# Bright source of spectrally uncorrelated polarization-entangled photons with
nearly single-mode emission

P. G. Evans<EMAIL_ADDRESS>Center for Quantum Information Science,
Computing and Computational Sciences Directorate, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  J. Schaake  Department of Physics
and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA  R. S.
Bennink  Center for Quantum Information Science, Computing and Computational
Sciences Directorate, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA  W. P. Grice  Center for Quantum Information Science, Computing and
Computational Sciences Directorate, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA  T. S. Humble  Center for Quantum Information Science,
Computing and Computational Sciences Directorate, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA

(December 13, 2010)

###### Abstract

We present results of a bright polarization-entangled photon source operating
at 1552 nm via type-II collinear degenerate spontaneous parametric down-
conversion in a periodically poled potassium titanyl phosphate crystal. We
report a conservative inferred pair generation rate of 123,000 pairs/s/mW into
collection modes. Minimization of spectral and spatial entanglement was
achieved by group velocity matching the pump, signal and idler modes and
through properly focusing the pump beam. By utilizing a pair of calcite beam
displacers, we are able to overlap photons from adjacent down-conversion
processes to obtain polarization-entanglement visibility of 94.7 +/- 1.1% with
accidentals subtracted.

###### pacs:

42.50.Dv 03.67.Bg 42.50.Ex 42.65.Lm

Spontaneous parametric down-conversion (SPDC) is the leading mechanism for
realizing photonic quantum states. For many applications, particularly those
that involve the interference of photons from multiple SPDC sources, the
individual photon pairs are required to be spectrally and spatially pure. This
is because multimode down-converted photons carry distinguishing information
that undermines the interference central to such applications pittman2003 . It
has been shown previously that spectral entanglement can be eliminated through
source engineering. This approach has led to an experimental demonstration of
heralded generation of spectrally pure single photons from type-II SPDC
mosley2008 ; migdall2010 , and has also inspired similar approaches of type-I
SPDC vicent2010 ; uren2005  and spontaneous four-wave mixing in fibers
soller2010 ; cohen2009 ; halder2009 .

In this Letter, we demonstrate the first bright SPDC source for polarization-
entangled biphoton states that are uncorrelated in both the spectral and
spatial degrees of freedom. As in the previous works, our source is engineered
for spectral purity through the selection of the pump properties and the
phase-matching characteristics of the SPDC medium. In addition, special
attention is paid to the spatial entanglement. Specifically, our source is
based on collinear type-II SPDC in noncritically phase-matched periodically
poled potassium titanyl phosphate (PPKTP), a geometry that eliminates walk-off
effects and maximizes the overlap of the pump, signal, and idler fields
fedrizzi2007 . In this configuration, a properly focused pump causes nearly
all of the photon to be emitted into a single spatial mode grice2010 ;
bennink2010 . The combination of these techniques for controlling the SPDC
output gives rise to a single-mode emission rate of 123,000 pairs/s/mW. The
source is further configured in a novel arrangement of calcite beam displacers
that separate and then recombine photons generated by either of the two
parallel pump beams.

The spectral and spatial properties of SPDC photons are determined by the pump
field and by the dispersive properties of the crystal. The two-photon
probability amplitude is the product of a pump function and a phase-matching
function. Entanglement can be eliminated only if this product yields no
correlations between the signal and idler photon properties. By careful
selection of the wavelength, pulse duration, and focus of the pump, as well as
the crystal material and length, it is possible to minimize entanglement in
these degrees of freedom. In general, the spectral and spatial properties are
not independent, but for the purposes of illustrating the design principles,
it is sufficient here to treat them separately.

To eliminate spectral entanglement, the shapes of the pump function and phase-
matching function must be chosen correctly grice2001 . The pump function,
which describes the range of energies available for down-conversion, generally
yields negatively correlated photon energies — the signal and idler energies
must sum to an energy somewhere in the pump spectrum, so a longer wavelength
for one photon is necessarily accompanied by a shorter wavelength for the
other. This correlation is strongest for a monochromatic pump, for which there
is but a single pump energy. Therefore, one requirement for the elimination of
spectral entanglement is a broad pump spectrum. However, this alone will not
eliminate spectral entanglement if the phase-matching function also leads to
negatively correlated energies.

Whereas the pump function describes the range of energies available for down-
conversion, the phase-matching function describes the ways that the pump
energies may be distributed to the signal and idler photons. The influence of
the phase-matching function on the spectral properties of the photons is
revealed by noting that the function has appreciable value only for
$\Delta{k}L\simeq 0$, where $L$ is the crystal length and where
$\Delta{k}=k_{p}-k_{s}-k_{i}$ is the wavevector mismatch. Using the
approximation $k\simeq k_{0}+\nu k^{\prime}$ and the fact that
$k_{p0}-k_{s0}-k_{i0}=0$ for a phase-matched interaction, we have
$\Delta{k}\simeq(\nu_{s}+\nu_{i})k_{p}^{\prime}-\nu_{s}k_{s}^{\prime}-\nu_{i}k_{i}^{\prime}$.
Here $\nu=\omega-\omega_{0}$ and $k^{\prime}=\partial{k}/\partial{\omega}$.
Imposing the requirement that $\Delta{k}\simeq 0$ yields

$\nu_{s}=-\nu_{i}\frac{k_{p}^{\prime}-k_{i}^{\prime}}{k_{p}^{\prime}-k_{s}^{\prime}}$
(1)

such that the phase-matching function leads to positively correlated photon
energies only if $k_{p}^{\prime}$ lies between $k_{s}^{\prime}$ and
$k_{i}^{\prime}$ or, since the group velocity $v_{g}=1/k^{\prime}$, if the
group velocity of the pump lies between the group velocities of the signal and
idler photons.

It is difficult to satisfy the group velocity matching condition as defined in
Eq. (1) for visible wavelengths with most materials since normal material
dispersion results in lower group velocities for the bluer pump wavelengths.
However, dispersion is more accommodating at longer wavelengths, and solutions
can be found for several type-II materials grice2001 . In particular, group
velocity matching can be achieved with type-II SPDC in KTP with a pump
wavelength range of 650-900 nm for degenerate down-conversion to 1.3-1.8
$\mu$m. Once the material and pump wavelength have been specified, the widths
of the pump and phase-matching functions must be chosen so that the resulting
probability amplitude exhibits neither positive nor negative correlations in
the photon energies. This requirement leads to a specific relationship between
the pump bandwidth and the crystal length. For the 20-mm PPKTP crystal used in
our source, our calculations predict that the spectral entanglement will be
minimized, with a spectral Schmidt number of 1.06, using a 776 nm pump with a
bandwidth corresponding to a transform-limited pulse duration of 1.3 ps. We
note that, with a longer crystal and/or a broader pump bandwidth, it is
possible to generate photon pairs having positive energy correlations, as has
been shown previously shimizu2009 .

The factors that must be considered to minimize the spatial (transverse
momentum) entanglement are similar to those pertaining to the spectral
entanglement. As in the spectral domain, the pump function leads to a tendency
toward negatively correlated transverse momenta — the signal and idler momenta
must sum to a momentum somewhere in the pump spectrum, so the emission
directions of the photons are negatively correlated. This correlation is
strongest when the pump transverse momentum spectrum is narrow, i.e., when the
pump is collimated. A necessary condition for the elimination of spatial
entanglement, therefore, is a pump with a broad transverse momentum spectrum,
a requirement that is easily met by focusing.

The requirement that the phase-matching function yield toward positively
correlated transverse momenta is satisfied in most down-conversion materials,
particularly when there is not spatial walk-off. This is the case for the
noncritically phase-matched PPKTP. All that remains then is to choose the
widths of the pump and phase-matching functions so as to eliminate the spatial
entanglement. For the 20-mm PPKTP crystal used in our source, calculations
predict the best performance with the pump having a divergence of 13.1 mrad
and focused at the center of the crystal. It is shown elsewhere grice2010
that the conditions that minimize spatial entanglement are the same conditions
that maximize coupling to single-mode collection optics. In our case, the
signal and idler photons are predicted to be emitted collinearly into single
modes with divergences of 18.4 and 18.1 mrad, respectively.

Our approach provides several advantages in comparison to traditional
multiphoton entanglement experiments: namely

  * •

the pump wavelength of 776 nm is accessible with a tunable pulsed Ti:Sapphire
laser, without the requirement for a second-harmonic generation crystal to
double the pump wavelength as in the usual UV $\rightarrow$ Vis down-
conversion schemes,

  * •

the minimization of spectral and spatial entanglement by the source removes
any need for interference and spatial filters to be used, with SPDC photons
emitted into a single spatial mode for optimal coupling to collection optics
and

  * •

776 nm $\rightarrow$ 1552 nm SPDC occurs in the technologically important
telecom band where optical fibers exhibit minimal attenuation and standard
telecoms equipment is readily available.

The experimental setup, which is a modification of the scheme first presented
in fiorentino2008 , is illustrated in Fig. 1. A 776 nm pump beam from a
Coherent Mira Ti:Sapphire laser is incident upon a lens, half-wave plate
(HWP1), birefringent wedge pair (BWP), and beam displacer (BD1). BD1 displaces
the orthogonal polarizations of the pump components by 4.2 mm; the vertically
polarized pump component is passed through the half-wave plate HWP2 oriented
with the fast axis at 45∘. The two pump beams incident on the PPKTP crystal
are horizontally polarized with a focus of 13.1 mrad to satisfy type-II phase
matching and to minimize spatial entanglement as described above. Both pump
waists are located midway along the length of the PPKTP crystal.

(a) Experimental setup

(b) Detail of beam displacer pair inset

Figure 1: (Color online) (a) Experimental setup. (b) Detail of the BD pair
showing the beam configuration on entry to BD2 (left diagram), the action of
the half-wave plates between BD2 and BD3 (middle diagram), and upon exit from
BD3 (right diagram). LPF: long-pass filter; PBS: polarization beam splitter.

Upon emergence from the PPKTP crystal, the signal and idler photons from the
two down-conversion regions are incident upon beam displacer BD2, which acts
to displace the signal and idler photons vertically, creating four beam paths,
shown in detail in Fig. 1b. Thin half-wave plates rotate the photon
polarizations in two of the paths before all four beams are incident on beam
displacer BD3, which acts to recombine photons such that both signal photons
are emergent in the upper path and both idler photons are emergent in the
lower path. Lenses following the beam displacers are used to match the spatial
mode of the signal and idler photons to the collection optics. A pair of long-
pass filters remove the residual pump from each arm and the photons are
coupled into single-mode fibers. Wave plates and polarizers are placed in the
two beams for analysis of the polarization-entangled state.

Photon detection is accomplished using two fiber-coupled idQuantique id200
InGaAs/InP avalanche photodiodes with a reported detection efficiency of
approximately 10%. Both detectors are set to use a 2.5 ns gate width and are
triggered at 4.75 MHz synchronized with the 76 MHz pulse train from the
Ti:Sapphire. Output pulses from the id200s, corresponding to photon detection
events, are input into ORTEC counting, delay and coincidence logic circuits
for computer readout.

The singles and coincidence counts are measured as a function of incident pump
power and are displayed in Fig. 2. The half-wave plates and polarization beam
splitters were removed from both collection arms for this measurement. The
single counts are linear with respect to increasing pump power; the
coincidence counts exhibit a slight nonlinear trend that we attribute to
multiple-pair generation at higher pump power wong2008 . By taking into
account Fresnel losses from the uncoated PPKTP crystal and calcite beam
displacers, plus attenuation by the pair of long-pass filters in each arm, we
estimate 38% transmission at 1552 nm through the system. In conjunction with a
conservative 50% free-space to SMF coupling efficiency and 10% detection
efficiency for each id200 detector, our detection efficiency for 1552 nm
photons is 1.9%. Given the actual pair generation rate is equal to the
measured coincidence rate divided by the singles detection efficiency squared,
we conservatively infer the pair generation rate of our source to be 123,000
pairs/s/mW pump. To the best of our knowledge, this is the brightest narrow-
band source operating in the telecom band reported to date.

Figure 2: (Color online) Singles $\sqrt{A.B}$ (left axis) and coincidence
counts (right axis) vs incident pump power. The dashed lines serve as guides
to the eye.

Polarization correlation measurements were carried out with the half-wave
plates and polarization beam splitters reinserted in both arms. Figure 3 shows
polarization correlation plots in the $\pm$45∘ basis. The incident pump power
was set to 16 mW and accidentals were subtracted. Curve fitting to the data
yields a visibility of 94.7 $\pm$ 1.1%.

Figure 3: Coincidence counts vs analyzer orientation angle in the $\pm$45∘
polarization basis showing a visibility of 94.7 $\pm$ 1.1%. The incident pump
power is 16 mW and accidental counts have been subtracted.

In order to examine the effect of multipair generation on the $\pm$45∘ basis
visibility, we conducted several polarization correlation measurements with
various incident pump powers. The results are presented in Fig. 4. The open
circles show the uncorrected data and exhibit a linear decrease with respect
to increasing pump power, providing evidence of multipair generation at higher
pump power. The filled circles represent the corrected data, i.e., raw data
with accidentals subtracted, which average 94% and are constant with respect
to incident pump power as one would expect. We considered several explanations
that would lead to our visibility being capped to 94%: differences in the
optical path lengths of the two down-conversion processes, small differences
in signal and idler photon wavelengths due to poling inhomogeneities, and
errors in wave plate orientation. Optical path differences were minimized by
positioning the birefringent wedge pair BWP to maximize visibility. We
measured the spectra of all four photons using a monochromator and found that
any wavelength differences were on the order of the monochromator resolution.
Thus we conclude that the likely source of the reduced visibility is
orientation error for the two wave plates placed between BD2 and BD3.

Figure 4: Raw ($\circ$) and subtracted accidentals ($\bullet$) visibility in
the $\pm$45∘ polarization basis as a function of incident pump power. The
dashed lines indicate linear fits to the data.

We performed joint spectral intensity measurements on our source using a
customized dual-slit scanning monochromator with 0.3 nm resolution. By
controlling the pump laser mode locking, we were able to adjust the pump pulse
duration and measure the joint spectral intensity at several different pump
pulse durations. Figure 5 shows the measured joint spectral intensity taken
with an incident pump power of 25 mW and 200 s counts per measurement and with
accidental coincidences subtracted, for an optimal pump laser pulse duration
of 1.3 ps [Fig. 5(a)] and 1.9 ps [Fig. 5(b)] respectively. Analysis of the raw
data yields spectral Schmidt numbers of 1.07 and 1.16, respectively, in
agreement to 1.06 as predicted by our calculations for the optimal case.
Furthermore, changes in the joint spectral intensity can clearly be seen
between the different pump pulse durations.

(a) 1.3 ps pump

(b) 1.9 ps pump

Figure 5: (Color online) Joint spectral intensities using two pump pulse
durations, 1.3 ps (a) and 1.9 ps (b). Analysis yields spectral Schmidt numbers
of 1.07 and 1.16, respectively.

In summary, we have demonstrated a unique source insofar as entanglement in
the spatial and spectral degrees of freedom has been minimized by appropriate
pump focusing and careful selection of pump bandwidth, wavelength, and phase
matching with type-II PPKTP. As a result, photons are emitted into single
spectral and spatial modes which require no spectral or spatial filtering to
observe high multiphoton visibility of 94.7 $\pm$ 1.1% with an inferred pair
generation rate of 123,000/s/mW pump. By using a novel arrangement of beam
displacers to recombine the signal and idler photons into distinct paths, we
have demonstrated a source design that can be extended to the construction of
multiphoton ($\geq$ 4) polarization-entangled states.

We thank M. Fiorentino and A. Migdall for enlightening discussions. Research
is sponsored by the Laboratory Directed Research and Development Program of
Oak Ridge National Laboratory, managed by UT-Battelle, LLC for the U. S.
Department of Energy under Contract No. De-AC05-00OR22725.

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