A Simple Scheme of Coherent Photon Addition to Two-mode Gaussian States
Pei Zhang
The entanglement between Gaussian entangled states can be increased by non-Gaussian operations. We firstly design a simple scheme of coherently adding a photon generated by spontaneous parametric down conversation process to Gaussian quadrature-entangled light pulses created by a nondegenerate optical parametric amplifier. According to this non-Gaussian operation, the entanglement is increased as an entanglement distillation.
paper1.tex
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\begin{document}

\title{A Simple Scheme of Coherent Photon Addition to Two-mode Gaussian
States}
\author{Pei Zhang}\address{Non-equilibrium Condensed Matter and Quantum Engineering
Laboratory, the Key Laboratory of Ministry of Education, Xi'an Jiaotong
University, Xi'an 710049, People's Republic of China}
\author{Fu-Li Li\footnote[1]{Electronic address: flli@mail.xjtu.edu.cn}}\address{Non-equilibrium Condensed Matter and Quantum Engineering
Laboratory, the Key Laboratory of Ministry of Education, Xi'an Jiaotong
University, Xi'an 710049, People's Republic of China}
\author{Hong-Rong Li}\address{Non-equilibrium Condensed Matter and Quantum Engineering
Laboratory, the Key Laboratory of Ministry of Education, Xi'an Jiaotong
University, Xi'an 710049, People's Republic of China}


\begin{abstract}
The entanglement between Gaussian entangled states can be increased
by non-Gaussian operations. We firstly design a simple scheme of
coherently adding a photon generated by spontaneous parametric down
conversation process to Gaussian quadrature-entangled light pulses
created by a nondegenerate optical parametric amplifier. According
to this non-Gaussian operation, the entanglement is increased as an
entanglement distillation.
\end{abstract}

\pacs{03.67.Mn, 03.65.Ud, 03.65.Wj, 42.50.Dv }\maketitle

% PACS, the Physics and Astronomy
% Classification Scheme.
%\keywords{Suggested keywords}%Use show keys class option if keyword
%display desired

\section{INTRODUCTION}

Quantum states of light are very important resource for quantum
communication and computation protocols \cite{Nilson00,knill01}. Especially,
entangled states play a key role in quantum information processing (QIP),
such as quantum teleportation \cite{ben93}, entanglement swapping \cite%
{zuk93}, remote state preparation \cite{ben01} and quantum repeaters \cite%
{bri98}. Quantum states of propagating light beams can be treated as
discrete single photon states or quantum continuous variables (QCV) states,
which can be analyzed either by photon counting or by homodyne detection,
respectively. QCV is a very interesting system of QIP \cite{bra03}, which
encoding the information in the quadratures $x$ and $p$ of traveling light
fields. In most studies the QCV states are limited to Gaussian states
because of their well understood theoretical structure and ability to be
easily generated experimentally \cite{bra03}. For example, Gaussian two-mode
squeezed state (QCV entangled light) can be efficiently produced via optical
parametric amplification (OPA), and the two-mode squeezing degree can be
easily varied by adjusting the pump power. However, non-Gaussian states are
also very important. They can be employed to fulfil optimal quantum cloning
and minimal disturbance measurement for coherent states \cite{cerf05}. And
there have been a few proposals to detect entanglement in non-Gaussian
states \cite{agar}. Furthermore, it has been shown that Gaussian
entanglement distillation requires non-Gaussian operations \cite{ei02}. With
the current experiment conditions, one of the practical ways to prepare
non-Gaussian states is the conditional subtraction\ (addition) of photons
from (to) Gaussian entangled beams \cite{sub and add}. And the properties of
these non-Gaussian states have been investigated, including the
non-classicality, inseparability and entanglement \cite{pro,our07,lhr07}.


We know that entanglement distillation allows one to produce stronger
entanglement between distant sites, which is essential for long-distance
quantum communications \cite{ben96}. In the discrete-variable systems,
entanglement distillation has been well studied and demonstrated by several
experiments recently \cite{pan}. While in the QCV systems, several proposals
have been put forward to realize QVC distillation \cite{duan00,ei04}. It has
been shown that the photon-subtracted and photon-added operations on
two-mode squeezed states can be applied to enhance the entanglement of QCV
states \cite{pro,our07,lhr07,ping08}. Recently, A. Ourjoumtsev \textit{et al.%
} have completed an experiment which increased entanglement between
Gaussian states by coherent photon subtraction (CPS) \cite{our07}.
They coherently subtracted single photons from Gaussian
quadrature-entangled light pulses, produced delocalized states with
negative Wigner functions and complex structures more entangled than
the initial states in terms of negativity, which is the quantity of
entanglement \cite{vidal02}. This experiment opens up a way to
coherently subtract one photon from two modes of light. Naturely, we
may ask how about coherent photon addition (CPA)? Can this
non-Gaussian operation enhance the entanglement? And how to
experimentally realize CPA? In this paper, we make a theoretical
analysis about CPA process and CPA states, find that it is similar to CPS state \cite%
{our07} whose entanglement, quantified by the negativity
\cite{vidal}, is significantly increased compared with the initial
state generated by OPA. We also give an experimental setup on how to
realize the CPA process under current technique conditions. Last, we
give a brief discussion and conclusion.

\section{THEORETICAL ANALYSIS}

The non-Gaussian state of coherently subtracting a photon from a two-mode
squeezed state has been investigated theoretically and experimentally \cite%
{our07}. For a simple theoretical analysis between CPS and CPA
below, we can get that the final states of CPS and CPA are almost in
a same form. This directly show us that the non-Gaussian operation
of CPA can enhance the entanglement.

The two-mode squeezed state can be expressed in the Fock state basis as
follows

\begin{equation}
\left\vert S(\lambda )\right\rangle =\sqrt{1-\lambda ^{2}}\overset{\infty }{%
\underset{n=0}{\sum }}\lambda ^{n}\left\vert n,n\right\rangle ,
\end{equation}%
where $\lambda =\tanh (r)$ and $r$ is the squeezing degree
parameter. Then the CPA state can be write as
\begin{equation}
\left\vert S(\lambda ,1)\right\rangle \propto (a_{1}^{\dag }+\mu a_{2}^{\dag
})\left\vert S(\lambda )\right\rangle .
\end{equation}%
$\mu $ is a complex parameter, $a_{1}^{\dag }$ and $a_{2}^{\dag }$ are
photon creation operators for modes 1 and 2. Without loss of generality, we
can consider the case of $\mu =1$. Under a detailed calculation, we get%
\begin{equation}
\left\vert S(\lambda ,1)\right\rangle =\frac{1-\lambda ^{2}}{\sqrt{2}}%
\overset{\infty }{\underset{n=0}{\sum }}\lambda ^{n}\sqrt{n+1}(\left\vert
n+1,n\right\rangle +\left\vert n,n+1\right\rangle ).
\end{equation}%
While for CPS process, we can define the operator as $a_{1}+\mu
a_{2}$. So coherently subtracting a photon from a two mode
squeezed state (Eq. (1)) can be written as%
\begin{equation}
\left\vert S(\lambda ,-1)\right\rangle \propto (a_{1}+\mu a_{2})\left\vert
S(\lambda )\right\rangle .
\end{equation}%
$a_{1}$ and $a_{2}$ are photon annihilation operators for modes 1 and 2. For
the same reason as above, we consider the case $\mu =1$. We can get%
\begin{eqnarray}
\left\vert S(\lambda ,-1)\right\rangle &=&\frac{1-\lambda ^{2}}{\sqrt{2}}%
\overset{\infty }{\underset{n=1}{\sum }}\lambda ^{n-1}\sqrt{n}(\left\vert
n-1,n\right\rangle +\left\vert n,n-1\right\rangle )  \notag \\
&=&\frac{1-\lambda ^{2}}{\sqrt{2}}\overset{\infty }{\underset{n^{\prime }=0}{%
\sum }}\lambda ^{n^{\prime }}\sqrt{n^{\prime }+1}(\left\vert n^{\prime
},n^{\prime }+1\right\rangle +\left\vert n^{\prime }+1,n^{\prime
}\right\rangle ),
\end{eqnarray}%
where $n^{\prime }=n-1$. We can see that Eq. (5) is almost same as
Eq. (3) except for different start point of $n$. This means that CPA
operation can act on squeezed vacuum state which CPS operation can
not. We know that the squeezed vacuum state is a very useful state
and in common use, so CPA operation should be more useful than CPA
operation.

CPS state has been experimental tested that the entanglement is
higher than the initial two-mode squeezed state with up to 3 dB of
squeezing, and even small experimental improvements should
significantly increase this limit \cite{our07}. From the simple
analysis above, it is interesting that CPA process can generate a
similar non-Gaussian state as CPS, and can be used as entanglement
distillation. Furthermore, we may combine CPS and CPA process
together, such as CPS then CPA or CPA then CPS, to study some new
phenomenon.



\begin{figure}[tbh]
\includegraphics[width=10cm]{2.eps}
\caption{Experimental scheme of coherent photon-addition to two-mode
squeezed states. The primary light source is a mode-locked Ti:sapphire laser
whose pulses are split to two spatial modes by a high-transmission beam
splitter (HT-BS), one is used for local oscillator and the other is
frequency-doubled by a second harmonic generation (SHG) to become the pump
for OPA and degenerated SPDC in a type-I BBO crystal. $\Phi_{1}$ and $%
\Phi_{2}$ are varied by the piezoelectric transducer (PZT). All the PBS are
translated vertical polarized light and reflected horizontal polarized
light. F is a combination of spectral and spatial filter made of a
interference filter and a single-spatial-mode optical fiber directly
connected to a single-photon counting module (SPCM). Additional optics and
computer control to realize the homodyne detection and coincidence count are
not shown here for the sake of clarity.}
\end{figure}

\section{EXPERIMENTAL SCHEME}

We have designed an experimental scheme for CPA process, which can
be realized in lab under current technique. The experimental scheme
is shown in Fig. 2. Femtosecond pulses produced by a Ti:sapphire
laser are split into two parts by a high-transmission beam splitter
(HT-BS). The small part (reflected part) is used as a local
oscillator for homodyne detection, and the main part (transmitted
part) is frequency doubled by a type I non-critically phase-matched
potassium niobate (KNbO$_{3}$) crystal. One of the frequency-doubled
beam pumps an identical crystal used as an OPA, generating Gaussian
quadrature-entangled pulses spatially separated by an angle of
$10\textordmasculine$ and also with orthogonal polarizations defined
as horizontal (H) polarization and vertical (V) polarization.
Another part of the frequency-doubled beam pumps a type-I
beta-barium borate (BBO) nonlinear crystal to generate entangled
photon pairs. The entangled Gaussian beams are spatially combined on
a polarizing beam splitter (PBS) cube but remain different
polarizations. Then this beam is injected in the BBO nonlinear
crystal as a seed field of signal mode, and the conditional
preparation of the photon-added state takes place every time that a
single photon is detected in the correlated idler mode. To achieve
adding one photon to the two mode coherently, we can adjust the
polarization of pump beam of BBO crystal by a half-wave plate (HWP)
in order to generate $45\textordmasculine$-polarized twin photons.
In this case, when the signal beam (mixed with signal photons and
the entangled Gaussian beams)
pass the second PBS, two orthogonal polarized beams are and the $45%
\textordmasculine$-polarized photons spatially separated and
combined with bright local oscillator beams. If the detector on the
idler route is clicked, we can get that one photon has been
coherently added to the Gaussian quadrature-entangled beam generated
by OPA. The intensities and the relative phase of local oscillator
can be easily adjusted by a quarter-wave plate (QWP), a HWP and a
piezoelectric transducer (PZT). At the end of this setup, we can
perform a homodyne detection to analyze the nonclassical character
of the CPA states.


In this setup, we mainly use polarization information of photons and pulses
to control their spatial routes, so the relative phases except $\Phi_{1}$
and $\Phi_{2}$ can be precisely adjusted by wave plates. $\Phi_{1}$ and $%
\Phi_{2}$ are controlled by the PZT, which can easily and precisely
realize a phase scan for experimental request. In our setup, we can
easily change the phase parameter $\mu $ of Eq. (2) by adjusting the
polarization of pump beam of BBO crystal. This can be achieved by
simply rotating the HWP before BBO crystal. Especially, when the
twin photons' polarization is at $45\textordmasculine$, we can
realize the CPA process mentioned above; when the twin photons'
polarization is at $0\textordmasculine$ ($90\textordmasculine$), we
can get that one photon is added in one of two modes definitely (normal photon
addition). So, this setup can be used to verify the nonclassical properties of
two-mode single-photon-added states from CPA to normal photon
addition. The precision of adding exactly a single photon is mainly
limited by the nonunit detection efficiency of SPCM trigger and the
higher-order emissions from the SPDC process in BBO crystal.
However, these limitations are not very restrictive for the present
application where the down-conversion should be operated in a regime
of low gain.

\section{CONCLUSION}

In conclusion, we firstly put forward a CPA protocol which allows
one to increase the entanglement between Gaussian states. This
non-Gaussian operation can generate similar non-Gaussian states whit
CPS even the initial state is squeezed vacuum state. And the
realization of CPA can make the further studies of combining CPS and
CPA. The final non-Gaussian CPA states can in principle be used as a
starting point for a "Gaussification" procedure \cite{ei04}. The
experimental setup we put forward here is simple and realizable
under current technique. It is useful for Gaussian entanglement
distillation, and it is also one of the key steps for long-distance
quantum communications with continuous variables.

\section{ACKNOWLEDGEMENT}

The authors thank Ai-Ping Fang and Yang Yang for interesting and
helpful discussion. This work was funded by the National Fundamental
Research Program, National Natural Science Foundation of China
(Grant No. and Grant No. 10774117).

\bigskip

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\end{thebibliography}

\end{document}

0909.3282.extracted_bib
\begin{thebibliography}{99}
\bibitem{Nilson00} M. A. Nielsen and I. L. Chuang, \textit{Quantum
Computation and Quantum Information} (Cambridge University Press, Cambridge,
2000).

\bibitem{knill01} E. Knill,\textit{\ }R. Laflamme and G. J. Milburn, Nature
(London) \textbf{409}, 46 (2001).

\bibitem{ben93} C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres
and W. K. Wootters, Phys. Rev. Lett. \textbf{70}, 1895 (1993).

\bibitem{zuk93} M. Zukowski, A. Zeilinger, M. A. Hone and A. K. Ekert, Phys.
Rev. Lett. \textbf{71}, 4287 (1993).

\bibitem{ben01} C. H. Bennett, D. P. Divincenzo, P. W. Shor, J. A. Smolin,
B. M. Terhal and W. K. Wootters, Phys. Rev. Lett. \textbf{87}, 077902 (2001).

\bibitem{bri98} H.-J. Briegel, W. D\"{u}r, J. I. Cirac and P. Zoller, Phys.
Rev. Lett. \textbf{81}, 5932 (1998); L.-M. Duan, M. D. Lukin, J. I. Cirac
and P. Zoller, Nature (London) \textbf{414}, 413 (2001).

\bibitem{bra03} S. L. Braunstein and A. K. Pati, \textit{Quantum Information
with Continuous Variables} (Kluwer Academic, Dordrecht, 2003); S. L.
Braunstein and P. van Look, Rev. Mod. Phys. \textbf{77}, 513 (2005).

\bibitem{cerf05} N. J. Cerf, O. Kr\"{u}ger, P. Navez, R. F. Werner and M. M.
Wolf, Phys. Rev. Lett. \textbf{95}, 070501 (2005); Ladislav Mi\v{s}ta, Jr.,
Phys. Rev. A \textbf{73}, 032335 (2006).

\bibitem{agar} G.S. Agarwal and A. Biswas, New J. Phys. \textbf{7}, 211
(2005); E. Shchukin and W. Vogel, Phys. Rev. Lett. \textbf{95}, 230502
(2005); M. Hillery and M.S. Zubairy, Phys. Rev. Lett. \textbf{96}, 050503
(2006).

\bibitem{ei02} J. Eisert, S. Scheel and M. B. Plenio, Phys. Rev. Lett.
\textbf{89}, 137903 (2002); J. Fiur\'{a}\v{s}ek, Phys. Rev. Lett.\textbf{\ 89%
}, 137904 (2002); G. Giedke and J. I. Cirac, Phys. Rev. A \textbf{66},
032316 (2002).

\bibitem{sub and add} G. S. Agarwal and K. Tare, Phys. Rev. A \textbf{43},
492 (1991); A. Zavatta, S. Viciani and M. Bellini, Science
\textbf{306}, 660 (2004); A. Zavatta, S. Viciani and M. Bellini,
Phys. Rev. A \textbf{72}, 023820 (2005); A. Zavatta, V. Parigi, and
M. Bellini, Phys. Rev. A \textbf{75}, 052106 (2007); V. Parigi,  A.
Zavatta, M. S. Kim and M. Bellini, Science \textbf{317}, 1890
(2007); M. S. Kim, H. Jeong, A. Zavatta, V. Parigi and M. Bellini,
Phys. Rev. Lett. \textbf{101}, 260401 (2008).

\bibitem{pro} A. R. Usha Devi, R. Prabhu and M. S. Uma, Eur. Phys. J. D
\textbf{40}, 133 (2006); M. S. Kim, E. Park, P. L. Knight and H. Jeong,
Phys. Rev. A \textbf{71}, 043805 (2005); A. Kitagawa, M. Takeoka, M. Sasaki
and A. Chefles, Phys. Rev. A \textbf{73}, 042310 (2006).

\bibitem{our07} A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri and P. Grangier,
Phys. Rev. Lett. \textbf{98}, 030502 (2007).

\bibitem{lhr07} H.-R. Li, F.-L. Li and S.-Y. Zhu, Phys. Rev. A \textbf{75},
062318 (2007).

\bibitem{ben96} C. H. Bennett, H. J. Bernstein, S. Popescu and B.
Schumacher, Phys. Rev. A \textbf{53}, 2046 (1996).

\bibitem{pan} P. G. Kwiat \textit{et al.}, Nature (London) \textbf{409},
1014 (2001); J.-W. Pan \textit{et al.}, Nature (London) \textbf{423}, 417
(2003); P. Walther \textit{et al.}, Phys. Rev. Lett. \textbf{94}, 040504
(2005).

\bibitem{duan00} L.-M. Duan \textit{et al.}, Phys. Rev. Lett. \textbf{84},
4002 (2000).

\bibitem{ei04} J. Eisert \textit{et al.}, Ann. Phys. (N.Y.) \textbf{311},
431 (2004).

\bibitem{ping08} P. Chang, B. Shao and G.-L. Long, Phys. Lett. A \textbf{372}%
, 7124 (2008).

\bibitem{vidal02} G. Vidal and R. F. Werner, Phys. Rev. A \textbf{65},
032314 (2002).

\bibitem{vidal} G. Vidal and R. F. Werner, Phys. Rev. A \textbf{65}, 032314
(2002).
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