energy
decreases to 5 MeV/nucleon. To clarify the physical nature of these two peaks,
we compared the calculated cross section 11Be+12C $\rightarrow^{10}$Be+n+12C
at 5 MeV/nucleon with the cross section 11Be+208Pb
$\rightarrow^{10}$Be+n+208Pb calculated in our previous work valiolda2022  at
the same energy (see Fig.6).

Figure 6: Breakup cross sections of 11Be impinging on light (12C) (multiplied
by 10) and heavy (208Pb) targets calculated within quantum-quasiclassical
approach at 5 MeV/nucleon.

In the latter case, the peak at 1.25 MeV is only slightly noticeable, although
the breakup cross section on the heavy target is significantly higher than the
cross section on the light target at the region below 2 MeV. However, the fact
that the positions of the 1.25 MeV and 0.3 MeV peaks do not depend on the type
of the target leads to the natural conclusion that they are related to the
spectral structure of 11Be nucleus. Since the quantum-quasiclassical
computational scheme allows one to manipulate the spectrum of 11Be, i.e. by
removing the excited state $1/2^{-}$ from the corresponding partial
interaction n-10Be and varying the binding energy of the ground state
$1/2^{+}$ valiolda2022 , this makes it possible to study the influence of
these bound states on the peak in the cross section about 0.3 MeV. The results
of this investigation are given in Fig.7. Here it can be seen that the
elimination of the excited state $1/2^{-}$ from the n-10Be interaction leads
to an increase in the peak at 0.3 MeV, since the breakup from the ground state
$1/2^{+}$ goes directly into the continuous spectrum without intermediate
population of the excited state $1/2^{-}$. A further change in the
interaction, leading to in an increase the binding energy of the ground state
of 11Be from $E_{1/2^{+}}$= -0.503 MeV, leads to the elimination of the peak
and reaching the plateau of the cross section at $E_{1/2^{+}}$= -1 MeV and
then to the elimination of the plateau at $E_{1/2^{+}}$ $\leq$ -2 MeV. The
influence of the position of the ground state energy level on the resonance
amplitude is also noticeable for $5/2^{+}$ resonance at 1.25 MeV.

Figure 7: Breakup cross sections of 11Be on 12C with different spectral
structure of 11Be nucleus calculated at 5 MeV/nucleon.

The study shows that the broad peak in the breakup cross section of 11Be on
12C near 0.3 MeV at 5 MeV/nucleon beam energy is caused by a weakly-bound
ground state of 11Be with the energy $E_{1/2^{+}}$= -0.503 MeV. The influence
of the excited state $1/2^{-}$\- ($E_{1/2^{-}}$= -0.183 MeV) to this peak is
reduced to its slight shift towards low energies and a decrease in its maximum
by an order of magnitude 30$\%$.

##  5 Conclusion

In this work the theoretical study is performed to the breakup of the 11Be
halo nuclei on a light target (12C) from intermediate (67 MeV/nucleon) to low
(5 - 30 MeV/nucleon) energies within quantum-quasiclassical approach, in which
the three dimensional time-dependent Schrödinger equation for halo nucleon was
integrated simultaneously with the classical Hamiltonian equations describing
relative projectile-target dynamics. In this hybrid quantum-quasiclassical
approach the time-dependent Schrödinger equation is integrated numerically
with a technique developed to study the Coulomb breakup of halo nuclei
melezhik1999 ; melezhik2003 .

In the frame of the quantum-quasiclassical approach we have calculated the
breakup cross sections of 11Be on a carbon target at energies 5 - 67
MeV/nucleon including Coulomb and nuclear interaction between projectile and
target. The performed research demonstrates that the model with the straight-
line projectile trajectories provides a satisfactory accuracy in calculating
the breakup cross sections of 11Be with decreasing the beam energy down to 5 -
10 MeV/nucleon. The uniqueness of our calculations lies in the inclusion of
low-lying resonances ($5/2^{+}$, $3/2^{-}$, $3/2^{+}$) in the breakup cross
section of the 11Be nucleus valiolda2022 ; valipepan2022 ; valejpfm2022 ;
valejpfm2023 . We also show that the analysis with our quantum-quasiclassical
approach allows to investigate spectral properties of 11Be nuclei from the
breakup reactions. Our results describe well the existing experimental data of
Fukuda et al. fukuda2004  at 67 MeV/nucleon and are in comparative agreement
with other existing calculations performed with alternative theoretical models
at 67 MeV/nucleon capel2004  and 20 MeV/nucleon hebborn2018 .

Summarizing, the results obtained by quantum-quasiclassical approach can
potentially be usefull in further investigations of breakup reactions at low
energies. In particular the region around 20 - 10 MeV/nucleon is of great
interest, since this is the energy range of HIE-ISOLD at CERN and the future
ReA12 at MSU, it has hardly been investigated theoretically so far.

## Acknowledgments

This research is funded by the Science Committee of the Ministry of Science
and higher education of the Republic of Kazakhstan (Grant No. AP13067742).
Additionally, authors note that the convergence of computational scheme and
its optimization was investigated with the algorithm developed with support of
the Russian Science Foundation under the Grant No. 20-11-20257.

## References

  * (1) I. Tanihata et al., Phys. Rev. Lett. 55, 2676 (1985)
https://doi.org/10.1103/PhysRevLett.55.2676

  * (2) G. Hansen, B. Jonson, Europhys. Lett. 4, 409 (1987)
https://doi.org/10.1209/0295-5075/4/4/005

  * (3) T. Nakamura et al., Phys. Lett. B 331, 296 (1994)
https://doi.org/10.1016/S0370-2693(96)01690-5

  * (4) R. Anne et al., Phys. Lett. B 304, 55 (1993)
https://doi.org/10.1016/0370-2693(93)91399-8

  * (5) N.A. Orr et al., Phys. Rev. C 51, 3116 (1995)
https://doi.org/10.1103/PhysRevC.51.3116

  * (6) C.A. Bertulani, K.W. McVoy, Phys. Rev. C 46, 2638 (1992)
https://doi.org/10.1103/PhysRevC.46.2638

  * (7) P. Banerjee, R. Shyam, Phys. Lett. B 349, 421 (1995)
https://doi.org/10.1016/0370-2693(95)00261-I

  * (8) T. Kido, K. Yabana, Y. Suzuki, Phys. Rev. C 50, R1276 (1994)
https://doi.org/10.1103/PhysRevC.50.R1276

  * (9) T. Kido, K. Yabana, Y. Suzuki, Phys. Rev. C 53, 2296 (1996)
https://doi.org/10.1103/PhysRevC.53.2296

  * (10) H. Esbensen, G. F. Bertsch, C. A. Bertulani, Nucl. Phys. A 581, 107 (1995)
https://doi.org/10.1016/0375-9474(94)00423-K

  * (11) V. S. Melezhik, D. Baye, Phys. Rev. C 59, 3232 (1999)
https://doi.org/10.1103/PhysRevC.59.3232

  * (12) P. Capel, D. Baye, V. S. Melezhik, Phys. Rev. C 68, 014612 (2003)
https://doi.org/10.1103/PhysRevC.68.014612

  * (13) J. A. Tostevin, F. M. Nunes, I. J. Thompson, Phys. Rev. C 63, 024617 (2001)
https://doi.org/10.1103/PhysRevC.63.024617

  * (14) M. Yahiro et al., Prog. Theor. Phys. 67, 1467 (1982)
https://doi.org/10.1143/PTP.67.1467

  * (15) R. J. Glauber, in High Energy Collision Theory, Lectures in Theoretical Physics, Vol. 1, edited by W. E. Brittin and L. G. Dunham (Interscience, New York, 1959), p. 315.
https://doi.org/10.1103/PhysRevC.63.024617

  * (16) D. Baye, P. Capel, G. Goldstein, Phys. Rev. Lett. 95, 082502 (2005)
https://doi.org/10.1103/PhysRevLett.95.082502

  * (17) G. Goldstein, D. Baye, P. Capel, Phys. Rev. C 73, 024602 (2006)
https://doi.org/PhysRevC.73.024602

  * (18) D. Valiolda, D. Janseitov, V.S. Melezhik, Eur. Phys. Jour. A 58, 34 (2022)
https://doi.org/10.1140/epja/s10050-022-00684-z

  * (19) D. Valiolda, D. Janseitov, V.S. Melezhik, Phys. of Part. and Nucl. Lett. 19, 477 (2022)
https://doi.org/10.1134/S154747712205044210.1134/S1547477122050442

  * (20) N. Fukuda et al., Phys. Rev. C 70, 054606 (2004)
https://doi.org/10.1103/PhysRevC.70.054606

  * (21) P. Capel, G. Goldstein, D. Baye, Phys. Rev. C 70, 064605 (2004)
https://doi.org/10.1103/PhysRevC.70.064605

  * (22) C. Hebborn, P. Capel, Phys. Rev. C 100, 054607 (2019)
https://doi.org/10.1103/PhysRevC.100.054607

  * (23) C. Hebborn, P. Capel, Phys. Rev. C 98, 044610 (2018)
https://doi.org/10.1103/PhysRevC.98.044610

  * (24) National Nuclear Data Center, https://www.nndc.bnl.gov/.
  * (25) S. N. Ershov, J. S. Vaagen, M. V. Zhukov, Phys. Atom. Nucl. 77(8), 989 (2014)
https://doi.org/10.1134/S1063778814070072

  * (26) D. Valiolda, D. Janseitov, V.S. Melezhik, Euras. Jour. of Phys. Func. Mater. 6, 165 (2022)
https://doi.org/10.32523/ejpfm.2022060301

  * (27) C.M. Perey, F.G. Perey, Atomic Data and Nuclear Data Tables 17, 1 (1976)
https://doi.org/10.1016/0092-640X(76)90007-3

  * (28) F.D. Bechetti, G.W. Greenless, Phys. Rev. 182, 1190 (1969)
https://doi.org/10.1103/PhysRev.182.1190

  * (29) B. Bonin et al., Nucl. Phys. A 445, 381 (1985)
https://doi.org/10.1016/0375-9474(85)90448-8

  * (30) J.S. Al-Khalili, J.A. Tostevin, J.M. Brooke, Phys. Rev. C 55, R1018 (1997)
https://doi.org/10.1103/PhysRevC.55.R1018

  * (31) V. S. Melezhik, J. S. Cohen, Chi-Yu Hu, Phys. Rev. A 69, 032709 (2004)
https://doi.org/10.1103/PhysRevA.69.032709

  * (32) V. S. Melezhik, Phys. Rev. A 103, 053109 (2021)
https://doi.org/10.1103/PhysRevA.103.053109

  * (33) V. S. Melezhik, D. Baye, Phys. Rev. C 64, 054612 (2001)
https://doi.org/110.1103/PhysRevC.64.054612

  * (34) V. S. Melezhik, J. Phys. A 56, 154003 (2023)
https://doi.org/10.1016/S0375-9601(97)00250-8

  * (35) V. S. Melezhik, Phys. Lett. A 230, 203 (1997)
https://doi.org/10.1016/S0375-9601(97)00250-8

  * (36) D. Valiolda, D. Janseitov, V.S. Melezhik, Euras. Jour. of Phys. Func. Mater. 7, 155 (2023)
https://doi.org/10.32523/ejpfm.2023070303


# A geometrical interpretation of foreground filters for HI intensity

Rahul Kothari1,2, Roy Maartens2,3,4

###### Abstract

We give a new geometrical interpretation of HI intensity mapping foreground
filters in harmonic space, for both single-dish and interferometer mode
surveys. We derive the foreground-filtered HI auto power spectrum and then
extend this to the cross-power spectrum of HI with CMB lensing. Foreground
filtering leads to a loss of isotropy in Fourier space, resulting in harmonic
space non-diagonal correlations, which we show are small compared to the
diagonal ones. On large scales, foreground filters lead to a major loss of
power in the HI$\,\times\,$CMB lensing correlations.

##  1 Introduction

We live in an exciting era of precision cosmology where cosmological models
can be robustly tested against the data. In this regard, the Cosmic Microwave
Background (CMB) and the Large Scale Structure (LSS) surveys are extremely
important probes. The LSS data, over a range of redshifts, contains
potentially more information as compared to CMB, that basically comprises of
photons coming from a fixed redshift $z_{*}\approx 1100$. In addition to
surveys that use galaxy number counts, some LSS surveys measure the galaxy
integrated spectral line emission [1, 2] as well. These surveys are
advantageous since they can cover large volumes quite rapidly as they do not
attempt to resolve individual galaxies, which becomes progressively harder as
$z$ increases. There are many such spectral lines like CO, HI etc. Out of
these, HI or 21cm line is especially intriguing as it comes from the neutral
hydrogen, the most abundant element in the Universe. In order to mine this
precious wealth of information, a number of surveys – SKAO [3], HIRAX [4, 5]
and PUMA [6, 7] are underway.

Although HI mapping is potentially a very promising cosmological probe, it
suffers from a serious problem of foreground contamination. Extensive work [8,
9, 10, 11, 12] is in progress to model and remove the HI foregrounds. In
addition to these foreground removal techniques, the method of foreground
filtering has also been adopted in the literature [13, 14, 15, 16, 17, 18].
These filters avoid certain regions in Fourier space where it is known that
the signal loss due to foregrounds would be non-negligible. We note that
foreground filtering has been described on the basis of a flat sky
approximation.

In this paper, we give for the first time a geometrical meaning of foreground
filtering in harmonic space and as an application, calculate and study the
foreground filtered auto correlation HI$\times$HI and cross-correlation of HI
and CMB lensing $\kappa$. The lensing of CMB anisotropies contains an imprint
of the LSS, projected from $z=0$ to $z=z_{*}$ [19]. The combination of CMB
lensing with HI can improve precision and break parameter degeneracies (see
[20] and references therein).

One might naïvely expect that unlike auto correlations HI$\times$HI, the
cross-correlation HI$\times\kappa$ will not be affected by the foregrounds. In
the previous studies [21, 22, 23, 24, 25, 26] on this cross correlation,
foreground effects were neglected. However, it is known that the cross-
correlation of HI with photometric galaxy surveys is strongly damped by the
foregrounds [27, 28]. CMB lensing is a more extreme case of radial precision
loss on small scales than photometric surveys. This suggests that the
correlation HI$\times\kappa$ would be suppressed even more. Thus it is
expected that foreground mitigation has a very important role to play in the
evaluation of the cross-power spectrum as well.

We start by giving a geometrical meaning to foreground avoidance. This is
followed by the evaluation of both auto and cross power spectra when
foreground filtering is applied. We discuss various attributes and
ramifications of the derived results. In order to keep numerical complications
under control, we only consider a Dirac window for the auto correlations. But
we also give an analytical expression of the auto power spectra for any given
window function . The cross correlation HI$\times\kappa$ is not affected by
window functions [29].

For this paper, we assume a fiducial $\Lambda$CDM cosmology Planck 2018
Cosmological Parameters [30]: $A_{\mathrm{s}}=2.101\times 10^{-9}$,
$n_{\mathrm{s}}=0.965$, $h=0.676$, $\Omega_{\mathrm{{b0}}}=0.049$,
$\Omega_{\mathrm{{c0}}}=0.261$. We have also checked that our results match
with Class [31, 32] when foregrounds are neglected.

Figure 1:  Foreground filters in Fourier space.

##  2 Implementing foreground filters

The intensity mapping surveys are operated in two modes,

  * •

Single Dish (SD): In this mode, auto-correlation signals from single dishes
are added. It is useful for probing large angular scales.

  * •

Interferometer (IF): Cross-correlated signals from the array elements are
combined, useful for probing small angular scales.

SKAO intensity mapping will be performed in SD while HIRAX and PUMA are
optimised for the IF mode.

The first of the foreground filters is applicable for both SD and IF modes.
This filter accounts for the loss of long wavelength radial modes due to the
smoothness of the foregrounds in frequency

$\big{|}k_{\|}\big{|}>k_{\rm fg}=0.01\,h\,{\rm Mpc}^{-1}\quad{\rm(SD+IF)}\,.$
(2.1)

This value of $k_{\mathrm{fg}}$ is often chosen for next-generation surveys.
Since $k_{\|}=k_{z}$, geometrically (2.1) means that we include only that
region of the Fourier space above $k_{z}=k_{\mathrm{fg}}$ and below the
$k_{z}=-k_{\mathrm{fg}}$ planes.

The second filter applies to IF mode only. A baseline with a given physical
length $L$ probes different angular scales $\ell$ at various frequencies. This
implies that intrinsically monochromatic emission from a foreground point
source can contaminate the spectrally non-smooth signal. If it is assumed that
the contamination only operates within one primary beam of each pointing, then
this leads to the exclusion of the primary beam ‘wedge’ region in $\mathbf{k}$
space and is estimated as [13, 14, 16, 17] (see Figure 1)

$\big{|}k_{\|}\big{|}>{\left[\frac{r(z)H(z)\sin\theta_{\rm
w}(z)}{(1+z)}\right]}\,k_{\perp}{\equiv\tan\alpha(z)\,k_{\perp}\,,}\quad{\rm(IF)}\,$
(2.2)

where $r(z)$ and $H(z)$ are respectively the comoving distance and Hubble
parameter at redshift $z$. Also $\theta_{\mathrm{w}}\approx
0.61\theta_{\mathrm{b}}$ with $\theta_{\mathrm{b}}$ the beam size, related to
dish diameter $D_{\mathrm{d}}$ and rest frame HI wavelength $\lambda_{21}$ as
$\theta_{\mathrm{b}}=1.22{\lambda_{21}(1+z)}/{D_{\mathrm{d}}}$. Further, for
both HIRAX and PUMA, $D_{\mathrm{d}}=6$m.

For the IF mode, we need to take the common region implied by conditions (2.1)
and (2.2). Note that the wedge condition is not strictly a foreground-
avoidance condition – in principle, the wedge can be removed with excellent
calibration of baselines [33], but we treat this as futuristic.

##  3 Angular power spectra

For a given field of interest, say $X(z,\mathbf{n})$, the spherical harmonic
expansion is performed as

$X(z,\mathbf{n})=\sum_{\ell\geq 0}\,\sum_{m=-\ell}^{\ell}a^{X}_{\ell
m}(z)\,Y_{\ell m}(\mathbf{n})\,,$ (3.1)

where $Y_{\ell m}(\mathbf{n})$ are spherical harmonics and $a^{X}_{\ell m}(z)$
are the spherical harmonic coefficients, which are expressible in terms of the
primordial curvature perturbations $\zeta_{\mathbf{k}}$ as

$a^{X}_{\ell m}(z)=4\pi
i^{\ell}\int_{\mathcal{D}_{X}}\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}Y^{*}_{\ell
m}\,(\hat{k})\zeta_{\mathbf{k}}\,\Delta^{X}_{\ell}(k,z)\,.$ (3.2)

Here $\mathcal{D}_{X}(z)$ is the region of integration in Fourier space and
$\Delta^{X}_{\ell}(z,k)$ are the standard angular transfer functions (see e.g.
[34, 35] for details). Thus we can study the effect of foreground filtering by
restricting the region of integration for (3.2) as in Figure 1 for SD and IF
modes. For later computation of angular power spectra, it is useful to write
these regions in an explicit form.

  1. 1.

First, since CMB lensing is not affected by HI foregrounds, we have
$\mathcal{D}_{\kappa}=\mathbb{R}^{3}$, i.e., the whole Fourier space.

  2. 2.

The region of integration $\mathcal{D}_{\text{HI}}$ can be found using Figure
1. For both survey modes, $0\leq\phi\leq 2\pi$ and $k>k_{\mathrm{fg}}$, while
$\theta$ is given as

$\displaystyle{0\leq\theta\leq\pi}$ $\displaystyle\text{SD}\,,$ (3.3)
$\displaystyle{0\leq\theta<\pi/2-\alpha~{}~{}\mbox{or}~{}~{}\pi/2+\alpha<\theta\leq\pi}$
$\displaystyle\text{IF}\,.$ (3.4)

Since $\alpha$ is $z$ dependent, so is the region of integration
$\mathcal{D}_{X}(z)$.

Figure 2: Effect on power in SD mode of $\Delta\ell$ at $z=4$. Upper (blue)
curves are the no-foreground case. Left: Auto power spectrum
$C_{\ell,\ell+\Delta\ell}^{\text{HI},\text{HI}}(z,z)$. Right: Cross power
$C_{\ell,\ell+\Delta\ell}^{\text{HI},\kappa}(z)$. (We use the Savgol filter
[36] of order 4 and window size 15 for smoothing purposes.) Note that we use
Class for the power spectra without foreground effects and our code when
including these effects. Our code is in agreement with Class, but because of
numerical complications, our code’s results are not as smooth as given by
Class, even in the absence of foregrounds.

With this, we can now calculate the two point correlation between harmonic
coefficients of various quantities $X,Y=\\{\text{HI},\kappa\\}$ of interest.

$\displaystyle\big{\langle}a^{X}_{\ell
m}(z)a^{Y*}_{\ell^{\prime}m^{\prime}}(z^{\prime})\big{\rangle}=(4\pi)^{2}i^{\ell-\ell^{\prime}}\int_{\mathcal{D}_{X}(z)}\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\int_{\mathcal{D}_{Y}(z^{\prime})}\frac{\mathrm{d}^{3}k^{\prime}}{(2\pi)^{3}}$
$\displaystyle\times Y^{*}_{\ell
m}(\hat{k})\,Y_{\ell^{\prime}m^{\prime}}(\hat{k}^{\prime})\,\Delta^{X}_{\ell}(k,z)\,\Delta^{Y}_{\ell}(k^{\prime},z^{\prime})\,\big{\langle}\zeta_{\mathbf{k}}\zeta_{\mathbf{k}^{\prime}}^{*}\big{\rangle}\,.$
(3.5)

We can simplify (3.5), using
$\big{\langle}\zeta_{\mathbf{k}}\zeta_{\mathbf{k}^{\prime}}^{*}\big{\rangle}=(2\pi)^{3}\delta^{(3)}(\mathbf{k}-\mathbf{k}^{\prime})P(k)$
where $P(k)$ is the primordial power spectrum:

$\big{\langle}a^{X}_{\ell
m}(z)\,a^{Y*}_{\ell^{\prime}m^{\prime}}(z^{\prime})\big{\rangle}=\frac{2}{\pi}i^{\ell-\ell^{\prime}}\int\mathrm{d}^{3}k\
Y^{*}_{\ell
m}(\hat{k})\,Y_{\ell^{\prime}m^{\prime}}(\hat{k})\Delta^{X}_{\ell}(k,z)\,\Delta^{Y}_{\ell}(k,z^{\prime})\,P(k)\,,$
(3.6)

where the integration is performed over the region common to both
$\mathcal{D}_{X}(z)$ and $\mathcal{D}_{Y}(z^{\prime})$. Notice that in the
absence of foreground filtering, the common region is the whole 3D space and
we can perform angular integration which gives $\delta_{\ell\ell^{\prime}}$
and $\delta_{mm^{\prime}}$ which is what we expect in the absence of
foregrounds. But in the presence of foregrounds this is no longer true.

Furthermore, since the case of auto correlation $\kappa\times\kappa$ is
trivial, in whatever follows, we only study the auto correlation
$\text{HI}\times\text{HI}$ and cross correlation $\text{HI}\times\kappa$.

###  3.1 HI auto power

In order to calculate the auto power, we need window functions, since
observations average over finite redshift bins. The harmonic coefficient for
any given window function $W(z,\tilde{z})$, centered at redshift $z$, follows
from (3.2):

$a^{\text{HI}}_{\ell m}(z)=4\pi
i^{\ell}\\!\int\\!\mathrm{d}\tilde{z}\,W(z,\tilde{z})\Delta^{\text{HI}}_{\ell}(k,\tilde{z})\\!\int_{\mathcal{D}_{\text{HI}}}\\!\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}Y^{*}_{\ell
m}(\hat{k})\zeta_{\mathbf{k}}.$ (3.7)

Then the final $\text{HI}\times\text{HI}$ power spectrum is found to be

$\tilde{C}_{\ell}^{\mathrm{HI,HI}}(z,z^{\prime})=\int\mathrm{d}\tilde{z}\
W(z,\tilde{z})\int\mathrm{d}\bar{z}\
W(z^{\prime},\bar{z})\,{C}_{\ell}^{\mathrm{HI,HI}}(\tilde{z},\bar{z})\,,$
(3.8)

where the unwindowed power spectra
$C_{\ell}^{\text{HI},\text{HI}}(z,z^{\prime})$ are defined in (4.1). Notice
that here we have only considered diagonal correlations.

From now on we avoid computational complications by showing our numerical
results using only a Dirac window. Integration over $\phi$ gives
$\delta_{mm^{\prime}}$ due to $\phi$ symmetry. Thus (3.6) takes the form

$\displaystyle\big{\langle}a^{\mathrm{\text{HI}}}_{\ell
m}(z)\,a^{\mathrm{\text{HI}}*}_{\ell^{\prime}m^{\prime}}(z^{\prime})\big{\rangle}=f^{m,m}_{\ell,\ell^{\prime}}\delta_{mm^{\prime}}\int_{Z(z,z^{\prime})}^{1}\mathrm{d}x\,P^{m}_{\ell}(x)\,P^{m}_{\ell^{\prime}}(x)$
$\displaystyle\times\int_{k_{\mathrm{fg}}/x}^{\infty}\mathrm{d}k\,k^{2}\,\Delta^{\mathrm{\text{HI}}}_{\ell}(k,z)\,\Delta^{\mathrm{\text{HI}}}_{\ell^{\prime}}(k,z^{\prime})\,P(k)\,,$
(3.9)

where $P^{m}_{\ell}(x)$ are associated Legendre polynomials, and

$\displaystyle Z(z,z^{\prime})$
$\displaystyle=\begin{cases}\pi/2&\text{SD,}\\\
\max{\big{\\{}\sin\alpha(z),\,\sin\alpha(z^{\prime})\big{\\}}}&\text{IF,}\end{cases}$
(3.10) $\displaystyle f^{m,m^{\prime}}_{\ell,\ell^{\prime}}$
$\displaystyle=\frac{i^{\ell-\ell^{\prime}}}{\pi}\Big{[}1+(-1)^{\ell+\ell^{\prime}+m+m^{\prime}}\Big{]}\left[{\frac{(2\ell+1)(2\ell^{\prime}+1)(\ell-m)!(\ell^{\prime}-m^{\prime})!}{(\ell+m)!(\ell^{\prime}+m^{\prime})!}}\right]^{1/2}.$
(3.11)

###  3.2 HI$\times\kappa$ cross power

For the cross correlation, as we discussed before, $\mathcal{D}_{\kappa}$ is
the whole of Fourier space. Thus (3.6) simplifies to

$\big{\langle}a^{\mathrm{\text{HI}}}_{\ell
m}(z)a^{\kappa*}_{\ell^{\prime}m^{\prime}}\big{\rangle}=f^{m,m}_{\ell,\ell^{\prime}}\delta_{mm^{\prime}}\int_{{\sin\alpha}}^{1}\mathrm{d}x\,P^{m}_{\ell}(x)\,P^{m}_{\ell^{\prime}}(x)\int_{k_{\mathrm{fg}}/x}^{\infty}\mathrm{d}k\,k^{2}\,\Delta^{\mathrm{\text{HI}}}_{\ell}(k,z)\,\Delta^{\kappa}_{\ell^{\prime}}(k)\,P(k).$

For SD mode, ${\alpha=0}$.

##  4 Numerical results and discussion

We highlight some of the common features shared by expressions (3.9) and
(3.2):

  * •

There is an explicit $m$ dependence and both expressions remain the same when
$m\to-m$. This is a consequence of $\phi$ symmetry in the Fourier space.

  * •

There are off-diagonal, i.e., $\Delta\ell=\ell^{\prime}-\ell\neq 0$,
correlations, due to the loss of $\theta$ symmetry in Fourier space (explicit
in Figure 1).

  * •

From (3.11), it is clear that $f^{m,m}_{\ell,\ell^{\prime}}\neq 0$ only when
$\ell+\ell^{\prime}$ is even. This implies that
$\Delta\ell=\ell^{\prime}-\ell$ is also even, ensuring that both expressions
are real.

Figure 3: Foreground filtering effects on the auto HI power in SD and IF modes
at $z=1,2,3,4$. Lower panels show the ratio of the power spectra with and
without foregrounds.

Figure 4: As in Figure 3 but for the cross power spectrum. (On account of
large oscillations at high $\ell$ we use a Savgol filter of larger window size
35 and order 4.)

Given an $m$ dependence and the presence of non-diagonal correlations in (3.9)
and (3.2), we can define the generalized angular power spectra for both auto
and cross correlations in the presence of foregrounds:

$C^{XY}_{\ell\ell^{\prime}}(z,z^{\prime})=\sum_{m=-L}^{L}\,\frac{\big{\langle}a^{X}_{\ell
m}(z)\,a^{Y*}_{\ell^{\prime}m}(z^{\prime})\big{\rangle}}{\sqrt{(2\ell+1)(2\ell^{\prime}+1)}}\,.$
(4.1)

Here $L=\min\\{\ell,\ell^{\prime}\\}$ and $\alpha$ is given by (2.2) for IF
mode and is 0 for SD mode. In the absence of foregrounds, i.e., when
$k_{\mathrm{fg}}\to 0$ and ${\alpha\to 0}$, this expression takes the expected
form,

$C_{\ell}^{XY}(z,z^{\prime})=\frac{2}{\pi}\int_{0}^{\infty}\mathrm{d}k\,k^{2}\,\Delta^{X}_{\ell}(k,z)\,\Delta^{Y}_{\ell}(k,z^{\prime})\,P(k)\,.$
(4.2)

In the left panel of Figure 2, we show the auto power spectrum
$C_{\ell,\ell+\Delta\ell}^{\text{HI},\text{HI}}(z,z)$, at $z=2$ for different
$\Delta\ell$. It is evident that non-diagonal correlations are small compared
to the diagonal ones. Furthermore, the magnitude of the power spectrum
decreases as we increase $\Delta\ell$. Thus for our further analysis, we
neglect the non-diagonal correlations. We also show the cross power spectrum
for $z=2$ in the right panel of Figure 2, where we see the same trend as was
observed in the auto power spectrum. Although the values corresponding to the
IF mode are smaller, similar behaviour is found for the IF mode in both auto
and cross power spectra.

Now we investigate both auto and cross power spectra in the presence of
foregrounds, in both SD and IF modes, and at redshifts ${z=1,2,3,4}$. Figure 3
shows the auto power case. We find that as the redshift increases, the power
loss due to IF mode becomes more and more severe. This can again be understood
from Figure 1, since increasing $z$ leads to increasing $\alpha$. This means
that the effects of foreground filtering on SD and IF modes will become more
and more prominent as $z$ increases.

The more interesting cross power results are shown in Figure 4. We notice the
following features.

  1. 1.

The HI$\,\times\,\kappa$ cross power is much more severely affected by
foregrounds as compared to the auto power HI$\,\times\,$HI.

  2. 2.

In particular, the HI$\,\times\,\kappa$ signal vanishes for $\ell<\ell_{\rm
min}$. Since $0\leq x\leq 1$ [see (3.2)], we have
$k>k_{\mathrm{fg}}/x>k_{\mathrm{fg}}$. Then a rough estimate of $\ell_{\rm
min}$ is

$\displaystyle\ell_{\rm min}(z)\sim
k_{\mathrm{fg}}r(z)\sim\big{(}23,36,44,50\big{)}~{}~{}\mbox{for}~{}~{}z=\big{(}1,2,3,4\big{)}.$
(4.3)

These match reasonably well with the values evident from Figure 4.

  3. 3.

The power loss in IF survey mode is larger than that in SD mode.

  4. 4.

Additionally, we notice a feature, clearly visible for $z=3$ and $z=4$ plots
but also present (to a lesser extent) for $z=2$. It can be seen that beyond a
certain $\ell=\ell_{\rm IF}$ value, the power spectra corresponding to the two
survey modes separate, with IF mode suppressed relative to SD mode. We can
estimate these $\ell_{\rm IF}$ values using a relation between Fourier and
angular harmonic scales. For an opening angle
$\Theta=\cos^{-1}(\mathbf{n}_{1}\cdot\mathbf{n}_{2})$, the angular multipole
is given by $\ell=2\pi/\Theta$. Basically, the Fourier wavelength refers to
the distance between successive maxima (or minima) [37]. On the contrary
$\Theta=\pi/\ell$ refers to the scales separating consecutive maxima and
minima. At equal redshifts, the comoving transverse wavelength is
$\lambda_{\perp}=2r\sin(\pi/\ell)$. So the comoving transverse wavenumber in
the flat sky regime ($\Theta\ll 1$) is

$k_{\perp}=\frac{\pi}{r\sin(\pi/\ell)}\approx\frac{\ell}{r}\,.$ (4.4)

Our $\ell$ values are obtained by using (4.4) in (2.2) and then comparing the
resulting equation with (2.1). After inserting parameter values, we obtain

${\ell_{\rm IF}}\approx\frac{0.01h}{0.027H(z)}\,.$ (4.5)

Using this equation, we find that

$\displaystyle{\ell_{\rm
IF}\approx(622,368,244,176)~{}~{}\text{for}~{}~{}z=(1,2,3,4)\,.}$ (4.6)

These numbers again match reasonably well with Figure 4.

##  5 Conclusion and outlook

In this paper, we developed a geometrical interpretation of foreground
filtering in harmonic space. As an application, we also studied the
foreground-filtered auto correlation $\text{HI}\times\text{HI}$ and cross
correlation $\text{HI}\times\kappa$, where $\kappa$ is the CMB convergence
field. We find that the foreground affects the cross power
$\text{HI}\times\kappa$ more severely as than the auto power
$\text{HI}\times\text{HI}$.

In the case of $\text{HI}\times\kappa$, we also find two interesting features:
(a) the power spectrum vanishes for $\ell<\ell_{\mathrm{min}}$ and
(b) beyond a certain $\ell=\ell_{\mathrm{IF}}$, the power spectra in single-
dish and interferometer modes begin to separate. We estimated the points
$\ell_{\mathrm{min}}$ and $\ell_{\mathrm{IF}}$ and these match reasonably well
with numerical plots.

In order to keep numerical complications under control, we only give numerical
results for the Dirac window in the auto power, but we an give analytical
expression for a general window function. The cross power
$\text{HI}\times\kappa$ is independent of window functions.

The techniques are general and may be useful to get analytical expressions in
the contexts where foreground filtering is applied, e.g., to calculate the HI
auto and cross bispectra.

## Acknowledgments

RK and RM acknowledge support from the South African Radio Astronomy
Observatory and National Research Foundation (Grant No. 75415). This work used
the HPC facilities at IIT Mandi and at South African Centre for High
Performance Computing, under the project Cosmology with Radio Telescopes,
ASTRO-0945. We thank Mario Ballardini, William Matthewson, Prabhakar Tiwari,
Sandeep and Richa for useful discussions. This paper used python packages
scipy, astropy and camb.

## References

  * Fonseca  _et al._ [2017] J. Fonseca, M. Silva, M. G. Santos, and A. Cooray, Mon. Not. Roy. Astron. Soc. 464, 1948 (2017), arXiv:1607.05288 [astro-ph.CO] .
  * Kovetz  _et al._ [2017] E. D. Kovetz _et al._ , (2017), arXiv:1709.09066 [astro-ph.CO] .
  * Bacon  _et al._ [2020] D. J. Bacon _et al._ (SKA), Publ. Astron. Soc. Austral. 37, e007 (2020), arXiv:1811.02743 [astro-ph.CO] .
  * Crichton  _et al._ [2022] D. Crichton _et al._ , J. Astron. Telesc. Instrum. Syst. 8, 011019 (2022), arXiv:2109.13755 [astro-ph.IM] .
  * Newburgh  _et al._ [2016] L. B. Newburgh _et al._ , Proc. SPIE Int. Soc. Opt. Eng. 9906, 99065X (2016), arXiv:1607.02059 [astro-ph.IM] .
  * Castorina  _et al._ [2020] E. Castorina _et al._ , (2020), arXiv:2002.05072 [astro-ph.IM] .
  * Slosar  _et al._ [2019] A. Slosar _et al._ (PUMA), Bull. Am. Astron. Soc. 51, 53 (2019), arXiv:1907.12559 [astro-ph.IM] .
  * Spinelli  _et al._ [2021] M. Spinelli, I. P. Carucci, S. Cunnington, S. E. Harper, M. O. Irfan, J. Fonseca, A. Pourtsidou, and L. Wolz, Mon. Not. Roy. Astron. Soc. 509, 2048 (2021), arXiv:2107.10814 [astro-ph.CO] .
  * Fornazier  _et al._ [2022] K. S. F. Fornazier _et al._ , Astron. Astrophys. 664, A18 (2022), arXiv:2107.01637 [astro-ph.CO] .
  * Liu and Shaw [2020] A. Liu and J. R. Shaw, Publ. Astron. Soc. Pac. 132, 062001 (2020), arXiv:1907.08211 [astro-ph.IM] .
  * Alonso  _et al._ [2015] D. Alonso, P. Bull, P. G. Ferreira, and M. G. Santos, Mon. Not. Roy. Astron. Soc. 447, 400 (2015), arXiv:1409.8667 [astro-ph.CO] .
  * Shaw  _et al._ [2015] J. R. Shaw, K. Sigurdson, M. Sitwell, A. Stebbins, and U.-L. Pen, Phys. Rev. D 91, 083514 (2015), arXiv:1401.2095 [astro-ph.CO] .
  * Ansari  _et al._ [2018] R. Ansari _et al._ (Cosmic Visions 21 cm), (2018), arXiv:1810.09572 [astro-ph.CO] .
  * Karagiannis  _et al._ [2020] D. Karagiannis, A. Slosar, and M. Liguori, JCAP 11, 052 (2020), arXiv:1911.03964 [astro-ph.CO] .
  * Bull  _et al._ [2015] P. Bull, P. G. Ferreira, P. Patel, and M. G. Santos, Astrophys. J. 803, 21 (2015), arXiv:1405.1452 [astro-ph.CO] .
  * Alonso  _et al._ [2017] D. Alonso, P. G. Ferreira, M. J. Jarvis, and K. Moodley, Phys. Rev. D 96, 043515 (2017), arXiv:1704.01941 [astro-ph.CO] .
  * Pober [2015] J. C. Pober, Mon. Not. Roy. Astron. Soc. 447, 1705 (2015), arXiv:1411.2050 [astro-ph.CO] .
  * Soares  _et al._ [2021] P. S. Soares, S. Cunnington, A. Pourtsidou, and C. Blake, Mon. Not. Roy. Astron. Soc. 502, 2549 (2021), arXiv:2008.12102 [astro-ph.CO] .
  * Lewis and Challinor [2006] A. Lewis and A. Challinor, Phys. Rept. 429, 1 (2006), arXiv:astro-ph/0601594 .
  * Chen  _et al._ [2021] S.-F. Chen, H. Lee, and C. Dvorkin, JCAP 05, 030 (2021), arXiv:2103.01229 [astro-ph.CO] .
  * Sarkar [2010] T. G. Sarkar, JCAP 02, 002 (2010), arXiv:0908.1840 [astro-ph.CO] .
  * Sarkar  _et al._ [2016] T. G. Sarkar, K. K. Datta, A. K. Pal, T. R. Choudhury, and S. Bharadwaj, J. Astrophys. Astron. 37, 26 (2016), arXiv:1610.08181 [astro-ph.CO] .
  * Dash and Sarkar [2021] C. B. V. Dash and T. G. Sarkar, JCAP 02, 016 (2021), arXiv:2010.05816 [astro-ph.CO] .
  * Tanaka  _et al._ [2019] S. Tanaka, S. Yoshiura, K. Kubota, K. Takahashi, A. J. Nishizawa, and N. Sugiyama, (2019), arXiv:1904.10363 [astro-ph.CO] .
  * Ballardini  _et al._ [2019] M. Ballardini, W. L. Matthewson, and R. Maartens, Mon. Not. Roy. Astron. Soc. 489, 1950 (2019), arXiv:1906.04730 [astro-ph.CO] .
  * Ballardini and Maartens [2022] M. Ballardini and R. Maartens, Mon. Not. Roy. Astron. Soc. 510, 4295 (2022), arXiv:2109.03763 [astro-ph.CO] .
  * Cunnington  _et al._ [2019] S. Cunnington, L. Wolz, A. Pourtsidou, and D. Bacon, Mon. Not. Roy. Astron. Soc. 488, 5452 (2019), arXiv:1904.01479 [astro-ph.CO] .
  * Guandalin  _et al._ [2022] C. Guandalin, I. P. Carucci, D. Alonso, and K. Moodley, Mon. Not. Roy. Astron. Soc. 516, 3029 (2022), arXiv:2112.05034 [astro-ph.CO] .
  * Matthewson and Durrer [2021] W. L. Matthewson and R. Durrer, JCAP 02, 027 (2021), arXiv:2006.13525 [astro-ph.CO] .
  * Aghanim  _et al._ [2020] N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020), arXiv:1807.06209 [astro-ph.CO] .
  * Lesgourgues [2011] J. Lesgourgues, “The cosmic linear anisotropy solving system (class) i: Overview,” (2011), arXiv:1104.2932 [astro-ph.IM] .
  * Blas  _et al._ [2011] D. Blas, J. Lesgourgues, and T. Tram, Journal of Cosmology and Astroparticle Physics 2011, 034 (2011).
  * Ghosh  _et al._ [2018] A. Ghosh, F. Mertens, and L. V. E. Koopmans, Mon. Not. Roy. Astron. Soc. 474, 4552 (2018), arXiv:1709.06752 [astro-ph.CO] .
  * Di Dio  _et al._ [2016] E. Di Dio, R. Durrer, G. Marozzi, and F. Montanari, JCAP 01, 016 (2016), arXiv:1510.04202 [astro-ph.CO] .
  * Di Dio  _et al._ [2019] E. Di Dio, R. Durrer, R. Maartens, F. Montanari, and O. Umeh, JCAP 04, 053 (2019), arXiv:1812.09297 [astro-ph.CO] .
  * Savitzky and Golay [1964] A. Savitzky and M. J. E. Golay, Analytical Chemistry 36, 1627 (1964).
  * Di Dio  _et al._ [2014] E. Di Dio, F. Montanari, R. Durrer, and J. Lesgourgues, JCAP 01, 042 (2014), arXiv:1308.6186 [astro-ph.CO] .


# Work done in a decoherence process

Gianluca Francica<EMAIL_ADDRESS>
###### Abstract

We investigate the thermodynamics of a quench realized by turning on an
interaction generating pure decoherence. We characterize the work probability
distribution function, also with a fluctuation relation, a lower bound of the
work and by considering some physical examples.

##  I Introduction

The interaction of an open quantum system with is environment can lead to the
phenomenon of decoherence zurek91 ; bookBreuer06 . Specifically, the
interaction creates correlations between the states of the system and the
environment and destroys superposition of these states in the course of time.
In the last decades the non-equilibrium thermodynamics has received a great
attention kosloff13 ; Vinjanampathy16 ; bookthermo18  and recently also the
thermodynamics of decoherence has been investigated Popovic21 .

In the same spirit of Ref. Popovic21 , in this paper we examine the
thermodynamics of a quench realized by turning on an interaction generating
pure decoherence. In this case the work done does not change the internal
energy of the system and is completely converted to heat. In particular, we
express the work probability distribution function as a convex combination of
probability distribution functions of the environment. Thus, we derive a
fluctuation relation characterizing the process and a lower bound of the work
done. We proceed with the investigation by considering some physical examples.

##  II Thermodynamics of the process

We consider a system interacting with its environment, such that for times
$t>0$ the total Hamiltonian is of the form

$H=H_{S}+H_{B}+H_{I}$ (1)

where $H_{S}$ and $H_{B}$ describe the free evolution of the system and the
environment, respectively. The interaction is taken to be

$H_{I}=\sum_{n}|n\rangle\langle n|\otimes B_{n}$ (2)

where $|n\rangle$ are the eigenstates of the system Hamiltonian $H_{S}$ and
$B_{n}$ are arbitrary environment hermitian operators. We recall that, due to
this kind of interaction, an initial state
$|\Psi(0)\rangle=\sum_{n}c_{n}|n\rangle\otimes|\phi\rangle$, where
$|\phi\rangle$ is an arbitrary environment state, evolves into the entangled
system-environment state
$|\Psi(t)\rangle=U_{t,0}|\Psi(0)\rangle=\sum_{n}c_{n}|n\rangle\otimes|\phi_{n}(t)\rangle$,
where the time evolution operator is $U_{t,0}=e^{-iHt}$. By considering the
reduced system density matrix
$\rho_{S}(t)=\text{Tr}_{B}\left\\{|\Psi(t)\rangle\langle\Psi(t)|\right\\}$, we
note that, since $\langle\phi_{n}(t)|\phi_{n}(t)\rangle=1$, the populations
are $\langle n|\rho_{S}(t)|n\rangle=\left|c_{n}\right|^{2}$ and are constant
in time. Conversely, the coherences are $\langle
n|\rho_{S}(t)|m\rangle=c_{n}c_{m}^{*}\langle\phi_{m}(t)|\phi_{n}(t)\rangle$,
with $n\neq m$, and typically the overlap
$\langle\phi_{m}(t)|\phi_{n}(t)\rangle$ rapidly decreases in time, generating
the phenomenon of pure decoherence. In particular it is useful to define the
decoherence function $\Gamma_{nm}(t)$ (for $n\neq m$) such that
$\left|\langle\phi_{n}(t)|\phi_{m}(t)\rangle\right|=e^{\Gamma_{nm}(t)}$.

Here, we take in exam the quench $H_{0}\to H$, where initially the interaction
is turned off, i.e. for times $t\leq 0$ the Hamiltonian reads

$H_{0}=H_{S}+H_{B}$ (3)

and the total system is prepared in the initial equilibrium state

$\rho(0)=\frac{e^{-\beta H_{0}}}{Z_{0}}=\rho_{S}(0)\otimes\rho_{B}(0)$ (4)

where the partition function is $Z_{0}=\text{Tr}\left\\{e^{-\beta
H_{0}}\right\\}$ and the initial reduced states are $\rho_{X}(0)=e^{-\beta
H_{X}}/Z_{X}$ with $Z_{X}=\text{Tr}\left\\{e^{-\beta H_{X}}\right\\}$, for
$X=S,B$. The energy difference of the system after a time $t$ is
$\langle\Delta
E_{S}\rangle=\text{Tr}\left\\{(U_{t,0}^{\dagger}H_{S}U_{t,0}-H_{S})\rho(0)\right\\}$
and, by noting that $[H,H_{S}]=0$, we obtain $\langle\Delta E_{S}\rangle=0$
for any time $t>0$. We note that, also for strong coupling, the internal
energy change, as defined in Ref. rivas20 , is equal to $\langle\Delta
E_{S}\rangle=0$ (see Appendix). From a thermodynamic point of view, this means
that the work done on the system is equal to the heat exchange with its
environment, i.e. $\langle w\rangle=\langle Q\rangle$. In particular, the work
reads

$\langle
w\rangle=\text{Tr}\left\\{(U_{t,0}^{\dagger}HU_{t,0}-H_{0})\rho(0)\right\\}=\text{Tr}\left\\{H_{I}\rho(0)\right\\}\,.$
(5)

It is worth noting that the work is produced suddenly at the instant in which
the interaction is turned on. Thus, during the decoherence process, i.e. for
$t>0$, no heat is exchanged with the environment. Anyway, we can realize a
pure decoherence process generated by a time-dependent Hamiltonian, for
instance by considering time-dependent operators $B_{n}(t)$ (such that
$B_{n}(0)=0$), such that work, and so heat (always equal to the work), are
produced during the decoherence process. By focusing on the interaction of Eq.
(2), with constant $B_{n}$, the work reads

$\langle
w\rangle=\sum_{n}p^{S}_{n}\text{Tr}\left\\{B_{n}\rho_{B}(0)\right\\}\,,$ (6)

where we have defined the system populations $p^{S}_{n}=\langle
n|\rho_{S}(0)|n\rangle$. Thus, the work is the weighted mean of the averages
of the environment operators $B_{n}$, calculated with respect to the state
$\rho_{B}(0)$, with weights $p^{S}_{n}$.

By considering a two-measurement scheme, the work can be expressed as $\langle
w\rangle=\int wp(w)dw$, where $p(w)$ is the work probability distribution
function (see, e.g., Ref. campisi11  for its general definition). By taking in
account the eigenvalue equation

$(H_{B}+B_{n})|E^{n}_{k}\rangle=E^{n}_{k}|E^{n}_{k}\rangle\,,$ (7)

the work probability distribution function can be expressed as the convex
combination

$p(w)=\sum_{n}p^{S}_{n}p_{n}(w)$ (8)

where $p_{n}(w)$ is the probability distribution function

$p_{n}(w)=\sum_{mk}p^{B}_{m}p^{B}_{nk|m}\delta(w-E^{n}_{k}+E_{m})$ (9)

where the transition probability is $p^{B}_{nk|m}=\left|\langle
E^{n}_{k}|E_{m}\rangle\right|^{2}$, $|E_{m}\rangle$ is the eigenstate of
$H_{B}$ with eigenvalue $E_{m}$ and the environment populations are
$p^{B}_{m}=\langle E_{m}|\rho_{B}(0)|E_{m}\rangle$. The proof of Eq. (8)
immediately follows from the definition of $p(w)$ by noting that the
eigenstates of $H$ are the states $|n\rangle\otimes|E^{n}_{k}\rangle$.
Physically, $p_{n}(w)$ represents the probability distribution function of the
environment energy change in the quench $H_{B}\to H_{B}+B_{n}$, starting from
the initial state $\rho_{B}(0)$. Since $\rho_{B}(0)$ is an equilibrium state,
we obtain the Jarzynski equality campisi11 ; Jarzynski97

$\langle e^{-\beta w}\rangle_{n}=e^{-\beta\Delta F_{B}^{n}}$ (10)

where the average $\langle\cdots\rangle_{n}$ is calculated with respect to
$p_{n}(w)$, and the free energy difference is $\Delta
F_{B}^{n}=F^{n}_{B}-F_{B}$ with $F_{B}^{n}=-\ln(Z_{B}^{n})/\beta$,
$Z_{B}^{n}=\text{Tr}\left\\{e^{-\beta(H_{B}+B_{n})}\right\\}$ and
$F_{B}=-\ln(Z_{B})/\beta$. Thus, from Eq. (10) we get the Jarzynski equality

$\langle e^{-\beta w}\rangle=\sum_{n}p^{S}_{n}e^{-\beta\Delta
F_{B}^{n}}=e^{-\beta\Delta F}$ (11)

where $\Delta F$ is the total free energy difference. In particular, in this
form, the Eq. (11) relates the work done to the free energy differences of the
environment. From Eq. (11), by using the Jensen’s inequality we obtain the
second law

$\langle w\rangle\geq\Delta
F=-\frac{1}{\beta}\ln\left(\sum_{n}p^{S}_{n}e^{-\beta\Delta F_{B}^{n}}\right)$
(12)

Since $f(x)=-\ln x$ is a convex function, we have

$\Delta F\leq\sum_{n}p^{S}_{n}F_{B}^{n}-F_{B}\,.$ (13)

Conversely, by applying the Jensen’s inequality to Eq. (10), we obtain

$\langle w\rangle_{n}\geq F_{B}^{n}-F_{B}\,,$ (14)

from which, since $\langle w\rangle=\sum_{n}p^{S}_{n}\langle w\rangle_{n}$, we
get the tighter lower bound of the work

$\langle w\rangle\geq\sum_{n}p^{S}_{n}F_{B}^{n}-F_{B}\geq\Delta F\,.$ (15)

Of course, from Eq. (15) we also get a lower bound of the irreversible work
$\langle w_{irr}\rangle=\langle w\rangle-\Delta F$. We note that if all the
operators $B_{n}$ commutate with the environment Hamiltonian $H_{B}$, i.e.
$[H_{B},B_{n}]=0$ for each $n$, the probability distribution function
$p_{n}(w)$ simplifies into
$p_{n}(w)=\sum_{m}p^{B}_{m}\delta(w-E^{n}_{m}+E_{m})$. Furthermore, we observe
that our results inherent in the work probability distribution function and
the lower bound can be immediately generalized to the case of time-dependent
operators $B_{n}(t)$ by taking the transition probability
$p^{B}_{nk|m}=\left|\langle
E^{n}_{k}(t)|e^{-iH_{B}t}V_{n}(t)|E_{m}\rangle\right|^{2}$, where we have
defined the unitary operator
$V_{n}(t)=\mathcal{T}e^{-i\int_{0}^{t}B^{I}_{n}(s)ds}$, the time-ordering
operator $\mathcal{T}$ and $B^{I}_{n}(t)=e^{iH_{B}t}B_{n}(t)e^{-iH_{B}t}$.
Thus, in this case $\langle w\rangle_{n}$ reads

$\displaystyle\langle w\rangle_{n}$ $\displaystyle=$
$\displaystyle\text{Tr}\left\\{(V_{n}^{\dagger}(t)H_{B}V_{n}(t)-H_{B})\rho_{B}(0)\right\\}$
(16)
$\displaystyle+\text{Tr}\left\\{V_{n}^{\dagger}(t)B^{I}_{n}(t)V_{n}(t)\rho_{B}(0)\right\\}\,.$

We recall that by considering the initial state
$\rho(0)=|\psi(0)\rangle\langle\psi(0)|\otimes\rho_{B}(0)$, the decoherence
function takes the form
$\Gamma_{nm}(t)=\ln\left|\text{Tr}\left\\{V_{m}^{\dagger}(t)V_{n}(t)\rho_{B}(0)\right\\}\right|$,
then there seems to be no strong relation between the decoherence and the work
done. We note that by switching off the interaction at the final time the work
will be equal to the environment energy change, such that we get as particular
case the situation investigated in Ref. Popovic21 .

##  III Physical examples

We consider a system that is a qubit with Hamiltonian
$H_{S}=-\omega\sigma_{z}/2$, where the matrix $\sigma_{z}$ is the third Pauli
matrix and the parameter $\omega$ is the difference of energy levels of the
system. We focus on the interaction $H_{I}=\sigma_{z}\otimes B$. By defining
the eigenstates $|0\rangle$ and $|1\rangle$ of $\sigma_{z}$ with eigenvalues
$1$ and $-1$, the interaction reads $H_{I}=|0\rangle\langle 0|\otimes
B_{0}+|1\rangle\langle 1|\otimes B_{1}$ with $B_{0}=-B_{1}=B$. We have the
work

$\langle w\rangle=\delta p^{S}\text{Tr}\left\\{B\rho_{B}(0)\right\\}$ (17)

where we have defined the population unbalance $\delta
p^{S}=p^{S}_{0}-p^{S}_{1}=\tanh(\beta\omega/2)$, and the second moment

$\langle w^{2}\rangle=\text{Tr}\left\\{B^{2}\rho_{B}(0)\right\\}\,.$ (18)

We start by considering a bosonic environment with Hamiltonian
$H_{B}=\sum_{k}\omega_{k}a_{k}^{\dagger}a_{k}$ and
$B=\sum_{k}g_{k}(a_{k}+a_{k}^{\dagger})$. We label with $k$ the modes of the
environment having frequency $\omega_{k}$ and creation and destruction
operators $a^{\dagger}_{k}$ and $a_{k}$, such that
$[a_{k},a_{k^{\prime}}^{\dagger}]=\delta_{kk^{\prime}}$ and the real
parameters $g_{k}$ are the coupling constants. We have that
$\text{Tr}\left\\{B\rho_{B}(0)\right\\}=0$, then $\langle w\rangle=0$ such
that we need a non-linear interaction for obtaining $\langle w\rangle\neq 0$.
Anyway, the higher work moments can be different from zero, for instance the
second moment reads $\langle
w^{2}\rangle=\sum_{k}g^{2}_{k}\coth(\beta\omega_{k}/2)$.

We proceed by considering a fermionic environment with Hamiltonians
$H_{B}=-t\sum_{j}(c_{j}^{\dagger}c_{j+1}+c_{j+1}^{\dagger}c_{j})-\mu\sum_{j}c_{j}^{\dagger}c_{j}$
and $B=\sum_{j}g_{j}c_{j}^{\dagger}c_{j}$. In particular, the parameter $t$ is
the hopping amplitude, $\mu$ is the chemical potential and $c^{\dagger}_{j}$
and $c_{j}$ are creation and destruction operators, such that
$\\{c_{j},c_{j^{\prime}}^{\dagger}\\}=\delta_{jj^{\prime}}$. In this case the
work can be non-zero. We focus on the particular case of a homogeneous
coupling $g_{j}=g$, such that we have $B=gN$ where the number operator is
$N=\sum_{j}c_{j}^{\dagger}c_{j}$. Since $[N,H_{B}]=0$, the work moments are
related to the moments of the number operator by the equations $\langle
w^{2n-1}\rangle=g^{2n-1}\delta
p^{S}\text{Tr}\left\\{N^{2n-1}\rho_{B}(0)\right\\}$ and $\langle
w^{2n}\rangle=g^{2n}\text{Tr}\left\\{N^{2n}\rho_{B}(0)\right\\}$. Then, by
performing a Fourier transformation and expressing the environment Hamiltonian
in the diagonal form $H_{B}=\sum\epsilon_{k}c_{k}^{\dagger}c_{k}$ with
$\epsilon_{k}=-\mu-2t\cos k$, it is easy to calculate the work, which reads
$\langle w\rangle=g\delta p^{S}\sum_{k}1/(e^{\beta\epsilon_{k}}+1)$. In
particular, the work can be positive or negative depending on the sign of the
coupling $g$. The lower bound of the work is
$\sum_{n}p^{S}_{n}F^{n}_{B}-F_{B}=e^{\beta\omega/2}F_{B}(\mu-g)/Z_{S}+e^{-\beta\omega/2}F_{B}(\mu+g)/Z_{S}-F_{B}(\mu)$,
where $Z_{S}=2\cosh(\beta\omega/2)$ and
$F_{B}(\mu)=-\sum_{k}\ln(1+e^{-\beta\epsilon_{k}})/\beta$. If $j=1,\ldots,L$
and $g=g^{\prime}/L$, in the limit $L\to\infty$ the bound is saturated, i.e.
$\langle w\rangle=\sum_{n}p^{S}_{n}F^{n}_{B}-F_{B}=g^{\prime}\delta
p^{S}/\pi\int_{0}^{\pi}dk/(e^{\beta\epsilon_{k}}+1)$. We emphasize that we get
the same bound for any time-dependent operator $B(t)$ such that at the final
time is equal to $B=gN$.

##  IV Conclusions

In summary, we have investigated the quench thermodynamics of a pure
decoherence process. In this case the work done on the system is equal to the
heat exchanged with the environment. We have characterized the work
probability distribution function, also with a fluctuation relation, a lower
bound of the work done and by considering some physical examples.

We note that our results are different from the ones of Ref. Popovic21
because we do not switch off the interaction at the final time. In particular,
in the switching off process an amount of work is suddenly produced such that
the total work is always non-negative, since the protocol becomes cyclic.

##  Appendix A Strong coupling

Since $H_{S}$ is constant, from Ref. rivas20  we have the internal energy at
the time $t$

$\langle
E_{S}(t)\rangle=\text{Tr}\left\\{(H^{*}_{S}(\beta,t)+\beta\partial_{\beta}H^{*}_{S}(\beta,t))\rho_{S}(t)\right\\}$
(19)

where
$\rho_{S}(t)=\text{Tr}_{B}\left\\{U_{t,0}\rho(0)U^{\dagger}_{t,0}\right\\}$
and

$H^{*}_{S}(\beta,t)=-\frac{1}{\beta}\ln\left(\text{Tr}_{B}\left\\{U_{t,0}e^{-\beta
H_{S}}\otimes\rho_{B}(0)U^{\dagger}_{t,0}\right\\}\right)$ (20)

We consider the eigenvalue equation $H_{S}|n\rangle=\epsilon_{n}|n\rangle$ and
the time evolved state
$U_{t,0}|n\rangle\otimes|E_{k}\rangle=|n\rangle\otimes|\phi^{n}_{k}(t)\rangle$,
then

$\displaystyle U_{t,0}e^{-\beta H_{S}}\otimes\rho_{B}(0)U^{\dagger}_{t,0}$
$\displaystyle=$ $\displaystyle\sum_{n}e^{-\beta\epsilon_{n}}|n\rangle\langle
n|\otimes\sum_{k}\frac{e^{-\beta E_{k}}}{Z_{B}}$ (21)
$\displaystyle\times|\phi^{n}_{k}(t)\rangle\langle\phi^{n}_{k}(t)|$

Since
$\text{Tr}\left\\{|\phi^{n}_{k}(t)\rangle\langle\phi^{n}_{k}(t)|\right\\}=1$,
by taking the partial trace we get

$\text{Tr}_{B}\left\\{U_{t,0}e^{-\beta
H_{S}}\otimes\rho_{B}(0)U^{\dagger}_{t,0}\right\\}=e^{-\beta H_{S}}$ (22)

then $H^{*}_{S}(\beta,t)=H_{S}$ and so the internal energy is $\langle
E_{S}(t)\rangle=\text{Tr}\left\\{U_{t,0}^{\dagger}H_{S}U_{t,0}\rho(0)\right\\}$.

## References

  * (1) W. H. Zurek, Physics Today 44, 36 (1991)
  * (2) The theory of open quantum systems, H.-P. Breuer and F. Petruccione, Oxford (2006)
  * (3) R. Kosloff, Entropy 15, 2100 (2013)
  * (4) S. Vinjanampathy and J. Anders, Contemp. Phys. 57, 545 (2016)
  * (5) Thermodynamics in the Quantum Regime - Fundamental Aspects and New Directions, (eds.) F. Binder, L. A. Correa, C.Gogolin, J. Anders, G. Adesso, Springer (2018)
  * (6) M. Popovic, M. T. Mitchison and J. Goold, arXiv:2107.14216
  * (7) Á. Rivas, Phys. Rev. Lett. 124, 160601 (2020)
  * (8) M. Campisi, P. Hänggi and P. Talkner, Rev. Mod. Phys. 83, 771 (2011)
  * (9) C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)