# Azimuthal Anisotropy at high transverse momentum in $p$-$p$ and $p$-$A$
collisions

Ismail Soudi<EMAIL_ADDRESS>Department of Physics and Astronomy,
Wayne State University, Detroit, MI 48201.  Abhijit Majumder
<EMAIL_ADDRESS>Department of Physics and Astronomy, Wayne State
University, Detroit, MI 48201.

###### Abstract

We explore the possibility that the initial transverse momentum distribution
of unpolarized and polarized partons within unpolarized nucleons, both with
and without the anisotropy of unpolarized hadrons produced in the
fragmentation of outgoing partons, could lead to the observed azimuthal
anisotropy of high transverse momentum (high-$p_{T}$) hadrons produced in high
energy proton-proton ($p$-$p$) or proton-ion ($p$-$A$) collisions. Including
simple Gaussian forms for transverse momentum dependent parton distribution
functions (PDF) and fragmentation functions, and assuming an $A^{1/3}$
enhancement of a PDF in $p$-$Pb$ collisions, we show that the observed
anisotropy, with _no modification_ to the angle integrated spectra
($R_{pA}\\!=\\!1$) for 5 GeV $\\!\lesssim\\!p_{T}\\!\lesssim\\!50$ GeV, can be
straightforwardly understood as arising from a few processes dominated by
gluon-gluon to gluon-gluon scattering.

_Introduction:_ Azimuthal anisotropy of hadrons with transverse momentum
$p_{T}\lesssim 3$ GeV Adams  _et al._ (2005); Adler  _et al._ (2003); Aamodt
_et al._ (2010); Chatrchyan  _et al._ (2013); Aad  _et al._ (2012), in a semi-
central relativistic heavy-ion collision, consistent with calculations of an
expanding elliptically asymmetric fluid dynamical system, with a small
specific shear viscosity [$\eta\lessapprox s/(2\pi)$, where $s$ is the entropy
density] is routinely proffered as evidence for the formation of a quark gluon
plasma (QGP) Romatschke and Romatschke (2007); Shen  _et al._ (2011); Song
_et al._ (2011); Everett  _et al._ (2021a, b). The high densities (and
associated gradients) that cause the collective flow of the QGP, inescapably
require the modification (energy-loss) of high-$p_{T}$ partons (quarks,
gluons) traversing the plasma Bjorken (shed); Gyulassy and Wang (1994); Baier
_et al._ (1997); Zakharov (1997); Gyulassy  _et al._ (2001); Wang and Guo
(2001); Arnold  _et al._ (2002). This _jet quenching_ is discerned via the
modification of high-$p_{T}$ jets Aad  _et al._ (2013); Chatrchyan  _et al._
(2011); Abelev  _et al._ (2014a) and hadrons Aad  _et al._ (2015); Chatrchyan
_et al._ (2012); Aamodt  _et al._ (2011); Adams  _et al._ (2003); Adcox  _et
al._ (2002) in central and semi-central heavy-ion collisions. Successful
comparisons with simulations, which utilize the expanding medium profile,
required for the bulk soft spectrum Kumar  _et al._ (2023), are considered a
corroborative signature of the QGP.

It is straightforward to expect that for semi-central collisions, where the
bulk medium may possess a component of elliptic deformation in the directions
transverse to the colliding beams, the quenching of jets in these directions
will also be different Wang (2004); Majumder (2007); Bass  _et al._ (2009). A
large, centrality dependent, suppression of jets or high-$p_{T}$ hadrons
should go hand-in-hand with an azimuthal anisotropy in the modification of
these jets or hadrons Noronha-Hostler  _et al._ (2016); Kumar  _et al._
(2020); He  _et al._ (2022). This is indeed what is observed in semi-central
heavy-ion collisions.

Recently, several measurements of high multiplicity $p$-$p$ and $p$-$Pb$
events Aaboud  _et al._ (2017a); Abelev  _et al._ (2014b); Khachatryan  _et
al._ (2015) have demonstrated the existence of an azimuthal anisotropy in the
soft sector ($p_{T}\\!\lesssim\\!3$ GeV), revealed in terms of the Fourier
coefficients of the azimuthal distribution of the yield of particles:

$\displaystyle\frac{dN}{d\phi}$ $\displaystyle\propto$ $\displaystyle
1+\sum\limits_{n=1}^{\infty}2v_{n}\cos\left[n(\phi-\Psi_{n})\right].$ (1)

In the equation above, $v_{n}$ are the $n^{\rm th}$ harmonic flow coefficients
with respect to the flow plane angle $\Psi_{n}$. While the existence of a
$v_{n}$ in these _small systems_ , comparable to those in $Pb$-$Pb$
collisions, are indicative of the formation of droplets of QGP in these
systems, no (angle integrated) modification of high-$p_{T}$ jets Adam  _et
al._ (2015); Khachatryan  _et al._ (2016) or hadrons ATL (2022); Acharya  _et
al._ (2022) has been found in these systems. Yet, $p$-$Pb$ (and even $p$-$p$)
events show an azimuthal anisotropy for hadrons at high-$p_{T}$ (measured up
to $50$ GeV in $p$-$Pb$), Aad  _et al._ (2014); Aaboud  _et al._ (2017b); Aad
_et al._ (2020). This indicates that a mechanism other than energy loss could
be the source of this asymmetry.

In this Letter, azimuthal anisotropies in high-$p_{T}$ hadron production will
be shown to arise from the asymmetric hard scattering between incoming partons
which possess a small transverse momentum relative to the $z$-axis, defined by
the opposing proton beams Field (1978). These Transverse Momentum-dependent
Distributions (TMDs) Collins and Soper (1982a, b, c) for both the incoming and
outgoing hadrons will display further angular asymmetries in the case when the
quarks possess transverse spin Boer and Mulders (1998) or the gluons possess
linear polarizations Mulders and Rodrigues (2001). The largest contributions
emanate from $g+g\rightarrow g+g$ scattering with polarization independent
gluon ($g$) TMDs, and from combinations of parton distribution functions
(PDFs) with a linearly polarized gluon within un-polarized protons (the Boer-
Mulders’ effect Boer and Mulders (1998); Mulders and Rodrigues (2001)),
followed by polarization dependent hard scattering and transverse momentum
dependent fragmentation to an un-polarized hadron from a linearly polarized
gluon (the Collins’ effect Collins (1993)).

The presence of $\cos{2\phi}$ asymmetries in the production of hadrons
involving the linear polarization of gluons (or transverse quark spin) is well
known in a variety of processes Collins (1993); Gamberg and Goldstein (2007);
Conway  _et al._ (1989); Falciano  _et al._ (1986); Aubert  _et al._ (1983);
Barone  _et al._ (2010); Boer  _et al._ (2011); Pisano  _et al._ (2013). In
this Letter, we will demonstrate that these $\cos{2\phi}$ asymmetries,
extended to $p$-$p$ Pisano  _et al._ (2013); Boer  _et al._ (2011); Anselmino
_et al._ (2005, 2006), lead to the elliptic anisotropy ($v_{2}$) observed in
$p$-$p$ and $p$-$Pb$, without modification of the angle integrated spectrum.

This Letter is organized as follows: We review the formalism for 2-to-2
scattering of hard partons at leading order leading twist, where the initial
and final hadrons are unpolarized. The intermediate partonic states may, or
may not, include a transverse spin-polarized quark and/or a linearly polarized
gluon. We then outline the phenomenological setup to calculate the
differential angle dependent cross section and outline how a $v_{n}$ can be
extracted from these calculations. This is followed by comparison of our
calculations with data from $p$-$p$ and $p$-$A$ collisions. While our
calculations will directly apply to the case of high-$p_{T}$ hadron production
in $p$-$p$ collisions, we will make phenomenological extensions to the case of
$p$-$A$ collisions (a rigorous calculation of spin-polarization dependent
multiple scattering is yet to be carried out).

#### Theoretical framework:

There have been extensive studies of high-$p_{T}$ hadron production in $p$-$p$
collisions, where the intrinsic transverse momentum of partons within the
incoming proton Boer  _et al._ (2011); Pisano  _et al._ (2013); den Dunnen
_et al._ (2014); Boer and den Dunnen (2014), and the transverse momentum of
the final hadron with respect to the high-$p_{T}$ parton Bacchetta  _et al._
(2000); Anselmino  _et al._ (2005, 2006); D’Alesio and Murgia (2004) have been
included. Following Anselmino  _et al._ (2005, 2006), the differential cross
section for an un-polarized $p$-$p$ collision leading to the production of an
un-polarized hadron [a pion ($\bm{\pi}$) with rapidity $y$ and transverse
momentum $P_{T}$], but including internal partonic states that may be
polarized, can be expressed as,

$\displaystyle\frac{d\sigma}{dyd^{2}\\!P_{T}}\\!\\!=\\!\\!\int\\!\\!\frac{dx_{a}dx_{b}dzd^{2}k_{\\!\bot
a}d^{2}k_{\\!\bot b}d^{3}k_{\\!\bot C}}{2\pi^{2}z^{3}s}\delta(\bm{k}_{\\!\bot
C}\\!\cdot\\!\hat{p}_{c})J(\bm{k}_{\\!\bot C})\>\>\>\;\;$ (2)
$\displaystyle\Gamma^{\sigma\mu}\\!(x_{a},\\!k_{\\!\bot
a}\\!)\Gamma^{\alpha\nu}\\!(x_{b},\\!k_{\\!\bot
b}\\!)\hat{M}_{\mu\nu\rho}\hat{M}^{*}_{\sigma\alpha\beta}\Delta^{\rho\beta}\\!(z,k_{\\!\bot
C}\\!)\delta(\hat{s}+\hat{t}+\hat{u}),$

where $J(\bm{k}_{\bot C})=\frac{(E_{C}^{2}+\sqrt{\bm{p}_{C}^{2}-\bm{k}_{\bot
C}^{2}})^{2}}{4(\bm{p}_{C}^{2}-\bm{k}_{\bot C}^{2})}$. All partons are
massless (ensured by the final $\delta$-function). The fourth parton is un-
hadronized allowing for any final state. A sum over all flavors is implied,
and the integral is over unconstrained components of the momenta. While the
generalized PDF $\Gamma^{\mu\nu}$ and fragmentation function (FF)
$\Delta^{\mu\nu}$ may describe quarks or gluons, in this Letter, we will focus
on the gluon-gluon partonic channel which dominates the cross section for pion
production at the $p_{T}$’s and $\sqrt{s}$ considered. The generalized gluon
PDF reads Mulders and Rodrigues (2001)

$\displaystyle{\mkern-36.0mu}\Gamma^{\mu\nu}_{P}(x,k_{\bot})$
$\displaystyle=\frac{-g_{T}^{\mu\nu}f(x,k_{\bot})+\left(\frac{k_{\bot}^{\mu}k_{\bot}^{\nu}}{M_{p}^{2}}+g_{T}^{\mu\nu}\frac{k_{\bot}^{2}}{2M_{p}^{2}}\right)h^{\bot}(x,k_{\bot}^{2})}{2x}\;,$
(3)

with
$g_{T}^{\mu\nu}\\!=\\!g^{\mu\nu}\\!\\!-n^{\mu}_{+}n^{\nu}_{-}\\!\\!-n^{\mu}_{-}n^{\nu}_{+}$.
The $f(x,k_{\perp}^{2})$ represents spin-polarization independent TMD-PDFs for
each incoming parton ($a,b$), with longitudinal momentum fraction $x_{(a,b)}$
and transverse momentum $k_{\bot(a,b)}$, relative to the $z$-axis defined by
the incoming beams. The distribution of linearly polarized gluons in nucleons
is given by the Boer-Mulders’ function $h_{1}^{\bot}(x,k_{\bot}^{2})$ Boer and
Mulders (1998). Using gluon polarization vectors, at leading order in
transverse momentum, given by
$\epsilon^{\lambda}_{\mu}=\nicefrac{{e^{-i\lambda\phi}\left(0,-\lambda,-i,0\right)}}{{\sqrt{2}}}+\mathcal{O}\left(\nicefrac{{k_{\bot}}}{{p_{z}}}\right)$,
one can project the gluon into correlated helicities,

$\displaystyle\mbox{}\\!\\!\\!\\!\Gamma^{\lambda_{1},\lambda_{2}}_{P}(x,k_{\bot}\\!)\\!=\\!\frac{-\delta^{\lambda_{1},\lambda_{2}}f(x,k_{\bot}^{2}\\!)+\delta^{\lambda_{1},-\lambda_{2}}\frac{k_{\bot}^{2}}{2M_{p}^{2}}h^{\bot}(x,k_{\bot}^{2}\\!)}{2x}.$
(4)

Similarly, for the fragmentation, the correlator is,

$\displaystyle\\!\\!\\!\Delta^{\lambda_{1},\lambda_{2}}(z,k_{\bot}\\!)\\!=\\!\frac{\\!-\delta^{\lambda_{1},\lambda_{2}}D(z,k_{\bot}^{2}\\!)+\delta^{\lambda_{1},-\lambda_{2}}\frac{k_{\bot}^{2}}{2M_{\pi}^{2}}H^{\bot}(z,k_{\bot}^{2}\\!)}{2/z}.$
(5)

Here $D(z_{c},k_{\bot C})$ represents the spin-polarization independent TMD-FF
for the outgoing parton (c) fragmenting to the detected $\bm{\pi}$, carrying
momentum $z\bm{p}_{c}+\bm{k}_{\bot C}$ (the first $\delta$-function in Eq. (2)
ensures that $\bm{k}_{\bot C}$ is perpendicular to the 3-momentum of the
outgoing parton $\hat{p}_{c}$). The distribution of fragmenting $\bm{\pi}$
from a linearly polarized gluon is given by the Collins’ function
$H^{\bot}(z,k_{\bot})$ Collins (1993).

Using the helicity projected correlators, the combination of matrix element
times complex conjugate with initial and final correlators can be written as

$\Sigma\equiv\sum\mbox{}_{\mbox{}_{\\{\lambda_{i}\\}}}\Gamma^{\lambda_{a},\lambda_{a}^{\prime}}\Gamma^{\lambda_{b},\lambda_{b}^{\prime}}\hat{M}_{\lambda_{a},\lambda_{b}}^{\lambda_{c},\lambda_{d}}\left(\hat{M}_{\lambda_{a}^{\prime},\lambda_{b}^{\prime}}^{\lambda_{c}^{\prime},\lambda_{d}^{\prime}}\right)^{*}\\!\\!\Delta^{\lambda_{c},\lambda_{c}^{\prime}}\;.$
(6)

The $\hat{M}_{\lambda_{a},\lambda_{b}}^{\lambda_{c},\lambda_{d}}$ represents
the matrix element for helicity dependent hard scattering, where the
helicities could be different in the complex conjugate. To obtain the familiar
formula for spin-polarization independent scattering, one sets each
$\lambda^{\prime}=\lambda$, and sums over the helicities, to obtain,

$\displaystyle\\!\\!\\!\\!\\!\frac{d\sigma^{0}}{dyd^{2}P_{T}}\\!\\!=\\!\\!\int\\!\\!\frac{dx_{a}dx_{b}dzd^{2}k_{\bot
a}d^{2}k_{\bot b}d^{3}k_{\bot
C}}{16\pi^{2}x_{a}x_{b}z^{2}s}\delta(\bm{k}_{\bot
C}\\!\cdot\\!\hat{p}_{c})J(\bm{k}_{\bot C})$ (7)
$\displaystyle\\!\\!\\!\\!\\!\hat{f}_{\\!\mbox{}_{a/A}}\\!(\\!x_{a},\\!k_{\bot
a}\\!)\hat{f}_{\\!\mbox{}_{b/B}}\\!(\\!x_{b},\\!k_{\bot
b}\\!)|M_{a,b}^{c,d}|^{2}\hat{D}^{\mbox{}^{C/c}}\\!(\\!z,k_{\bot
C}\\!)\delta(\hat{s}+\hat{t}+\hat{u}).$

While the equation above was derived for gluon-gluon scattering, it also holds
for all other partonic scatterings, with a simple replacement of different
distributions.

Eqns. (2,7) may be combined:
$\nicefrac{{d\sigma}}{{[dyd^{2}P_{T}]}}\\!\\!=\\!\nicefrac{{d\sigma^{0}}}{{[dyd^{2}P_{T}]}}\,+\,\nicefrac{{d\Delta\sigma}}{{[dyd^{2}P_{T}]}}$.
The $\Delta\sigma$ explicitly refers to contributions where the helicity of
any parton in the amplitude is opposite to its helicity in the complex
conjugate. Both $d\sigma^{0}$ and $d\Delta\sigma$ will contribute to azimuthal
anisotropies.

Figure 1: Color Online: Left: Illustration of the initial state of a $p$-$p$
collision. Blue proton with a solid boundary is coming out of page, red proton
with dotted boundary goes into page. Hard parton is shown as a small dot
(blue/red) with a small transverse momentum indicated with small arrow,
balanced by the transverse momentum of the artificially separated remaining
matter, indicated by the blob. Right: Net transverse momentum of the final
state hard di-jet balanced with the net momentum of the produced remainder
(gray arrows).

#### Azimuthal Anisotropy:

Consider a hard parton which carries a non-negligible fraction $x_{a}$ of the
momentum of proton $A$ and a transverse momentum $k_{\bot,a}$. This is
indicated as the solid blue dot in Fig. 1. The remainder of the nucleon
(bounded by outer blue circle) is indicated by the blue blob and contains both
hard partons and soft bulk. Due to momentum conservation, the net transverse
momentum of the remainder is $-k_{\bot,a}$ (opposing solid blue arrows in the
figure). For a hard 2-to-2 scattering, a similar configuration must arise in
the opposing proton indicated by the red dashed boundary in the figure: The
transverse momentum of a hard parton $k_{\bot,b}$ is balanced by the remainder
(red blob in Fig. 1), and indicated by the dashed red arrows. The net momentum
of the participating hard partons is balanced by the remainders, indicated by
the opposing yellow arrows.

The hard partons scatter to produce 2 back-to-back jets in the right panel of
Fig. 1. The entire remainder of the protons does not interact in the
collision. The portion that does, is indicated as the orange blob in the right
panel of Fig. 1. The net momentum of the remainder does not exactly balance
that of the di-jets. However, there will be a strong correlation between the
opposing net momentum of the interacting remainder and the participating di-
jets (the di-jet system is deflected towards the right and the bulk particle
production is boosted towards the left). Soft hadrons are produced in the
hadronization of the remainder, as well as from the fragmentation of the jets.
The correlated opposing momenta of the two systems will form an approximate
_event plane_.

The above picture can be realized with a PYTHIA simulation Bierlich  _et al._
(2022): A single hard scattering event without initial or final state parton
shower is set up. The Lund model relies on beam remnants, which are the
partons that do not participate in the hard scattering but continue traveling
along the $\pm z-$direction. For each remnant we sample a transverse momentum
$\bm{k}_{\perp i}$, following a Gaussian distribution with an average momentum
square $\langle k_{\perp}^{2}\rangle$ and random transverse direction. The
total transverse momentum of all remnants $\bm{q}_{T}=\sum_{i}\bm{k}_{\perp
i}$ is compensated by applying a Lorentz boost to the center of mass frame of
the hard scattering with the velocity
$\bm{v}=\nicefrac{{\bm{q}_{T}}}{{\sqrt{(E_{a}+E_{b})^{2}+\bm{q}_{T}^{2}}}}$.
By applying a boost to the center of mass frame, we ensure that the scattering
cross-section is not altered.

After hadronization, using the $Q$-vector method Borghini  _et al._ (2001)
with the azimuthal angles $\phi_{j}$ of the outgoing hadrons, we determine the
$n^{\rm th}$-order event plane
$\Psi_{n}\\!\\!=\\!\arg[\sum_{j}\\!e^{in\phi_{j}}\\!]/n$. The sum over $j$
includes all hadrons in the event. The $\Psi_{n}$ are found to be tightly
correlated with the direction of the $\bm{q}_{T}$ vector, $\phi_{\bm{q}_{T}}$,
as demonstrated in the right panel of Fig. 2, which shows the distribution of
event fraction for the difference between the $\phi_{\bm{q}_{T}}$ and the
first or second order event plane $\Psi_{1},\Psi_{2}$, for $\langle
k_{\perp}^{2}\rangle^{1/2}\\!=\\!3$ GeV.

Figure 2: Left: Template fits [Eq. (8)] to the azimuthal distribution of
hadrons with $p_{T}\\!\geq\\!5$ GeV with respect to the event angle
$\phi_{\bm{q}_{T}}$ set by the vector $\vec{q}_{T}$ (see text). Three
different choices of the mean intrinsic transverse momentum $\langle
k_{\perp}^{2}\rangle^{1/2}$ are presented: $1$ GeV is the value used for
$p$-$p$ collisions in Fig. 3, progressively higher values of $3,5$ GeV
visually illustrate the escalation of an elliptic anisotropy. Right: The event
fraction distribution of the difference between the direction of the di-jet
angle $\phi_{\bm{q}_{T}}$ and the event plane angles $\Psi_{n}$ (see text).

A high-$p_{T}$ azimuthal anisotropy will result if there is an angular
($\phi$) correlation between the direction of the highest $p_{T}$ hadrons,
projected on the azimuthal plane (of Fig. 1), roughly equal to
$\vec{p}_{\bot,c}-\vec{p}_{\bot,d}$ and the azimuthal direction of the event
plane ($\phi_{\bm{q}_{T}}$), given by
$\bm{q}_{T}\simeq\vec{p}_{\bot,c}+\vec{p}_{\bot,d}$ (one can approximately
view this as the angle between $p_{c}$ and the right pointing gray arrow in
Fig. 1).

Using the events generated in our PYTHIA simulation, we compute the azimuthal
angle distribution of hadrons with momentum $p_{T}\\!\geq\\!5$ GeV, relative
to the direction $\phi_{\bm{q}_{T}}$. We obtain the distribution shown in the
left panel of Fig. 2, where the green circles are for the typical intrinsic
transverse momentum of $1$ GeV. The blue cross and red square points are
results for $\langle k_{\perp}^{2}\rangle^{1/2}=3,5$ GeV respectively. With
increasing $\langle k_{\perp}^{2}\rangle^{1/2}$, one visually notes the
appearance of an elliptic anisotropy. This notion can be quantified by fitting
these points with the form

$f(\phi)=1+2v_{1}\cos(\phi-\phi_{\bm{q}_{T}})+2v_{2}\cos[2(\phi-\phi_{\bm{q}_{T}})]\;.$
(8)

The azimuthal anisotropy ($v_{2}$) increases, as the mean transverse momentum
of the initial hard parton is increased. In the Supplement sup , we
demonstrate using the same PYTHIA simulation, that a larger transverse energy
($E_{T}$), at forward rapidity, is produced with increasing $\langle
k_{\perp}^{2}\rangle$, leading to the identification of these as more central
events. Thus, “more central” events, have a larger initial intrinsic $\langle
k_{\perp}^{2}\rangle$ and a resulting higher $v_{2}$.

In addition to the $\cos(2\phi)$ correlation from polarization dependent cross
sections ($d\Delta\sigma$) Boer  _et al._ (2011); Pisano  _et al._ (2013);
Conway  _et al._ (1989); Falciano  _et al._ (1986); Aubert  _et al._ (1983);
Barone  _et al._ (2010), spin independent cross sections (i.e., $\sigma^{0}$)
also produce angular asymmetries: Partonic cross sections will contain terms
such as $\nicefrac{{\hat{s}}}{{\hat{t}}}$ or
$\nicefrac{{\hat{s}}}{{\hat{u}}}$. Considering the $\nicefrac{{1}}{{\hat{t}}}$
for the special case that the incoming parton has
$p_{a}=\left[p_{a},p_{a}\theta_{a},0,p_{a}(1-\theta_{a}^{2}/2)\right]$ (with
$\theta_{a}\\!\rightarrow\\!0$), and the outgoing parton has
$p_{c}=\left[p_{c},p_{c}\cos(\phi_{c}),p_{c}\sin(\phi_{c}),0\right]$ (at mid-
rapidity), we obtain,

$\displaystyle\frac{\hat{s}}{\hat{t}}=\frac{-\hat{s}}{2p_{c}\cdot
p_{a}}=\frac{-\hat{s}}{2p_{c}p_{a}[1-\theta_{a}\cos(\phi_{c})]}$ (9)
$\displaystyle=\frac{-\hat{s}}{2p_{c}p_{a}}\left[1+\frac{\theta_{a}^{2}}{4}+\theta_{a}\cos(\phi_{c})+\frac{\theta_{a}^{2}\cos(2\phi_{c})}{4}+\ldots\right].$

Thus, the inclusion of initial state transverse momentum, leading to a non-
zero $\theta_{a}\simeq\nicefrac{{k_{\perp,a}}}{{p_{a}}}$, results in a variety
of anisotropies $v_{1}[\cos(\phi_{c})$], $v_{2}[\cos(2\phi_{c})]$ etc. The
inclusion of spin-polarization dependent states, with transverse polarized
quarks and/or linearly polarized gluons, leads to additional sources of
asymmetries, which will dominate over these power suppressed terms at high
$p_{T}$ sup . Other power suppressed terms leading to azimuthal asymmetries
have been pointed out before in deep-inelastic scattering and Drell-Yan
processes Cahn (1978); Berger (1980).

#### Phenomenology and comparison with data:

In this Letter, comparisons will be made with the ATLAS data on the elliptic
anisotropy $(v_{2})$ in $p$-$p$ and $p$-$Pb$ collisions, presented in Refs.
Aaboud  _et al._ (2017b); Aad  _et al._ (2020) respectively. The experimental
data is limited to $p_{T}\\!\lesssim\\!50$ GeV. Our calculations will only
include the partonic process of $g+g\rightarrow g+g$ (both unpolarized and
linearly polarized). Subdominant processes, that prevail at higher $p_{T}$,
can be found in Ref. Soudi and Majumder (2023).

Since the linearly polarized portion of the gluon correlator in Eqns. (4-5)
flips the helicity between the matrix element and its complex conjugate, the
only allowed scattering involves two linearly polarized gluons. While
scatterings of two linearly polarized gluons into unpolarized gluons (double
Boer-Mulders’ functions) are allowed, we will focus on the dominant
contribution for the second momentum anisotropy $(v_{2})$, which is the
coupling of one Boer-Mulders’ function with one Collins’ function ($BM\otimes
C$). In this case, Eq. (6) simplifies as,

$\displaystyle\Sigma^{\rm BM\otimes
C}{\mkern-4.0mu}={\mkern-14.0mu}\sum_{\lambda=\pm
1}{\mkern-12.0mu}H^{\bot(1)}(z,k_{\bot
C}){\mkern-4.0mu}\left[h^{\bot(1)}(x_{a},k_{\bot a}^{2})f(x_{b},k_{\bot
b}^{2})\delta_{\alpha,1}\right.$ $\displaystyle\left.+f(x_{a},k_{\bot
a}^{2})h^{\bot(1)}(x_{b},k_{\bot
b}^{2})\delta_{\alpha,-1}\right]{\mkern-4.0mu}\hat{M}^{\lambda,\lambda}_{\lambda,\lambda}\left(\hat{M}_{-\alpha\lambda,\alpha\lambda}^{\lambda,\lambda}\right)^{*},$
(10)

where we define $h^{\bot(1)}\equiv(k_{\bot}^{2}/2M_{p}^{2})h^{\bot}$ and
$H^{\bot(1)}\equiv(k_{\bot}^{2}/2M_{\pi}^{2})H^{\bot}$. The color and spin
averaged matrix elements (times complex conjugate) can be expressed as,

$\displaystyle\hat{M}_{\lambda,\lambda}^{\lambda,\lambda}\left(\hat{M}_{-\lambda,\lambda}^{-\lambda,\lambda}\right)^{*}\\!\\!\\!\\!=g_{s}^{4}\frac{N^{2}}{N^{2}-1}\frac{t^{2}+tu+u^{2}}{t^{2}}e^{4\lambda
i(\phi_{ab}-\phi_{bc})},$ (11)
$\displaystyle\hat{M}_{\lambda,\lambda}^{\lambda,\lambda}\left(\hat{M}_{\lambda,-\lambda}^{-\lambda,\lambda}\right)^{*}\\!\\!\\!\\!=g_{s}^{4}\frac{N^{2}}{N^{2}-1}\frac{t^{2}+tu+u^{2}}{u^{2}}e^{4\lambda
i(\phi_{ab}-\phi_{ac})},$ (12)

where, using partonic momenta in spherical coordinates
$\bm{p}_{i}=(p_{i},\theta_{i},\phi_{i})$, the phases are given by

$\displaystyle\\!\\!\\!\\!\tan\phi_{ij}=\tan\frac{\phi_{j}-\phi_{i}}{2}\\!\left(\sin\frac{\theta_{j}+\theta_{i}}{2}\right)\\!\\!\bigg{/}\\!\\!\left(\sin\frac{\theta_{j}-\theta_{i}}{2}\right){\mkern-2.0mu}.~{}~{}~{}~{}$
(13)

Constrains on the gluon correlator lead to a bound on the spin-dependent
distributions known as the Soffer bound: $|h^{\bot(1)}(x,k_{\bot}^{2})|\leq
f(x,k_{\bot}^{2})$ and $|H^{\bot(1)}(x,k_{\bot}^{2})|\leq D(x,k_{\bot}^{2})$
Soffer (1995); Mulders and Rodrigues (2001). While the Soffer bound only
constrains the spin-polarization dependent distributions to be limited by the
regular distributions, the actual magnitudes of the Boer-Mulders’ and Collins’
functions are poorly constrained. In this effort, we take the spin-polarized
distribution to be positive and proportional to the spin-polarization
independent distribution, i.e.,

$h^{\bot(1)}\\!(x,k_{\bot}^{2}\\!)\\!=\\!b\\!\cdot\\!\\!f(x,k_{\bot}^{2}\\!)\;;H^{\bot(1)}\\!(x,k_{\bot}^{2}\\!)\\!=\\!B\\!\cdot\\!\\!D(x,k_{\bot}^{2}\\!),$
(14)

where, $b$, $B$ are fractions assumed independent of $x,k_{\perp}$.

Recently, parametrizations of hard scale ($Q^{2}$) dependent joint spin-
independent TMDPDFs [$f(x,k_{\perp}^{2},Q^{2})$, where the $x$ and $k_{\perp}$
dependence is not factorized] have appeared Bermudez Martinez  _et al._ (2019,
2021). These can be directly used for $p$-$p$ collisions. However, there are
no parametrizations of un-factorized TMDPDFs including nuclear effects, that
can be used for $p$-$Pb$ collisions. To obtain such a distribution,
calculations of parton branching in the presence of multiple scattering will
have to be carried out. In the absence of such calculations, we employ a
simple two step model: The TMDPDFs and TMDFFs will be assumed to be factorized
with a Gaussian _Ansatz_ Gamberg and Goldstein (2007); D’Alesio and Murgia
(2004):

$\displaystyle f(x,k_{\bot},Q^{2})=\frac{4\pi}{\langle
k_{\bot}^{2}\rangle}e^{-\nicefrac{{k_{\bot}^{2}}}{{\langle
k_{\bot}^{2}\rangle}}}f(x,Q^{2})\;,$ (15)

and the mean transverse momentum $\langle k_{\bot}^{2}\rangle$ in $p$-$A$
collisions, for the hard parton in the proton, is enhanced by $A^{1/3}$ (from
multiple scattering) compared to $p$-$p$ as,

$\displaystyle\langle k_{\perp}^{2}\rangle_{pA}\simeq A^{1/3}\langle
k_{\perp}^{2}\rangle_{pp}\;.$ (16)

This extra transverse momentum originates in multiple scattering of the hard
parton prior to the hardest scattering which generates the di-jet, with
$\Delta\langle k_{\perp}^{2}\rangle\\!=\\!\langle
k_{\perp}^{2}\rangle_{pA}\\!-\\!\\!\langle
k_{\perp}^{2}\rangle_{pp}\simeq\hat{q}L$, where $\hat{q}$ is the transverse
momentum transport coefficient in confined matter Baier  _et al._ (1997), and
$L\propto A^{1/3}\\!-\\!1$ is the average length traversed prior to hard
scattering. This $A^{1/3}$ enhancement, demonstrated for spin-independent
multiple scattering Guo (1998); Fries  _et al._ (2000); Fries (2003); Majumder
and Muller (2008), has not been established for polarization dependent
scattering. While transverse momentum is enhanced by multiple scattering, we
expect vanishing initial state energy loss. This can be immediately understood
from the simple schematic that energy loss $\Delta E\\!\propto\\!\hat{q}L^{2}$
sup . For a Lorentz contracted nucleus, a finite $\Delta\langle
k_{\perp}^{2}\rangle$ is consistent with $\Delta E\rightarrow 0$, ensuring the
$R_{pA}$ remains unmodified.

We employ the nCTEQ parametrization Kovarik  _et al._ (2016) for integrated
PDFs [$f(x,Q^{2})$] and leading order KKP Kniehl  _et al._ (2000) for FFs. We
present results with a constant value of $\langle
k_{\bot}^{2}\rangle_{pp}=1$GeV2 and $\langle
k_{\bot}^{2}\rangle_{pA}=A^{1/3}$GeV2. While the $\langle
k_{\perp}^{2}\rangle$ is still not $Q^{2}$ dependent, we present results with
a range of values for the bound $b\cdot B$ ($0.1\leq b\cdot B\leq 0.4$) to
encompass a variety of uncertainties. For the pion TMD fragmentation, we
employ a constant value of $\langle k_{\bot}^{2}\rangle^{1/2}=0.8$GeV. We have
explicitly checked that all these formulations produce almost no change in the
angle integrated spectrum Soudi and Majumder (2023).

Figure 3: Color Online: Azimuthal anisotropy coefficient $v_{2}$ as a function
of pion transverse momentum $p_{T}$ for $p$-$p$ collisions at $\sqrt{s}=13$
TeV (bottom) and $p$-$Pb$ at $8.16$ TeV (top).

The elliptic coefficient $v_{2}$ is obtained from the average:

$\langle\cos[2(\phi_{\bm{q}_{T}}-\phi_{\pi})]\rangle\equiv\frac{\int
d\phi_{\bm{q}_{T}}\cos[2(\phi_{\bm{q}_{T}}-\phi_{\pi})]\frac{d\sigma}{dP_{T}}}{\int
d\phi_{\bm{q}_{T}}\frac{d\sigma}{dP_{T}}},$ (17)

where, $\phi_{\pi}$ is the direction of the $\bm{\pi}$, and
$\phi_{\bm{q}_{T}}$ is the direction of the reaction plane (Fig. 1,2). Results
for $p$-$p$ collisions at $\sqrt{s}=13$ TeV are shown in the bottom panel of
Fig. 3, where ATLAS results Aaboud  _et al._ (2017b) are used to constrain the
magnitude of the product of the TMDPDF and TMDFF to $10\%-40\%$ of the Soffer
bound, i.e. $0.1\leq b\cdot B\leq 0.4$ (14). Using the same parameters, we
obtain $v_{2}$ for $p$-$Pb$ collisions at $8.16$ TeV in the top panel of Fig.
3 and compare to ATLAS results Aad  _et al._ (2020).

We reiterate that the only difference between the calculations for $p$-$p$ and
those for $p$-$A$ is the $A^{1/3}$ factor enhancing the $\langle
k_{\perp}^{2}\rangle$. The results in Fig. 3 include both polarization
dependent and independent contributions. A decomposition into separate
portions is presented in the Supplement sup , and shows that the polarization
dependent piece dominates at $p_{T}\gtrsim 10$ GeV.

#### Summary:

In this Letter, we have presented the first successful post-diction of an
elliptic anisotropy in high $p_{T}$ particle production in $p$-$p$ and
$p$-$Pb$ collisions with no modification to the angle integrated spectrum. We
posit the source of this anisotropy to be entirely from the initial state, and
_not_ from energy loss in the final state.

Incoming PDFs and outgoing FFs can have intrinsic transverse momenta, which is
enhanced for one PDF in $p$-$A$. Transverse momenta in the PDF selects out a
preferred direction of the outgoing partons in the hard scatterings and
correlates this with the net transverse momentum $\bm{q}_{T}$ of the two
incoming hard partons. Conservation of momentum, within each incoming nucleon,
correlates the net momentum of the bulk portion of the collision with the hard
partonic scattering (Fig. 1). Intrinsic transverse momentum allows for spin-
polarization dependent partonic contributions, even with the incoming and
outgoing hadrons being unpolarized (Boer-Mulders’ and Collins’ functions).
These provide the dominant contributions to the anisotropy. These processes
may also play a role in the high-$p_{T}$ anisotropy in $A$-$A$ collisions.

The calculations in this Letter can be improved upon: Joint $x,k_{\perp}$
distributions, available for $p$-$p$, need to be calculated for $p$-$A$. The
use of factorized $k_{\perp}$ distributions requires further analysis and
constraints from independent experimental data. The $A^{1/3}$ enhancement
needs to be demonstrated for the case of spin-polarization dependent multiple
scattering. Further, higher twist terms Boer  _et al._ (2003) and
contributions from Generalized TMDs Boer  _et al._ (2018) will have to be
evaluated. While gluon-gluon scattering was dominant in the considered $p_{T}$
range, quark initiated processes will gain prominence at higher $p_{T}$, where
mass effects may introduce additional corrections Pisano  _et al._ (2013).
Finally, semi-analytical calculations need to be enhanced via extensive
simulations, where correlations between the hard and soft sectors in these
asymmetric collisions can be further investigated. All calculations in this
Letter have assumed factorization between hard and soft sectors in
high-$p_{T}$ hadron production. This is done with full knowledge that no
generalized factorization theorem exists for processes with TMDs in $p$-$p$
collisions Rogers and Mulders (2010).

#### Acknowledgments:

Both authors are members of the JETSCAPE collaboration and have benefited from
internal discussions. This work is supported in part by the U.S. DOE under
Grant No. DE-SC0013460.

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##  I Supplemental materials for: ‘Azimuthal Anisotropy at high transverse
momentum in $p$-$p$ and $p$-$A$ collisions’

###  I.1 Results from an $x$-dependent $\langle k_{\perp}^{2}\rangle$

Figure 4: Color Online: Average transverse momentum of gluons as a function of
their lightcone momentum $x\sqrt{s}/2$. Green curves are results for proton-
proton collisions, while red curves are multiplied by $A^{1/3}$ to represent
proton-lead collisions. Dot-dashed lines display a constant $\langle
k_{\perp}^{2}\rangle$, while solid lines display the $x$-dependent $\langle
k_{\perp}^{2}\rangle$ from Eq. (I.1).

$x$-independent values of $\langle k_{\bot}^{2}\rangle$ are not entirely
physical: Small $x$-partons can end up with a larger transverse momentum than
their light-cone momentum, as shown by the intersection of the dot-dashed
lines with the blue solid line in Fig. 4. An $x$-dependent $\langle
k_{\perp}^{2}\rangle$ _ansatz_ was presented in Ref. D’Alesio and Murgia
(2004), which vanishes at both $x\\!=\\!0$ and $x\\!=\\!1$. However, it rises
to large values in between, especially when enhanced by $A^{1/3}$ (shown by
the light dashed lines in Fig. 4). Calculations of the transport coefficient
$\hat{q}$, in deconfined matter, using perturbative QCD, always contain a
$\log{(E/E_{0})}$ from the integration over phase space. $E$ is the energy of
the hard parton and $E_{0}$ is a soft screening scale Arnold and Xiao (2008).
This motivates a modification of the $x$-dependent form of Ref. D’Alesio and
Murgia (2004) as 111For momentum fractions in the range $x_{0}\leq x\leq
10x_{0}$, we use a hyperbolic tangent
$\left(\tanh\frac{x-x_{0}}{x_{0}}\right)$ to smoothly match the logarithmic
behavior to the power-law.,

$\displaystyle\langle k_{\bot}^{2}\rangle^{1/2}(x)=\langle
k_{\bot}^{2}\rangle_{0}^{1/2}\left((1-x)\sqrt{\frac{s}{s_{0}}}\right)^{0.15}A^{1/6}$
$\displaystyle\times\begin{cases}\left(\frac{x}{x_{0}}\sqrt{\frac{s}{s_{0}}}\right)^{0.8}\;,&\text{
if }x\leq x_{0}\\\
\left[1+b_{0}\log\left(\frac{x}{x_{0}}\sqrt{\frac{s}{s_{0}}}\right)\right]^{1/2}\;,&\text{
if }x>x_{0}.\end{cases}$ (18)

We take $\langle k_{\bot}^{2}\rangle_{0}\\!=\\!5$ GeV2, $x_{0}\\!=\\!10^{-2}$,
$\sqrt{s_{0}}\\!=\\!8.16$ TeV and $b_{0}\\!=\\!2$ (solid red line in Fig. 4).
The constants are chosen to maintain the form of Ref. D’Alesio and Murgia
(2004) at $x\leq x_{0}$, turning over at larger $x$, and yielding a mean
($x$-integrated) $\langle k_{\perp}^{2}\rangle$ comparable to the
$x$-independent value sup . Strict $A^{1/3}$ scaling between $p$-$p$ and
$p$-$A$ is maintained by setting $A=1$ for $p$-$p$ (solid green line in Fig.
4).

###  I.2 Contribution from polarized and unpolarized scatterings, and power
corrections

In the Letter above, we found that both unpolarized and polarized parton
scattering can lead to azimuthal anisotropies at high-$p_{T}$. However, there
is a difference: azimuthal anisotropies from unpolarized parton scattering are
a power correction compared to polarized parton scattering. This is clearly
demonstrated for the case of a constant $\langle k_{\perp}^{2}\rangle^{1/2}=1$
GeV, for the case of $p$-$p$ collisions in Fig. 5, with $b\cdot B=0.4$. One
clearly notices the blue hatched area that represents the polarization
independent term falling sharply compared to the polarization dependent term
in solid green, with increasing $p_{T}$.

Figure 5: Azimuthal anisotropy coefficient $v_{2}$ as a function of the pion
transverse momentum $p_{T}$ for pp at $13$ TeV. The filled green represents
the $\rm BM\otimes C$ contribution, the blue hatched represents polarization
independent contributions.

Extending the fixed $\langle k_{\perp}^{2}\rangle$ to $p$-$Pb$ by setting
$\langle k_{\perp}^{2}\rangle=A^{1/3}$ GeV, we obtain the plot in Fig. 6,
still with $b\cdot B=0.4$. The contribution from the polarization independent
term is somewhat larger at $p_{T}\lesssim 10$ GeV. However, even in this case,
it is suppressed at higher $p_{T}$ compared to the polarization dependent
term.

Figure 6: Azimuthal anisotropy coefficient $v_{2}$ as a function of the pion
transverse momentum $p_{T}$ for $p$-$Pb$ at $8.16$ TeV. The filled green
represents the $\rm BM\otimes C$ contribution, the blue hatched represents
polarization independent contributions.

Using the $x$-dependent $\langle k_{\bot}^{2}\rangle$, reduces the suppression
of the polarization independent terms, as they effectively produce a larger
$\langle k_{\perp}^{2}\rangle$ at larger $x$, and a much smaller $\langle
k_{\perp}^{2}\rangle$ at smaller $x$, as shown in Fig. 4. Using the form of
$\langle k_{\perp}^{2}\rangle^{1/2}$ from the previous subsection, we present
the decomposition of the $x-$dependent $\langle k_{\perp}^{2}\rangle$
contribution into the polarization dependent and independent terms for $p$-$p$
and $p$-$Pb$ collisions in Figs. 7 and 8 respectively.

Figure 7: Azimuthal anisotropy coefficient $v_{2}$ as a function of the pion
transverse momentum $p_{T}$ for $p$-$p$ at $13$ TeV. The filled green
represents the $\rm BM\otimes C$ contribution, the blue hatched represents
polarization independent contributions.

In both Fig. 7 and 8, the decomposition of contributions to $v_{2}$ from spin-
polarization independent terms in blue hatched area and $\rm BM\otimes C$
terms in filled green for $p$-$p$ at 13TeV and $p$-$Pb$ at $8.16$ TeV show a
similar trend. Despite taking the magnitude of the product of TMD functions to
be $40\%$ of the Soffer bound, we find the $\rm BM\otimes C$ contribution to
be larger than the polarization independent term at $p_{T}\gtrsim 10$ GeV.

Figure 8: Azimuthal anisotropy coefficient $v_{2}$ as a function of the pion
transverse momentum $p_{T}$ for pPb at $8.16$ TeV. The filled green represents
the $\rm BM\otimes C$ contribution, the blue hatched represents polarization
independent contributions.

###  I.3 Centrality selection of events

Figure 9: Average transverse energy $\langle E_{T}\rangle=E\sin\theta$ in the
forward direction $3\leq|\eta|\leq 10$ as a function of the average transverse
momentum of the initial hard parton $\langle k_{\perp}^{2}\rangle^{1/2}$

Experimental studies of heavy-ion collisions perform centrality selections of
the events based on the total transverse energy $E_{T}$ in the forward
calorimeter (FoCal) (see e.g. Adams  _et al._ (2005); Adler  _et al._ (2003);
Aamodt  _et al._ (2010); Chatrchyan  _et al._ (2013); Aad  _et al._ (2012)).
Events are considered to be more central when the transverse energy is large.
If we consider the total hard partonic scattering to be compensated by the
remaining softer scatterings, a large transverse energy leads to a larger
number of these partons deflected from the $z$-axis towards the FoCal
rapidities. We employ the Monte Carlo event generator PYTHIA 8 Bierlich  _et
al._ (2022) to study the effect of the initial transverse momentum on the
correlation between hard and soft partons.

We set up a single hard scattering event without initial or final state parton
shower and focus on the effect of initial intrinsic transverse momentum on the
Lund model based fragmentation. The Lund model relies on beam remnants, which
are the partons that do not participate in the hard scattering but continue
traveling along the $z-$direction. For each remnant parton we sample a
transverse momentum $\bm{k}_{\perp i}$ following a Gaussian distribution with
an average momentum square $\langle k_{\perp}^{2}\rangle$ and random
transverse direction. The total transverse momentum of all remnants
$\bm{q}_{T}=\sum_{i}\bm{k}_{\perp i}$ is compensated by applying a Lorentz
boost to the center of mass frame of the hard scattering with the velocity

$\displaystyle\bm{v}=\frac{\bm{q}_{T}}{\sqrt{(E_{a}+E_{b})^{2}+\bm{q}_{T}^{2}}}\;.$
(19)

By applying a boost to the center of mass frame, we ensure that the scattering
cross-section is not altered. However, this will entail that the initial hard
parton will acquire a slightly different mean transverse momentum than the
beam remnants. We ignore this effect as it should not alter our conclusions.

We compute the transverse energy within a range of forward rapidities between
$3\leq|\eta|\leq 10$,

$\displaystyle E_{T}=\sum_{i}E_{i}\sin\theta_{i}\;,$ (20)

where $E_{i}$ is the energy of the final state hadron and $\theta_{i}$ is the
angle between the hadron momentum and the $z$-axis. Only hadrons that lie
within the selected pseudo-rapidity range are included in the sum above. The
average transverse energy $\langle E_{T}\rangle$ is shown in Fig. 9 as a
function of the average transverse momentum of the initial hard parton
$\langle k_{\perp}^{2}\rangle^{1/2}$. We find a clear linear relation between
the initial transverse momentum of the hard parton and the transverse energy
in the forward direction. We conclude that centrality selection based on the
transverse energy in the forward direction, where larger transverse energy is
indicative of a more central event, leads to a selection of events where more
central events are those with a large initial transverse momentum.

###  I.4 Mean transverse momentum squared

Figure 10: Mean transverse momentum squared $\langle k_{\perp}^{2}\rangle$ of
the initial hard parton as a function of the scale Q of the PDF.

Due to the $x$-dependence of the mean transverse momentum squared in the
Gaussian _Ansatz_ of Eq. (I.1), the factorization between transverse and
longitudinal momenta distributions is broken. Effectively, the mean transverse
momentum of the partons is not directly the function $\langle
k_{\bot}^{2}\rangle(x)$. Instead, to obtain the effective $\langle
k_{\perp}^{2}\rangle$ on needs to integrate the $x$-dependent $\langle
k_{\bot}^{2}\rangle$ weighted by the PDF, as follows

$\displaystyle\langle k_{\perp}^{2}\rangle(Q)=\frac{\int dx_{a}\langle
k_{\bot}^{2}\rangle f(x_{a},Q)}{\int dx_{a}f(x_{a},Q)}\;,$ (21)

where $Q$ is the scale where the PDF is evaluated.

Using the nCTEQ PDF for gluon in the proton, we obtain the mean transverse
momentum squared of the initial hard parton as a function of the scale $Q$ in
Fig. 10. We see that, effectively, the transverse momentum does not exceed
$2$GeV at low scales and as the scale increases the transverse momentum
plateaus at around $\sim 0.6$GeV.

###  I.5 Energy Loss from initial state scattering.

The probability of radiating a gluon induced by initial state scattering prior
to a hard event was calculated in Ref. Fries  _et al._ (2000), the mirror
calculation for a gluon produced by final state rescattering was carried out
in Ref. Wang and Guo (2001). Using a formula derived from their expressions,
and including the interference of double hard and hard-soft terms, we obtain
the probability for a quark with energy-momentum $(Q/2,0,0,Q/2)$ to radiate a
gluon, prior to hard scattering, as

$\frac{dN}{dy}\\!\\!=\\!\\!\int\limits_{0}^{Q^{2}}\\!\\!\frac{\alpha_{S}}{2\pi}\frac{dl_{\perp}^{2}}{l_{\perp}^{4}}P(y)\\!\\!\\!\int\limits_{-L^{-}}^{0}\\!\\!\\!d\tau^{-}\hat{q}\left[2-2\cos\left(\frac{l_{\perp}^{2}}{Q}\tau^{-}\right)\right].$
(22)

In the above equation, $\hat{q}$ is the transport coefficient in cold nuclear
matter, $\tau^{-}$ is the light-cone location of the soft scattering, which
ranges from the location of the hard scattering at $0$ to a light-cone
distance $-L^{-}$. The splitting function $P(y)$, where $y$ is the light-cone
momentum fraction carried by the radiated gluon can be approximated in the
soft gluon limit as $P(y)\simeq 2/y$. The transverse momentum of the radiated
gluon is $l_{\perp}$.

The energy lost by a single gluon emission from a quark with energy $Q/2$ can
be estimated as

$\Delta E=\int_{0}^{1}dy\left(y\frac{Q}{2}\right)\frac{dN}{dy}$ (23)

The integral of this expression yields a $\Delta E\sim{L^{-}}^{3}$ at very
small distances $L^{-}\ll 1/Q$ and a $\Delta E\sim{L^{-}}^{2}$ at larger
distances. As a result, for a Lorentz contracted nucleus in the lab frame, a
parton traverses $L^{-}_{Lab}\\!\\!=\\!\\!L^{-}/\gamma_{Lab}$ with
$\hat{q}_{Lab}\\!=\\!\gamma_{Lab}\hat{q}$, where $\gamma_{Lab}$ is the boost
factor. Thus, for finite $\Delta\langle
k_{\perp}^{2}\rangle=\hat{q}L^{-}=\hat{q}_{Lab}L^{-}_{Lab}$ and
$\gamma_{Lab}\gtrsim 1000$, one has a $\Delta E\rightarrow 0$, ensuring the
angle integrated cross section (and $R_{pA}$) remains unmodified.


# Photon-echo-based quantum memory for optical squeezed states

Miao-Xin Wu, Ming-Feng Wang, and Yi-Zhuang Zheng† School of Physics ${\&}$
Electronics Information Engineering, Wenzhou University, Wenzhou 325027,
People’s Republic of China<EMAIL_ADDRESS>
###### Abstract

The ability to efficiently realize storage and readout of optical squeezed
states plays a key roll in continuous-variables quantum information
processing. Here we study the quantum memory (QM) for squeezed state of
propagating light in atoms based on the hybrid photon echo re-phasing (HYPER).
The optical quantum state is recorded in two sublevels of the ground state of
an atomic ensemble to realize long-lived QM. Taking into account the noise
effect due to atomic decay, our estimation indicates that, with currently
available technique, high fidelities larger than the classical fidelity
threshold 81.5 ${\%}$ are obtainable. Our work provides some practical
guidance for realization of efficient and faithful photon-echo-based memory
for squeezed light.

##  1 Introduction

QM for light plays a key role in quantum-information processing and has been
considered as a basic ingredient for quantum repeaters [1] and scalable all-
optical quantum computers [2]. Nowadays, much effort has been devoted to such
QM through different approaches, including Faraday rotation [3, 4],
electromagnetically induced transparency (EIT) [5, 6], off-resonant Raman
transitions [7, 8], and photon echoes [9-12]. Photon-echo-based QMs are
currently attracting particular attention, not only for their abilities to
implement in solid-state materials, but also for their successful achievement
in storage efficiency [13, 14] and multimode-memory capacity [15]. By now, in
this area, many impressive results have been achieved. Gisin et al. showed
[16] that photon-echo techniques can be used to store and retrieve time-bin
qubits [17]. Clausen et al. demonstrated [18] the storage of entangled photons
using atomic frequency comb (AFC) technique. Recently, high efficient storage
of optical coherent state (about 87${\%}$) based on gradient echo memory (GEM)
has also been demonstrated [14]. Despite these achievements, however, optical
squeezed-state storage based on photon-echo techniques, as far as we know, has
not yet been studied both theoretically and experimentally.

Squeezed state of light (SSL), which provides a reduction in one of its
quadratures below the standard quantum limit, plays a fundamental role in
quantum-information processing. More specifically, a QM for SSL has been shown
to be especially valuable for interferometry [19], high-precision measurement
[20], and quantum information with continuous variables [21]. During the last
decade, a variety of approaches concerning the storage of SSL have been
proposed. One of them proposes to transfer the squeezed state from non-
classical light to the excited atomic states [22]. Another approach to map the
SSL onto atomic ensemble based on stimulated Raman scattering has been
developed by Kozhekin et al. [8]. In particular, storage and retrieval of SSL
with EIT was also investigated [6]. Very recently, Jensen et al. demonstrated
[4] the QM for entangled two-mode squeezed states via Faraday rotation.

In this paper, we study the long-lived QM for SSL based on HYPER—a variant of
the two-pulse photon echo (2PE) technique [23]. 2PE is well known to be
unsuitable for serving as a QM technique, because of the irritating collective
emission of photons (amplified spontaneous emission). In order to overcome
this problem, McAuslan et al. developed a new photon-echo technique called
HYPER, which uses electrical field gradients to eliminate the unwanted gain
created by the strong re-phasing pulse ($\pi$-pulse [23] or frequency-chirped
pulse [24]). Such technique has the outstanding characteristic of avoiding
spectral hole-burning, which enable us to implement large-bandwidth and high-
efficiency QMs. Here, utilizing this approach, we detailedly derived the
processes for storage and retrieval of SSL. We find that, for the ideal case,
the squeezed-quadrature variance (SQV) of the retrieved optical field relies
only on the optical depth of atomic ensemble. The larger the optical depth,
the more squeezing of the input mode will be preserved. Taking into account
the noise effect due to atoms coupled to thermal reservoirs, the SQV now
depends not only on the optical depth but also on the decay rate. We show that
substantial squeezing of the input mode can also be preserved even with
currently available technique [23]. The rest of this paper is organized as
follows. In Sec. II, we present the basic equations governing the evolution of
the quantized atomic and photonic fields based on photon-echo techniques.
Next, storage and retrieval of SSL is considered. In Sec. IV, we discuss the
possibility of implementing the technique in the solid-state materials.
Section V gives our conclusions.

##  2 The storage process

###  2.1 The basic evolutions

Our storage material consists of a large number of atoms with an excited state
${\left|e\right\rangle}$ and two lower states ${\left|g\right\rangle}$,
${\left|s\right\rangle}$, see Fig. 1(a). Let us first study the absorption
process. Consider a quantized optical pulse being injected into the storage
medium to couple ${\left|g\right\rangle}$ and ${\left|e\right\rangle}$. In the
one-dimensional light propagation model, the negative frequency part of the
optical field can be decomposed in forward and backward modes

$\hat{a}\left({z,t}\right)=\varepsilon{\hat{a}_{f}}\left({z,t}\right){e^{-i\left({{\omega_{0}}t-kz}\right)}}+\varepsilon{\hat{a}_{b}}\left({z,t}\right){e^{-i\left({{\omega_{0}}t+kz}\right)}}\;,$
(1)

where ${\varepsilon=\sqrt{\hbar\omega_{0}/\left(\varepsilon_{0}V\right)}}$,
${\omega_{0}}$ is the central frequency of optical pulse, ${\varepsilon_{0}}$
is the vacuum permeability, and ${V}$ is the quantized volume.
${\hat{a}_{k}\left({z,t}\right)}$ (where ${k=f,b}$) represents the optical
operators with the commutation
relation${\left[{{{\hat{a}}_{k}}\left({z,t}\right),\hat{a}_{k}^{\dagger}\left({z,t^{\prime}}\right)}\right]=\delta\left({t-t^{\prime}}\right)}$.
For atomic operators, the mean field per atom is defined as [25]

${\hat{\sigma}_{ge}}\left({z,t,\Delta}\right)=\frac{1}{{N\left({\Delta,z}\right)}}\sum\limits_{n=1}^{N\left({\Delta,z}\right)}{{{\left|g\right\rangle}_{nn}}\left\langle
e\right|}\;,$ (2)

where ${\Delta=\omega_{eg}-\omega_{0}}$ is the detuning from resonance, with
the atomic transition frequency ${\omega_{eg}}$. In the above sum, the index
${n}$ runs over all atoms ${N\left({\Delta,z}\right):=\rho\delta
zg\left(\Delta\right)\delta\Delta}$ within the infinitesimal slice ${\delta
z}$ and ${\delta\Delta}$. The notation ${\rho}$ denotes the number density of
atoms, and ${g(\Delta)}$ is the normalized atomic spectral distribution. In
analogy with the optical field, the negative-frequency part of the atomic
operator can be described by the two counter-propagating contributions

${\hat{\sigma}_{ge}}\left({z,t,\Delta}\right)={\hat{\sigma}_{f}}\left({z,t,\Delta}\right){e^{-i\left({{\omega_{0}}t-kz}\right)}}+{\hat{\sigma}_{b}}\left({z,t,\Delta}\right){e^{-i\left({{\omega_{0}}t+kz}\right)}}\;.$
(3)

In the low-excitation limit, if the atoms are initially prepared in the ground
state, the operators associated to the atomic coherence ${\hat{\sigma}_{k}}$
have the following commutation relation [26]

$\left[{{{\hat{\sigma}}_{k}}\left({z,t,\Delta}\right),\hat{\sigma}_{k}^{\dagger}\left({z^{\prime},t,\Delta^{\prime}}\right)}\right]=\frac{1}{{\rho
g\left(\Delta\right)}}\delta\left({z-z^{\prime}}\right)\delta\left({\Delta-\Delta^{\prime}}\right)\;.$
(4)

With these definitions, the dynamic of the optical absorbing process can be
described by the following equations

$\left({\frac{1}{c}{\partial_{t}}+{\partial_{z}}}\right){\hat{a}_{f}}\left({z,t}\right)=ig\rho\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right){\hat{\sigma}_{f}}\left({z,t,\Delta}\right)\;,$
(5)
${\partial_{t}}{\hat{\sigma}_{f}}\left({z,t,\Delta}\right)=\left({i\Delta-\frac{\Gamma}{2}}\right){\hat{\sigma}_{f}}\left({z,t,\Delta}\right)+ig{\hat{a}_{f}}\left({z,t}\right)+{\hat{f}_{1}}\left({z,t,\Delta}\right)\;,$
(6)

where ${g}$ is the coupling constant of the light-atoms interactions, and we
have considered the decoherence of atomic system with ${\Gamma}$ the decay
rate of the excited state. ${{\hat{f}_{1}}\left({z,t,\Delta}\right)}$ is the
Langevin noise operator, satisfying the following correlation functions [27]

$\left\langle{{{\hat{f}}_{1}}\left({z,t,\Delta}\right)\hat{f}_{1}^{\dagger}\left({z^{\prime},t^{\prime},\Delta^{\prime}}\right)}\right\rangle=\frac{\Gamma}{{\rho
g\left(\Delta\right)}}\delta\left({z-z^{\prime}}\right)\delta\left({t-t^{\prime}}\right)\delta\left({\Delta-\Delta^{\prime}}\right)\;,$
(7)
$\left\langle{\hat{f}_{1}^{\dagger}\left({z^{\prime},t^{\prime},\Delta^{\prime}}\right){{\hat{f}}_{1}}\left({z,t,\Delta}\right)}\right\rangle=0\;.$
(8)

The solution of Eq. (6) can easily be found to be

$\displaystyle{\hat{\sigma}_{f}}\left({z,t,\Delta}\right)$ $\displaystyle=$
$\displaystyle{\hat{\sigma}_{f}}\left({z,{t_{0}},\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-{t_{0}}}\right)}}+ig\int_{{t_{0}}}^{t}{ds}{\hat{a}_{f}}\left({z,s}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}$
(9)
$\displaystyle+\int_{{t_{0}}}^{t}{ds}{\hat{f}_{1}}\left({z,s,\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}\;,$

where ${\hat{\sigma}_{f}\left({z,{t_{0}},\Delta}\right)}$ is the initial
atomic operator. To derive the time evolution of optical field, we insert Eq.
(9) into Eq. (5) to generate

$\displaystyle{\partial_{z}}{{\hat{a}}_{f}}\left({z,t}\right)$
$\displaystyle=$
$\displaystyle-{g^{2}}\rho\int_{{t_{0}}}^{t}{ds}{{\hat{a}}_{f}}\left({z,s}\right){e^{-\Gamma\left({t-s}\right)/2}}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right){e^{i\Delta\left({t-s}\right)}}$
(10)
$\displaystyle+ig\rho\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right)\times{{\hat{\sigma}}_{f}}\left({z,{t_{0}},\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-{t_{0}}}\right)}}$
$\displaystyle+ig\rho\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right)\int_{{t_{0}}}^{t}{ds}{{\hat{f}}_{1}}\left({z,s,\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}\;.$

Here, we have neglected the temporal derivative in Eq. (5), since we consider
the regime ${\tau\gg L/c}$ (${\tau}$ is the temporal length of the incident
optical pulse; ${L}$ is the length of medium), which allows us to ignore the
temporal retardation effect of light. In order to find the solution of Eq.
(10), we assume the atomic spectral distribution is uniform, that is
${g\left(\Delta\right)\sim 1/\gamma}$ (where ${\gamma}$ is the bandwidth of
the ensemble), which is justified since, for natural inhomogeneous broadening
systems, such as rare-earth-ion doped crystals, the ratio between
inhomogeneous line-width and homogeneous line-width can be approximately
achieved 6 or more orders of magnitude [28]. Furthermore, the duration of the
absorption process is essentially governed by the temporal length of the
signal optical pulse, which ensures the values of ${t-s}$ of order ${\tau}$.
Hence, in the limit ${\gamma\tau\gg 1}$ the Fourier transform
${\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right){e^{i\Delta\left({t-s}\right)}}}$
in Eq. (10) will act like a ${\delta}$ function. With these assumptions, Eq.
(10) can be directly solved to give

$\displaystyle{{\hat{a}}_{f}}\left({z,t}\right)={{\hat{a}}_{f}}\left({0,t}\right){e^{-\alpha
z/2}}$
$\displaystyle+ig\rho\int_{0}^{z}{dz^{\prime}}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right){{\hat{\sigma}}_{f}}\left({z^{\prime},{t_{0}},\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-{t_{0}}}\right)}}{e^{\alpha\left({z^{\prime}-z}\right)/2}}$
$\displaystyle+ig\rho\int_{0}^{z}{dz^{\prime}}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right)\int_{{t_{0}}}^{t}{ds}{{\hat{f}}_{1}}\left({z,s,\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}{e^{\alpha\left({z^{\prime}-z}\right)/2}}\;,$

where ${\hat{a}_{f}\left({0,t}\right)}$ is the input photonic field, and we
have defined the absorption coefficient ${\alpha=2\pi{g^{2}}\rho/\gamma}$. The
first term describes the exponential decay of the incoming optical pulse.

###  2.2 Long-lived memory for SSL

Figure 1: (Color online) (a) Atomic-level structure of storage medium. (b)
Pulse sequence of the long-lived HYPER protocol. In region 1, the optical data
is absorbed by the storage medium at time ${t_{1}}$. Subsequently, a pair of
control fields (${\pi}$ pulses or frequency-chirped pulses) are applied to the
atoms to transform the atomic coherence
${\hat{\sigma}_{ge}}\rightarrow\hat{\sigma}_{gs}$ and
${\hat{\sigma}_{gs}}\rightarrow\hat{\sigma}_{ge}$. Region 2 realizes the
rephasing of the stored light, and finally in region 3 the optical pulse comes
out.

Now, we consider a long-time storage protocol based on HYPER technique. This
protocol consists of three separate time regions, as shown in Fig. 1(b). In
region 1 ${(t_{0}<t<t_{4})}$, just after absorbing the signal optical field, a
linear electrical field gradient is applied to the sample to cause an
additional atomic phase change depending on the position of the atoms. Soon
after this field, a strong control optical field coupling to
${\left|e\right\rangle}$ , ${\left|s\right\rangle}$ is turned on to transfer
the population from ${\left|e\right\rangle}$ to ${\left|s\right\rangle}$.
Obviously, information about signal field is now stored in the spin wave.
After a time interval ${(t_{3}-t_{2})}$, another strong optical field
propagating in the same direction are applied to drive the population back
onto the excited state. In region 2 ${(t_{4}<t<t_{5})}$, two ${\pi}$ pulses
sandwiched with an electrical field are applied to the atoms to rephase the
input optical mode. In region 3 ${(t>t_{5})}$, the ensemble will emits an
expected echo at ${t_{6}}$.

During the storage process, for simplicity, we assume that the first
electrical field is short enough, which enables us to ignore the decay of the
excited state during the time interval ${(t_{1}<t<t_{2})}$. We also neglect
spontaneous decay of the state ${\left|s\right\rangle}$, which is reasonable
since we assume the time interval ${(t_{3}-t_{2})}$ is much smaller than the
lifetime of ${\left|s\right\rangle}$. Furthermore, the time spent in the
region 2 is also temporally short, which allows us to reduce the dynamics of
this region to be an instantaneous operation at ${t_{5}}$ [23]. Besides, if
the two pulses used in region 2 propagate in opposite directions, the system
will satisfy the condition of the backward retrieval resulting in the
capability for achieving 100 ${\%}$ QM efficiency [25]. After applying the
second pulse in region 2, the atoms start to rephase. At
${t_{6}=2t_{5}-2t_{4}+t_{3}-t_{2}+t_{1}}$, an echo propagating along negative
${z}$-axis through the medium occurs, which can be described by the following
equations

$-{\partial_{z}}{\hat{a}_{b}}\left({z,t}\right)=ig\rho\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right){\hat{\sigma}_{b}}\left({z,t,\Delta}\right)\;,$
(12)
${\partial_{t}}{\hat{\sigma}_{b}}\left({z,t,\Delta}\right)=\left({i\Delta-\frac{\Gamma}{2}}\right){\hat{\sigma}_{b}}\left({z,t,\Delta}\right)+ig{\hat{a}_{b}}\left({z,t}\right)+{\hat{f}_{2}}\left({z,t,\Delta}\right)\;,$
(13)

where ${\hat{f}_{2}\left({z,t,\Delta}\right)}$ is the Langevin noise of the
retrieval process. In deriving the above equations, we have used the boundary
condition of the atomic field created at ${t_{5}}$

${\hat{\sigma}_{b}}\left({z,t=0;\Delta}\right)={\hat{\sigma}_{f}}\left({z,t=0;\Delta}\right){e^{-i\Delta
T}}\;,$ (14)

where we have set ${t_{5}=0}$, leading to the QM time ${T=0-t_{1}}$. In
accordance with (9), (12), and (14), one can easily derive the output optical
field

$\displaystyle{{\hat{a}}_{b}}\left({0,t}\right)={{\hat{a}}_{b}}\left({L,t}\right){e^{-\alpha
L/2}}+\alpha{e^{-\Gamma
T/2}}\int_{L}^{0}{dz}{{\hat{a}}_{f}}\left({z,t-T}\right){e^{-\alpha z/2}}$
$\displaystyle-
ig\rho\int_{L}^{0}{dz}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right){{\hat{\sigma}}_{f}}\left({z,{t_{0}},\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-{t_{0}}}\right)}}{e^{-i\Delta
T}}{e^{-\alpha z/2}}$ $\displaystyle-
ig\rho\int_{L}^{0}{dz}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right)\int_{{t_{0}}}^{0}{ds}{{\hat{f}}_{1}}\left({z,s;\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}{e^{-i\Delta
T}}{e^{-\alpha z/2}}$ $\displaystyle-
ig\rho\int_{L}^{0}{dz}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right)\int_{0}^{t}{ds}{{\hat{f}}_{2}}\left({z,s;\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}{e^{-\alpha
z/2}}\;,$ (15)

where ${\hat{a}_{b}\left({L,t}\right)}$ is the vacuum input of the backward
retrieval optical mode, and we have used the fact that the optical field is
out at ${z=0}$. Next, Eq. (11) is substituted into Eq. (15), and after
interchanging the order of the associated double integral
${\int_{0}^{L}{dz}\int_{0}^{Z}{dz^{\prime}}\to\int_{0}^{L}{dz^{\prime}}\int_{z^{\prime}}^{L}{dz}}$,
we will obtain

$\displaystyle{{\hat{a}}_{b}}\left(t\right)={{\hat{a}}_{b}}\left({L,t}\right){e^{-\alpha
L/2}}+{e^{-\Gamma T/2}}\left({{e^{-\alpha
L}}-1}\right){{\hat{a}}_{f}}\left({0,t-T}\right)$
$\displaystyle+ig\rho\int_{0}^{L}{dz}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right){{\hat{\sigma}}_{f}}\left({z,{t_{0}},\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-{t_{0}}}\right)}}{e^{-i\Delta
T}}{e^{\alpha z/2}}{e^{-\alpha L}}$
$\displaystyle+ig\rho\int_{0}^{L}{dz}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right)\int_{{t_{0}}}^{t-T}{ds}{{\hat{f}}_{1}}\left({z,s;\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}{e^{-i\Delta
T}}{e^{\alpha z/2}}{e^{-\alpha L}}$
$\displaystyle+ig\rho\int_{0}^{L}{dz}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right)\int_{t-T}^{0}{ds}{{\hat{f}}_{1}}\left({z,s;\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}{e^{-i\Delta
T}}{e^{-\alpha z/2}}$
$\displaystyle+ig\rho\int_{0}^{L}{dz}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right)\int_{0}^{t}{ds}{{\hat{f}}_{2}}\left({z,s;\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}{e^{-\alpha
z/2}}\;,$ (16)

where ${t-T}$ is the time when the incident pulse is completely absorbed. The
second term shows that information about the input optical mode is now printed
onto the output mode. The last three noises terms are independent of each
other, since they correspond to the Langevin noises at different times.
Integrating Eq. (16) over times of the duration of the input pulse, one
obtains the dimensionless optical operator

$\displaystyle{{\hat{a}}_{b}}={\tau^{-1/2}}\int_{0}^{\tau}{dt}{{\hat{a}}_{b}}\left(t\right)$
$\displaystyle={\tau^{-1/2}}{e^{-\alpha
L/2}}\int_{0}^{\tau}{dt}{{\hat{a}}_{b}}\left({L,t}\right)$
$\displaystyle+{\tau^{-1/2}}{e^{-\Gamma{t_{d}}/2}}\left({{e^{-\alpha
L}}-1}\right)\int_{0}^{\tau}{dt}{{\hat{a}}_{f}}\left({0,t-T}\right)$
$\displaystyle+i\sqrt{\alpha}\int_{0}^{L}{dz}\left\\{{\left[{\hat{D}\left(z\right)+{{\hat{F}}_{11}}\left(z\right)}\right]{e^{\alpha
z/2}}{e^{-\alpha
L}}+\left[{{{\hat{F}}_{12}}\left(z\right)+{{\hat{F}}_{2}}\left(z\right)}\right]{e^{-\alpha
z/2}}}\right\\}\;,$

where we defined the new operators for the initial atomic operator and
Langevin noise operators

$\displaystyle\hat{D}\left(z\right)=\sqrt{\frac{{\rho\gamma}}{{2\pi\tau}}}\int_{0}^{\tau}{dt}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right){\hat{\sigma}_{f}}\left({z,{t_{0}},\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-{t_{0}}}\right)}}{e^{-i\Delta
T}},$
$\displaystyle{\hat{F}_{11}}\left(z\right)=\sqrt{\frac{{\rho\gamma}}{{2\pi\tau}}}\int_{0}^{\tau}{dt}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right)\int_{{t_{0}}}^{t-T}{ds}{\hat{f}_{1}}\left({z,s;\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}{e^{-i\Delta
T}},$
$\displaystyle{\hat{F}_{12}}\left(z\right)=\sqrt{\frac{{\rho\gamma}}{{2\pi\tau}}}\int_{0}^{\tau}{dt}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right)\int_{t-T}^{0}{ds}{\hat{f}_{1}}\left({z,s;\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}{e^{-i\Delta
T}},$
$\displaystyle{\hat{F}_{2}}\left(z\right)=\sqrt{\frac{{\rho\gamma}}{{2\pi\tau}}}\int_{0}^{\tau}{dt}\int_{-\infty}^{\infty}{d\Delta}g\left(\Delta\right)\int_{0}^{t}{ds}{\hat{f}_{2}}\left({z,s;\Delta}\right){e^{\left({i\Delta-\Gamma/2}\right)\left({t-s}\right)}}.$

According to Eqs. (4), (8) and (9), the above operators obey

$\left\langle{\hat{D}\left(z\right){{\hat{D}}^{\dagger}}\left({z^{\prime}}\right)}\right\rangle=A{e^{\Gamma{t_{0}}}}\delta\left({z-z^{\prime}}\right)\;,$
(18)
$\left\langle{{{\hat{F}}_{11}}\left(z\right)\hat{F}_{11}^{\dagger}\left({z^{\prime}}\right)}\right\rangle=\left({{e^{-\chi}}-A{e^{\Gamma{t_{0}}}}}\right)\delta\left({z-z^{\prime}}\right)\;,$
(19)
$\left\langle{{{\hat{F}}_{12}}\left(z\right)\hat{F}_{12}^{\dagger}\left({z^{\prime}}\right)}\right\rangle=\left({A-{e^{-\chi}}}\right)\delta\left({z-z^{\prime}}\right)\;,$
(20)
$\left\langle{{{\hat{F}}_{2}}\left(z\right)\hat{F}_{2}^{\dagger}\left({z^{\prime}}\right)}\right\rangle=\left({1-A}\right)\delta\left({z-z^{\prime}}\right)\;.$
(21)

Here we have defined the dimensionless factor
${A=\frac{1}{{\Gamma\tau}}\left({1-{e^{-\Gamma\tau}}}\right)}$ and
${\chi=\Gamma T}$. Eq. (17) is the main result of this paper. Obviously, in
contrast to the input light state the output one is severely distorted. In
order to measure how well the output state compares to the input one, we
introduce the fidelity
${F=\left\langle\varphi\right.\left|{{{\hat{\rho}}_{out}}}\right.\left|\varphi\right\rangle}$
for light, where ${\left|\varphi\right\rangle}$ is the input state, and
${\hat{\rho}_{out}}$ is the output state. Note that the atomic and optical
states involved here are all Gaussian. Furthermore, we assume that the input
squeezed state is with zero displacement. For such states, the fidelity is
given by ${F=\frac{1}{2}\sqrt{\det{{\left({M+M^{\prime}}\right)}^{-1}}}}$ [29]
where ${M}$ and ${M^{\prime}}$ stand for the covariance matrix of the input
and output states, respectively. In accordance with Eqs. (17) - (21), one may
directly derive the covariance matrix, and we finally get

Figure 2: (Color online) Storage fidelities versus optical depth in the
presence of atomic decay for 7 dB squeezing input.

${F=\frac{2}{{\sqrt{\left[{\left({1+{e^{-2r}}}\right)+{e^{-\chi}}{{\left({1-{e^{-\alpha
L}}}\right)}^{2}}\left({{e^{-2r}}-1}\right)}\right]\left[{\left({1+{e^{2r}}}\right)+{e^{-\chi}}{{\left({1-{e^{-\alpha
L}}}\right)}^{2}}\left({{e^{2r}}-1}\right)}\right]}}}}$.

In deriving Eq. (22), we have used the fact that
${\left\langle{{{\hat{X}}_{f}}\left({t-{t_{d}}}\right){{\hat{X}}_{f}}\left({t^{\prime}-{t_{d}}}\right)}\right\rangle={e^{-2r}}\delta\left({t-t^{\prime}}\right)}$
${/4}$, which means that the ${x}$-quadrature of the input optical mode are
initially squeezed with the squeezing parameter ${r}$. Note that, for the
ideal case ${\chi=0}$ and the certain squeezing input, the fidelity depends
only on the optical depth. The larger the optical depth, the higher the
fidelity. For the case of ${\chi\neq 0}$, in Fig. 2 we show the fidelity in
its dependence on the optical depth for different decay parameter ${\chi}$,
where the dashed line denotes classical fidelity threshold (CFT) [30]. One can
see from the figure that, for optical depth large than 2, the fidelity is well
above CFT, and the maximal fidelity is mainly limited by the atomic decay.
Fidelity vs optical depth for different squeezing is also depicted in Fig. 3,
showing that the more highly squeezed the input mode, the less we can tolerate
the losses of the memory process. It should be noted that, for low-squeezed
light, its fidelity is much higher than CFT even within small optical depth,
which reflects that the quantum features of low-squeezed states are much more
harder to be reconstructed via a measure-and-prepare strategy [30].

Figure 3: (Color online) The memory fidelities as a function of optical depth
for ${\chi}$ = 0.01.

##  3 Implementation

We now focus on a possible experimental implementation for the present
protocol. We propose to use praseodymium ions doped into an Y2SiO5 crystal.
This system has been used for implementing an efficient frequency-selective
population transfer (qubit distillation) [31] and a long-lived (storage time
above 40s) EIT-based memory [32]. In particular, in such system, high
efficient photon-echo-based QMs [13] have also been demonstrated. We consider
the 3H4 to 1D2 transition (605.977nm) for site 1 in a 0.02 ${\%}$ Pr3+:Y2SiO5
crystal [28]. Its homogeneous and inhomogeneous line-width are about
${\gamma}$ = 4.4 GHz and ${\Gamma}$ = 1 KHz, respectively. For light, we
suggest preparing the input squeezed optical pulse with duration ${\tau}$ =
${1~{}\mu s}$, which ensures ${\gamma\tau\gg 1}$. If we set the storage time
${T=10~{}\mu s}$, the value of ${\chi}$ can be determined to be 0.01. For
experimental feasible parameter ${\alpha L=2.5}$ [23], one can achieve ${F}$ =
98.16 ${\%}$, 89.65 ${\%}$, and 78.67 ${\%}$ for 3 dB, 7 dB, and 10 dB
squeezing input, respectively.

##  4 Conclusion

We have studied the long-lived QM for squeezed state of propagating light in a
solid-state system based on HYPER. We found that, the HYPER technique has a
good performance in storage of SSL. Just as the storage of classical field,
the limitation of the QM for SSL mainly comes from the optical depth. The
larger the optical depth, the more squeezing of the input mode will be
preserved. Besides, the atomic decay is another factor that limits the
achievable fidelities, while we show that its influence on the storage process
can be suppressed via sacrificing the storage time. In conclusion, our work
provides hopeful possibilities for the practical realization of efficient QM
for SSL based on photon echoes.

##  5 Acknowledgments

We acknowledge support from the Natural Science Foundation of Zhejiang
province (Grants No. LY12A05001), the department of education of Zhejiang
province (Y201120838) and the Natural Science Foundation of China (Grant No.
11074190).

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