Conditional generation of path-entangled optical NOON states
APS/123-QED
Conditional generation of path-entangled optical NOON states
Anne E. B. Nielsen and Klaus Mølmer
Lundbeck Foundation Theoretical Center for Quantum System Research,
Department of Physics and Astronomy, University of Aarhus, DK-8000 Århus C, Denmark
(Dated: November 4, 2018)
We propose a measurement protocol to generate path-entangled NOON states conditionally from
two pulsed type II optical parametric oscillators. We calculate the fidelity of the produced states
and the success probability of the protocol. The trigger detectors are assumed to have finite dead
time, and for short pulse trigger fields they are modeled as on/off detectors with finite efficiency.
Continuous-wave operation of the parametric oscillators is also considered.
PACS numbers: 03.65.Wj, 03.67.-a, 42.50.Dv
I. INTRODUCTION
Nonclassical states of light have many applications,
and a number of different protocols exist for the genera-
tion of various classes of states. The two-mode maximally
entangled N -photon states
|NOON〉 =
|N, 0〉+ eiφ|0, N〉
, (1)
the so-called NOON states, are particularly interesting
because a single-photon phase shift of χ induced in one of
the two components changes the relative phase of the two
terms by Nχ. This special property of NOON states may
be utilized to enhance spatial resolution in (quantum) mi-
croscopy and lithography [1], and in interferometry it has
been shown that a certain measurement strategy, using
NOON states, leads to a phase estimation error scaling
as L−1/4N
T if the phase to be estimated is known to
lie within an interval from −π/L to π/L, where NT is the
total number of photons used in the measurements [2].
This is better than the classical shot noise limit ofN
and NOON states are thus useful to perform accurate
measurements and may be a valuable field resource in
sensors. NOON states are also a source of entanglement
with applications in quantum information protocols and
in fundamental studies such as tests of Bells inequality
It is thus of great interest to be able to produce NOON
states, and various NOON state generation schemes have
been suggested theoretically [4, 5, 6, 7, 8, 9] and studied
in experiments [7, 8, 10, 11]. The N = 1 and N = 2
NOON states may be generated by combining either a
single photon and a vacuum state or two single-photon
states on a 50 : 50 beam splitter, but this simple approach
is not directly extendable to N > 2, and we shall thus
mainly be concerned with generation of N = 3 NOON
states in the present paper, even though the suggested
protocol is, in principle, applicable for all N . Mitchell,
Lundeen, and Steinberg have generated NOON states
with N = 3 from a pair of down converted photons
and a local oscillator photon using certain polarization
transforming components and post-selection [10]. In this
experiment, however, the successful generation of the
NOON state is witnessed by a destructive detection of
the state. In the present paper we propose and analyze
in detail a nondestructive generation protocol, which con-
ditions the successful generation of the N -photon NOON
state on the registration of N photo detection events in
other field modes, and which uses as resource only linear
optics and the output from two optical parametric oscil-
lators (OPOs). The protocol does not rely on efficient
photo detection. The analysis is carried out in terms of
Wigner function formalism, and effects of finite detector
efficiency and finite detector dead time are considered.
Conditional generation of nonclassical states occupying
a single mode has been investigated both experimentally
and theoretically [12, 13, 14, 15, 16, 17, 18, 19, 20]. With
the correlated output from a single nondegenerate OPO
it is, for instance, possible to generate n-photon Fock
states of light in the signal beam conditioned on n photo
detections in the idler (trigger) beam [16, 19, 20], and
in principle the entanglement of a highly squeezed two-
mode field from an OPO makes it possible to prepare any
state in the signal beam that can be either measured as
an eigenstate of a suitable observable of the idler beam or
produced as the final state of a generalized measurement.
The basic idea of the protocol proposed in the present
paper is to mix the output from two OPOs and employ
the entanglement to prepare a two-mode state in two of
the output beams by detection of the desired output state
in the remaining beams.
In Sec. II we explain the NOON state generation pro-
tocol in detail. In Sec. III we analyze the performance
of the protocol quantitatively for pulsed OPO sources.
We provide the fidelity of the generated states and the
success probability. In Sec. IV we consider production of
NOON states from continuous-wave OPO sources, and
Sec. V concludes the paper.
II. EXPERIMENTAL SETUP FOR NOON
STATE GENERATION
The experimental setup is illustrated in Fig. 1. Two
pulses of two-mode squeezed states are generated by two
identical OPOs via type II parametric down conversion.
The field mode operators of the modes generated by the
http://arxiv.org/abs/0704.0397v2
first OPO are denoted â+ and â−, respectively, while the
field mode operators of the modes generated by the sec-
ond OPO are denoted b̂+ and b̂−, respectively. For def-
initeness, we assume that the plus modes are vertically
polarized and that the minus modes are horizontally po-
larized. The modes are separated spatially by the first
two polarizing beam splitters, and the third polarizing
beam splitter combines the â− and b̂+ modes, which are
subsequently subjected to the NOON state measurement
proposed in [21] and illustrated for N = 3 in Fig. 1. The
idea behind this measurement is to apply the highly non-
linear operator ÂN = â
− − (b̂+eiθ)N to the state. The
result is only nonzero if either the â− mode or the b̂+
mode contains at least N photons. On the other hand,
if the squeezing is sufficiently small, it is unlikely to have
more than a total of N photons in the two trigger modes,
and by conditioning on the successful application of ÂN ,
we select the pulses of the system where N photon pairs
are generated in one OPO and zero photon pairs in the
other. It is equally probable that the photons originate
from the first OPO or from the second OPO, and, as we
shall see in detail below, the result is that a NOON state
is generated conditionally in the output modes â+ and
As stated in [21], ÂN can be rewritten as a simple
product of single photon annihilation operators
âN− −
â− −
ei2πn/N
, (2)
and it is thus possible to implement ÂN by means of
beam splitters and photo detectors. We first consider odd
values of N . Beam splitters are used to divide the input
into N distinct spatial modes labeled by n = 1, . . . , N .
The beam splitter reflectivities are chosen to obtain the
same expectation value of the intensity in each of the
modes. The vertically polarized modes are then phase
shifted by the factor ei2πn/N+iπ relative to the horizon-
tally polarized modes, i.e., b̂+ → −b̂+ei2πn/N , and finally
polarizing beam splitters with principal planes oriented
at 45◦ relative to the horizontal polarization transform
â− and −b̂+ei2πn/N into (â−− b̂+ei2πn/N )/
2 (the trans-
mitted mode) and (â− + b̂+e
i2πn/N )/
2 (the reflected
mode) [22]. The annihilation of a photon in each of
the modes transmitted by the beam splitters witnesses
the overall application of the operator ÂN . If one ob-
serves both reflected and transmitted modes simultane-
ously, one conditions on detection events in all the trans-
mitted modes and no detection events in all the reflected
modes. If detection events are instead observed in all the
reflected modes and in none of the transmitted modes,
an operator of the form (2) is also obtained, but θ is ef-
fectively transformed into θ + π due to the phase shift
at the polarizing beam splitter, and the value of φ of the
generated NOON states is changed by Nπ (see below).
The success probability is thus increased by a factor of
two if both outcomes are accepted.
FIG. 1: Experimental setup for NOON state generation.
OPO, optical parametric oscillator; PBS, polarizing beam
splitter; PS, phase shifter; and APD, avalanche photo diode.
The part of the setup enclosed in the dashed box performs the
NOON state measurement, and here it is shown for N = 3.
Note that the polarizing beam splitters inside the box are ori-
ented at 45◦. The numbers denote beam splitter reflectivities
of 1/3 and 1/2, and the three phase shifters transform b̂+
into −b̂+e
2πin/3, where n = 1, 2, 3, respectively. See text for
details.
For even values of N a similar measurement scheme
is applicable, but it is sufficient to divide the field into
N/2 spatial modes initially, and in this case the NOON
state generation is conditioned on detection events in
both transmitted and reflected modes (see [21]).
III. PERFORMANCE OF THE PROTOCOL
After this presentation of the basic idea and the phys-
ical setup we now consider the actual outcome of the
detection process. For short pulse OPO output the dead
time of the photo detectors may typically be longer than
the pulse duration, and we shall thus assume that it is
impossible to obtain more than a single detection event
per detector per pulse, i.e., if the detector efficiency is
unity, the detectors are only able to distinguish between
vacuum and states different from the vacuum state. Such
detectors are denoted on/off detectors, and they are dis-
cussed in detail in Ref. [23]. The finite dead time of the
detectors is not severe to the measurement procedure de-
scribed in [21] because the on/off detector model and the
conventional photo detector model, represented by the
annihilation operator, lead to identical signal states if the
total number of photons in the idler modes is guaranteed
to be less than or equal to the number of conditioning
detection events, i.e., N .
We analyze the performance of the setup using Gaus-
sian Wigner function formalism [15, 19, 20], which is
applicable because the squeezed states generated by the
OPOs and the vacuum states coupled into the system
via the beam splitters are all Gaussian. In general, the
Wigner function of an n-mode Gaussian state with zero
mean field amplitude takes the form
WV (x1, p1, . . . , xn, pn) =
det(V )
TV −1y, (3)
where y ≡ (x1, p1, . . . , xn, pn)T and V is the 2n × 2n
covariance matrix. If ĉi denotes the field mode annihi-
lation operator of mode i, the elements of V are given
in terms of the real and imaginary parts of the expecta-
tion values 〈ĉ†i ĉj〉 and 〈ĉiĉj〉. Note that for a multi-mode
Gaussian state we are free to include only the modes of
interest in (3) because the partial trace operation over un-
observed modes is equivalent to integration over the cor-
responding quadrature variables. A unit efficiency ‘on’
detection in mode i projects mode i on the subspace of
Hilbert space that is orthogonal to the vacuum state, i.e.,
the Wigner function is multiplied by (1− 2πW0(xi, pi)),
whereW0(x, p) = exp(−x2−p2)/π is the Wigner function
of the vacuum state, the variables xi and pi are integrated
out, and the state is renormalized. Since the Gaussian
nature of a state is preserved under linear transforma-
tions, and since a detector with single-photon efficiency
η is equivalent to a beam splitter with transmission η fol-
lowed by a unit efficiency detector [23], effects of non-unit
detector efficiency are easily included in the covariance
matrix.
To calculate 〈ĉ†i ĉj〉 and 〈ĉiĉj〉 explicitly we note that
the state generated by the OPOs is [24]
|ψi〉 = (1− r2)
rn+m|n, n,m,m〉, (4)
where r is the squeezing parameter and the modes are
listed in the order: â+, â−, b̂+, b̂−. We assume that N is
odd and consider the transmitted trigger modes (which
we number from 1 to N), the â+ mode (mode N + 1),
and the b̂− mode (mode N + 2). By expressing the field
operators of the trigger modes (those observed by the
unit efficiency detectors) in terms of â−, b̂+, and field
operators representing vacuum states we find
〈ĉ†j ĉk〉 = 〈ψi|
− − e−2πij/N b̂
(â− − e2πik/N b̂+eiθ)|ψi〉
1 + e2πi(k−j)/N
, (5)
where j ∈ {1, 2, . . . , N}, k ∈ {1, 2, . . . , N}, λ ≡ ηr2/(1−
r2), and we allow of a constant phase shift θ of b̂+ relative
to â−. Furthermore
〈ĉ†N+1ĉN+1〉 = 〈ψi|â
+â+|ψi〉 = r2/(1− r2), (6)
〈ĉ†N+2ĉN+2〉 = 〈ψi|b̂
−b̂−|ψi〉 = r2/(1− r2), (7)
〈ĉk ĉN+1〉 = 〈ψi|
(â− − e2πik/N+iθ b̂+)â+|ψi〉
1− r2
, (8)
〈ĉk ĉN+2〉 = 〈ψi|
(â− − e2πik/N+iθ b̂+)b̂−|ψi〉
1− r2
e2πik/N+iθ , (9)
〈ĉj ĉk〉 = 〈ĉN+1ĉN+1〉 = 〈ĉN+2ĉN+2〉 = 〈ĉN+1ĉN+2〉 =
〈ĉ†N+1ĉN+2〉 = 〈ĉ
k ĉN+1〉 = 〈ĉ
k ĉN+2〉 = 0. (10)
For even values ofN the factors
η/(2N) are replaced by
η/N . Note that loss in the signal beam may be taken
into account by performing the transformations â+ →√
ηsâ+ and b̂− →
ηsb̂− in the above expressions, where
1− ηs is the loss.
The NOON state fidelity FN of the signal state con-
ditioned on N photo detection events in the transmitted
trigger modes is
WNOON(xN+1, pN+1, xN+2, pN+2)
(1− 2πW0(xi, pi))
WV (x1, p1, . . . , xN+2, pN+2)
dxidpi
, (11)
whereWNOON is the Wigner function of the NOON state
(1), and
(1− 2πW0(xi, pi))
WV (x1, p1, . . . , xN+2, pN+2)
dxidpi
, (12)
is the success probability, i.e., the probability to obtain
the conditioning detection events and produce the NOON
state in a given pulse of the OPO system. We expand
the product
(1− 2πW0(xi, pi)) =
(−2πW0(xi, pi))di ,
where the sum is over all d ≡ (d1, d2, . . . , dN ) with
di ∈ {0, 1}, and define the diagonal matrix Jd =
diag(d1, d1, d2, d2, . . . , dN , dN ) and the n×n identity ma-
trix In. Furthermore, we divide the covariance matrix
into four parts
Vtt Vts
V Tts Vss
, (14)
where Vtt is the 2N×2N covariance matrix of the trigger
modes, Vss is the 4 × 4 covariance matrix of the signal
modes, while Vts contains the correlations between the
trigger and the signal modes, and we define the vector
ys = (xN+1, pN+1, xN+2, pN+2)
T and the matrix
Ud = Vss − V Tts Jd(JdVttJd + I2N )−1JdVts. (15)
This allows us to write Eqs. (11) and (12) in the following
compact forms [25]
det(I2N + JdVtt)
WNOON(ys)WUd(ys)dys, (16)
det(I2N + JdVtt)
. (17)
Since WNOON is a product of a polynomial and a Gaus-
sian the integral in Eq. (16) may be evaluated analytically
and for N = 3 and η = 1 we find
(1− r2)2(2− r2)2(3− 2r2)(6 − 5r2)
18(4− 3r2)
, (18)
where the optimal value φ = Nθ + π + 2πn, n ∈ Z,
is assumed. Expressions for PN are given in table I for
N = 1, 2, 3, and 4, and FN and PN are plotted for N = 3
in Figs. 2 and 3, respectively. We observe that high prob-
abilities are only found in the parameter regime, where
the fidelity is low. If, for instance, we want a NOON
state fidelity of at least 0.9, we choose r = 0.14, and if
η = 0.25, P3 is of order 10
−8. With a repetition rate
of order 106 s−1 (see [16]) one state is produced every
second minute on average. The production rate is very
dependent on detector efficiency, and if η is increased to
unity, the rate is increased by approximately a factor of
For odd values of N we may observe both reflected
and transmitted trigger modes and condition on detec-
tion events in all the transmitted trigger modes and no
detection events in all the reflected trigger modes, or, vice
versa. In this case we also include the reflected trigger
modes in the covariance matrix, which we now denote by
V +. By a similar analysis as above we obtain the success
probability
P+N = 2
2N (−2)
det(I4N + J
, (19)
N PN P
(λ+1)2
(λ+1)2
λ3(λ+4)
(λ+2)2(λ+3)(λ+6)
2λ3(3λ+4)
(λ+1)2(λ+2)2(2λ+3)(5λ+6)
λ4(λ2+6λ+6)
(λ+1)2(λ+2)2(λ2+8λ+8)
TABLE I: Success probabilities calculated from Eqs. (17) and
(19). λ ≡ ηr2/(1− r2).
0 0.2 0.4 0.6 0.8 1
FIG. 2: NOON state fidelity F3 (solid lines) and F
3 (dashed
lines) as a function of squeezing parameter r for η = 1 (upper
lines), η = 0.25 (middle lines), and η → 0 (lower lines). Note
that in the latter case F3 = F
where J+d ≡ diag(d1, d1, . . . , dN , dN , 1, 1, . . . , 1, 1), while
the NOON state fidelity F+N is given by Eq. (11) with V
replaced by the matrix
V − (V +R )
T (V +RR + I2N )
−1V +R , (20)
where V +RR is the covariance matrix of the reflected trig-
ger modes and V +R consists of the correlations between
the reflected trigger modes and the signal and transmit-
ted trigger modes. Explicit results for P+N are given in
table I for N = 1 and 3. F+3 and P
3 are compared to
F3 and P3 in Figs. 2 and 3, and it is observed that F
and P+3 are both larger than F3 and P3 if r is not large
(and η > 0). For r → 1, P+3 → 0 because in this limit it
is very unlikely to obtain no detection events in all the
reflected or in all the transmitted trigger modes.
In the limit of very small detector efficiency a simple
expression for the NOON state fidelity for the case of N
trigger detectors is easily derived without using Wigner
function formalism. In general, if the state of interest
is expressed in the photon number basis, the mathemat-
ical operation corresponding to an ‘on’ detection is to
multiply each term by
1− (1 − η)n, where n is the
number of photons in the mode observed by the non-
unit efficiency detector, trace out the detected mode,
and renormalize. If nη ≪ 1 for all contributing terms,
1− (1− η)n ≈ √nη ∝
n, and the on/off detector
0 0.2 0.4 0.6 0.8 1
FIG. 3: Success probability P3 (solid lines) and P
3 (dashed
lines) as a function of squeezing for η = 1 (upper lines) and
η = 0.25 (lower lines). The dotted lines represent the approx-
imate expression (23).
model becomes equivalent to the photo detector model.
In this case the density operator of the output state is
obtained as
ρ = M
〈p|〈q|(âN− − (b̂+eiθ)N )|ψi〉
〈ψi|((â†−)N − (b̂
−iθ)N )|q〉|p〉
(1− r2)N+2
2N !r2N
(r2)n+m
(n−N)! |n,m〉〈n,m|
−e−iNθ
n!(m+N)!
(n−N)!m!
|n,m〉〈n−N,m+N |
(r2)n+m
(m−N)!
|n,m〉〈n,m|
−eiNθ
(n+N)!m!
n!(m−N)!
|n,m〉〈n+N,m−N |
where M is a normalization constant and the traces are
over the â− and b̂+ modes. This leads to the NOON state
fidelity
N = 〈NOON|ρ|NOON〉 = (1− r
2)N+2, (22)
where again φ = Nθ + π + 2πn, n ∈ Z, is assumed.
It is interesting to compare this result with the fidelity
(1 − r2)N+1 obtained for production of N -photon states
from a single two-mode squeezed state by conditioning
on N detection events in the idler beam and using detec-
tors with very small efficiency. If a single-photon state is
produced by this method and transformed into an N = 1
NOON state as explained in the Introduction, the NOON
state fidelity is F1,s = (1−r2)2, and the success probabil-
ity is P1,s = λ/(λ + 1). Choosing squeezing parameters
such that F1,s = F1, we find that P
1 = (4/3)P1,s in the
high fidelity limit. It is thus possible to achieve a higher
success probability using the scheme with two OPOs, but
the price to pay is a more technically involved setup,
and NOON states with two different values of φ are pro-
duced. For N = 2 the present protocol and combination
of two single-photon states on a 50 : 50 beam splitter,
each produced conditionally from a single OPO, lead to
identical fidelities and success probabilities. Finally we
note that the photo detector model underestimates FN
for η > 0 because 1 − (1 − η)n = η
i=0 (1 − η)i < nη
for n = 2, 3, . . . while 1− (1− η)n = nη for n = 0, 1, i.e.,
the ‘wrong’ terms containing more than N photons are
given a too large weight. This is also what we observe in
Fig. 2.
In the limit of small r and for odd values of N the
success probability is given approximately by the simple
expression
〈ψi|((â†−)N − (b̂
−iθ)N )
(âN− − (b̂+eiθ)N )|ψi〉 =
(2N)N
λN (N odd). (23)
Again η/(2N) must be replaced by η/N to obtain PN for
even values of N . The approximation to P3 is shown in
Fig. 3.
IV. NOON STATES FROM
CONTINUOUS-WAVE OPO SOURCES
Our protocol is not limited to pulsed fields, and for
completeness we now briefly consider NOON state gen-
eration from continuously driven OPOs. We assume
N = 3. For continuous-wave fields each of the three
detected trigger beams and the two signal beams are
described by time dependent field operators ĉi(t). The
trigger detections take place in particular modes local-
ized around the three detection times tc1, tc2, and tc3,
and we want to determine the NOON state fidelity of
an output state occupying one temporal mode in each
signal beam. Following the general multimode formal-
ism in Refs. [15, 20], the five relevant modes are spec-
ified by the mode functions fi(t), and the correspond-
ing five single mode operators are ĉi =
fi(t)ĉi(t)dt.
In general, we are free to choose the two output mode
functions at will, and in the present case it is natural to
choose the mode function which gives rise to the largest
three-photon state fidelity when three-photon states are
generated from a single type II continuous-wave OPO.
Since we are mainly interested in the parameter region
where the squeezing is small and the NOON state fidelity
is large, we use the optimal three-photon state mode
function derived for very low beam intensity in [20], i.e.,
f4(t) = f5(t) =
k=1 ck
γ/2 exp(−γ|t− tck|/2), where
0 0.05 0.1 0.15
FIG. 4: NOON state fidelity as a function of ǫ/γ for states
generated from a pair of continuous-wave OPO sources when
conditioning on three simultaneous trigger detection events
tc1 = tc2 = tc3.
0 0.5 1 1.5 2
FIG. 5: Fidelity of NOON states from continuous-wave OPO
sources as a function of separation between trigger detection
events (tc3 − tc1)γ for N = 3, tc3 − tc2 = tc2 − tc1, and ǫ/γ =
0.01.
the coefficients ck are functions of the intervals between
the detection times and γ is the leakage rate of the OPO
output mirror. We furthermore assume that the trigger
mode functions are nonzero only in an infinitesimal time
interval centered at the detection time, which is valid if
the trigger detections take place on a time scale much
shorter than γ−1. Since we consider a low intensity con-
tinuous beam, and since we formally assume that the
trigger detectors only register the light field in infinitesi-
mal time intervals around the detection times, the anni-
hilation operator detector model is perfectly valid in this
case and detector dead time is insignificant.
We may now proceed as above and eliminate all the
irrelevant modes from the analysis by writing down the
Gaussian Wigner function of the five interesting modes.
The only difference is that this time 〈ĉ†i ĉj〉 and 〈ĉiĉj〉 are
expressed in terms of the two time correlation functions
of the OPO output. Also, the operators applied to the
Wigner function to take conditioning into account are
different because the annihilation detector model is used.
The reader is referred to Refs. [15, 20] for details.
The resulting fidelity is shown as a function of ǫ/γ in
Fig. 4, where ǫ is the nonlinear gain in the OPO, and as a
function of the temporal distance between the condition-
ing detection events in Fig. 5. As in the pulsed case the
fidelity decreases when the degree of squeezing increases.
The fidelity also decreases when the temporal distance
between the conditioning detection events increases from
zero, but it is permissible to have a small time interval
between the trigger detection events. We note that the
curves represent a lower limit to the theoretically achiev-
able fidelity since a better fidelity may be obtained for
another choice of output state mode functions.
V. CONCLUSION
In conclusion we have analyzed a method to gener-
ate path entangled NOON states from the output from
two optical parametric oscillators. The method relies
on the joint detection of photons in a number of trig-
ger beams, and we presented a theoretical analysis of the
role of detector efficiency and dead time for the fidelity
of the states obtained and the success probability of the
protocol. Our specific NOON state protocol applies for
general photon numbers of the states, but in practice it
is not realistic to go beyond the case of N = 3, studied
here. This is due to the reduction of the fidelity due to
unwanted contributions from higher number states, when
the OPO output power gets too high, combined with the
severe reduction of the probability to obtain the number
of conditioning detection events needed when the OPO
output power is too low. The N = 3 NOON states, which
can be produced at 90% fidelity at the rate of one state
produced every 10− 100 seconds, seem to be at the limit
of realistic experiments of the proposed kind. Finally, we
also determined the NOON state fidelity for continuous-
wave fields, where the best NOON state occupies a pair
of suitably selected temporal mode functions, and where
we find high fidelities as long as the trigger events occur
within a short time window compared to the lifetime of
the OPO cavity field.
We presented this analysis for the production of op-
tical NOON states, but we note that recent theoreti-
cal proposals and experiments with four wave mixing
of matter waves [26], engineered quadratic interactions
among trapped ions [27], and entanglement between field
and atomic degrees of freedom [28, 29] bring promise for
similar conditional generation of atomic and interspecies
atom-field NOON states.
This work was supported by the European Integrated
project SCALA.
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http://arxiv.org/abs/quant-ph/0508158
http://arxiv.org/abs/quant-ph/0612154
