# The Equation of State with EPOS model

Maria Stefaniak  Warsaw University of Technology, pl. Politechniki 1, 00-661
Warsaw, Poland
GSI Helmholtzzentrum für Schwerionenforschung, Planckstr. 1, 64291 Darmstadt,
Germany
Klaus Werner
Johannès Jahan  SUBATECH, University of Nantes – IN2P3/CNRS – IMT Atlantique,
Nantes, France  Hanna Zbroszczyk  Warsaw University of Technology, pl.
Politechniki 1, 00-661 Warsaw, Poland

###### Abstract

Transitions between different states of matter and their thermodynamical
properties are described by the Equation of State (EoS). A universal
representation of the equation of state of QCD for the whole phase diagram has
not yet been determined. Expectation of the systems to undergo various types
of transitions depending on the temperature, the chemical potential $\mu_{B}$
and other theromodymnamical features makes that puzzle even more challenging
to solve. Furthermore, it is not apparent which experimentally measurable
observables could provide the useful information for the determination of EoS.
The application of different EoS for hydrodynamical evolution was introduced
in the EPOS generator, what allowed to study the effect of its changing on the
final observables. The family of EoS proposed by the BEST Collaboration was
implemented. The Critical Point position and the strength of criticality
variations were investigated with particle yield, transverse momentum spectra,
flow, and moments of the net-proton distributions.

††preprint: APS/123-QED

##  I Introduction

The determination of the Equation of State (EoS) is crucial for the complete
description and understanding the QCD phase diagram. The rules of physics have
to be universal for all the systems existing in nature. The relations between
thermodynamical quantities characterizing different states of nuclear matter
are depicted in the construction of EoS. The substantial topic of present
research is investigation of various transitions between partonic and hadronic
medium. It is expected to undergo diverse processes depending on the
temperature and baryon-chemical potential ($\mu_{B}$) of the medium. EoS
expresses the relations between various matter parameters such as pressure,
temperature, energy density, speed of sound, and former. It is not trivial to
characterise the whole phase diagram area. Consequently, such features of QCD
as chiral symmetry breaking or confinement, are not fully understood. There
are various attempts performed to generate the EoS, which will allow one to
characterise the whole QCD phase diagram, starting from $\mu_{b}=0$ ending on
cold neutron stars [1]. These EoS introduce the first-order phase transition
for finite baryochemical potential and relatively lower temperatures.
Nevertheless, some of them provide information about the Critical Point
location and properties of this transition [2, 3].

##  II BEST Equation of State

Collaboration Beam Energy Scan Theory (BEST) proposed a family of Equations of
State describing the same region of QCD phase diagram as studied in the BES
program [3, 4]. It covers the region of chemical potential in range
$0\leq\mu_{B}\leq 450$ MeV and T between $30$ MeV and $800$ MeV. The equations
respect the lattice QCD results up to $\mathcal{0}(\mu_{B}^{4})$. They
consider existence of cross-over transition, first order phase transition and
give possibility to choose the location of Critical Point on the QCD phase
diagram $\\{T,\mu_{B}\\}$.

In the proposed EoS, the consistency with the derived from the first principle
calculations is preserved. However, the problem with the limitation in the
extraction of information about the medium characterized by the finite
$\mu_{B}$ is still present. Although the Taylor series used to expend the
region studied with the QCD EoS never goes beyond the $\mu_{B}$ higher than
one at Critical Point, the universality class to which the Critical Point (CP)
belongs is known, and it is possible to study the behaviour of EoS in some
limited neighbourhood of CP.

The calculations applied by BEST are based on a strategy descibed in [3, 4]:

  1. 1.

In the region close to CP, the suitable parametrization is used in the
description of EoS universal scaling behaviour in the 3D Ising model;

  2. 2.

The 3D Ising model phase diagram is mapped using a parametric change of
variables (not universal) onto the QCD one;

  3. 3.

The Ising model thermodynamics is used to estimate the critical contribution
to Lattice QCD expansion coefficients;

  4. 4.

The full pressure is reconstructing - ensuring compatibility with Lattice QCD
at $\mu_{B}=0$ and consisting the proper critical behaviour.

Figure 1: The linear transformation is used in mapping the 3D Ising model
diagram on the QCD one [4].

In the Fig. 1 the non-universal mapping procedure is illustrated. Using six
parameters the critical thermodynamics is linearly transferred to QCD (Ising
variables to QCD coordinates: $(h,r)\mapsto(T,\mu_{B})$).

Despite the possibility to locate the CP on the thermodynamics and
hydrodynamic simulations, the proposed approach depends on the choice of
parameters in the mapping of Ising $\rightarrow$ QCD, which is not arbitrary.
It is a consequence of a need to match Lattice QCD data which puts some
limitations on the CP position.

###  II.1 Selection of the EoS parameters

The primary studies possible due to the modifications and development of EPOS
model allow one to study the impact on final observables of the changes
between the variety of EoS. As mentioned in Section II, the BEST EoS is, in
reality, the ”family” of EoS, the set of various EoS tables. To obtain the EoS
one needs to choose the set of parameters corresponding to mapping properties
and locating the CP onto the QCD phase diagram.

The composition of the input parameters is crucial in setting the strength of
the ”criticality” of the transitions of the matter, moreover via changing the
position of the CP one can expect the cross-over or the first-order
transitions in the evolution of examined simulated systems. The structure of
the parameter input file is:

$MODE\quad
T_{0}\quad\kappa\quad\mu_{BC}\quad\Delta_{\alpha_{1},\alpha_{2}}\quad\omega\quad\rho$

where:

  * •

MODE - corresponds to the CP location and choice of following parameters. In
this studies the CP lies on parabola parallel to the chiral transition line -
what reports to MODE = PAR;

  * •

$T_{0}$ \- the value of temperature at which the parabolic pseudo-critical
line crosses the T axis;

  * •

$\kappa$ \- the curvature of the transition line at the T axis;

  * •

$\mu_{BC}$ \- the chemical potential at the critical point: since both
critical temperature $T_{C}$ and $\alpha_{1}$ are determined by this choice,
they do not need to be input;

  * •

$\Delta_{\alpha_{1},\alpha_{2}}$ \- the difference between two angles (see
Fig. 1)

  * •

$\omega$ \- the global scaling parameter in the mapping (the higher $\omega$
the less criticality in transitions of matter);

  * •

$\rho$ \- the relative scaling in the mapping [4];

The $T_{C}$, $\alpha_{1,2}$ can be easily calculated from the given
parameters:

$\displaystyle T_{C}=T_{0}+\kappa/T_{0}\mu_{BC}^{2},$ (1a)
$\displaystyle\alpha_{1}=180/\pi\bigg{|}\arctan(-\frac{2\kappa}{T_{0}\mu_{BC}})\bigg{|},$
(1b) $\displaystyle\alpha_{2}=\alpha_{1}+\Delta{\alpha_{1},\alpha_{2}}$ (1c)

##  III EPOS Model

EPOS is an abbreviation of Energy conserving quantum mechanical multiple
scattering approach, based on Partons (parton ladders), Off-shell remnants,
and Saturation of parton ladders.

The model consists of several phases of evolution:

  * •

initial stage (based on the Parton Gribov-Regge theory) [5, 6],

  * •

core/corona division [7, 8, 9]

  * •

hydrodynamical evolution [8],

  * •

hadronization based on the given EoS,

  * •

hadron rescattering (based on UrQMD [10, 11]),

  * •

resonance decays.

In the model’s theoretical framework, the crucial element is the sophisticated
treatment of the hadron-hadron scattering and the initial stage of the
collisions at ultra-relativistic energies. It is highly relevant in the
understanding of possible parton-hadron phase transition. The merged approach
of Gribov-Regge Theory (GRT) and the eikonalised parton model is utilised to
provide proper treatment of the first interactions happening just after the
collision - satisfying conservation laws and equal treatment of subsequent
Pomerons [6, 5, 12].

If the density of the strings is very high, they cannot decay independently,
and this situation is characteristic for the heavy-ions and the high-
multiplicity $p-p$ collisions. In EPOS the dynamical process of division of
the strings segments into core and corona is introduced in order to deal with
this issue [7, 8, 9].

The separation is based on the abilities of a given string segment to leave
the ”bulk matter”. As the criteria for deciding, if it goes to a given group,
the transverse momentum of the element and the local string density is taken
into account. If the string segment belongs to the very dense area, it will
not escape but will build the core, which will be driven in the next step by a
hydrodynamical evolution. When the segment originates from the part of the
string close to a kink, which is characterized by the high transverse momenta,
it escapes the bulk matter and join the corona and consequently will show up
as a hadron (jet-hadrons). There is also a possibility that the string segment
is close to the surface of the dense part of the medium, and its momentum is
high enough to leave it; it also becomes a corona particle. The following
equation is used for the determination of core and corona:

$p^{new}_{t}=p_{t}-f_{Eloss}\int_{\gamma}\rho dL$ (2)

where: $\gamma$ is the trajectory of the segment, $\rho$ the string density,
and $f_{Eloss}$ a non-zero constant for $p_{T}>p_{T,1}$, null for
$p_{T}<p_{T,2}$ and interpolated linearly between $p_{T,1}$ and $p_{T,2}$. If
the $p^{new}_{t}$ is positive for a given segment, it escapes and becomes a
corona particle; in the opposite case, it contributes to the core.

Figure 2: The Monte Carlo simulation: full circles correspond to string
segments contributing to core and open to corona. The big circles are just for
the eye-guidance showing the surface of collided nuclei [9]

As it has been studied [13, 14, 15, 16], the QGP does not expand like an ideal
fluid, and the effect of the bulk viscosity has to be taken into account in
the simulations. In EPOS, the 3D+1 viscous hydrodynamics is applied, providing
an excellent description of the collective expansion of the matter [8].

In the simulations, the highly important is the definitive treatment of
individual events - the generalization in considering smooth initial
conditions for all events is not applied. The event-by-event (ebe) approach in
hydrodynamical evolution is based on the random flux tube initial conditions
[8]. It has a relevant impact on the final observables such as spectra or
various harmonics of flow.

The final part of the simulation uses a so-called hadronic afterburner \-
UrQMD [10, 11].

When the density is very high and the mean free paths of constituent particles
are small about any macroscopic length scale, the hydrodynamic description can
be used - in the initial phase of the QGP evolution. With the cooling of the
system, the density and the mean free paths decrease; oppositely, the $\eta/s$
increases. Finally, the differences in the mean free path of various particle
species become relevant, and the collective description of the system becomes
not adequate. When the density and the temperature are low enough, the kinetic
theory is applied using the UrQMD code [10, 11].

The particles can interact only when they leave the hyper-surface of the
freeze-out. The $2\rightarrow n$ hadronic scattering is performed according to
the measure reaction cross sections [17]. The 60 different baryonic species
and their anti-particles, about 40 mesonic states are considered [11, 10].
There are implemented such interactions between hadrons as [18]:

  * •

elastic scattering,

  * •

string excitations,

  * •

resonance excitations,

  * •

strangeness exchange reactions

The hadronic scattering has a significant impact on the final observables [19,
20].

In this project, the possibility to introduce the new EoS was applied giving
the possibility to study its impact on the simulation results.

##  IV Results and discussion

###  IV.1 Simulations

The two collision systems were studied: Au+Au collisions at
$\sqrt{s_{NN}}=7.7$ GeV and $27$ GeV (the lowest and the medium collision
energy form Beam Energy Scan I at RHIC). The simulations using EPOS 3.4 model
were performed using computer clusters at the Centre de Calcul de l’IN2P3/CNRS
in Lyon. This is the preliminary version of the model, which is still under
development. In the simulations the given Equations of State were applied:

  * •

X3F cross-over, 3 flavour conservation [REF]

  * •

BEST EoS with various parameters listed in Table 1.

Table 1: Sets of input parameters for the construction of nine BEST EoS used in the EPOS simulations. Number | $MODE$ | $T_{0}$ | $\kappa$ | $\mu_{BC}$ | $\Delta_{\alpha_{1,2}}$ | $\omega$ | $\rho$ | $T_{C}$ | $\mu_{BC}$ | $\alpha_{1}$ | $\alpha_{2}$ | $\omega T_{C}$ | $\rho\omega T_{C}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
BEST 1: | PAR | 155 | -0.0149 | 350 | 90 | 1 | 2 | 143 | 350 | 3 | 93 | 143 | 286
BEST 2: | PAR | 155 | -0.0149 | 350 | 90 | 4 | 1 | 143 | 350 | 3 | 93 | 572 | 572
BEST 3: | PAR | 155 | -0.0149 | 420 | 90 | 0.75 | 2 | 138 | 420 | 4 | 94 | 103 | 207
BEST 4: | PAR | 155 | -0.0149 | 350 | 90 | 10 | 1 | 143 | 350 | 3 | 93 | 1432 | 1432
BEST 5: | PAR | 169 | -0.0149 | 420 | 90 | 0.25 | 2 | 165 | 3 | 86 | 179 | 23599 | 6749k
BEST 6: | PAR | 169 | -0.0149 | 420 | 90 | 1 | 1 | 153 | 420 | 4 | 94 | 153 | 153
BEST 7: | PAR | 169 | -0.0149 | 420 | 90 | 0.5 | 1 | 153 | 420 | 4 | 94 | 76 | 76
BEST 8: | PAR | 174 | -0.0149 | 440 | 90 | 1 | 1 | 157 | 440 | 4 | 94 | 157 | 157
BEST 9: | PAR | 178 | -0.0149 | 300 | 90 | 1 | 1 | 170 | 300 | 2 | 92 | 170 | 170

The only element of the whole simulation performed by EPOS model which changes
is the Equation of State. Concluding, all the presented data sets can be used
in the direct comparison of the proposed EoS. The ”BEST 5” set of parameters
was excluded due to the non-physical output.

###  IV.2 Production of particles

The production mechanism highly depends on the potentials characterising the
system. The higher the $\mu_{B}$, the more produced baryons than antibaryons.
Such relations are illustrated in Figure 3, where the most central $0-5\%$
collisions of Au+Au simulated with EPOS 3.4. model are compared with STAR data
published in [21]. Various EoS sets of parameters were used in performed
simulations; the numbers of EPOS’ data sets correspond to these listed in the
Table 1. The relations between particles’ and antiparticles’ production is
reflected using ratios in Figure 4.

The higher number of produced baryons than antibaryons proves that in EPOS
simulations, the impact of non-zero baryon potential is kept for all the
proposed EoS. The model reflects the experimental data reasonably, except for
pions. The number of these both negative and positive mesons is overestimated.
Nonetheless, their ratio is kept.

The possible reason for such discrepancies could be the too wide rapidity
distribution of simulated data. In the experimental analysis, the selection of
particles characterised by the $|y|<0.1$ is very narrow. In such a case, even
a small deviation in the rapidity distribution strongly affects the performed
investigation.

In both Fig. 3 and 4 no relevant differences between simulations obtained with
various EoS are observed. Notwithstanding the $X3F$ Equation of State
corresponds to the cross-over transition, which is not expected to happen for
cooling systems created in collisions of Au+Au at $\sqrt{s_{NN}}=7.7$ GeV.

Figure 3: Particle yields for Au+Au most central $0-5\%$ collisions at
$\sqrt{s_{NN}}=7.7$ GeV simulated with EPOS 3.4. model using various EoS and
compared with STAR data [21]. Figure 4: Particle ratios for Au+Au most central
$0-5\%$ collisions at $\sqrt{s_{NN}}=7.7$ GeV simulated with EPOS 3.4. model
using various EoS and compared with STAR data [21].

###  IV.3 Moments of particle distributions

The non-monotonic behaviour in the event-by-event fluctuations of globally
conserved quantities is treated as one of the signatures of the presence of CP
[22, 23]. The moments of distributions characterising the given fluctuations
are:

  * •

the mean ($M$)

  * •

the standard deviation ($\sigma$)

  * •

the skewness ($S$), which corresponds to the asymmetry of the distribution

  * •

the kurtosis ($\kappa$), reflecting the degree to which the distribution is
peaked relative to the normal distribution.

They are linked with the corresponding higher-order thermodynamic
susceptibilities, and the system’s correlation length [24, 25], which are
expected to deviate for large samples in equilibrium at the CP. As a result of
slowing down in the vicinity of CP, in reality, the system is driven away from
the thermodynamic equilibrium, and the maximum value of correlation length
attains 1.5-3 fm [25]. During the fireball evolution after the hadronisation
stage, the freeze-out signal information can dissipate [26]. However, if it
survives, the higher moments can become useful in studies of CP’s location. As
the CP’s location is changed in various used EoS, the moments of particle
distributions are expected to be a useful tool in the performed investigation.
In Figures 5 and 6 the comparison between two examined collision energies
$\sqrt{s_{NN}}=7.7$ and $27$ GeV are shown for $S\sigma$ and
$\kappa\sigma^{2}$ as a function of number of participants. These combinations
of moments are related directly with second and fourth order cumulants [27],
which carry the information about possible critical behaviour of the system.
Only BEST9 data set does not show the energy dependence, where the lower
energy correspond to the higher values of $S\sigma$, what is the expected
trend [28]. The significant disparities are present for various EoS for the
$\sqrt{s_{NN}}=7.7$ GeV data even with the presence of high statistic
uncertainties. Both $S\sigma$ and $\kappa\sigma^{2}$ for the
$\sqrt{s_{NN}}=27$ does not show any multiplicity dependence.

Figure 5: The $S\sigma$ for Au+Au collisions at $\sqrt{s_{NN}}=7.7$ (full
markers) and $27$ GeV (empty markers) as a function of number of participants.
The shifts of the point on x-axis are applied for better visualisation.
Figure 6: The $\kappa\sigma^{2}$ for Au+Au collisions at $\sqrt{s_{NN}}=7.7$
(full markers) and $27$ GeV (empty markers) as a function of number of
participants. The shifts of the point on x-axis are applied for better
visualisation.

Figures 7 and 8 show the $S\sigma$ and $\kappa\sigma^{2}$ integrated overall
centralities as a function of the collision energy. The significant energy
dependence is present for $S\sigma$ for all EoS. All the points at given
energy are within the statistic errors, so no clear statement about the
discrepancies between the EoS can be made. $\kappa\sigma^{2}$ shows bigger
variations between different EoS data sets. At $\sqrt{s_{NN}}=7.7$ GeV the
highest point corresponds to the EoS where the CP is located at high T and low
$\mu_{B}$ and the simulated system is expected to go through the first-order
transition, while the negative value is related with BEST 4, where the
criticality is the less pronounced. For data sets simulated at
$\sqrt{s_{NN}}=27$ GeV the differences are smaller; however, the BEST9 value
is the highest. The energy dependence is not definite.

Figure 7: The $S\sigma$ for Au+Au collisions at $\sqrt{s_{NN}}=7.7$ and $27$
GeV  as a function of the collision energy $\sqrt{s_{NN}}$. The zoomed window
corresponds to the points got collisions at $\sqrt{s_{NN}}=27$ GeV. The shifts
of the point on x-axis are applied for better visualisation. Figure 8: The
$\kappa\sigma^{2}$ for Au+Au collisions at $\sqrt{s_{NN}}=7.7$ and $27$ GeV
as a function of the collision energy $\sqrt{s_{NN}}$. The zoomed window
corresponds to the points got collisions at $\sqrt{s_{NN}}=27$ GeV. The shifts
of the point on x-axis are applied for better visualisation.

The measurements of the net-proton distributions’ moments show the differences
between data simulated using various Equations of State. They are more
pronounced in peripheral collisions where we do not expect the big
contribution form the ”core” particles, so consequently less dependent on the
EoS.

###  IV.4 Particles’ dynamics

The differences between the EoS were searched in the dynamics of the expanding
matter. The listed below observables were investigated:

  * •

transverse momentum spectra of identified hadrons: p, $\bar{p}$, $K^{\pm}$,
$\pi^{\pm}$ (Au+Au at $\sqrt{s_{NN}}=7.7$ and $27$ GeV, $0-5\%$ and $60-80\%$
centrality ranges),

  * •

elliptic flow of identified hadrons: p, $\bar{p}$, $K^{\pm}$, $\pi^{\pm}$
(Au+Au at $\sqrt{s_{NN}}=7.7$ and $27$ GeV, $0-80\%$ centrality),

  * •

event-by-event $v_{2}^{obs}$, not corrected with resolution of event plane
(Au+Au at $\sqrt{s_{NN}}=7.7$ GeV, $0-80\%$ centrality).

None of studied obeservables related to the expansion of the matter showed the
variations between different EoS.

##  V Conclusion

The studies of various Equations of State implemented in the EPOS model were
described. The development of the generator’s code by introducing new EoS gave
a unique possibility to investigate the impact of EoS on the final
observables. Apart of the EoS, the whole structure of the model remained
unchanged. The moments of net protons distributions gave important information
about the variations of EoS. It makes the investigation of fluctuation of the
particle distributions the most adequate observable for studies of the EoS.

##  VI Acknowledgments

I would like to express the gratitude to Yurii Karpenko and Gabriel Sophys for
fruitful discussions. This work was supported by the Grant of National Science
Centre, Poland, No:  2021/41/B/ST2/02409 and  2020/38/E/ST2/00019 . Studies
were funded by IDUB-POB-FWEiTE-3, project granted by Warsaw University of
Technology under the program Excellence Initiative: Research University (ID-
UB), Deutsche Akademische Austauschdienst, and GET$\\_$INvolved Programme.

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# High visibility on-chip quantum interference of single surface plasmons

Yong-Jing Cai, Ming Li, Xi-Feng<EMAIL_ADDRESS>Chang-Ling Zou, Xiao
Xiong, Hua-Lin Lei, Bi-Heng<EMAIL_ADDRESS>Guo-Ping
<EMAIL_ADDRESS>and Guang-Can Guo  Key Lab of Quantum Information,
University of Science and Technology of China, CAS, Hefei, Anhui, 230026,
China.  Synergetic Innovation Center of Quantum Information $\&$ Quantum
Physics, University of Science and Technology of China, Hefei, Anhui 230026,
China

###### Abstract

Quantum photonic integrated circuits (QPICs) based on dielectric waveguides
have been widely used in linear optical quantum computation. Recently, surface
plasmons have been introduced to this application because they can confine and
manipulate light beyond the diffraction limit. In this study, the on-chip
quantum interference of two single surface plasmons was achieved using
dielectric-loaded surface-plasmon-polariton waveguides. The high visibility
(greater than 90%) proves the bosonic nature of single plasmons and emphasizes
the feasibility of achieving basic quantum logic gates for linear optical
quantum computation. The effect of intrinsic losses in plasmonic waveguides
with regard to quantum information processing is also discussed. Although the
influence of this effect was negligible in the current experiment, our studies
reveal that such losses can dramatically reduce quantum interference
visibility in certain cases; thus, quantum coherence must be carefully
considered when designing QPIC devices.

Photonic integrated circuits (PICs), in which multiple photonic functional
components comprise a single chip, have attracted considerable attention owing
to their small footprints, scalability, reduced power consumption, and
enhanced processing stability. In addition to their wide application in
classical information processing, integrated photonic quantum logic gates and
Shor’s quantum factoring algorithm have been demonstrated on these chips
Obrien08 ; Obrien09 ; thus, they show great feasibility and high operation
fidelity. More recently, much effort has been dedicated to surface plasmon
polaritons (SPPs), which are electron density waves excited at the interface
between a metal and a dielectric material Ebbesen03 to further condense PICs
beyond the diffraction limit. Not only can SPPs confine light at the nanoscale
Sch , they are also useful for integrated polarization-controlling devices PBS
; Polarizer . Studies using periodic metallic hole arrays provided the first
experimental evidence that quantum entanglement can be preserved in the
photon-SPP-photon conversion processAlt ; energy ; ren06 . Furthermore, the
non-classical statistics of SPPs have been demonstrated using basic quantum
Hong-Ou-Mandel (HOM) interference Hong , in both long-range plasmonic
waveguides (weakly confining waveguide) Fuji  and sub-wavelength metal
plasmonic waveguides Rei . These studies indicate that assembling quantum PICs
(QPICs) using plasmonic components is possible.

However, two obstacles remain that hinder the development of SPP-based QPICs.
The first is that the experimental raw visibility of the quantum interference
realized in plasmonic waveguides is below 50% Rei , which is the boundary
between classical and quantum interference. This low visibility is not
compelling evidence that single plasmons are usable for quantum information
processing. Interference visibility is so important that higher quantum
interference visibility implies higher operation fidelity and a higher
probability of success. For example, the HOM interference with 95% visibility
that can be achieved in QPICs based on dielectric waveguides is used to
realize quantum controlled-NOT gates Obrien08 . Second, loss is unavoidable in
QPICs that are based on plasmonic waveguides Sch , owing to the absorption of
metals. Although the properties of lossy quantum channels have been recently
observed and studied in the context of free-space quantum optics Bar ; Liu ,
the influence of such losses on quantum processing using QPICs remains unknown
Ste ; Gupta .

The current study experimentally achieved the on-chip quantum interference of
single plasmons using dielectric-loaded plasmonic waveguides at telecom
wavelengths. The visibility was as high as $95.7\pm 8.9\%$, which
unambiguously demonstrates the bosonic nature of single plasmons and paves the
way for the performance of basic quantum operations in plasmon-based QPICs.
Furthermore, an SPP waveguide might provide a perfect testing ground for
studying lossy photonic devices owing to the relatively high loss of such a
devices compared with dielectric devices. Our analysis indicates that sub-
wavelength plasmonic components can be used as quantum devices for QPICs only
when the loss effect is carefully addressed because loss can significantly
reduce quantum interference visibility. Because loss is inevitable in
waveguides, it is necessary to consider its influence on quantum coherence
when designing QPIC structures.

Figure 1: (a) Schematic of the experimental setup. Photon pairs are generated
via a degenerate type-II non-collinear spontaneous parametric down-conversion
(SPDC) process. A 1.5-W pump laser (775 nm, Coherent Inc.) is focused on a
1-mm-long BBO crystal. The produced twin photons (1550 nm) are separated in
free space by a $6^{\circ}$ angle based on the phase-matching condition and
directed to different optical single-mode fibers. A motorized delay line in
one of the arms allows the optical-path-length difference between the two
photons to be controlled with 100-nm resolution. Using the fiber taper
connected to the single-mode fiber, the single photons are converted into
single plasmons in the plasmonic waveguide, which then interfere with each
other. We collected the photons from P2 of the on-chip directional coupler
(DC) and sent them to a second 50/50 fiber BS. Coincidence measurements
revealed the quantum properties of single plasmons. (b) A scanning electron
microscope (SEM) image of part of a typical plasmonic DC structure. (c) A CCD
image showing the coupling of the fiber taper and the SPP waveguide. A single-
mode fiber was used to collect the photons from P2 using the end-fire coupling
method.

## Results

The principle of quantum interference. HOM interference, a basic type of
quantum interference that reflects the bosonic properties of a single
particle, is generally used to test the quantum properties of single plasmons
Hong . In addition to its fundamental importance within quantum physics, the
HOM effect underlies the basic entanglement mechanism in linear optical
quantum computing Klm  because two-qubit quantum gates, which form the core of
linear optical quantum computing, can be obtained via classical and quantum
interference (HOM interference) effects followed by a measurement-induced
state projection.

HOM interference can be described as follows: when two indistinguishable
photons enter a 50/50 beam splitter (BS) from different sides at the same
time, according to the exchange symmetry of photons (bosons), a 50% chance
exists of obtaining two photons in output port 1 (P1); furthermore, a 50%
chance probability exists of obtaining two photons in output port 2 (P2).
However, the two photons will never be in different output ports. The twin
photon state $|1,1\rangle$ is converted into a quantum superposition state
$1/\sqrt{2}(|2,0\rangle+|0,2\rangle)$. This phenomenon is a signal of photon
bunching and can only be explained using a quantum mechanism Lim . Experiments
typically control the arrival times of two photons by adjusting the path-
length difference between them and measure the photon coincidence of P1 and
P2. When two indistinguishable photons completely overlap at the BS, they give
rise to the maximum interference effect, and no coincidence exists. Visibility
is defined as $V_{1}=(C_{max}-C_{min})/C_{max}$, where $C_{max}$ is the
maximum coincidence and $C_{min}$ is the minimum coincidence. For perfect
quantum interference, $C_{min}=0$ and $V_{1}=1$, whereas for a classical
coherent laser, $V_{1}=50\%$. Consequently, to prove that destructive
interference is due to two-photon quantum interference, the visibility must be
greater than 50%. Here, we used a modified HOM interferometer (see Figure 1a).
We collected the photons from P2 of the first BS, sent them to the second
50/50 BS, and then measured the coincidence. According to quantum interference
theory, a 25% chance should exist for us to record a click when HOM
interference occurs and a 12.5% chance otherwise. In this case, visibility was
modified as follows: $V_{2}=(C_{max}-C_{min})/C_{min}$. For perfect quantum
interference,, $C_{max}=2C_{min}$ and $V_{2}=1$. Our modified interferometer
is capable of reflecting the indistinguishability of the input particles and
can tell us whether these plasmons are bosons.

Experimental design. In the current experiment, we chose a dielectric-loaded
SPP waveguide (DLSPPW) Kumar  to test the bosonic properties of the single
plasmons. A DLSPPW is a typical sub-wavelength plasmonic waveguide that is
formed by placing a dielectric ridge on top of a thin metal layer. Among the
various plasmonic-waveguide structures, DLSPPWs are promising for enriching
the functional portfolio of plasmonics owing to their dielectric-loading
properties, which have been demonstrated in practice. They can confine the
lateral size of propagating modes to the sub-wavelength scale and
simultaneously transmit photons and electrons in the same component. In
addition, because the energy is mostly confined to the surface of the metal,
highly efficient control of the waveguide-mode characteristics is possible.
For example, power-monitoring Kumar2  and switching Kala  elements with high
response speeds have been experimentally demonstrated in DLSPPWs. We used
nanofabrication techniques to prepare our plasmonic waveguide. Specifically,
our waveguide was constructed of polymethyl methacrylate (PMMA) and placed on
top of a 45-nm-thick gold layer deposited on a $SiO_{2}$ substrate. Figure 1b
shows a scanning electron microscope (SEM) image of part of the fabricated
sample.

Based on our calculations, the lateral size of the single-mode DLSPPW for
photons at 1,550 nm was 600nm$\times$600nm Theory , because such a waveguide
supports only one fundamental mode (see Figure 2b). The BS is realized using a
directional coupler (DC), which is composed of two waveguides. In the coupling
region, the evanescent fields of the two waveguide modes couple with each
other and exchange energy. As a result, two new coupling eigenmodes, the
symmetric (Figure 2c) and anti-symmetric (Figure 2d) superpositions of the two
waveguide modes, are generated. Owing to the different effective refractive
indices of these two modes, the beating of the two modes leads to a BS-like
function. By controlling the coupling strength, the amount of output at the
two waveguide ports (the splitting ratio) can be tuned. Using the engineered
waveguide gap, we obtained a coupling profile with a splitting ratio of
approximately 1:1.

Figure 2: (a) Three-dimensional simulation of field distribution on our
plasmonic DC structure. (b) Field distribution in single mode plasmonic
waveguide with lateral size of 600nm$\times$600nm. (c) Field distribution of
symmetric eigenmode in coupling section. (d) Field distribution of anti-
symmetric eigenmode in coupling section.

The coupling efficiencies among our SPP circuit, the external source, and the
detectors were particularly crucial because the quantum signals were weak
(approximately 7,000 photons pairs per second in our experiment). However, it
is difficult to directly connect our plasmonic waveguide to a single-mode
optical fiber because its lateral-mode field area is much smaller than that of
the fiber (diameter $6.8\mu m$, 980HP, Thorlabs Inc.). Therefore, we adopted
the alternative adiabatic method PBS ; Zou  to excite the plasmons using fiber
taper Dong ; Tong . As Figure 1c shows, the photons in the fiber are
adiabatically squeezed into the microfiber via the taper region and coupled to
the plasmon waveguide when the microfiber approached the waveguide. Owing to
the high efficiency conversion and evanescent field coupling, the ideal
conversion efficiency might have been higher than 99%. Under the limitations
imposed by the experimental conditions, the efficiency of our fiber taper
coupling system was estimated to be approximately 30%. Importantly, the
alignment direction of the fiber taper was vertical to the collection fiber,
thereby avoiding the collection of directly scattered photons from the end of
the fiber taper.

Figure 3: (a) HOM interference of the down-converted photon pairs measured
using a fiber 50/50 BS; the visibility was $95.5\pm 1.0\%$ and the optical
coherence length was $162.6\pm 5.0\mu m$. (b) Modified HOM interference of the
down-converted photon pairs measured using two fiber 50/50 BSs; the visibility
was $96.5\pm 3.1\%$, and the optical coherence length was $173.9\pm 5.7\mu m$.
(c) Quantum interference of single plasmons on DLSPPWs: For Sample 1, the
visibility was $95.7\pm 8.9\%$, and the optical coherence length was $191.6\pm
17.6\mu m$; for Sample 2, the visibility was $93.6\pm 6.7\%$, and the optical
coherence length was $193.0\pm 13.0\mu m$. All results are at the level of
single photons.

Quantum-interference results. The 1,550-nm quantum photon pairs were generated
via the spontaneous parametric down-conversion (SPDC) Burn  process of a BBO
crystal (Type-II phase matching, non-collinear) pumped by a 775-nm-wavelength
laser (Coherent Inc.; see Figure 1a). The down-converted twin photons
consisted of one photon in the horizontal (H) polarization and one in the
vertical (V) polarization. The photons were separated into two paths, each of
which contained a prime reflector (PR), a half-wave plate (HWP, 1,550 nm), a
long-pass filter (LP; 830 nm), and a narrow-band filter (IF, 1,550 nm, 8.8 nm
FWHM). After these components, the two photons, which now had the same
polarization, were guided into two separate single-mode fibers. One of the
fiber couplers was installed on a motorized stage to adjust the optical path.

As shown in Figure 3a, the indistinguishability of the produced photon pairs
was first characterized using a standard HOM interferometer with a fiber BS.
The dip represented the quantum interference of two photons that arrived at
the BS simultaneously, and the coherence of the photons determined its width.
The quantum-interference results were fit using $N_{HOM}=C\cdot[1-V\cdot
e^{-(\Delta\omega\cdot\Delta\tau)^{2}}]$Hong , where $N_{HOM}$ is the measured
coincidence count, $C$ is a fitting constant, $V$ is the quantum-interference
visibility, $\Delta\omega$ is the bandwidth of the photons, and $\Delta\tau$
is the optical time delay. For perfect quantum interference of
indistinguishable photon pairs, the visibility should be unity. Here, we
obtained a visibility of $V=95.5\pm 1.0\%$ and an optical coherence length of
$c/\Delta\omega=162.6\pm 5.0\mu m$, where $c$ is the speed of light. The
deviation of the visibility from $100\%$ was attributed to the polarization
distortion of the photons during propagation in the fiber, photon source
variability , or both. We also tested the modified quantum interferometer, in
which photons from one output port were divided using a second fiber BS and
detected with two single-photon detectors. Fitting the experimental results
(Figure 3b) using the function $N_{M}=C\cdot[1+V\cdot
e^{-(\Delta\omega\cdot\Delta\tau)^{2}}]$, we obtained a visibility of $96.5\pm
3.1\%$ and a coherence length of $173.9\pm 5.7\mu m$. These values were
consistent with standard HOM interference.

Finally, we observed the quantum interference of single plasmons using the
modified quantum interferometer, in which two single photons from the fiber
excited plasmon pairs in separate waveguides, and quantum interference
occurred in the coupling section. We sought to collect the two plasmons from
the two output ports and record their coincidence with a standard HOM
interferometer; to do so, we required two additional fiber tapers to collect
the signal. To avoid this requirement, we simplified the experimental design
by collecting the photons scattered from P2 using an end-fire-coupled single-
mode fiber. Using the second fiber BS, we divided the collected photons into
two ports and measured the coincidence. Three samples were measured, and the
visibilities were $95.7\pm 8.9\%$, $93.6\pm 6.7\%$, and $93.1\pm 16.5\%$.
These values are well above the classical limitation of $50\%$. The coherence
lengths of the plasmons were also calculated using the experimental data,
yielding $191.6\pm 17.6\mu m$,$193.0\pm 13.0\mu m$ and$146.4\pm 9.4\mu m$,
which were similar values to those of the photons. Our results demonstrate
that although the electron is a fermion, a single plasmon (i.e., the quasi-
particle of a collective electron-density wave) acts as a boson. The high
visibility also suggests that plasmonic structures can be used in QPICs.

Discussion. In this section, we address the second question: what is the
influence of loss on quantum interference visibility? The inevitable loss of
SPPs attenuates the amplitude of light; thus, gain materials are often used to
compensate for this loss. In addition, the first-order coherence of the
photons is destroyed during absorption and re-emission processes. It is
necessary to determine how this loss influences second-order quantum
interference visibility and under what conditions these losses are tolerable.
The following discussion provides a detailed account of the two-photon quantum
interference of lossy channels based on our plasmonic DC structure.

The operation of a four-port DC can be described as follows:

$\begin{pmatrix}b_{1}^{\dagger}\\\
b_{2}^{\dagger}\end{pmatrix}=\begin{pmatrix}r&t\\\
t&r\end{pmatrix}\begin{pmatrix}a_{1}^{\dagger}\\\
a_{2^{\dagger}}\end{pmatrix},$ (1)

where $a_{1}^{\dagger}$ and $a_{2}^{\dagger}$ as well as $b_{1}^{\dagger}$ and
$b_{2}^{\dagger}$ are the creation operators of the input and output boson
particles, and $r$ and $t$ are the amplitudes of the reflection and
transmission coefficients. The output state of the input twin-particle state
$|1,1\rangle$ is

$|\Phi\rangle_{out}=\sqrt{2}rt|2,0\rangle+\sqrt{2}rt|0,2\rangle+(r^{2}+t^{2})|1,1\rangle$
(2)

multiplied by a normalization factor. Here, we discard the terms that
represent the loss of one or two particles because only the coincidence counts
were recorded in the experiment. The probability of finding two particles in
the same mode (proportional to the HOM interference visibility) is

$P=\dfrac{4|rt|^{2}}{4|rt|^{2}+|r^{2}+t^{2}|^{2}}.$ (3)

For a lossless system, the DC is characterized by its classical transmission
and reflection coefficients, $|r|^{2}$ and $|t|^{2}$. Thus, designing a DC
with $|r|^{2}=|t|^{2}$ should optimize quantum interference. However, for a
lossy system, the structures and microscopic transport process of the DC will
determine the second-order quantum coherence. In our DLSPPW DC, when plasmons
were propagated in the coupling region of the two waveguides, they were in
coherent superpositions of symmetric and anti-symmetric modes (see Figures 2c
and 2d). The precise microscopic losses can be included using the coefficients
Loss

$\displaystyle r=\dfrac{e^{in_{2}k_{0}L}}{2}(e^{i\mathrm{Re}(\Delta
n)k_{0}L}e^{-\mathrm{Im}(\Delta n)k_{0}L}+1)$ (4) $\displaystyle
t=\dfrac{e^{in_{2}k_{0}L}}{2}(e^{i\mathrm{Re}(\Delta
n)k_{0}L}e^{-\mathrm{Im}(\Delta n)k_{0}L}-1)$ (5)

Here, $\Delta n=n_{1}-n_{2}$, where $n_{1(2)}$ is the effective refractive
index of the symmetric mode (the anti-symmetric mode), $k_{0}$ is the wave
vector in free space, and $L$ is the coupling length. The imaginary portion of
$n_{1(2)}$ corresponds to the propagation loss of the plasmons and leads to a
non-unitary operation matrix for the DC.

By substituting Eqs. (4) and (5 ) into Eq. (3), we obtain $P$, which is
related to the loss difference between the two intermediate eigenmodes
($\propto\mathrm{Im}(\Delta n)$) and the coupling length $L$ . When $L$ is
sufficiently large, the energy in the eigenmode with higher loss approximates
0 and can therefore be neglected compared with the lower-loss eigenmode. In
this case, $P$ decreases to 0.5, which corresponds to a classical random
process.

Figure 4 illustrates the relationships among $P$ and $L$ for a lossless DC
(black dots), our DLSPPW DC (blue dots), and a metal-strip DC (red dots) in
which we selected the $L$ that corresponded to a 50/50 splitter . For a
lossless DC, $P=1$ for any selected $L$. In our sample, $P$ slowly decreased
as $L$ increased. This result is because the difference between $n_{1}$
($1.318-0.00426i$) and $n_{2}$ ($1.150-0.00437i$) is small; therefore, we were
able to achieve a high interference visibility for a small $L$. For the metal-
strip DC used in Rei , $P$ fastly decreased as $L$ increased because the
difference between $n_{1}$ ($2.036-0.02i$) and $n_{2}$ ($1.841-0.01i$) was
much larger, especially in the imaginary portions.

The influence of loss on quantum coherence defined the limitations of lossy
QPIC devices. The high-order quantum interference of photons should be
considered when designing integrated photonic components because the
microscopic processes of photons in these devices might deviate from the
expected unitary evolution.

In summary, we experimentally demonstrated that single plasmons can be used as
qubits to perform on-chip quantum information processing. The discussion
presented here regarding loss also introduces a platform for using plasmonic
structures to investigate the on-chip quantum-decoherence phenomenon.
Additional investigations should consider using single plasmons as qubits to
carry quantum information and achieve on-chip linear optical computations or
quantum simulations.

Figure 4: The relationship between $P$ and the coupling length $L$. The black,
blue, and red dots represent the theoretical calculations for a lossless DC,
our DLSPPW DC and a metal-strip DC Rei , respectively. $R$ decreases as $L$
increases and converges to 50% for sufficiently large $L$ in lossy DCs. Here,
we used the $L$ values that corresponded to a 50/50 splitter.

## Acknowledgments

This work was funded by NBRP (grant nos. 2011CBA00200 and 2011CB921200), the
Innovation Funds from the Chinese Academy of Sciences (grant no. 60921091),
NNSF (grant nos.11374289, 10934006, 11374288, and 11104261), and NCET. We
thank Prof. Fang-Wen Sun, Bao-Sen Shi and Zheng-Wei Zhou for useful discussion
and Mrs Jun-Yi Xue from Qingdao No.2 High School of Shandong Province for her
help in optical measurement.

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