# Effect of Blue Component Stars in parameters of NGC 6866

Gireesh C. Joshi<EMAIL_ADDRESS>P.P.S.V.M.I. College,
Nanakmatta, U.S. Nagar-262311, Uttarakhand, India

###### Abstract

The open star clusters (OSC) are important tracers for understanding the
Galactic evolution. The parametric study of these astronomical-objects are
crucial task due to the appearing sequence of the members of OSC. These
members are defined through the various approaches such as photometric,
statistical, kinematics etc. In the present paper, we have been using the
photometric colours of the identified stars for categorized them into the blue
and red component groups and identification of these groups is possible
through (B-V) vs V colour magnitude diagram (CMD). Furthermore, the
influence/effect of these groups is also examined in the estimation of cluster
parameters. The stellar enhancement of cluster NGC 6866 is found through the
blue-component-stars (BCS) and the linear solution of best fitted values of
King Models of the radial-density-profile (RDP) gives the core radius as
$5.22{\pm}0.29~{}arcmin$. The good agreement of present estimated parameters
of the cluster with the literature seems to be an effective evidence to
consider BCS as the true representative of the cluster. The stellar
distribution of the cluster shows continuous phenomena of the mass
segregation. An effect of incompleteness of photometric data is well
understood in the term of the mass-function slope values, which is found to be
$-3.80{\pm}0.11$ and also shows increment nature with the incompleteness.

###### keywords:

(Galaxy): Open star cluster; individual: NGC 6866, blue-component; Most
probable members

††journal: New Astronomy

##  1 Introduction

The observational study of intermediate-age open clusters plays an important
role to constrain the theories of stellar and Galactic evolution (Sharma et
al., 2006). Though, the cluster region is also influenced by field stars,
therefore, the identification of probable members of open-star-clusters (OCs)
becomes a crucial task for parametric analysis. The influence of field stars
within the cluster may be occurred due to embedded position of the cluster on
the Galactic disk, which easily seems through the stellar distribution on
colour-magnitude diagram (CMD) and the proper motion plane. Such influence of
field stars known as contamination, which depends on the location of a
particular cluster on Galactic plane (Joshi et al., 2014). As a result,
scattered stellar distribution of stars may be occurred in the CMD and
definitely produced the photometric broadening of members in the sequence of
stellar evolution. Such broadening in appeared stellar sequence of the CMD may
be produce the uncertainty in exact identification of the stellar sequence,
which leads to estimation error in the fundamental parameters (reddening,
distance and age) of the cluster. It is believed that the accurate measurement
of all above said parameters will be possible after decontamination of the
field stars from cluster region. The above said decontamination would be
carried out through either the photometric statistical criteria (Sharma et
al., 2008) or the statistical cleaning in CMDs (Joshi et al., 2014). Recently,
Joshi et al. (2015) have been introduced the statistical cleaning procedure of
field star decontamination through the colour-magnitude distance for field
star decontamination, which is dependent on the grid size around the field
star. The size of grid would be fixed on the basis of comparison of cluster
and field stars in the term of their magnitude and colour (Joshi et al.,
2015). After field stars decontamination from the observed CMDs, it becomes
very effective tool to derive the precise values of fundamental parameters
with the comprehensive results of the Luminosity-function (LF) and the Mass-
Function (MF).
The outlines of this paper is described as following. The quality of available
photometric data of cluster has been prescribed in Section 2. The separation
procedure of blue and red component stars of this cluster is described in
Section 3. The RDPs and SNDPs of both type stars have been described in
Section 4 and Section 5, respectively. The identification procedure of giant
star sequence within cluster have been discussed in Section 6. The estimation
of reddening, distance and age are described in Sections 7 and 8. The MF study
and examination of nature of total-to-selective-extinction are done in
Sections 9 and 10 respectively. Finally, we have been summarized our results
and discussed their importance in Section 11.

##  2 The previous studies and data

Recently, three photometric catalogue of deep CCD observations are available
for this cluster. These catalogues are constructed by Joshi et al. (2012),
Janes et al. (2014) and Bostanci et al. (2015). The field of view of these
observations are about 13.5$\times$13.5 arcmin2, 42.8$\times$42.8 arcmin2 and
21.5$\times$21.5 arcmin2, respectively. All authors of these catalogue have
been derived the fundamental parameters of cluster based on their adopted
algorithm procedure. Bostanci et al. (2015) have been shown comparison of
their photometry with the Joshi et al. (2012) and Janes et al. (2014) in their
manuscript at Figure 4 and Figure 5 respectively. On the basis of above
prescribed figures, it is concluded that their photometry and Janes et al.
(2014) are well matched to each other. Since, Janes et al. (2014) is contained
the more field of view of the cluster region rather than others, therefore, it
is used to further analysis of influence of blue-component-stars (BCS) in the
estimation of cluster parameters. The first part of present manuscript
represents the role of BCS to constrain the dynamics of cluster.

Figure 1: a) Comparison of photometry catalogues between Joshi et al. (2012)
and Janes et al. (2014). b) The comparison of $(U-B)$ vs $(B-V)$ two-colour
diagrams of these catalogues. c) Here, We have also shown $(B-V)$ vs $V$ CMDs
from both cataloges. The black and blue dots of panels (b) and (c) are
represent stars from Joshi et al. (2012) catalogue and Janes et al. (2014)
catalogue respectively. The red dots of panel (b) are those stars of Joshi et
al. (2012) catalogue, which having $V$-magnitude less than 18 mag.

To see the difference of stellar magnitude of common stars between Janes et
al. (2014) and Joshi et al. (2012), we are cross-matched the photometric
magnitude of stars within scatter of $1~{}arcsec$. After comparison, we have
found 2092 common stars between both catalogues. Furthermore, we have been
obtained a slope in residual magnitudes of both catalogues, which has been
shown in the residual magnitude vs $V-$magnitudes plots (as depicted in the
left panels of Figure 1). A deep investigation of figures 4 and 5 of
manuscript of Bostanci et al. (2015) has clearly shown that their photometry
shows approximate zero residual with Janes et al. (2014) and non zero
magnitude residual with Joshi et al. (2012). On the behalf of these figures,
it seems to be some technical problems with the procedure of estimation of
stellar magnitude of the work of Joshi et al. (2012). For further
investigation for it, we have been constructed the plot of $(B-V)$ vs $(U-B)$
TCD (as shown in Figure 1-b). The stellar spread is clearly observed in this
TCD, which has been produced by those stars which having $V-$magnitude greater
than 18 mag. In this figure, we have also seen a upward shift of the stellar
distribution (depicted by red dots) through Joshi et al. (2012) catalogue in
the comparison of catalogue of Janes et al. (2014). This shifted colour values
of stars increase the reddening value of the cluster, which is easily noticed
by the comparison of resultant values of reddening from the both works.
In the Figure 1-c, we have been depicted the stellar distribution on the CMD
through the magnitude extracted from the both catalogue. The black and blue
dots of above described figure represents the stars from Joshi et al. (2012)
and Janes et al. (2014) respectively. It is seems to be approximate similar
stellar distribution in CMD except lower end of CMD, which leads to the
slightly change in the value of apparent distance modulus.The value of
reddening, $E(B-V)$, is required for calculation of cluster distance.
Reddening value of cluster is obtained by fitting Zero-Age-Main-Sequence
(ZAMS, Schmidt-Kalar (1982)) in the $(U-B)$ vs $(B-V)$ diagram as shown in
Figure 2. In this figure, the black curve shows the reddened ZAMS which is
obtained after the adding a shift 0.12 mag in $E(B-V)$ values of normal ZAMS.

Figure 2: The blue and red component stars of cluster is depicted by blue and
red dots respectively. The solid black and green line represent the best
fitted ZAMS on blue component stars and best linear fit in red component stars
of cluster. The dashed arrow shows normal reddening law.

The reddened ZAMS fitting is occurred for BCS while the RCS are show a linear
trend. We have fitted a linear equation to obtained linear relationship
between two colours of RCS. The solution of this linear equation is found to
be as,

$(U-B)=(1.47{\pm}0.03)(B-V)-0.78{\pm}0.04.$

Though, the RCS stars, having ($B-V{\geq}1.8~{}mag$), found to be below this
linear solution. As a result, the slope of $(U-B)/(B-V)$ is found to be very
high for RCS compare to expected slope for the clusters.
It is noticeable that the scattering for $A_{0}$ type stars found to be less
and increases with ($B-V$) after $A_{0}$ type stars. The arrow indicates the
normal colour excess law ($\frac{(U-B)}{B-V}=0.72$, which is used for adding a
shift in colour-excess $(U-B)$. Since, some RCS are also giant members of
cluster, therefore, the shifting value chosen in such a way that the redden
ZAMS also follow the pattern of RCS within the two colour region of BCS. Some
RCS are embedded in the distribution of BCS on the TCD. Mostly such type RCS
are shows below position from best fitted ZAMS on the TCD and they have lesser
scatter. then they give the opportunity to find best fitted reddened ZAMS
together with $A_{0}$ type stars of the cluster. Thus, the reddening of
cluster is found to be $0.12{\pm}0.01$ mag which is slightly high compare to
reddening (0.10 mag) estimated by Joshi et al. (2012) and shows close
agreement to reddening ($0.16{\pm}0.04$)as derived by Janes et al. (2014)
through Bayesian analysis.

##  3 Separation procedure of Blue and Red component stars

The stellar distribution seems to be spread out on (B-V) vs V CMD (as shown in
Figure 3 and Figure 11). The dense stellar distribution is obtained to be
broad and sloppy line as shown in the left side of the above prescribed CMD,
which may be occurred due to the evolutionary processes of the probable main
sequence (MS) of the cluster. The comparison of known literature
characteristic with the cluster characteristics through this sequence
(detailed discussion in at Section 4) is verified the fact that said sequence
is the MS of present studied cluster. Other hand, such linear trend does not
appear for the stellar distribution of right side of CMD but the space is
sparsely fulfilled by other detected stars of the cluster. The said sloppy
sequence of the cluster is clearly identified through visually inspection of
the distribution of massive and fainter stars. Here, We are used following
magnitude-colour relation to separate this probable MS from the stellar
distribution of the observed field of cluster,

$V_{o}=7.66{\times}(B-V)_{o}+7.32,$

where $V_{o}$ and $(B-V)_{o}$ are the observed magnitude and colour of stars.
The above prescribed linear equation is represented by a black dashed arrow on
$(B-V)$ vs $V$ CMD of the cluster as depicted in Figure 3.

Figure 3: The blue and red component stars of the cluster have been shown by
blue and red dots. The black dashed arrow represents the separation boundary
of these two type components.

The left sided stellar distribution of this arrow is referred as the
distribution of blue component stars (BCS), whereas stars of right side of
this arrow is considered to be red component stars (RCS) of the cluster. The
blue and red dots in Figure 3 are represented the BCS and RCS of the cluster.
The obtained RCS may be either super-giant stars of the cluster or field
stars. The super-giant stars are those cluster members which are evaporated
from the main sequence through the dynamical evolution phenomena of the
cluster. A total of 4088 stars and 1250 stars are found to be blue component
and red component stars, respectively. This separation procedure is applied
for those stars which are brighter than 18.5 mag in V-band due to the
following fact. The CMDs in Figure 11 is direct indicate that BCS and RCS are
not easily separated to each other for those stars, which having V-magnitude
greater than 18.5 mag. Although it seems that RCS of $(V-I)/V$ CMD are visibly
separated from the MS stars but are not seen separately in the $(B-V)/V$ CMD.

##  4 Cluster extent due to blue component Stars

The finding chart is very effective to check out that whether the BCS and RCS
are present on a particular sky distribution of cluster region or not. The
stellar distribution of BCS and RCS are shown in Figure 4. In this figure, the
size of dots is the function of the stellar brightness and increases towards
to the lower magnitudes of the stars. The smallest size of dots is represented
those stars which having V-magnitude to be 18.5 mag. Our deep analysis shows
that the model limiting radius is greater than that of the observed cluster
field by Janes et al. (2014). The procedure of estimation of limiting radius
is prescribed as below.

Figure 4: The inner and outer circles represent the core radius and limiting
radius of clusters. The size of the dots is proportional to their brightness.
The blue and red dots are shown to be blue and red component stars
respectively.

The cluster center is taken in such way that we have obtained best RDP through
it and follow the condition of maximum count within the cluster region. The
detail procedure of above said approach is prescribed by the Joshi et al.
(2015). In the present case, the cluster center $(\alpha,~{}\delta)$ is found
to be
($20^{h}:03^{m}:54.9^{s},~{}+44^{o}:09^{{}^{\prime}}:28.2^{{}^{\prime\prime}}$)
in $RA-DEC$ plane. This coordinate is slightly shifted from the value of
cluster center as listed at SIMBAD database and various concentric rings are
constructed around the center to determine the radius of the cluster. The
radial densities of both components are computed for each ring which are used
to construct the radial density profiles for both components as shown in
Figure 5.

Figure 5: The blue and red component stellar density of the cluster has been
shown by blue and red dots respectively.

The above prescribed profile of stars of both component is shown direct
evidence of cluster characteristics through BCS. The following empirical
formula of King Model (King, 1962) is fitted on the RDP through BCS of
cluster,

$\rho(r)=\rho(0)+\frac{f_{0}}{1+(\frac{r}{r_{c}})^{2}},$

where $f_{0}$, $r_{c}$, $\rho(0)$ and $\rho(r)$ are peak stellar density,
core-radius, background stellar density and instantaneous stellar density,
respectively. The value of $f_{0}$, $r_{c}$ and $\rho(0)$ for the BCS are
found to be $5.21{\pm}0.29$, $8.39{\pm}0.85$ and $0.89{\pm}0.16$,
respectively. The resultant fit cuts the $\rho(0)$ at the outside of range of
determined radial densities, therefore, we are not capable to give the value
of cluster radius. The exact cluster radius would be defined after the
complete observation of the field view of the cluster. Other hand, the value
of limiting radius
[$r_{limiting}=r_{c}\sqrt{\frac{f_{0}}{3\sigma}-1},~{}\sigma$ is the
estimation error in background stellar density] can be obtained without any
cut point. Hence, we obtained the limiting radius of the cluster which comes
as $26.48{\pm}0.32~{}arcmin$. The values of radial densities of RCS did not
find in the order of continue decreasing as seems for the BCS. The RDP of RCS
seem to be made by two plateau regions, such as, $0<r{\leq}7~{}arcmin$ and
$10<r{\leq}17~{}arcmin$. The visual inspection of stellar densities of these
regions of RCS shows that the first one (core) is higher stellar density than
that of the later one (outside). The higher stellar density of RCS at the core
region may be appears due to the enhancement of probable super giant members
of the cluster. Furthermore, the sloppy region of RCS is found to be at the
radial distance range of $7<r{\leq}10~{}arcmin$, which may be the core-corona
transition region of the cluster. Similarly, the model core radius limits,
$7.54<r_{c}<9.24$, may also be clue for the existence of such type region.

##  5 Surface number density tracer of mass segregation

Surface number density profile (SNDP) is very effective to shown the stellar
variation of various magnitude bin with the radial distance. Such information
will be becomes an ideal tracers to construct the evolutionary models of
stellar evolution, which is beneficial to understand the stellar dynamics of
the cluster. In addition, SNDPs of various radial zones are effectively
explain the possible stellar distribution according to their masses.
Similarly, this profile also helpful to understand the over-density inside the
core-radius of cluster and possible cause of constant stellar density of RCS
after the core region of cluster. The SNDPs of both components have been
depicted in Figure 5.

Figure 6: The upper and lower panels are shown SNDP for BCS and RCS
respectively. The surface number density is defined as the number of stars per
magnitude interval in unit area (Sung et al., 1996).

The SND values of the BCS indicates that mostly massive stars are concentrated
at the innermost region ($0<r{\leq}5~{}arcmin$) of the cluster in the
comparison of the fainter stars. The adjacent SNDPs of BCS of two innermost
ring regions shows clear separation between them and this separation
continuously decreases upto 16 mag of $V-$band whereas it increases for those
members which having stellar magnitude greater than above prescribed magnitude
limit. This result shows the evidence of mass-segregation phenomena within
cluster. The SNDPs of stars (having stellar $V-$magnitude less than
$14~{}$mag) of outermost region indicate that the SND values of massive stars
do not distinguish from each other. For other stars, these profiles are
distinguishable and their SND values decrease in the order of number of rings
from the cluster center.
The SNDPs of RCS of core region shows clear separation between them and both
profiles is stellar enhancement compare to the SNDPs of coronal region of the
cluster. The profiles of coronal region are not so much visually separated as
seen for former profiles. It is also noticeable fact that the values of SND of
massive stars are obtained to be close to each other for both category (BCS
and RCS). Such stellar distribution gives an indication of similar birthing
environment of massive stars of both components. Due to this fact, it may be
concluded that some massive stars are evaporated from the MS evolution of the
cluster but they are still belong to the cluster. These so called stars may be
the super-giant members. It can be understandably fact that the density of
massive stars of field region too high as for massive members of the cluster.
The higher density of massive members is possible due to the addition of
super-giant members of the cluster with field stars. The slightly high value
of radial density of RCS of the core with compare to the densities than that
for corona (as depicted in Figure 4), may be confirm the mixing of super-giant
members of the cluster with the field stars. On the behalf of this discussion,
it may be concluded that the all RCS of the coronal region are field stars due
to the absence of massive stars.

##  6 Identification of cluster giant stars through statistical distance
method

Since, the RCS of core region of clusters contains the super-giant stars with
filed stars whereas the RCS are coronal region are only field stars. We are
assumed that the field stars of RCS are uniformly distributed over all the
cluster region. To identify the possible giant stars of the cluster, we have
been applied the statistical distance algorithm on the RCS [detailed procedure
has been prescribed by Joshi et al. (2015)]. The core
($0<r{\leq}8.39~{}arcmin$) is considered to be giant stars region and the
region, having radial distance range ($10<r{\leq}13.05~{}arcmin$) and
equivalent area to giant star region, considered as the filed region. Here, we
are used $V$, $B$ and $I$ filters instead of $J$, $H$ and $K$ filters and the
magnitude-colour distance is taken as,

$D_{mc}=\sqrt{\Delta_{V}^{2}+\Delta_{BV}^{2}+\Delta_{VI}^{2}},$

where $\Delta_{V}$, $\Delta_{BV}$ and $\Delta_{VI}$ are the differences of $V$
magnitude , $B-V$ colour and $V-I$, respectively, between stars of field
region and giant (core of cluster) region. For it, the $\Delta_{V}$,
$\Delta_{BV}$ and $\Delta_{VI}$ have been considered to be $0.50$, $0.15$ and
$0.15$ respectively. After applying this procedure, we have found 129 giant
stars among 264 RCS of the core region. The giant star vertical sequence
presents at the colour, ($V-K$)= 4.0 mag of Figure 8. In this Figure, a
parallel giant star sequence of main sequence stars is also being appeared.
These both giant sequences are depicted by red dots. The position of these
sequence stars is indicated that the giant stars are born in a different time
span and different dynamical evolution processes.
It is also noticeable fact that all BCS members are not main sequence of stars
and some of them may be field stars. These stars are not possible to separate
from MS due to following facts. Firstly, the available photometric data not
cover all cluster region. Secondary, photometrc criteria (Sharma et al., 2008)
does not tell about the field stars, which having colour and magnitude values
in the similar order of the MS of cluster.

##  7 No clue for distinguish of stars through Proper Motion

The members of the cluster are loosely gravitational bound and each members
shows its individual motion, therefore, members change their location in sky
with respect to each other. The exact determination of velocity of stars are
not possible through observations but parallax used to determine the proper
motion (angular displacement) of stars by using the datasets of two different
time span. Here, the proper motion values extracted from the catalogue of
Roeser et al. (2010). This catalogue provides position in the International
Celestial Reference System (accuracy: 80-300 mas) and absolute proper motion
(accuracy: 4-10 mas yr-1) of about 900 million stars, derived from the optical
USNOB1.0 and near-infrared 2MASS catalogues. We have been estimated the mean
proper motion of BCS and RCS as ($-2.02{\pm}0.15~{}mas~{}yr^{-1}$,
$-4.52{\pm}0.16~{}mas~{}yr^{-1}$) and ($-2.18{\pm}0.25~{}mas~{}yr^{-1}$,
$-4.24{\pm}0.27~{}mas~{}yr^{-1}$), respectively. The mean proper motion values
of both components have been computed through utilizing the proper motion
value of 973 stars among 1093 BCS and 291 stars among 342 RCS, respectively.
Their vector-point diagrams are shown in Figure 6.

Figure 7: The proper motion distribution of BCS and RCS is depicted by blue
and red dots respectively.

We have not found any clear trend separating members and field stars, such
type result is also found by Bellini et al. (2010) through the study of nearby
well-studied open star clusters. The mean proper motion of RCS has slightly
differ from that of BCS which happened due to influence of field stars in RCS
and we have already shown in Section 6 that RCS of core region are mixture of
the cluster giant and field stars.
Since, it is well known fact that the influence of field stars is high in
coronal region compared to core. Due to the high membership probability, we
have also estimated the mean proper motion of the BCS of core using proper
motion values of 411 stars which are left over after applying an iteration
procedure (as suggested by Joshi et al. (2014)) on 458 blue component members
of the core-region. These value are coming to be
$-1.78{\pm}0.28~{}mas~{}yr^{-1}$ and $-4.58{\pm}0.25mas~{}yr^{-1}$ in Right
Ascension (RA) and Declination (DEC) directions, respectively, and members are
depicted by green dots in Figure 7. The mean proper motion values of core and
whole cluster is found to be similar within uncertainty which suggested that
the BCS of the cluster are less influenced by field stars.

##  8 Distance and Age of the cluster

The distance and age of any cluster is estimated through colour-magnitude
diagrams (CMDs). These parameters of cluster had been already estimated by
many authors (Joshi et al., 2012; Gunes et al., 2012, 2013; Janes et al.,
2014). Janes et al. (2014) are computed the metallicity $Z=0.014{\pm}0.005$
for this cluster through Bayesian analysis, which shows close agreement with
solar metallacity within error. Thus, we have adopted Marigo’s theoretical
isochrones of solar metallicity (Marigo et al., 2008) for further analysis. By
taking the new reddening value ($0.12~{}mag$) and solar metallicity, we are
fitted various isochrones of various age group nearby the literature ages of
the cluster on ($V-K$ vs $V$) CMD. It is noticeable fact for any cluster that
the scattering of stars is increases with the gap of effective wavelength of
filters. We are familiar with the low detection rate of stars in $U$ and $B$
filters. In this background, $V$ band of the optical photometry and $K$ band
of the 2MASS photometry were taken for constructing the CMD of the cluster.
Our best fitted isochrone solution provides the cluster log(age) and apparent
distance modulus as $8.85{\pm}0.05$ and $10.75{\pm}0.10mag$ respectively. The
resultant distance modulus shows close agreement with the
$(m-M)_{V}=10.98{\pm}0.24~{}mag$ (Janes et al., 2014) while it is low compared
to $11.15~{}mag$ (Joshi et al., 2012) and $11.33~{}mag$ (Karchenko et al.,
2005). The best fitted isochrone has been shown by blue solid line in Figure
8.

Figure 8: The blue and red solid lines represent the boundary for binary
stars. The blue line is also show the best fitted isochrone of the cluster
whine the red line represents the red sequence of the binary stars.

The red solid line in this Figure represents the red sequence which get by the
adding $0.75~{}mag$ and $0.105~{}mag$ in magnitude $(m-M)_{V}$ and colour
($V-K$) respectively. This shifting is taken account due to the unresolved MS
binaries. The heliocentric distance of the cluster is found to be
$1.19{\pm}0.04~{}kpc$ through the following relation,

$d=10^{\left[(m-M)_{V}-R_{V}E(B-V)\right]/{5}+1}~{}kpc.$

Whenever, we have change distance modulus 10.75 mag to 11.25 mag, we have
noticed that stellar numbers between sequences is increases in $(B-V)/V$ and
$(V-I)/V$ CMDs (see stellar distribution between dashed lines in Figure 10)
but deceases in $(V-K)/V$ CMD (233 stars instead of 538 stars). This result
leads an idea that the data incompleteness increases between $(B-V)/V$ and
$(V-K)/V$ CMDs with the increasing distance modulus. The decreasing distance
modulus shift the blue sequence in RCS region of $(B-V)/V$ CMD. Thus, the
resultant apparent distance cluster is justified through this prescribed
procedure.

##  9 Blue Component Stars and Mass Function

To understand the effect of incompleteness of data, we are used two type of
CMDs for studying the mass function (MF) of the cluster. The MF is the
relative number of stars per unit mass which is determined due to the absence
a procedure of measurement of the initial mass function (IMF i.e. stellar mass
distribution in per unit volume in a star formation event). The MF can be
defined by a power law as $Nlog(M){\propto}M^{\Gamma}$ and its slope $\Gamma$
can be determined through

$\Gamma=\frac{dlog~{}N~{}(log~{}m)}{d~{}log~{}m},$

where $N~{}log(m)$ is the stellar number in per unit logarithm mass. Firstly,
We are determined the MF slope of the cluster using the BCS fall between blue
and red sequences. A total of 1318 stars in $(B-V)$ vs $V$ CMD and and 538
stars in $(V-K)$ vs $V$ CMD are found between these sequences. The observed MF
values have been given in Table 1 and shown in Figure 9.

Table 1: The MF of the cluster NGC 6866 derived from BCS between blue and red
sequences in (A) $(B-V)$ vs $V$ CMD and (B) $(V-K)$ vs $V$ CMD (represented by
suffix A and B respectively).

V range | Mass range | $\bar{m}$ | $log(\bar{m})$ | $N_{A}$ | $N_{B}$
---|---|---|---|---|---
(mag) | $M_{\odot}$ | $M_{\odot}$ |  |  |
11.5-12.5 | 2.235-1.854 | 2.045 | 0.311 | 20 | 04
12.5-13.5 | 1.854-1.524 | 1.689 | 0.228 | 42 | 16
13.5-14.5 | 1.524-1.267 | 1.395 | 0.145 | 127 | 22
14.5-15.5 | 1.267-1.070 | 1.168 | 0.068 | 246 | 74
15.5-16.5 | 1.070-0.912 | 0.991 | -0.004 | 328 | 110
16.5-17.5 | 0.912-0.787 | 0.850 | -0.071 | 338 | 173
17.5-18.5 | 0.787-0.677 | 0.732 | -0.135 | 212 | 135

V range | $log(\Phi)_{A}$ | $e_{log(\Phi)_{A}}$ | $log(\Phi)_{B}$ | $e_{log(\Phi)_{B}}$
---|---|---|---|---
11.5-12.5 | 2.392 | 0.224 | 1.693 | 0.500
12.5-13.5 | 2.693 | 0.154 | 2.274 | 0.250
13.5-14.5 | 3.200 | 0.089 | 2.438 | 0.213
14.5-15.5 | 3.525 | 0.064 | 3.004 | 0.116
15.5-16.5 | 3.675 | 0.055 | 3.200 | 0.095
16.5-17.5 | 3.723 | 0.054 | 3.432 | 0.076
17.5-18.5 | 3.511 | 0.069 | 3.315 | 0.086

Figure 9: The blue and red solid lines represent the boundary for binary stars
in each CMD (a & b). The MF slope values estimated through these CMDs have
been shown in below of them (c & d).

The least stellar number in $(V-K)/V$ CMD indicates the data incompleteness.
The $\Gamma$ values of BCS between sequences are computed to be
$-3.81{\pm}0.51$ and $-2.83{\pm}0.69$ through $(B-V)/V$ and $(V-K)/V$ CMDs
respectively. In present case, the incompleteness does not significantly
altered the $\Gamma$ value of cluster though the mass function values are
significantly decrease due to incompleteness of the data. The $\Gamma$ value
through $(V-K)/V$ CMD is slightly increases which suggested that $\Gamma$
value increases due to the incompleteness of data.
The $\Gamma$ values are further estimated by all BCS through these CMDs. We
are found 1111 BCS in $(V-K)/V$ CMD which are low compare to 4088 BCS in
$(B-V)/V$ CMD. The MF values through these members have been listed in Table 2
and shown in Figure 10.

Figure 10: The MF slope values estimated through $(B-V)/V$ and $(V-K)/V$ CMDs. Table 2: The MF of the cluster NGC 6866 derived from BCS found in (A) $(B-V)$ vs $V$ CMD and (B) $(V-K)$ vs $V$ CMD (represented by suffix C and D respectively). V-mag | $N_{C}$ | $N_{D}$ | $log(\Phi)_{C}$ | $log(\Phi)_{D}$
---|---|---|---|---
11.5-12.5 | 48 | 11 | $2.772{\pm}0.114$ | $2.132{\pm}0.302$
12.5-13.5 | 113 | 41 | $3.123{\pm}0.094$ | $2.683{\pm}0.156$
13.5-14.5 | 262 | 63 | $3.514{\pm}0.062$ | $2.895{\pm}0.126$
14.5-15.5 | 460 | 125 | $3.797{\pm}0.047$ | $3.231{\pm}0.089$
15.5-16.5 | 800 | 205 | $4.062{\pm}0.035$ | $3.470{\pm}0.070$
16.5-17.5 | 1265 | 300 | $4.296{\pm}0.028$ | $3.671{\pm}0.058$
17.5-18.5 | 1121 | 355 | $4.234{\pm}0.030$ | $3.735{\pm}0.053$

The ${\Gamma}$ values for all BCS are found to be $-3.80{\pm}0.11$ and
$-3.51{\pm}0.19$ through $(B-V)/V$ and $(V-K)/V$ CMDs respectively. These
results are show similar characteristic as mentioned above. To investigate why
the mass function values in fainter end of both CMD have low value, we have
saw the characteristic of the data of cluster. For this purpose, we have drawn
$(B-V)/V$ CMD and $(V-I)/V$ for cluster through all archive data of Janes et
al. (2014) and these CMDs are shown in Figure 11.

Figure 11: The $(B-V)/V$ CMD and $(V-I)/V$ CMDs. The solid lines represent the
boundary of blue and red sequence whenever the apparent distance modulus is
10.75 mag while dashed lines represent same for apparent distance modulus is
11.25 mag.

The very low stars are appeared between 18 to 19 mag of V-band as seen in
Figure 11. Therefore, the MF values found to be low compare to expected value.
We have not found different ${\Gamma}$ for all BCS compare to that for BCS
between sequences. Thus, we have considered the ${\Gamma}$ value for cluster
$-3.80{\pm}0.11$ which is very high from to normal ${\Gamma}$ value (-2.35)
for clusters. We are also concluded that MF study is more reliable through BCS
compare to the binary stars sequences as comes from the photometric criteria.

##  10 Nature of total-to-selective-extinction

We are estimated the IR interstellar extinction through the $(V-K)/(J-K)$ TCD
as shown in Figure 12.

Figure 12: The $(V-K)$ vs $(J-K)$ TCD for the cluster. The black solid arrows
represent the normal reddening vector $\frac{E(J-K)}{E(V-K)}=0.173$. The
horizontal and vertical green lines represent the errors in colours, (V-K) and
(J-K).

In this figure, the blue solid line represents the solar metallicity
isochrones of log(age)=8.85 while the red solid line is obtained by adding the
0.45 mag and 0.07 mag shift in $E(V-K)$ and $E(J-K)$, respectively. Thus, the
colour-excess, $E(V-K)$ and $E(J-K)$ for the cluster are found to be 0.45 mag
and 0.07 mag, respectively, which are give the value of normal reddening
vector ($\frac{E(J-K)}{E(V-K)}$) as 0.16. This vector value is low than
expected value i.e. 0.173. This low value is occurred due to the presence of
interstellar clouds and dust in the direction of cluster field. The reddening
value of cluster is found to be 0.16 mag through the Mathis (1990) relation,
$\frac{E(V-K)}{E(B-V)}=2.75$. This resultant reddening value is slightly high
compare to 0.12 mag as estimated through $(U-B)/(B-V)$ TCD. These two
different value of reddening through different CMD suggests that the total-to-
selective-extinction value for the cluster is alter from the normal value.

##  11 Conclusion

On the comparison of various photometric catalogue of this cluster, the
photometric procedure of Joshi et al. (2012) for estimation of the stellar
magnitude seems to be less precise. Such least precision is occurred due to
either the sky condition on the observational night or adopted standardization
procedure. The photometric catalogue of Janes et al. (2014) is used to
estimate the fundamental parameters of this cluster due to large observational
field. Our deep inspection indicate that the radial over density of stars of
cluster are occurred due to the BCS stars which is further supplemented by
SNDs in various radial zones. The core-corona transition region is found as
$7-10~{}arcmin$, which would be very effective to understand the evolutionary
interaction between core and corona regions of the cluster. Though the BCS and
RCS are neither distinguishable in RA-DEC plane nor distinct proper motions,
whereas they have been separated on the basis of their location on $CMD$
plane. We have been also found that the ZAMS are not fitted on the RCS,
whereas, RCS followed a linear trend. Our analysis shows that some RCS of
core-region found to be Giant members of the cluster. The value of MF slope
shows its dependency on the incompleteness of data. This manuscript is a mile-
stone step to understand the exact members of the cluster through their known
parametric values.

## Acknowdege

This research has made use of the VizieR catalogue access tool, CDS,
Strasbourg, France. The original description of the VizieR service was
published in A&AS 143, 23. GCJ is also thankful to APcyber Zone (Nanakmatta)
for providing computer facilities. GCJ is also thankful to Shree Nilamber
Joshi for providing the friendly environment, which becomes mile stone of my
research work.

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# Wigner function for SU(1,1)

U. Seyfarth  Max-Planck-Institut für die Physik des Lichts, Staudtstraße 2,
91058 Erlangen, Germany  A. B. Klimov  Departamento de Física, Universidad de
Guadalajara, 44420 Guadalajara, Jalisco, Mexico  H. de Guise  Department of
Physics, Lakehead University, Thunder Bay, Ontario P7B 5E1, Canada  G. Leuchs
Max-Planck-Institut für die Physik des Lichts, Staudtstraße 2, 91058 Erlangen,
Germany  Institute for Applied Physics,Russian Academy of Sciences, 630950
Nizhny Novgorod, Russia  L. L. Sánchez-Soto  Max-Planck-Institut für die
Physik des Lichts, Staudtstraße 2, 91058 Erlangen, Germany  Departamento de
Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain

###### Abstract

In spite of their potential usefulness, Wigner functions for systems with
SU(1,1) symmetry have not been explored thus far. We address this problem from
a physically-motivated perspective, with an eye towards applications in modern
metrology. Starting from two independent modes, and after getting rid of the
irrelevant degrees of freedom, we derive in a consistent way a Wigner
distribution for SU(1,1). This distribution appears as the expectation value
of the displaced parity operator, which suggests a direct way to
experimentally sample it. We show how this formalism works in some relevant
examples.

Dedication: While this manuscript was under review, we learnt with great
sadness of the untimely passing of our colleague and friend Jonathan Dowling.
Through his outstanding scientific work, his kind attitude, and his inimitable
humor, he leaves behind a rich legacy for all of us. Our work on SU(1,1) came
as a result of long conversations during his frequent visits to Erlangen. We
dedicate this paper to his memory.

##  1 Introduction

Phase-space methods represent a self-standing alternative to the conventional
Hilbert-space formalism of quantum theory. In this approach, observables are
$c$-number functions instead of operators, with the same interpretation as
their classical counterparts, although composed in novel algebraic ways.
Quantum mechanics thus appears as a statistical theory on phase space, which
can make the corresponding classical limit emerge in a natural and intuitive
manner.

The realm of the method was established in the groundbreaking work of Weyl [1]
and Wigner [2]. Later, Groenewold [3] and Moyal [4] established a solid
foundation that has developed over time into a complete discipline useful in
many diverse fields [5, 6, 7, 8].

The main ingredient of this approach is a _bona fide_ mapping that relates
operators to functions defined on a smooth manifold, endowed with a very
precise mathematical structure [9]. However, this mapping is not unique:
actually, a whole family of functions can be consistently assigned to each
operator. In particular, quasiprobability distributions are the functions
connected with the density operator [10, 11, 12, 13]. For continuous
variables, such as Cartesian position and momentum, the quintessential example
that fuelled the interest for this field, the most common choices are the $P$
(Glauber-Sudarshan) [14, 15], $W$ (Wigner) [2], and $Q$ (Husimi) [16]
functions, respectively.

This formalism has been applied to other different dynamical groups 111We
adhere to the usual convention that a Lie group $G$ (with Lie algebra
$\mathfrak{g}$) is a dynamical group if the Hamiltonian of the system under
consideration can be expressed in terms of the generators of $G$ (that is, the
element of the Lie algebra $\mathfrak{g}$) [17].. Probably, the most
widespread example beyond the harmonic oscillator is that of SU(2), with the
Bloch sphere as associated phase space [18, 19, 20]; this case is of paramount
importance in dealing with spinlike systems [21, 22, 23, 24, 25, 26]. Suitable
results have also been found for the Euclidean group E(2), this time with the
cylinder as phase space [27, 28, 29, 30]; this is of primary importance in
treating the orbital angular momentum of twisted photons [31, 32]. Additional
applications to more general dynamical groups have also appeared in the
literature [33, 34, 35, 36]. Moreover, the basic notions have been
successfully extended to discrete qudits, where the phase space is a finite
grid [37, 38, 39, 40, 41, 42, 43].

Surprisingly, the phase-space description of systems having SU(1,1) symmetry
has received comparatively little attention [44, 45], in part because the
representation theory of this group is not as familiar as SU(2) or even E(2).
However, SU(1,1) plays a major role in connection with what can be called two-
photon effects [46, 47, 48, 49]. The topic is experiencing a revival in
popularity due to the recent realization of a nonlinear SU(1,1) interferometer
[50, 51]. According to the pioneering proposal of Yurke et al. [52], this
device would allow one to improve the phase measurement sensitivity in a
stunning manner [53].

In spite of the importance of these systems, the mathematical complexity of
the group SU(1,1) [54] has prevented a proper phase-space description. In this
paper we approach this question resorting to a physics-based approach. For
SU(2), one can model the description in terms of a superposition of two
harmonic modes. In technical terms, this corresponds to the Jordan-Schwinger
bosonic realization of the algebra $\mathfrak{su}(2)$ [55]. Here, we propose a
similar way to deal with $\mathfrak{su}(1,1)$: starting with two orthonormal
modes, and using the standard tools for continuous variables, we eliminate the
spurious degrees of freedom and we get a description on the upper sheet of a
two-sheeted hyperboloid, which is the natural arena to represent the physics
associated to these systems.

Our final upshot is that the Wigner function can be expressed as the average
value of the displaced parity operator. This is reassuring, for it is also the
case for continuous variables [56]. Moreover, as this property has been
employed for the direct sampling of the Wigner function for a quantum field
[57, 58, 59], our result opens the way for the experimental determination of
the Wigner function for SU(1,1).

##  2 Phase-space representation of a single mode

To keep the discussion as self-contained as possible, we first briefly
summarize the essential ingredients of phase-space functions for a harmonic
oscillator that we shall need for our purposes.

We consider the standard oscillator described by annihilation and creation
operators $\hat{a}$ and $\hat{a}^{\dagger}$, which obey the bosonic
commutation relation

$[\hat{a},\hat{a}^{\dagger}]=\hat{\mathbbm{1}}\,.$ (1)

They are the generators of the Heisenberg-Weyl algebra, which has become the
hallmark of noncommutativity in quantum theory [60]. The classical phase space
is here the complex plane $\mathbb{C}$.

These complex amplitudes are expressed in terms of the quadrature operators
$\hat{x}$ and $\hat{p}$ as

$\hat{a}=\frac{1}{\sqrt{2}}(\hat{x}+i\hat{p})\,,\qquad\qquad\hat{a}^{\dagger}=\frac{1}{\sqrt{2}}(\hat{x}-i\hat{p})\,,$
(2)

and the commutation relation (1) reduces then to the canonical form
$[\hat{x},\hat{p}]=i\,\hat{\mathbbm{1}}$.

A central role in what follows will be played by the unitary operator

$\hat{D}(\alpha)=\exp(\alpha\hat{a}^{\dagger}-\alpha^{\ast}\hat{a})\,,\qquad{\alpha\in\mathbb{C}\,,}$
(3)

which is called the displacement operator for it displaces a state localized
in phase space at $\alpha_{0}$ to the point $\alpha_{0}+\alpha$. The Fourier
transform of the displacement is the Cahill-Glauber kernel [61]

$\hat{w}(\alpha)=\frac{1}{\pi^{2}}\int_{\mathbb{C}}\,\exp(\alpha\beta^{\ast}-\alpha^{\ast}\beta)\,\hat{D}(\beta)\,d\beta\,,$
(4)

which is an instance of a Wigner-Weyl quantizer [62].

The operators $\hat{w}(\alpha)$ constitute a complete trace-orthonormal set
that transforms properly under displacements; that is

$\hat{w}(\alpha)=\hat{D}(\alpha)\,\hat{w}(0)\,\hat{D}^{\dagger}(\alpha)=2\;\hat{D}(\alpha)\,(-1)^{\hat{a}^{\dagger}\hat{a}}\,\hat{D}^{\dagger}(\alpha)\,,$
(5)

where $\hat{w}(0)=\int_{\mathbb{C}}\hat{D}(\beta)\,d\beta=2\hat{P}$, and

$\hat{P}=(-1)^{\hat{a}^{\dagger}\hat{a}}\,.$ (6)

In this way, $\hat{w}(\alpha)$ appear as the displaced parity operator [56].

If $\hat{A}$ is an arbitrary (trace-class) operator acting on the Hilbert
space of the system, the Wigner-Weyl quantizer allows one to associate to
$\hat{A}$ a function $W_{\hat{A}}(\alpha)$ representing the action of the
corresponding dynamical variable in phase space:

$W_{\hat{A}}(\alpha)=\mathop{\mathrm{Tr}}\nolimits[\hat{A}\,\hat{w}(\alpha)]\,.$
(7)

The function $W_{\hat{A}}(\alpha)$ is the symbol of the operator $\hat{A}$.
Such a map is one-to-one, so we can invert it to get the operator from its
symbol through

$\hat{A}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{C}}\hat{w}(\alpha)\,W_{\hat{A}}(\alpha)\,\,d\alpha\,.$
(8)

We focus on what follows on the Wigner function, although the discussion can
be immediately extended to any other quasiprobability. Actually, the Wigner
function is nothing but the symbol of the density matrix $\hat{\varrho}$.
Consequently,

$\displaystyle
W_{\hat{\varrho}}(\alpha)=\mathop{\mathrm{Tr}}\nolimits[\hat{\varrho}\,\hat{w}(\alpha)]\,,$
(9)
$\displaystyle\hat{\varrho}=\displaystyle\frac{1}{(2\pi)^{2}}\int_{\mathbb{C}}\hat{w}(\alpha)W_{\hat{\varrho}}(\alpha)\,d\alpha\,.$

The $W_{\hat{\varrho}}(\alpha)$ defined in (9) fulfills the basic properties
required for any good probabilistic description [33]. First, due to the
Hermiticity of $\hat{w}(\alpha)$, it is real for Hermitian operators. Second,
the probability distributions for the canonical variables can be obtained as
the corresponding marginals. Third, $W_{\hat{\varrho}}(\alpha)$ is
translationally covariant, which means that for the displaced state
$\hat{\varrho}^{\prime}=\hat{D}(\alpha^{\prime})\,\hat{\varrho}\,\hat{D}^{\dagger}(\alpha^{\prime})$,
one has

$W_{\hat{\varrho}^{\prime}}(\alpha)=W_{\hat{\varrho}}(\alpha-\alpha^{\prime})\,,$
(10)

so that the Wigner function follows displacements rigidly, without changing
its form, reflecting the fact that physics should not depend on any choice of
the origin.

Finally, the overlap of two density operators is proportional to the integral
of the associated Wigner functions

$\mathop{\mathrm{Tr}}\nolimits(\hat{\varrho}\,\hat{\varrho}^{\prime})\propto\int_{\mathbb{C}}W_{\hat{\varrho}}(\alpha)W_{\hat{\varrho}^{\prime}}(\alpha)\,d\alpha\,,$
(11)

provided the integral converges. This property (known as traciality) offers
practical advantages, because it allows one to predict the statistics of any
outcome, once the Wigner function of the measured state is known.

To conclude, we mention that the displacements also constitute a basic
ingredient in the concept of coherent states. If we choose a fixed normalized
reference state $|\Psi_{0}\rangle$, we have [63]

$|\alpha\rangle=\hat{D}(\alpha)\,|\Psi_{0}\rangle\,,$ (12)

so they are parametrized by phase-space points. These states have a number of
remarkable properties inherited from those of $\hat{D}(\alpha)$. The standard
choice for the fiducial vector $|\Psi_{0}\rangle$ is the vacuum $|0\rangle$
(or, more generally, a highest or lowest weight state).

##  3 Phase-space representation of two modes

Next, we consider the superposition of two modes in two orthogonal directions,
say $x$ and $y$, with momenta $p_{x}$ and $p_{y}$, respectively. Since they
are kinematically independent, the complex amplitudes of these modes (denoted
by $\hat{a}$ and $\hat{b}$) commute ($[\hat{a},\hat{b}]=0$) and the total
Wigner-Weyl quantizer can be expressed as the product of the corresponding
ones for each mode:

$\hat{w}(\alpha,\beta)=\hat{w}(\alpha)\,\hat{w}(\beta)\,.$ (13)

With the form given in Eq. (5) and disentangling the exponentials, we get

$\hat{w}(\alpha,\beta)=4\exp[-2(|\alpha|^{2}-|\beta|^{2})](-1)^{\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}}\exp[-2(\alpha\hat{a}^{\dagger}-\beta^{\ast}\hat{b})]\,\exp[2(\alpha^{\ast}\hat{a}-\beta\hat{b}^{\dagger})]\,.$
(14)

As we can see, this kernel depends on the four real variables
$\alpha=(x,p_{x})$ and $\beta=(y,p_{y})$. As a consequence, the resulting
Wigner function $W(\alpha,\beta)$ contains all the information on the two
modes, but it is hard to grasp any physical flavor from it: in particular, it
cannot be plotted, which is always a big advantage in depicting complex
phenomena. To avoid this drawback we use the parametrization

$\alpha=re^{i(\chi+\varphi)/2}\cosh(\tau/2)\,,\qquad\beta=re^{i(\chi-\varphi)/2}\sinh(\tau/2)\,,$
(15)

where the radial variable $r^{2}=|\alpha|^{2}-|\beta|^{2}$ represents the
difference in intensities between the two modes. We can safely take
$|\alpha|>|\beta|$, for the opposite case can be obtained by just a
relabelling of modes, with no physical consequences. The parameters $\chi$ and
$\tau$ can be interpreted as azimuthal and “polar” angles on a two-sheeted
hyperboloid $\mathbb{H}_{2}$ [64]. A similar parametrization as in (15),
wherein the hyperbolic functions are replaced with trigonometric ones, maps
two complex modes into the Bloch sphere $\mathbb{S}_{2}$. This is an instance
of a Hopf fibration [65]. Therefore, the hyperboloid $\mathbb{H}_{2}$ can be
properly called the Bloch hyperboloid and the map (15) is a noncompact Hopf
fibration [66].

After some lengthy algebra, the kernel can be recast in the form

$\hat{w}(r,\chi,\tau,\varphi)=4\exp(-2r^{2})\left(-1\right)^{\hat{N}}\hat{S}(\zeta)\;\exp(-2re^{i\chi}e^{i\varphi}\hat{a}^{\dagger})\exp(2re^{-i\chi}e^{-i\varphi}\hat{a})\,\hat{S}^{\dagger}(\zeta)\,,$
(16)

where $\hat{N}=\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}$ is the total
number and we have introduced the two-mode squeeze operator

$\hat{S}(\zeta)=\exp(\zeta\hat{K}_{+}-\zeta^{\ast}\hat{K}_{-})\,,$ (17)

with $\zeta=\tfrac{1}{2}\tau e^{i\chi}$. This operator is defined in terms of
the two-mode realization of the $\mathfrak{su}(1,1)$ algebra

$\hat{K}_{+}=\hat{a}^{\dagger}\hat{b}^{\dagger}\,,\qquad\hat{K}_{-}=\hat{a}\hat{b}\,,\qquad\hat{K}_{0}=\tfrac{1}{2}(\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}+\mathbbm{1})\,,$
(18)

with commutation relations

$[\hat{K}_{0},\hat{K}_{\pm}]=\pm\hat{K}_{\pm}\,,\qquad[\hat{K}_{-},\hat{K}_{+}]=2\hat{K}_{0}\,.$
(19)

Using the Baker-Campbell-Hausdorff formula, one can check that

$\displaystyle\hat{S}(\zeta)\,\hat{a}\,\hat{S}^{\dagger}(\zeta)=\hat{a}\cosh|\zeta|-\hat{b}^{\dagger}\,e^{i\arg\zeta}\sinh|\zeta|\,,$
(20)

Notice that the transformation $\hat{S}(\zeta)$ depends only on the sum of the
phases $e^{i\chi}$. This makes the phase $\varphi$ irrelevant and,
consequenly, we proceed to integrate over $\varphi$ to get

$\displaystyle\hat{w}(r,\zeta)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}d\varphi\
\hat{w}(r,\chi,\tau,\varphi)$ (21) $\displaystyle=$ $\displaystyle
4\exp(-2r^{2})(-1)^{\hat{N}}\hat{S}(\zeta)\
\sum_{k=0}^{\infty}\frac{(2r)^{2k}(-1)^{k}}{k!^{2}}\hat{a}^{\dagger
k}\hat{a}^{k}\ \hat{S}^{\dagger}(\zeta)\,.\ $

Finally, we integrate over $r$, which is tantamount to averaging over
intensity information:

$\hat{w}(\zeta)=2\int_{0}^{\infty}\
dr\,r\,\hat{w}(r,\zeta)=2(-1)^{\hat{N}}\hat{S}(\zeta)\,\sum_{k=0}^{\infty}\frac{(-2)^{k}}{k!}\hat{a}^{\dagger
k}\hat{a}^{k}\;\hat{S}^{\dagger}(\zeta)\,.$ (22)

If we realize that

$\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\hat{a}^{\dagger
k}\hat{a}^{k}=(z+1)^{\hat{a}^{\dagger}\hat{a}}\,,$ (23)

we arrive at our central result

$\boxed{\hat{w}(\zeta)=2\hat{S}(\zeta)\,(-1)^{\hat{K}_{0}}\,\hat{S}^{\dagger}(\zeta)\,}$
(24)

Since $(-1)^{\hat{K}_{0}}$ is the SU(1,1) parity and $\hat{S}(\zeta)$ is a
displacement operator, this shows that the Wigner function for SU(1,1) can be
understood much in the same way as for the harmonic oscillator: just a
displaced parity.

The optical parity operator have been considered as a candidate for
approaching the highest level of sensitivity in the detection of small phase
shifts via optical interferometry [67]. This idea is adaption of a proposal by
Bollinger _et al._ [68] in the context of spectroscopy for a collection of
maximally entangled two-level trapped ions, in which parity is determined via
counting the number of ions of the sample that populate the excited state.
This detection has been recently proposed as a scheme to beat the Heisenberg
limit in SU(1,1) interferometry [69, 70, 71, 72]. Bear in mind though that the
SU(1,1) parity is not, in general, the parity of the photon numbers [73].

The operator $\hat{S}(\zeta)$ displaces by a complex number
$\zeta\in\mathbb{C}$. Moreover, there is a one-to-one correspondence between
$\zeta\in\mathbb{C}$ and the upper sheet of the hyperboloid, usually denoted
by $\mathbb{H}_{2}$: it is established via stereographic projection from the
south pole, so that

$\xi=\tanh(\tau/2)e^{i\chi}\Leftrightarrow\mathbf{n}=(\cosh\tau,\sinh\tau\cos\chi,\sinh\tau\sin\chi)\,,$
(25)

where $\mathbf{n}$ is a unit vector on $\mathbb{H}_{2}$, with the metric
$\mathbf{n}^{2}=n_{0}^{2}-n_{1}^{2}-n_{2}^{2}$. Note that $\zeta$ are points
in $\mathbb{C}$, whereas $\xi$ are points in $\mathbb{H}_{2}$, but both are
equivalent. This construction provides a complex structure on the upper sheet
of the hyperboloid $\mathbb{H}_{2}$, which can be treated as a noncompact
complex manifold.

##  4 Explicit form of the Wigner function for SU(1,1)

To gain further insights into this formalism, we will obtain the structure of
the Wigner function for SU(1,1) in more details.

Before going ahead we recall that the irreducible representations (irreps) of
SU(1,1) are labeled by the eigenvalues of the Casimir operator

$\hat{K}^{2}=\hat{K}_{0}^{2}-\hat{K}_{1}^{2}-\hat{K}_{2}^{2}=k(k-1)\mathbbm{1}\,,$
(26)

where $\hat{K}_{\pm}=\pm i(\hat{K}_{1}\pm i\hat{K}_{2})$. The irrep $k$ is
carried by a Hilbert space spanned by the common eigenstates of
$\hat{K}^{2}$and $\hat{K}_{0}$: $\\{|k,\mu\rangle\,:\mu=k,k+1,\ldots\\}$. All
unitary irreps are infinite dimensional. There are several different series of
irreps for SU(1,1) fixed by the domains of the eigenvalues $k$ [74]. For
representations in the positive discrete series, where $2k=1,2,3,\ldots$ and
including the two limit of discrete series with $k=1/4$ and $3/4$, the action
of the generators $\\{\hat{K}_{0},\hat{K}_{\pm}\\}$ therein is

$\displaystyle\hat{K}_{0}|k,\mu\rangle$ $\displaystyle=$
$\displaystyle\mu|k,\mu\rangle$ $\displaystyle\hat{K}_{\pm}|k,\mu\rangle$
$\displaystyle=$ $\displaystyle\sqrt{(\mu\pm k)(\mu\mp k\pm 1)}\,|k,\mu\pm
1\rangle\,.$

This carrier space is denoted by $\mathcal{D}_{k}^{+}$.

If the number of excitations in modes $a$ and $b$ are $n_{a}$ and $n_{b}$,
respectively, then $k$ and $\mu$ satisfy

$k=\tfrac{1}{2}(|n_{a}-n_{b}|+1)\,,\qquad\mu=\tfrac{1}{2}(n_{a}+n_{b}+1)\,.$
(28)

As discussed before, $n_{b}>n_{a}$ can be obtained from $n_{a}>n_{b}$ by just
a relabelling of modes, with no physical consequences. Therefore , we consider
$\pm(n_{a}-n_{b})$ to be equivalent irreps. The total Hilbert space of the two
oscillators decomposes then as

$\mathcal{H}_{a}\otimes\mathcal{H}_{b}=\mathcal{D}_{\frac{1}{2}}^{+}\oplus\mathcal{D}_{1}^{+}\oplus\mathcal{D}_{\frac{3}{2}}^{+}\oplus\cdots.$
(29)

This decomposition allows us to expand any (pure) state in an
SU(1,1)-invariant way; viz,

$|\Psi\rangle=\sum_{k}\sum_{\mu}\Psi_{k\mu}|k,\mu\rangle\,,$ (30)

where $\Psi_{k\mu}=\langle k,\mu|\Psi\rangle$. The Wigner function reads

$W(\zeta)=\langle\Psi|\hat{w}(\zeta)|\Psi\rangle=\sum_{k}\sum_{\mu,\mu^{\prime}}\Psi_{k\mu}^{\ast}\Psi_{k\mu^{\prime}}[d^{(k)}_{\mu^{\prime}\mu}(\tau)]^{2}\;(-1)^{\mu}\;e^{2i(\mu-\mu^{\prime})\chi}\,,$
(31)

where $d_{\mu\mu^{\prime}}^{(k)}(\tau)$ are the $d$-functions for SU(1,1),
which are the hyperbolic counterparts of the Wigner $d$ functions for SU(2)
[75]; that is,

$d_{\mu\mu^{\prime}}^{(k)}(\tau)=\langle
k,\mu|e^{i\tau\hat{K}_{y}}|k,\mu^{\prime}\rangle\,.$ (32)

They can be expressed as [76, 77]

$\displaystyle d_{\mu\mu^{\prime}}^{(k)}(\tau)$ $\displaystyle=$
$\displaystyle\left[\frac{\Gamma(\mu+k)\Gamma(\mu-k+1)}{\Gamma(\mu^{\prime}+k)\Gamma(\mu^{\prime}-k+1)}\right]^{1/2}\frac{1}{\Gamma(\mu-\mu^{\prime}+1)}\;{}_{2}F_{1}\biggl{(}\begin{array}[]{@{
}c@{ }}k-\mu^{\prime}\mskip 8.0muk+\mu\\\
\mu-\mu^{\prime}+1\end{array};\tanh^{2}(\tau/2)\biggr{)}$ (35)
$\displaystyle\times$
$\displaystyle\left[\cosh(\tau/2)\right]^{-2k+\mu^{\prime}-\mu}\left[\sinh(\tau/2)\right]^{\mu-\mu^{\prime}}\,,$
(36)

where ${}_{2}F_{1}$ is the hypergeometric function [78]. In the final
expression (31), we have made use of the fact that
$d_{\mu\mu^{\prime}}^{(k)}(\tau)$ are real and

$d_{\mu\mu^{\prime}}^{(k)}(\tau)=(-1)^{\mu^{\prime}-\mu}\,d_{\mu^{\prime}\mu}^{(k)}(\tau)\,.$
(37)

Equation (31) is a closed expression for the SU(1,1) Wigner function we were
looking for. Alternatively, one can rewrite the Wigner kernel $\hat{w}(\zeta)$
in the form

$\hat{w}(\zeta)=\exp\left(i\pi[\hat{K}_{0}\cosh\tau-\tfrac{1}{2}(e^{i\chi}\hat{K}_{+}+e^{-i\chi}\hat{K}_{-})\sinh\tau]\right)\,,$
(38)

which can be disentangled as

$\hat{w}(\zeta)=e^{\gamma_{-}\hat{K}_{-}}\;e^{i\pi\hat{K}_{0}}e^{\ln\gamma_{0}\hat{K}_{0}}e^{\gamma_{+}\hat{K}_{+}}\,,$
(39)

with

${\gamma_{\pm}=e^{\pm i\chi}\tanh\tau\,,\qquad\gamma_{0}=1/\cosh^{2}\tau}\,.$
(40)

The action of this operator in the basis $\\{|k,\mu\rangle\\}$ can be easily
found by expanding the exponentials. After a lengthy calculation the final
result coincides with (36).

Figure 1: Density plots of the Wigner function on the upper sheet of the
hyperboloid $\mathbb{H}_{2}$ and the associated distribution in the unit disk,
obtained from the upper sheet by stereographic projection from the south pole.
Left panel is a two-mode squeezed vacuum (42) with $k=1/2$ and $\xi=0.485$.
Right panel is a factorized state (43) of a coherent and a single-mode
squeezed state with $\alpha=1$ and $\xi=4+i/2$. The colormap goes from dark
blue (corresponding to 0) to red (corresponding to the maximum value).

A word of caution is in order here. Strictly speaking, Wigner functions can be
properly defined only for a single irrep, where the concept of phase space is
uniquely defined. Nonetheless,since the Hilbert space of our original two-mode
problem splits as in (29), our Wigner function appears as a sum over all the
irreps in the discrete positive series.

Let us consider the relevant example of SU(1,1) coherent states, defined as
[63]

$|\xi,k\rangle=\hat{S}(\zeta)|k,k\rangle\,,$ (41)

where $\xi=\tanh(\tau/2)e^{i\chi}$ and $\zeta=\tfrac{1}{2}\tau\,e^{i\chi}$.
These states live in the irrep $k$ and for $k=1/2$ they are nothing but two-
mode squeezed vacuum states, which in the photon-number basis read

$|\Psi\rangle=\sqrt{1-|\xi|^{2}}\sum_{n=0}^{\infty}\xi^{n}\;|n\rangle_{a}|n\rangle_{b}\,.$
(42)

Using this form in our general formula, we get an involved expression. The
result appears in Fig. 1. As we can appreciate, the squeezing appears here as
a displacement (and not merely as a deformation, as in the case of continuous
variables). The limit of infinite squeezing corresponds to displacing the
state to the infinity in the upper sheet, which means that the function tends
to the border of the unit disk. Note that the metric in the unit disk is not
Euclidean, but hyperbolic. This means, that as we approach the boundary, big
squeezing translates in small displacements.

As a second example, we consider the factorized state

$|\Psi\rangle=|\alpha\rangle_{a}|\xi\rangle_{b}\,,$ (43)

where $|\alpha\rangle_{a}$ is a coherent state in mode $a$ and
$|\xi\rangle_{b}$ a single-mode squeezed state in mode $b$. The decomposition
(31) now involves a sum over all irreps. This sum can be split into integer
and half-integer values, which translates in the presence of two peaks in the
corresponding Wigner function. The displacement from the origin of these peaks
is related to the squeezing, as before, and can be appreciated in Fig. 1.

##  5 Application to an SU(1, 1) interferometer

To check how the Wigner formalism developed thus far works, let us consider a
typical SU(1,1) interferometer, as sketched in Fig. 2. Two input modes
interact via an optical parametric amplifier (OPA) with gain parameter
$\mathcal{G}$. The action of the OPA is given by the squeeze operator
$\hat{S}(\zeta)$, defined in Eq. (17), with $\zeta=\mathcal{G}e^{i\vartheta}$.
After the first OPA, the upper path undergoes a phase shift $\phi_{1}$ and the
lower path undergoes a phase shift $\phi_{2}$. We assume a balanced
configuration, where the two OPAs have a fixed phase difference of $\pi$ and
the same gain factor. In consequence, the action of the interferometer, which
transforms input modes (labeled 0) into output modes (labeled 2), can be
concisely expressed as

$\hat{T}=\hat{S}(\zeta)\;e^{i(\phi_{1}\hat{a}^{\dagger}\hat{a}+\phi_{2}\hat{b}^{\dagger}\hat{b})}\;\hat{S}^{\dagger}(\zeta)\,.$
(44)

The important observation is that, up to a constant phase, in each subspace
with a fixed difference $\hat{a}^{\dagger}\hat{a}-\hat{b}^{\dagger}\hat{b}$,
this can be written in an compact SU(1,1) notation; viz

${\hat{T}=\hat{S}(\zeta)\;e^{i\Phi\hat{K}_{0}}\;\hat{S}^{\dagger}(\zeta)}\,,$
(45)

with $\Phi=\phi_{a}+\phi_{b}$.

The action of $\hat{T}$ (and, hence, of the interferometer) on the input state
is
$\hat{\varrho}_{\mathrm{out}}=\hat{T}\,\hat{\varrho}_{\mathrm{in}}\;\hat{T}^{\dagger}$
and on the Wigner function reads

${W_{\mathrm{out}}(\zeta)=\mathop{\mathrm{Tr}}\nolimits[\hat{\varrho}_{\mathrm{in}}\;\hat{T}\hat{w}(\zeta)\hat{T}^{\dagger}]=\mathop{\mathrm{Tr}}\nolimits[\hat{\varrho}_{\mathrm{in}}\,\hat{w}(g^{-1}\zeta)]=W_{\mathrm{in}}(g^{-1}\zeta)\,,}$
(46)

where $g$ is the group element corresponding to $\hat{T}$, which has the form

$g=\left(\begin{array}[]{cc}\cos(\Phi/2)+i\sin(\Phi/2)\cosh\tau&ie^{i\chi}\sin(\Phi/2)\sinh\tau\\\
-ie^{-i\chi}\sin(\Phi/2)\sinh\tau&\cos(\Phi/2)-i\sin(\Phi/2)\cosh\tau\end{array}\right)\,.$
(47)

The action is via Möbius transformations [63]; i.e.,

$g^{-1}\zeta=\frac{-\alpha^{\ast}\zeta+\beta}{\beta^{\ast}\zeta-\alpha}\,,$
(48)

where $\alpha$ and $\beta$ are the matrix elements of $g$ in (47)

$g=\left(\begin{array}[]{cc}\alpha&\beta\\\
\beta^{\ast}&\alpha^{\ast}\end{array}\right),\qquad|\alpha|^{2}-|\beta|^{2}=1\,.$
(49)

We thus have a closed formula that allows one to compute the Wigner function
of the output state for any input state in the interferometer.

As an example, we show the action of an interferometer with gain
$\mathcal{G}=0.5$ and $\Phi=\pi/2$, on an input state that consists of a two-
mode squeezed vacuum (42), with $k=1/2$ and $\xi=0.5$. The interferometer
action can be clearly identifies: it squeezes and the rotates the state. This
phase-space approach allows one to understand this action in a very clear and
intuitive way.

Figure 2: Schematic diagram of an SU(1,1) interferometer, which consists in a
Mach-Zehnder configuration in which the beam splitters have been replaced with
optical parametric amplifiers (OPAs). Each arm of the interferometer undergoes
a different phase shift. On the left we show the Wigner function on the unit
disk corresponding to an input state that is a two-mode squeezed vacuum (42),
with $k=1/2$ and $\xi=0.5$, whereas on the right we see the corresponding
Wigner function for the output. We have assumed a gain $\mathcal{G}=0.5$ and
$\Phi=\pi/2$.

##  6 Concluding remarks

Quantum phenomena must be depicted in the proper phase space. This is
unanimously recognized for continuous variables (with the complex plane as
phase space), for spinlike systems (with the Bloch sphere as the underlying
manifold), for orbital angular momentum (represented in the cylinder), and for
other systems. Surprisingly, the physics related to the SU(1,1) symmetry is
not displayed on the hyperboloid, the natural arena for these phenomena.

What we have accomplished here is to provide a practical framework to
represent SU(1,1) states in an appropriate way. Apart from the intrinsic
beauty of the formalism, our compelling arguments should convince the
community of the benefits that arise using the proper phase-space tools to
deal with these systems.

## Acknowledgments

We acknowledge financial support from the Mexican Consejo Nacional de Ciencia
y Tecnologia (CONACYT) (Grant 254127), the Spanish Ministerio de Ciencia e
Innovación (MICINN) (Grant PGC2018-099183-B-I00) and the Natural Sciences and
Engineering Research Council (NSERC) of Canada

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