High purity bright single photon source
J. S. Neergaard-Nielsen, B. Melholt Nielsen, H. Takahashi⋆,
A. I. Vistnes† and E. S. Polzik
QUANTOP, Danish National Research Foundation Center for Quantum Optics,
Niels Bohr Institute, University of Copenhagen, DK 2100, Denmark
⋆ Permanent address: Department of Applied Physics, The University of Tokyo 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-8656, Japan, and National Institute of Information and
Communications Technology, 4-2-1 Nukui-Kita, Koganei, Tokyo 184-8795, Japan
† Permanent address: Department of Physics, University of Oslo, Postboks 1048, Blindern,
0316 Oslo, Norway
jneergrd@nbi.dk
Abstract: Using cavity-enhanced non-degenerate parametric down-
conversion, we have built a frequency tunable source of heralded single
photons with a narrow bandwidth of 8 MHz, making it compatible with
atomic quantum memories. The photon state is 70% pure single photon
as characterized by a tomographic measurement and reconstruction of the
quantum state, revealing a clearly negative Wigner function. Furthermore, it
has a spectral brightness of ∼1,500 photons/s per MHz bandwidth, making
it one of the brightest single photon sources available. We also investigate
the correlation function of the down-converted fields using a combination
of two very distinct detection methods; photon counting and homodyne
measurement.
© 2021 Optical Society of America
OCIS codes: (120.2920) Homodyning; (230.6080) Sources; (270.5290) Photon statistics.
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1. Introduction
Pure single photon states produced efficiently and at a high rate are highly desirable for prac-
tical implementations of various quantum information processing protocols, in particular in
quantum cryptography [1], quantum computing with linear optics [2], and for testing quantum
memories [3, 4]. The latter applications require at the same time compatibility with some kind
of a quantum memory. Different approaches towards generation of a single photon state have
been implemented in a number of physical systems. It should be noted that in many instances
a source is claimed to be ”a single photon source” based just on the property of antibunch-
ing, i.e., on the low rate of two-photon contribution compared to a single photon part. Such
http://arxiv.org/abs/quant-ph/0609033
property should be combined with the likewise low contribution of the vacuum state, in order
to claim a high-purity truly single photon source which is the aim of the present work. Single
emitters usually suffer from low purity due to small collection efficiency for light. For example,
quantum dot based sources [5, 6], color centres in diamond [7, 8], single molecules [9], or a
single atom [10] have the detection efficiency/purity at best at a few percent level. Placing sin-
gle emitters inside high-Q cavities improves the purity dramatically. However, even complex
state-of-the-art experiments still have limited overall collection efficiency and thus low purity.
The best results with quantum dots [11] show 8% collection efficiency/purity. Besides, quan-
tum dots usually emit light in a several GHz bandwidth. The best efforts with cavity-QED with
atoms or ions yield 30-40% efficiency just outside the cavity and the overall efficiency/purity at
10-20% level [12, 13, 14]. Recently atomic ensembles have been used to produce non-classical
light [15, 16, 17], however, the light collection efficiency does not exceed a few percent even
when atoms are placed inside a cavity [16]. Parametric down-conversion in free space non-
linear crystals or waveguides [18, 19, 20, 21, 22, 23] has been widely used for generation of
heralded photon pulses. The major disadvantage of parametric down-conversion is the random
arrival time of the photons – the source is not deterministic. However, this is compensated by
many attractive properties like well-defined wavelength, high collection efficiency, and non-
cryogenic experimental setups. The standard pulsed, single-pass down-conversion process suf-
fers from a limited photon generation rate which must be kept low to avoid pulses containing
two photons, which is detrimental for quantum information applications. The bandwidth of the
down-conversion is typically several nanometers, which means that the spectral brightness (the
number of photons per MHz per second) is below one. This poses a serious limitation to the
feasibility of interaction with atomic systems, where linewidths are on the MHz scale. To over-
come this problem, the nonlinear crystal can be placed inside an optical cavity which serves to
enhance the down-conversion process and limit the bandwidth of the output to that of the cav-
ity [24]. Furthermore, the spatial field mode is defined by the cavity as well, so no additional
spatial filtering is needed. Various studies on this type of setup have been performed, and the
results do indeed show a marked increase in the attained spectral brightness [24, 25, 26].
In this paper we present our scheme for generation of heralded single photons with a very
high purity and spectral brightness, and we perform homodyne tomography on these photons,
which gives a complete image of the state of the source. Tomographic measurements of single
photons have previously been performed in the pulsed regime [20, 27, 28] but not for continu-
ously pumped systems. In overview, we operate an optical parametric oscillator (OPO) with a
pump level which is far below the oscillation threshold; in effect this is just cavity-enhancement
of the spontaneous parametric down-conversion of the nonlinear crystal. The ordinary phase
matching bandwidth of the down-conversion process is several nm, but the cavity effectively
inhibits down-conversion into frequencies which do not fulfill the resonance condition. Thus,
the output of the OPO consists of several narrow-band frequency modes separated by the free
spectral range (FSR) of the cavity. By appropriately filtering the output, we can obtain with
high efficiency a single photon in a specific one of those modes conditioned on the detection of
a trigger photon. The bandwidth of this photon will then be the cavity bandwidth which is very
narrow compared to the phase matching bandwidth.
2. Experiment
The OPO, as well as the rest of the setup, is depicted in Fig. 1. It is a bow-tie type cavity with
a length of 81 cm corresponding to a FSR of 370 MHz. Centered between two 5 cm curva-
ture mirrors is a 10 mm long PPKTP crystal which is periodically poled for noncritical phase
matching around 860 nm. The output coupler has a transmission of T = 12.5%, and the total
internal losses are L = 0.4%, giving a cavity HWHM bandwidth of γ1/2 = 2π 4.0 MHz and an
Fig. 1. (Color online) Setup diagram. The second harmonic generator (SHG) pumps the
optical parametric oscillator (OPO). The filter cavities should allow only a single mode
(at frequency ω−) to reach the single photon counting avalanche photo diode (APD). Two
acousto optic modulators (AOM) shift the main frequency to ω− and ω+ - the latter is used
for the local oscillator (LO) of the homodyne measurement, the former for an alignment
beam, which is used to bring all cavities resonant with ω− but which is blocked during
measurement.
escape efficiency ηesc = T/(T +L) = 0.97. With an effective nonlinearity ENL ≈ 0.020 W−1,
the threshold pump power for oscillation is around Pthr = (T +L)
2/4ENL = 210 mW. The blue
pump (430 nm) is generated by frequency doubling the main Ti:Sapph laser in a second har-
monic generator (SHG) of similar geometry as the OPO, but with a KNbO3 crystal as the
nonlinear medium. For single photon generation the pump should be rather weak to inhibit
the population of higher photon numbers. The pumping strength is quantized as the pump pa-
rameter ε =
Pb/Pthr, where Pb is the blue pump power. This pump parameter is most easily
inferred by observing the parametric gain, G= 1/(1−ε)2 of a beam of half the pump frequency
seeded into the OPO.
The frequency spectrum of the OPO is illustrated in Fig. 2. With no seed beam, the output
field in the degenerate frequency mode (half pump frequency) is quadrature-squeezed vacuum,
whereas the non-degenerate modes taken individually are thermal states. They are, however,
pairwise correlated symmetrically around the degenerate frequency. In the weak pump regime
this means that for each down-converted photon in the ω− mode one FSR below the degenerate
frequency, there is a twin photon in the ω+ mode one FSR above. In the time domain, the field
operator correlations for the two modes are given by [29]:
〈â±(t)â∓(t ′)〉 =
λ 2 − µ2
e−µ|t−t
e−λ |t−t
〈â†±(t)â±(t ′)〉 =
λ 2 − µ2
e−µ|t−t
−λ |t−t′|
〈â±(t)â±(t ′)〉 = 〈â†±(t)â∓(t ′)〉= 0 ,
λ = γ1/2(1+ ε) , µ = γ1/2(1− ε) .
Fig. 2. (Color online) Schematic illustration of the frequency mode spectrum of the OPO
(blue). The pump at frequency 2ω0 induces down-conversion into these and several other
neighbouring modes. The ω− and ω+ modes are correlated, and they are separated on the
first filter cavity which is resonant on ω− and reflects ω+ (red). Subsequent filters, of which
one is depicted (orange), serves to further suppress uncorrelated modes in the trigger arm.
Thus, if we can spatially separate the two frequency modes and detect the ω− photon on a
single photon detector we have heralded the existence of an ω+ photon within a temporal mode
determined by these correlations. This separation is done using an empty cavity which works
as a frequency filter; the FSR is four times that of the OPO, so with the cavity resonant on ω−,
the ω+ mode will be almost completely reflected. With this scheme the non-degenerate OPO
has previously been used to produce highly quadrature entangled EPR beams [30]. Because of
the wide phase matching bandwidth, many other modes than ω− will slip through the first filter
cavity. Hence we need two more filter cavities with different FSR and a 0.3 nm interference
filter on the way towards the photon counting avalanche photo diode (APD). If any photons
uncorrelated with the ω+ photons arrive at the APD, the ω+ state conditioned on these “false”
detections will be the original thermal state instead of a single photon. The spectral arrangement
of these filters is illustrated in Fig. 2. The lengths of the cavities are 210 mm, 3.7 mm, and 12
mm, and the FWHM bandwidths are roughly 48 MHz, 270 MHz, and 96 MHz, respectively.
To keep all cavities (OPO + filters) on resonance with the ω− frequency, we monitor the total
transmission (the APD click rate) and keep it on maximum using individual error signals from
each cavity obtained by dithering them at different frequencies. We recently employed the
same series of filter cavities and APD to herald the generation of a photon subtracted squeezed
vacuum state (a ”Schrödinger kitten“) [31].
With the ω− and ω+ modes thus separated and with the APD click heralding an ω+ photon,
the existence of this photon must be confirmed. Instead of just measuring the arrival of the
photons on another APD, we do a homodyne measurement of the field by mixing it on a 50/50
beam splitter with a strong local oscillator (LO) and subsequently recording the difference of
the photocurrents measured in the two arms. The LO has been shifted by 370 MHz to the center
frequency of the ω+ mode by sending part of the main laser beam through an AOM (acousto
optic modulator). The detector employs two Hamamatsu photo diodes (special production of
the S5971 type) with a specified quantum efficiency of 98%. It has a bandwidth of more than
100 MHz, and with 1.5 mW light on each diode the shot noise is 10 dB above the electronic
noise floor. The output of the detector goes to a fast digital oscilloscope which samples the
signal at 500 MS/s for a period of 2 µs around each APD trigger event. By repeating the state
generation and measurement several thousand times, statistics about the quadrature distribution
of the output state is build up. We scan the phase of the local oscillator to observe all quadrature
phases, but as expected the distribution is completely phase invariant.
In the post-processing of the recorded noise, we have to extract the conditional quadrature
information from the thermal state background. This is done by applying a temporal mode
function filter, fs(t), to the noise traces and afterwards integrate the traces over time. This
leaves us with a single mode quadrature value corresponding to the operator
âs =
ηsâ+(t ′)+
1−ηsâ+,vac(t ′)
dt ′ , (2)
where ηs is the total generation and detection efficiency of the signal, and the vacuum mode is
added to maintain the commutator relations. For the very low gain regime (ε ≪ 1), the optimal
field mode function for high single photon fidelity is simply the double-sided exponential [24,
fs,opt(t) =
γ1/2e
−γ1/2|t−tc| , (3)
with tc the time of the trigger event. For high gains the problem of finding the optimal mode
function becomes somewhat more involved – see [32] – but this is not a big concern at the low
gains at which we operate. Due to the filtering of the trigger photon, the correlations between
trigger and signal will be smeared out, so our optimal mode function should be a bit wider and
rounded off. However, since the narrowest trigger filter cavity has a bandwidth 6 times wider
than that of the OPO, the effect is not very significant, and using just the first approximation to
the optimal mode function above, we obtain fidelities quite similar to those obtained using more
precise mode functions. The procedure of post-processing the homodyne photo current with
temporal filtering is equivalent to performing the homodyne measurement with a pulsed LO of
the same shape as the mode function. However, the shaping of the LO will have to be initiated
by the trigger photon detection, and until the shaped LO is ready, the signal photon must be
delayed so that the two fields reach the beam splitter simultaneously. See [33] for a detailed
account of the problem of temporal/spectral mode matching in continuous-wave homodyning.
3. Analysis
For the data presented in this paper, we performed a total number of 180,000 genera-
tions/measurements of the single photon state. The measured parametric gain was about
G ≈ 1.2, corresponding to a pump parameter ε ≈ 0.09 ≪ 1 (the effective blue pump power
is ≈ 1.7 mW). Applying the mode function (3) to the noise traces, we get the quadrature dis-
tribution shown in Fig. 3(a,b). A simple fit to the single photon quadrature distribution admixed
with vacuum, η |〈q|1〉|2+(1−η)|〈q|0〉|2, shows that our data is consistent with a single photon
state which has been detected with an efficiency of η = 62%. Based on the measured quadrature
values and corresponding phases, we have reconstructed the density matrix and Wigner func-
tion of the generated state, using the maximum likelihood tomographic reconstruction method
[34]. The results are presented in Fig. 3(c,d). We see that our state consists almost entirely
of n = 0 and n = 1 number states, with an n = 1 population of 61%. There is, however, a
tiny contribution of the n = 2 and even higher number states. This is unavoidable in down-
conversion based single photon sources; there is a finite probability that neighbouring photon
pairs are produced so close to each other that they overlap, thus giving a higher average pho-
ton number than 1 within the mode function. For very low gain these higher photon number
components become insignificant, but at the same time, of course, the generation rate goes
towards zero. With the current gain we have achieved a good compromise between genera-
tion rate and low 2-photon contribution to the state. The average trigger detection rate in this
measurement series was 12,800s−1. Corrected for the trigger photon losses – the APD detec-
tion efficiency of 44%, the total trigger beam path transmission of 14%, and the OPO escape
efficiency of 97% – the estimated photon production rate was Robserved ≈ 215,000s−1. This
figure is close to the theoretically expected production rate in each cavity mode, which, from
Fig. 3. (Color online) a) Part of the recorded quadrature data set with corresponding phases.
b) Histogram of the distribution of all 180,000 conditional quadrature points (blue) and
40,000 vacuum points (red). The superimposed curves are the theoretical vacuum state
distribution, and the single photon distribution fitted to the data with the total efficiency η
as the only parameter. The fitted value is η = 0.625±0.002. The dashed curve is the ideal
(η = 1) single photon distribution. c) The density matrix of the state, reconstructed via a
maximum likelihood method, and in d) the corresponding Wigner function.
eq. (1), is Rtheory = 〈â†±(t)â±(t)〉= γ1/2ε2/(1− ε2)≈ 200,000s−1. Both of these numbers are,
however, too uncertain to be the basis for an estimate of the number of false clicks. The given
photon production rate corresponds to a spectral brightness of 1500 photons/s per MHz within
the 8 MHz FWHM bandwidth.
The inferred total detection efficiency, η = 62%, does not fit too well with the calculated
value based on independent estimates of the various loss/efficiency contributions, which was
as follows. The already mentioned escape efficiency of the OPO was 97%, the transmission
towards the homodyne detector was 92%, and the visibility with the LO was 97% leading to
an overlap efficiency of (97%)2. On top of these purely optical loss contributions come a spec-
ified diode quantum efficiency of 98% and a contribution from the electronic noise of 91% (in
the frequency range concerned, the electronic noise level is 10.5 dB below vacuum noise). In
total, the estimated efficiency of generation and measurement is η = 75% – but this number
is far from what we observe. A likely explanation for part of this discrepancy might be an in-
sufficient suppression of the uncorrelated frequency modes in the series of trigger filters. As
already mentioned, this would lead to a statistical admixture of the thermal state rather than
vacuum. There is already a small amount of thermal state admixed due to the dark counts of the
APD (∼ 100s−1), but since the thermal state for the low gain is almost indistinguishable from
the vacuum, the effect of the thermal state admixture is basically identical to losses (vacuum
admixture). Hence, it is also difficult to assert whether the discrepancy between expected and
observed efficiency is due to insufficient filtering or unknown sources of loss. Such an addi-
tional source of loss could possibly be the effect of low-frequency classical laser noise which
is not completely balanced out in the homodyne setup, since the state selection done by the
mode function integration includes all frequencies within the OPO bandwidth. A beginning
diode saturation due to too high intensity on the tiny diodes might be another cause. Finally,
the temporal mode function, chosen as (3), is not ideally matched to the single photon field and
hence some vacuum is admixed to the state on this account. Any fluctuations in the arrival time
of the photons would have the same effect.
The Wigner function in Fig. 3(d) clearly has the shape of the single photon Fock state, al-
though mixed with some vacuum. The negative dip has a value of W (0,0) = −0.070 – a clear
signature of a non-classical state measured with high efficiency. If we correct the state for the
purely measurement related losses (detector quantum efficiency and noise), we get a 70% pure
state with a Wigner function dip of W (0,0) =−0.12. This state is what we obtain after mixing
on the beam splitter with the local oscillator, and as such is the state which would be relevant
for the storage in an atomic memory, where the quantum state to be stored must be mixed with
a strong interaction field [3].
4. Correlation function measurement
Now we demonstrate how the cross correlation function between the two modes of the down-
converted field can be extracted from the recorded data. Usually correlation functions are meas-
ured via coincidence clicks on photon counting detectors. In [35] each photon counter is re-
placed with a homodyne detection setup and the g(2)(τ) correlation function is calculated from
the continuous frequency sideband measurements of the field quadratures. The scheme pre-
sented here, which in the essence is similar to the work by Foster et al. [36], is a combination
of these two approaches, where one mode is detected by a photon counter and the other by a
homodyne setup. We use exactly the same setup and the same data as for the single photon
generation and measurement. Figure 4 shows the point-wise variance of the 180,000 2µs long
quadrature noise traces, together with a similar variance for the vacuum state. Before taking
the variance, the traces have been low-pass filtered by a Lorentz-shaped filter with a 30 MHz
cut-off. The increased variance of the conditioned state around the trigger time is evident and
consistent with the expected quadrature variance of a single photon state which – in the ideal
case – is 3/2 in the normalization where the vacuum variance is 1/2. The reason for the lower
peak value is partly the limited detection efficiency, but also the effect of including frequency
components far outside the OPO bandwidth where there is nothing but vacuum (a frequency
filter much narrower than the 30 MHz would decrease the contribution from this vacuum and
hence increase the variance, but it would also widen the temporal shape). This signal mode
variance conditioned on a trigger photon detection at time tc is
〈∆q̂s(τ)2〉
〈â†t (tc)(q̂s(tc + τ))2ât(tc)〉
〈â†t ât〉
〈â†t (tc)â†s (tc + τ)âs(tc + τ)ât(tc)〉
〈â†t ât〉
, (4)
where we used that the input states are Gaussian with zero mean, and that many correla-
tion terms of the non-degenerate OPO are vanishing, cf. (1). The quadrature operator q̂ is
defined as q̂ ≡
âe−iθ + â†eiθ
2, where the phase is made implicit due to the phase-
invariance of the state. In order to calculate explicitly the expected variance, the signal and
trigger modes, âs and ât , must include the detection efficiencies and any transformations –
optical or electronic – applied to them, as done by the mode function in (2). For example,
the filtering of the trigger field by the filter cavities must be taken into account. For uncorre-
lated modes (for instance far away from tc), the variance reduces to the thermal state variance
〈∆q̂s(τ)2〉
uncond
= 〈∆q̂s(τ)2〉
thermal
= 1/2+ 〈â†s âs〉. The g
ts (τ) cross-correlation function is
Fig. 4. (Color online) a) Variance of the recorded quadrature noise traces for the signal
conditioned on a trigger event at t = 0 (red) and for vacuum (black). The traces have been
low-pass filtered with a bandwidth of 30 MHz to suppress most of the detector output which
lies outside of the field bandwidth. b) The cross correlation function, g
ts (t − tc) (with the
trigger time tc = −29ns), calculated from the variances in a). Far away from the trigger
time the value is 1, but the large values around tc demonstrate a strong correlation between
the trigger and signal fields. The black curves are the theoretically expected functions for
single photon state contents (versus thermal state) of 1, 0.8, and 0.6 (the lowest).
now easily seen to be a simple expression of the quadrature variances
ts (τ) ≡
〈â†t (tc)â†s (tc + τ)âs(tc + τ)ât(tc)〉
〈â†t ât〉〈â†s âs〉
〈∆q̂s(τ)2〉
− 1/2
〈∆q̂s(τ)2〉|uncond − 1/2
. (5)
In Fig. 4(b), this g
ts function has been calculated from the variances in Fig. 4(a), where the
thermal state variances have been calculated as the mean values of the traces far away from the
trigger time. The expression (5) does no longer depend on the signal efficiency, which means
that high frequency vacuum contributions play no role in the shape and size of the correlation
function. In the figure are also plotted three expected g
ts functions, calculated from a pump
parameter ε = 0.09 and a statistical single photon content of 1, 0.8, and 0.6, respectively, where
the remaining parts are made up of the thermal state. The optical filtering of the trigger mode (24
MHz, according to the bandwidth of the narrowest filter cavity) and the digital 30 MHz signal
mode filtering have been included in these plots. In principle, since the correlation functions
are independent on the signal detection efficiency but depend on the amount of thermal state
admixture, it should be possible to find this amount by fitting these theoretical curves to the
measurements. However, the uncertainty in the value of the thermal state variance (which is
very close to 1/2) turns into a huge uncertainty in the derived g
ts function – the error bars are
so large that they are not displayed in the figure – so that this estimation is meaningless. More
precise values would have been attainable if we had made a large number of measurements of
the unconditioned thermal state.
5. Conclusion
In conclusion, we have presented a high-purity and high-spectral brightness source of her-
alded single photons, whose narrow bandwidth and clean spatial mode make it very suitable
for interactions with an atomic memory. The generated single photons were characterized by
homodyne tomography, showing a clearly negative Wigner function. Finally, we demonstrated
how the correlation function between two fields can be measured by a combination of instan-
taneous photon counting and continuous homodyne measurement. The 70% purity of the state
demonstrated here should not be the limit. With some effort a purity of 90% is within reach for
the present setup.
This research has been funded in part by EU grants QAP and COVAQIAL. We acknowledge
enlightening discussions with Anne E. B. Nielsen and Klaus Mølmer, and the kind assistance
of Akira Furusawa in supplying us with the PPKTP crystal.
	Introduction
	Experiment
	Analysis
	Correlation function measurement
	Conclusion

  Using cavity-enhanced non-degenerate parametric downconversion, we have built
a frequency tunable source of heralded single photons with a narrow bandwidth
of 8 MHz, making it compatible with atomic quantum memories. The photon state
is 70% pure single photon as characterized by a tomographic measurement and
reconstruction of the quantum state, revealing a clearly negative Wigner
function. Furthermore, it has a spectral brightness of ~1,500 photons/s per MHz
bandwidth, making it one of the brightest single photon sources available. We
also investigate the correlation function of the down-converted fields using a
combination of two very distinct detection methods; photon counting and
homodyne measurement.

Introduction
Pure single photon states produced efficiently and at a high rate are highly desirable for prac-
tical implementations of various quantum information processing protocols, in particular in
quantum cryptography [1], quantum computing with linear optics [2], and for testing quantum
memories [3, 4]. The latter applications require at the same time compatibility with some kind
of a quantum memory. Different approaches towards generation of a single photon state have
been implemented in a number of physical systems. It should be noted that in many instances
a source is claimed to be ”a single photon source” based just on the property of antibunch-
ing, i.e., on the low rate of two-photon contribution compared to a single photon part. Such
http://arxiv.org/abs/quant-ph/0609033
property should be combined with the likewise low contribution of the vacuum state, in order
to claim a high-purity truly single photon source which is the aim of the present work. Single
emitters usually suffer from low purity due to small collection efficiency for light. For example,
quantum dot based sources [5, 6], color centres in diamond [7, 8], single molecules [9], or a
single atom [10] have the detection efficiency/purity at best at a few percent level. Placing sin-
gle emitters inside high-Q cavities improves the purity dramatically. However, even complex
state-of-the-art experiments still have limited overall collection efficiency and thus low purity.
The best results with quantum dots [11] show 8% collection efficiency/purity. Besides, quan-
tum dots usually emit light in a several GHz bandwidth. The best efforts with cavity-QED with
atoms or ions yield 30-40% efficiency just outside the cavity and the overall efficiency/purity at
10-20% level [12, 13, 14]. Recently atomic ensembles have been used to produce non-classical
light [15, 16, 17], however, the light collection efficiency does not exceed a few percent even
when atoms are placed inside a cavity [16]. Parametric down-conversion in free space non-
linear crystals or waveguides [18, 19, 20, 21, 22, 23] has been widely used for generation of
heralded photon pulses. The major disadvantage of parametric down-conversion is the random
arrival time of the photons – the source is not deterministic. However, this is compensated by
many attractive properties like well-defined wavelength, high collection efficiency, and non-
cryogenic experimental setups. The standard pulsed, single-pass down-conversion process suf-
fers from a limited photon generation rate which must be kept low to avoid pulses containing
two photons, which is detrimental for quantum information applications. The bandwidth of the
down-conversion is typically several nanometers, which means that the spectral brightness (the
number of photons per MHz per second) is below one. This poses a serious limitation to the
feasibility of interaction with atomic systems, where linewidths are on the MHz scale. To over-
come this problem, the nonlinear crystal can be placed inside an optical cavity which serves to
enhance the down-conversion process and limit the bandwidth of the output to that of the cav-
ity [24]. Furthermore, the spatial field mode is defined by the cavity as well, so no additional
spatial filtering is needed. Various studies on this type of setup have been performed, and the
results do indeed show a marked increase in the attained spectral brightness [24, 25, 26].
In this paper we present our scheme for generation of heralded single photons with a very
high purity and spectral brightness, and we perform homodyne tomography on these photons,
which gives a complete image of the state of the source. Tomographic measurements of single
photons have previously been performed in the pulsed regime [20, 27, 28] but not for continu-
ously pumped systems. In overview, we operate an optical parametric oscillator (OPO) with a
pump level which is far below the oscillation threshold; in effect this is just cavity-enhancement
of the spontaneous parametric down-conversion of the nonlinear crystal. The ordinary phase
matching bandwidth of the down-conversion process is several nm, but the cavity effectively
inhibits down-conversion into frequencies which do not fulfill the resonance condition. Thus,
the output of the OPO consists of several narrow-band frequency modes separated by the free
spectral range (FSR) of the cavity. By appropriately filtering the output, we can obtain with
high efficiency a single photon in a specific one of those modes conditioned on the detection of
a trigger photon. The bandwidth of this photon will then be the cavity bandwidth which is very
narrow compared to the phase matching bandwidth.
2. Experiment
The OPO, as well as the rest of the setup, is depicted in Fig. 1. It is a bow-tie type cavity with
a length of 81 cm corresponding to a FSR of 370 MHz. Centered between two 5 cm curva-
ture mirrors is a 10 mm long PPKTP crystal which is periodically poled for noncritical phase
matching around 860 nm. The output coupler has a transmission of T = 12.5%, and the total
internal losses are L = 0.4%, giving a cavity HWHM bandwidth of γ1/2 = 2π 4.0 MHz and an
Fig. 1. (Color online) Setup diagram. The second harmonic generator (SHG) pumps the
optical parametric oscillator (OPO). The filter cavities should allow only a single mode
(at frequency ω−) to reach the single photon counting avalanche photo diode (APD). Two
acousto optic modulators (AOM) shift the main frequency to ω− and ω+ - the latter is used
for the local oscillator (LO) of the homodyne measurement, the former for an alignment
beam, which is used to bring all cavities resonant with ω− but which is blocked during
measurement.
escape efficiency ηesc = T/(T +L) = 0.97. With an effective nonlinearity ENL ≈ 0.020 W−1,
the threshold pump power for oscillation is around Pthr = (T +L)
2/4ENL = 210 mW. The blue
pump (430 nm) is generated by frequency doubling the main Ti:Sapph laser in a second har-
monic generator (SHG) of similar geometry as the OPO, but with a KNbO3 crystal as the
nonlinear medium. For single photon generation the pump should be rather weak to inhibit
the population of higher photon numbers. The pumping strength is quantized as the pump pa-
rameter ε =
Pb/Pthr, where Pb is the blue pump power. This pump parameter is most easily
inferred by observing the parametric gain, G= 1/(1−ε)2 of a beam of half the pump frequency
seeded into the OPO.
The frequency spectrum of the OPO is illustrated in Fig. 2. With no seed beam, the output
field in the degenerate frequency mode (half pump frequency) is quadrature-squeezed vacuum,
whereas the non-degenerate modes taken individually are thermal states. They are, however,
pairwise correlated symmetrically around the degenerate frequency. In the weak pump regime
this means that for each down-converted photon in the ω− mode one FSR below the degenerate
frequency, there is a twin photon in the ω+ mode one FSR above. In the time domain, the field
operator correlations for the two modes are given by [29]:
〈â±(t)â∓(t ′)〉 =
λ 2 − µ2
e−µ|t−t
e−λ |t−t
〈â†±(t)â±(t ′)〉 =
λ 2 − µ2
e−µ|t−t
−λ |t−t′|
〈â±(t)â±(t ′)〉 = 〈â†±(t)â∓(t ′)〉= 0 ,
λ = γ1/2(1+ ε) , µ = γ1/2(1− ε) .
Fig. 2. (Color online) Schematic illustration of the frequency mode spectrum of the OPO
(blue). The pump at frequency 2ω0 induces down-conversion into these and several other
neighbouring modes. The ω− and ω+ modes are correlated, and they are separated on the
first filter cavity which is resonant on ω− and reflects ω+ (red). Subsequent filters, of which
one is depicted (orange), serves to further suppress uncorrelated modes in the trigger arm.
Thus, if we can spatially separate the two frequency modes and detect the ω− photon on a
single photon detector we have heralded the existence of an ω+ photon within a temporal mode
determined by these correlations. This separation is done using an empty cavity which works
as a frequency filter; the FSR is four times that of the OPO, so with the cavity resonant on ω−,
the ω+ mode will be almost completely reflected. With this scheme the non-degenerate OPO
has previously been used to produce highly quadrature entangled EPR beams [30]. Because of
the wide phase matching bandwidth, many other modes than ω− will slip through the first filter
cavity. Hence we need two more filter cavities with different FSR and a 0.3 nm interference
filter on the way towards the photon counting avalanche photo diode (APD). If any photons
uncorrelated with the ω+ photons arrive at the APD, the ω+ state conditioned on these “false”
detections will be the original thermal state instead of a single photon. The spectral arrangement
of these filters is illustrated in Fig. 2. The lengths of the cavities are 210 mm, 3.7 mm, and 12
mm, and the FWHM bandwidths are roughly 48 MHz, 270 MHz, and 96 MHz, respectively.
To keep all cavities (OPO + filters) on resonance with the ω− frequency, we monitor the total
transmission (the APD click rate) and keep it on maximum using individual error signals from
each cavity obtained by dithering them at different frequencies. We recently employed the
same series of filter cavities and APD to herald the generation of a photon subtracted squeezed
vacuum state (a ”Schrödinger kitten“) [31].
With the ω− and ω+ modes thus separated and with the APD click heralding an ω+ photon,
the existence of this photon must be confirmed. Instead of just measuring the arrival of the
photons on another APD, we do a homodyne measurement of the field by mixing it on a 50/50
beam splitter with a strong local oscillator (LO) and subsequently recording the difference of
the photocurrents measured in the two arms. The LO has been shifted by 370 MHz to the center
frequency of the ω+ mode by sending part of the main laser beam through an AOM (acousto
optic modulator). The detector employs two Hamamatsu photo diodes (special production of
the S5971 type) with a specified quantum efficiency of 98%. It has a bandwidth of more than
100 MHz, and with 1.5 mW light on each diode the shot noise is 10 dB above the electronic
noise floor. The output of the detector goes to a fast digital oscilloscope which samples the
signal at 500 MS/s for a period of 2 µs around each APD trigger event. By repeating the state
generation and measurement several thousand times, statistics about the quadrature distribution
of the output state is build up. We scan the phase of the local oscillator to observe all quadrature
phases, but as expected the distribution is completely phase invariant.
In the post-processing of the recorded noise, we have to extract the conditional quadrature
information from the thermal state background. This is done by applying a temporal mode
function filter, fs(t), to the noise traces and afterwards integrate the traces over time. This
leaves us with a single mode quadrature value corresponding to the operator
âs =
ηsâ+(t ′)+
1−ηsâ+,vac(t ′)
dt ′ , (2)
where ηs is the total generation and detection efficiency of the signal, and the vacuum mode is
added to maintain the commutator relations. For the very low gain regime (ε ≪ 1), the optimal
field mode function for high single photon fidelity is simply the double-sided exponential [24,
fs,opt(t) =
γ1/2e
−γ1/2|t−tc| , (3)
with tc the time of the trigger event. For high gains the problem of finding the optimal mode
function becomes somewhat more involved – see [32] – but this is not a big concern at the low
gains at which we operate. Due to the filtering of the trigger photon, the correlations between
trigger and signal will be smeared out, so our optimal mode function should be a bit wider and
rounded off. However, since the narrowest trigger filter cavity has a bandwidth 6 times wider
than that of the OPO, the effect is not very significant, and using just the first approximation to
the optimal mode function above, we obtain fidelities quite similar to those obtained using more
precise mode functions. The procedure of post-processing the homodyne photo current with
temporal filtering is equivalent to performing the homodyne measurement with a pulsed LO of
the same shape as the mode function. However, the shaping of the LO will have to be initiated
by the trigger photon detection, and until the shaped LO is ready, the signal photon must be
delayed so that the two fields reach the beam splitter simultaneously. See [33] for a detailed
account of the problem of temporal/spectral mode matching in continuous-wave homodyning.
3. Analysis
For the data presented in this paper, we performed a total number of 180,000 genera-
tions/measurements of the single photon state. The measured parametric gain was about
G ≈ 1.2, corresponding to a pump parameter ε ≈ 0.09 ≪ 1 (the effective blue pump power
is ≈ 1.7 mW). Applying the mode function (3) to the noise traces, we get the quadrature dis-
tribution shown in Fig. 3(a,b). A simple fit to the single photon quadrature distribution admixed
with vacuum, η |〈q|1〉|2+(1−η)|〈q|0〉|2, shows that our data is consistent with a single photon
state which has been detected with an efficiency of η = 62%. Based on the measured quadrature
values and corresponding phases, we have reconstructed the density matrix and Wigner func-
tion of the generated state, using the maximum likelihood tomographic reconstruction method
[34]. The results are presented in Fig. 3(c,d). We see that our state consists almost entirely
of n = 0 and n = 1 number states, with an n = 1 population of 61%. There is, however, a
tiny contribution of the n = 2 and even higher number states. This is unavoidable in down-
conversion based single photon sources; there is a finite probability that neighbouring photon
pairs are produced so close to each other that they overlap, thus giving a higher average pho-
ton number than 1 within the mode function. For very low gain these higher photon number
components become insignificant, but at the same time, of course, the generation rate goes
towards zero. With the current gain we have achieved a good compromise between genera-
tion rate and low 2-photon contribution to the state. The average trigger detection rate in this
measurement series was 12,800s−1. Corrected for the trigger photon losses – the APD detec-
tion efficiency of 44%, the total trigger beam path transmission of 14%, and the OPO escape
efficiency of 97% – the estimated photon production rate was Robserved ≈ 215,000s−1. This
figure is close to the theoretically expected production rate in each cavity mode, which, from
Fig. 3. (Color online) a) Part of the recorded quadrature data set with corresponding phases.
b) Histogram of the distribution of all 180,000 conditional quadrature points (blue) and
40,000 vacuum points (red). The superimposed curves are the theoretical vacuum state
distribution, and the single photon distribution fitted to the data with the total efficiency η
as the only parameter. The fitted value is η = 0.625±0.002. The dashed curve is the ideal
(η = 1) single photon distribution. c) The density matrix of the state, reconstructed via a
maximum likelihood method, and in d) the corresponding Wigner function.
eq. (1), is Rtheory = 〈â†±(t)â±(t)〉= γ1/2ε2/(1− ε2)≈ 200,000s−1. Both of these numbers are,
however, too uncertain to be the basis for an estimate of the number of false clicks. The given
photon production rate corresponds to a spectral brightness of 1500 photons/s per MHz within
the 8 MHz FWHM bandwidth.
The inferred total detection efficiency, η = 62%, does not fit too well with the calculated
value based on independent estimates of the various loss/efficiency contributions, which was
as follows. The already mentioned escape efficiency of the OPO was 97%, the transmission
towards the homodyne detector was 92%, and the visibility with the LO was 97% leading to
an overlap efficiency of (97%)2. On top of these purely optical loss contributions come a spec-
ified diode quantum efficiency of 98% and a contribution from the electronic noise of 91% (in
the frequency range concerned, the electronic noise level is 10.5 dB below vacuum noise). In
total, the estimated efficiency of generation and measurement is η = 75% – but this number
is far from what we observe. A likely explanation for part of this discrepancy might be an in-
sufficient suppression of the uncorrelated frequency modes in the series of trigger filters. As
already mentioned, this would lead to a statistical admixture of the thermal state rather than
vacuum. There is already a small amount of thermal state admixed due to the dark counts of the
APD (∼ 100s−1), but since the thermal state for the low gain is almost indistinguishable from
the vacuum, the effect of the thermal state admixture is basically identical to losses (vacuum
admixture). Hence, it is also difficult to assert whether the discrepancy between expected and
observed efficiency is due to insufficient filtering or unknown sources of loss. Such an addi-
tional source of loss could possibly be the effect of low-frequency classical laser noise which
is not completely balanced out in the homodyne setup, since the state selection done by the
mode function integration includes all frequencies within the OPO bandwidth. A beginning
diode saturation due to too high intensity on the tiny diodes might be another cause. Finally,
the temporal mode function, chosen as (3), is not ideally matched to the single photon field and
hence some vacuum is admixed to the state on this account. Any fluctuations in the arrival time
of the photons would have the same effect.
The Wigner function in Fig. 3(d) clearly has the shape of the single photon Fock state, al-
though mixed with some vacuum. The negative dip has a value of W (0,0) = −0.070 – a clear
signature of a non-classical state measured with high efficiency. If we correct the state for the
purely measurement related losses (detector quantum efficiency and noise), we get a 70% pure
state with a Wigner function dip of W (0,0) =−0.12. This state is what we obtain after mixing
on the beam splitter with the local oscillator, and as such is the state which would be relevant
for the storage in an atomic memory, where the quantum state to be stored must be mixed with
a strong interaction field [3].
4. Correlation function measurement
Now we demonstrate how the cross correlation function between the two modes of the down-
converted field can be extracted from the recorded data. Usually correlation functions are meas-
ured via coincidence clicks on photon counting detectors. In [35] each photon counter is re-
placed with a homodyne detection setup and the g(2)(τ) correlation function is calculated from
the continuous frequency sideband measurements of the field quadratures. The scheme pre-
sented here, which in the essence is similar to the work by Foster et al. [36], is a combination
of these two approaches, where one mode is detected by a photon counter and the other by a
homodyne setup. We use exactly the same setup and the same data as for the single photon
generation and measurement. Figure 4 shows the point-wise variance of the 180,000 2µs long
quadrature noise traces, together with a similar variance for the vacuum state. Before taking
the variance, the traces have been low-pass filtered by a Lorentz-shaped filter with a 30 MHz
cut-off. The increased variance of the conditioned state around the trigger time is evident and
consistent with the expected quadrature variance of a single photon state which – in the ideal
case – is 3/2 in the normalization where the vacuum variance is 1/2. The reason for the lower
peak value is partly the limited detection efficiency, but also the effect of including frequency
components far outside the OPO bandwidth where there is nothing but vacuum (a frequency
filter much narrower than the 30 MHz would decrease the contribution from this vacuum and
hence increase the variance, but it would also widen the temporal shape). This signal mode
variance conditioned on a trigger photon detection at time tc is
〈∆q̂s(τ)2〉
〈â†t (tc)(q̂s(tc + τ))2ât(tc)〉
〈â†t ât〉
〈â†t (tc)â†s (tc + τ)âs(tc + τ)ât(tc)〉
〈â†t ât〉
, (4)
where we used that the input states are Gaussian with zero mean, and that many correla-
tion terms of the non-degenerate OPO are vanishing, cf. (1). The quadrature operator q̂ is
defined as q̂ ≡
âe−iθ + â†eiθ
2, where the phase is made implicit due to the phase-
invariance of the state. In order to calculate explicitly the expected variance, the signal and
trigger modes, âs and ât , must include the detection efficiencies and any transformations –
optical or electronic – applied to them, as done by the mode function in (2). For example,
the filtering of the trigger field by the filter cavities must be taken into account. For uncorre-
lated modes (for instance far away from tc), the variance reduces to the thermal state variance
〈∆q̂s(τ)2〉
uncond
= 〈∆q̂s(τ)2〉
thermal
= 1/2+ 〈â†s âs〉. The g
ts (τ) cross-correlation function is
Fig. 4. (Color online) a) Variance of the recorded quadrature noise traces for the signal
conditioned on a trigger event at t = 0 (red) and for vacuum (black). The traces have been
low-pass filtered with a bandwidth of 30 MHz to suppress most of the detector output which
lies outside of the field bandwidth. b) The cross correlation function, g
ts (t − tc) (with the
trigger time tc = −29ns), calculated from the variances in a). Far away from the trigger
time the value is 1, but the large values around tc demonstrate a strong correlation between
the trigger and signal fields. The black curves are the theoretically expected functions for
single photon state contents (versus thermal state) of 1, 0.8, and 0.6 (the lowest).
now easily seen to be a simple expression of the quadrature variances
ts (τ) ≡
〈â†t (tc)â†s (tc + τ)âs(tc + τ)ât(tc)〉
〈â†t ât〉〈â†s âs〉
〈∆q̂s(τ)2〉
− 1/2
〈∆q̂s(τ)2〉|uncond − 1/2
. (5)
In Fig. 4(b), this g
ts function has been calculated from the variances in Fig. 4(a), where the
thermal state variances have been calculated as the mean values of the traces far away from the
trigger time. The expression (5) does no longer depend on the signal efficiency, which means
that high frequency vacuum contributions play no role in the shape and size of the correlation
function. In the figure are also plotted three expected g
ts functions, calculated from a pump
parameter ε = 0.09 and a statistical single photon content of 1, 0.8, and 0.6, respectively, where
the remaining parts are made up of the thermal state. The optical filtering of the trigger mode (24
MHz, according to the bandwidth of the narrowest filter cavity) and the digital 30 MHz signal
mode filtering have been included in these plots. In principle, since the correlation functions
are independent on the signal detection efficiency but depend on the amount of thermal state
admixture, it should be possible to find this amount by fitting these theoretical curves to the
measurements. However, the uncertainty in the value of the thermal state variance (which is
very close to 1/2) turns into a huge uncertainty in the derived g
ts function – the error bars are
so large that they are not displayed in the figure – so that this estimation is meaningless. More
precise values would have been attainable if we had made a large number of measurements of
the unconditioned thermal state.
5. Conclusion
In conclusion, we have presented a high-purity and high-spectral brightness source of her-
alded single photons, whose narrow bandwidth and clean spatial mode make it very suitable
for interactions with an atomic memory. The generated single photons were characterized by
homodyne tomography, showing a clearly negative Wigner function. Finally, we demonstrated
how the correlation function between two fields can be measured by a combination of instan-
taneous photon counting and continuous homodyne measurement. The 70% purity of the state
demonstrated here should not be the limit. With some effort a purity of 90% is within reach for
the present setup.
This research has been funded in part by EU grants QAP and COVAQIAL. We acknowledge
enlightening discussions with Anne E. B. Nielsen and Klaus Mølmer, and the kind assistance
of Akira Furusawa in supplying us with the PPKTP crystal.
	Introduction
	Experiment
	Analysis
	Correlation function measurement
	Conclusion
