Multipartite entanglement in the optical frequency comb of a depleted-pump optical parametric oscillator

Abstract

We study the effect of pump depletion in a single type-I optical parametric oscillator (OPO) and find that pump-mediated, signal-field quantum correlations generate multipartite entanglement in the optical frequency comb of the OPO.

Full text

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Multipartite entanglement in the optical frequency comb of a depleted-pump optical
parametric oscillator
Reihaneh Shahrokhshahi∗and Olivier Pfister†
Department of Physics, University of Virginia, 382 McCormick Rd., Charlottesville, VA 22903, USA
We theoretically analyzed the generation of multipartite continuous-variable entanglement in a
single optical parametric oscillator (OPO) well above threshold. In this system, the quantum dynam-
ics of the strongly depleted pump field is responsible for the onset of multipartite entanglement, over
the initially independent pairs of two-mode squeezed, bipartite entangled, OPO signal fields. We
verify the multipartite nature of the entanglement by evaluating the van Loock-Furusawa criterion.
I.INTRODUCTION
The generation of massively entangled states is of great interest for quantum information. For quantum communica-
tion, good examples are multiparty quantum teleportation [1] and quantum secret sharing [2]. For measurement-based
quantum computing, cluster state [3, 4] generation is known to enable one-way quantum computing [5, 6]. It was
shown theoretically that massive cluster-state entanglement can be obtained in the optical frequency comb, i.e. the set
of resonant modes of a single optical cavity [7, 8]. The cavity must contain a nonlinear medium, thereby constituting
an optical parametric oscillator (OPO), in order to provide entanglement between the comb quantum modes, a.k.a.
“Qmodes” (in analogy with Mermin’s more harmonious spelling of “Qbit” [9]). Recently, the simultaneous generation
of 15 quadripartite “square” cluster states over 60 Qmodes was realized experimentally in a single OPO [10].
In the aforementioned experimental work, as well as other proposals to generate cluster states [7, 8, 11], the
OPO’s nonlinear crystal must simultaneously phasematch two or three nonlinear interactions [12]. Here we consider
the simplest possible case of an OPO with a single, nondegenerate nonlinear interaction. Below the OPO emission
threshold, such a system is known to emit two-mode squeezed fields which demonstrate the Einstein-Podolsky-Rosen
(EPR) paradox [13, 14]. In this case the interaction Hamiltonian is
Hint= 2i?χβ
n
?
i=1
(a†
ia†
−i− aia−i),(1)
where β is the classical (real) and constant (undepleted) pump field (in practice a stable, narrow-linewidth, continuous-
wave laser) and a±iare the photon annihilation operators of entangled Qmodes ±i, of frequencies
ω±i=ωp
2
±
?
i +1
2
?
∆,(2)
where ωp= ωi+ ω−iis the pump frequency (see Fig. 1) and ∆ is the free spectral range of the OPO cavity.
∗rs2vw@virginia.edu
†opfister@virginia.edu
arXiv:1110.6450v1 [quant-ph] 28 Oct 2011

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FIG. 1. Optical frequency comb defined by the resonant modes of the OPO cavity, spaced by free spectral range ∆. The green
arrow symbolizes the pump field, placed at half its frequency for clarity.
Just above the OPO threshold, the generation of EPR states has also been shown to be possible, theoretically [15]
and experimentally [16–19]. However, the undepleted classical pump approximation then breaks down as the external
pump power is increased significantly above threshold. In that case, the pump must be treated as an actual quantum
field in the three-wave mixing interaction
Hint= 2i?χ
n
?
i=1
(pa†
ia†
−i− p†aia−i),(3)
where p is the annihilation operator of the pump field. Recently, it was predicted [20] and experimentally demonstrated
[21] that the pump field participates in three-way entanglement in this case. In another interesting case, a theoretical
analysis showed that the signal fields from two OPOs pumped by the same field could become entangled [22]. Here,
we extend this analysis to the optical frequency comb of a single OPO, in which a vast number of different Qmode
pairs are already known to be entangled by their parametric downconversion from the pump field [10, 23]. In the
undepleted pump approximation, all EPR pumps are independent. However, when one considers the OPO well above
threshold, it is straightforward to see that strong (ideally total) pump depletion will entail strong correlations between
the EPR fields, since a pump photon downconverting into one Qmode pair will necessarily not be downconverted into
any other pair. This should yield multipartite, rather than bipartite, quantum correlations and the goal of this paper
is to ascertain whether they result in multipartite entanglement, which they do.
The paper is organized as follows. In Section II we solve the Heisenberg equation for an undepleted quantum pump,
in the limit of linearized quantum fluctuations. We are certainly aware that more sophisticated treatments exist
[24–28] and may indeed be interesting to use in order to explore this system further. However, our simple approach
already yields interesting new results: in Section III, we use the multipartite inseparability criterion derived by van
Loock and Furusawa [29] to establish the existence of multipartite entanglement in the optical frequency comb of a
single OPO. We then conclude.
II.THE QUANTUM OPTICAL FREQUENCY COMB OF A SINGLE OPO ABOVE THRESHOLD
As mentioned before, we consider the simplest possible case of an OPO cavity with a single pump mode and a
single nondegenerate interaction in its nonlinear crystal. Such a crystal implements the Hamiltonian of Eq. (3). We
assume a single two-mirror standing wave cavity, with one mirror of reflectivity R
an output coupler of R±i= 1−T±i< 1. Taking into account the vacuum modes Ain
output coupler, the input-output theory [30, 31] can be used to derive the equations of motion for the internal cavity
modes
?
±i= 1 for all modes and the other
ithat enter the cavity through its
˙ ai= 2χpa†
˙ a−i= 2χpa†
−i− kiai+
i− k−ia−i+
n
?
?
2kiAin
?2k−iAin
i
(4)
−i
(5)
˙ p = −2χ
i
aia−i− kpp +?2kppin.(6)

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Here k±i= T±i/2τ are the loss rates of the cavity mirror for mode i, τ being the cavity round trip time. In order to
solve Eqs. (4-6) we first rewrite field operators as centered fluctuations about their expectation value
ai= αi+ δai
Ain
p = ? + δp
pin= ?in+ δpin.
(7)
i= δAin
i
(8)
(9)
(10)
A.Classical steady-state solutions
The semiclassical, or mean value, equations follow directly from Eqs. (4-6):
˙ αi= 2χ?α∗
˙ α−i= 2χ?α∗
−i− kiαi
i− k−iα−i
n
?
(11)
(12)
˙ ? = −2χ
i
αiα−i− kp? +?2kp?in.(13)
Now, considering same cavity losses for all signal modes, ki= k−i= ka, the stationary solutions of the two coupled
equations Eqs. (11-12) for semi-classical mean values are
2χ?α0∗
2χ?α0∗
−i= kaα0
i = kaα0
i
(14)
−i.(15)
this results in |α0
α0
stationary solution for the pump field’s mean value can then be written as
i| = |α0
−i| , |?0| = ka/2χ and φ0
i+ φ0
−i− φ0= 0 where φ0
±iand φ0are the respective phases of
±iand ?0. For simplicity we take ?inreal and positive therefore based on Eq. (13), φ0= 0 and φ0
i= −φ0
−i. The
4χ2
n
?
i
|α0
i|2= kakp(√σ − 1),(16)
where
σ =
?
2χ
ka
?
2
kp?in
?2
. (17)
Clearly, the right-hand side of Eq. (16) must be positive and σ = 1 defines the threshold pump field
?th
in=ka
2χ
?
kp
2,(18)
Hence, σ = (?in/?th
Eq. (16).
in)2is also the pump to threshold power ratio. The classical signal amplitudes are weakly set by
B.Stability analysis
The stability of the steady-state solution can be determined by a linearized analysis for small perturbations:
αi= α0
?i= ?0
i+ δαi
(19)
i+ δ?i. (20)

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Substituting Eqs. (19-20) into Eqs. (11-13), we get
...
δ ˙ αi= kaδα∗
α−i= kaδα∗
...
−i+ 2χα0
i+ 2χα0
iδ? − kaδαi
−iδ? − kaδα−i
(21)
δ ˙(22)
δ ˙ ? = −2χ
n
?
i
(α0
iδα−i+ α0
−iδαi) − kpδ?, (23)
where i = 1,...,n, n being the number of signal and idler pairs considered inside the cavity.
?... δαi δα∗




δ?∗
...0
−2χα0∗
We derived the eigenvalues of the matrix in Eq. (24) for n = 1,2,3. In all these cases, the eigenvalue sets have the
following form:
Defining δA =
iδα−i δα∗
−i... δ? δ?∗?TWe can rewrite Eqs. (21-23) in block matrix form:




−i
d
dt









...
δαi
δα∗
δα−i
δα∗
...
δ?
i
−i









=













......
0
...
0
.........
0
...
...
...
...
−ka
0
0
ka
...
ka
0
0
−ka
...
0
−2χα0∗
...
...
... 2χα0
...
2χα0
0
i
−ka
ka
0
...
0
ka
−ka
0
...
−2χα0
0
2χα0∗
0
2χα0∗
...
0
−kp
i
−i
0
...
−i
... −2χα0
−i
i
...
...
−kp
0
i


























...
δαi
δα∗
δα−i
δα∗
...
δ?
δ?∗
i
−i













.(24)
{λ} = {0,...,0
? ?? ?
2n−1
,−2ka,...,−2ka
?
?
?
??
κ?κ − 8(√σ − 1)??
κ + 2 ±
?
2n−1
,λ1,λ2,λ3,λ4}, (25)
where, posing the pump-signal loss ratio κ = kp/ka,
λ1,2= −1
λ3,4= −1
2ka
κ ±
?
(26)
2ka
?
(κ + 2)2− 8n√σ
?
.(27)
Because of the particular symmetry of the problem—namely the block structure of the matrix in Eq. (24), we argue
that it is reasonable to postulate that Eq. (25) is the general eigenvalue set, ∀n, even though a complete inductive
proof is formally required. For certain initial conditions all 2(2n+1) eigenvalues of the matrix in Eq. (25) can only be
zero or negative, which ensures the stability of the stationary solution presented in Eq. (16). Equations (26-27) show
that, as the number of times above threshold σ increases, one can always find negative values for λ1,...,4by increasing
the pump-signal loss ratio κ, thereby tending towards the doubly resonant OPO, which is always stable.
C.Quantum fluctuations
Now, we rewrite Eqs. (4-6) for the quantum fluctuations around these classical mean values. Notice α∗
−i= αi,
˙ δai= 2χ(δpαieiφi+ ?δa†
˙
δa−i= 2χ(δpαie−iφi+ ?δa†
n
?
−i) − kaδai+
i) − kaδa−i+
?
2kaδAin
?
i
(28)
2kaδAin
−i
(29)
δ ˙ p = −2χ
i
(αieiφiδa−i+ αie−iφiδai) − kpδp +?2kpδpin.(30)
We introduce the generalized field quadrature operators as Qi= (eiφia†
solve these coupled equations we can use the symmetry of the equations in the exchange of the two signal modes and
i+e−iφiai) and Pi= i(eiφia†
i−e−iφiai). Then

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introduce the new variables [32]
Qi+= Qi+ Q−i
Qi−= Qi− Q−i
Pi+= Pi+ P−i
Pi−= Pi− P−i.
(31)
(32)
(33)
(34)
The equations of motion for these quadratures are
δ˙Qi+= 4χαiδQp+
δ˙Qi−= −2kaδQi−+
δ˙Qp= −2χ
?
2kaδQin
?
i+
(35)
2kaδQin
i−
(36)
n
?
i
αiδQi+− kpδQp+?2kpδQin
?
2kaδPin
i−
δ˙Pp= −2χ
i
p
(37)
δ˙Pi+= 4χαiδPp− 2kaδPi++
δ˙Pi−=
n
?
2kaδPin
i+
(38)
?
(39)
αiδPi+− kpδPp+?2kpδPin
p.(40)
As seen from Eqs. (35-40), the equations for the antisymmetric modes are decoupled from the pump and the ones for
the symmetric modes can be easily solved in the frequency domain [32]:
?
iΩδ˜Pi+(Ω) = 4χαiδ˜Pp(Ω) − 2kaδ˜Pi+(Ω) +
iΩδ˜Qp(Ω) = −2χ
i
n
?
These equations can be easily solved for pump and signal- idler pairs. The output quadratures are finally determined
using input-output relations:
?
δPout
±
δQout
p
=
δPout
p
=
iΩδ˜Qi+(Ω) = 4χαiδ˜Qp(Ω) +2kaδ˜Qin
i+(Ω) (41)
?
2kaδ˜Pin
i+(Ω)(42)
n
?
αiδ˜Qi+(Ω) − kpδ˜Qp(Ω) +?2kpδ˜Qin
αiδ˜Pi+(Ω) − kpδ˜Pp(Ω) +?2kpδ˜Pin
p(Ω)(43)
iΩδ˜Pp(Ω) = −2χ
i
p(Ω). (44)
δQout
± =
2kaδQ±− δQin
2kaδP±− δPin
?2kpδQp− δQin
±
=
?
?2kpδPp− δPin
±
p
p.(45)
The solutions of Eqs. (41-44) are
δ˜Qout
+i(Ω) = −

1 +
?
2ka
?
Ω
kpΩ + i[Ω2+ 8χ2(α2
?
16iχ2kaαi
−ikpΩ + Ω2− 8χ2?
iΩ
2ka+ iΩδ˜Qin
?
kpΩ + i
i−?
jα2
?
j)]
?
−ikpΩ + Ω2− 8χ2?
jα2
j

δ˜Qin
+,i(Ω)
−
Ω
jα2
j
?
?
j?=i
αjδ˜Qin
+,j(Ω) −
8χ?kakpαi
−ikpΩ + Ω2− 8χ2?
jα2
j
δ˜Qin
p(Ω) (46)
δ˜Qout
−i(Ω) = −
−i(Ω)(47)
δ˜Qout
p (Ω) =
kpΩ − iΩ2− 8χ2?
jα2
j
?
?
Ω2− 8χ2?
jα2
j
?δ˜Qin
p(Ω) +
4iχ?kakp
kpΩ + i
?
Ω2− 8k2?
jα2
j
?
n
?
j=1
αjδ˜Qin
+,j(Ω)(48)

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δ˜Pout
+i(Ω) =
?
−1 +
2ka
2ka+ iΩ−
16χ2kaα2
i
(2ka+ iΩ)[(2ka+ iΩ)(kp+ iΩ) + 8χ2?
16χ2kaαi
(2ka+ iΩ)[(2ka+ iΩ)(kp+ iΩ) + 8χ2?
(2ka+ iΩ)(kp+ iΩ) + 8χ2?
?
jα2
j]
?
δ˜Pin
+,i(Ω)
−
jα2
j]
?
j?=i
αjδ˜Pin
+,j(Ω)
+
8χ?kakpαi
jα2
j
δ˜Pin
p(Ω)(49)
δ˜Pout
−i(Ω) =
−1 −2ika
Ω
?
δ˜Pin
−i(Ω) (50)
δ˜Pout
p
(Ω) =
2ka(kp− iΩ) + ikpΩ + Ω2− 8χ2?
4k√ka
2ka(kp+ iΩ) + ikpΩ − Ω2+ 8k2?
jα2
jα2
j
2ka(kp+ iΩ) + ikpΩ − Ω2+ 8k2?
−
j
δ˜Pin
p(Ω)
?kp
jα2
j
n
?
i=1
αiδ˜Pin
+,i(Ω).(51)
Substituting the classical solutions Eq. (16) in Eqs. (46-51) and taking Ω = 0, these equations yield:
δQout
δQout
δPout
−i−→ ∞
δPout
+i−→ ∞
−i−→ 0
(52)
(53)
(54)
+i=4χ?kakpαi
kakp√σ
δPin
p−
4χ2α2
kakp√σδPin
i
+,i−
4χ2αi
kakp√σ
?
j?=i
αjδPin
+j
(55)
1.Two-mode squeezing
In order to quantitatively study squeezing behavior, we assumed equal classical mean values for all pairs. However,
we can assume any ratio between the classical mean values, as long as they satisfy?n
V (Q+i) =?(δQ+i)2?−→ ∞
V (P−i) =?(δP−i)2?−→ ∞
i|αi|2=
kakp(√σ−1)
4χ2
= const,
Eq. (16). Therefore the variances of squeezed and antisqueezed quadratures are
(56)
V (Q−i) =?(δQ−i)2?−→ 0
V (P+i) =?(δP+i)2?=2(σ − 1)
(57)
(58)
nσ
,(59)
which yields the classic EPR result at threshold (σ = 1), the generation of n independent entangled (i,−i) pairs.
However, if the OPO is above threshold (σ > 1), the variance of the phase sum, Eq. (59) will increase from zero
[15, 32] and will eventually stop being squeezed. It states the well-known fact that for the OPO operating well above
the emission threshold, twin pairs are not independent EPR pairs due to the pump statistics, unless kp? ka[15].
This is precisely the mechanism that we rely upon to create multipartite entanglement in this work. We give two
preliminary examples before turning to the evaluation of precise multimode entanglement criteria.
2.Multimode squeezing
We give two examples of squeezed multimode operators which will be useful in the next section.
First off, specifically combining phase sum operators of two pairs i and j yields
αi(Pj+ P−j) − αj(Pi+ P−i) −→ 0.(60)

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FIG. 2. Plot of nV (Pi+ P−i), in red, and V [?n
not squeezed at threshold, subsequently drops from the shot noise level (of value 1 in this graph) and shows squeezing. In this
particular graph, the minimum of V [?n
i=1(Pi+ P−i) − xPp], in blue, for n = 3 and x = σ. When σ increases, the
variance of Pi+ P−i increases from zero and approaches the shot noise level. However, the variance of?n
i=1(Pi+ P−i) − xPp] occurs at σ = 1.18 but, in general, the value of σ for which the
blue curve reaches its minimum, as well as the value of the minimum itself, depends on the choice of x.
i=1(Pi+ P−i) − xPp,
Even though phase sum operators for each pair become noisier with increasing input pump, this noise can be canceled
appropriate linear combinations.
Another interesting example is that of operator?n
and the pump is a sensible one. Having established this, we turn to directly testing the existence of multipartite
entanglement in our system.
i=1(Pi+P−i)−xPp. In Fig. 2 we plot its variance as a function
of σ, for x = σ. This graph clearly shows that the assumption of the existence of a correlation between all modes
III. MULTIPARTITE ENTANGLEMENT IN THE OPO WELL ABOVE THRESHOLD
A. The van Loock-Furusawa (vLF) inseparability criterion [29]
This multipartite entanglement criterion is the multipartite generalization of the Duan [33]-Simon [34] criterion,
itself the continuous-variable formulation of the Peres [35]-Horodecki [36] positive partial transpose criterion. A
partially separable density operator can be written as the convex sum
?
which indicates that modes (k1,...,km) are separable from modes (km+1,...,kn). If we define two quadrature operators
with arbitrary real parameter sets {hi}iand {gi}i,
u = h1Q1+ h2Q2+ ··· + hnQn
v = g1P1+ g2P2+ ··· + gnPn,
then the separable density operator of Eq. (61) must verify the vLF inequality [29]
Vρ(u) + Vρ(v) ? 2?|hk1gk1+ ··· + hkmgkm| + |hkm+1gkm+1+ ··· + hkngkn|?,
whose violation implies the existence of entanglement between mode sets (kr,...,km) and (ks,...,kn).
ˆ ρ =
i
ηiˆ ρi,k1.....km⊗ ˆ ρi,km+1...,kn, (61)
(62)
(63)
(64)
B. Multipartite entanglement in a single, depleted-pump OPO
In order to demonstrate multipartite entanglement, we examine the conditions for violation of all possible vLF in-
equalities, corresponding to all possible respective mode partitions such as Eq. (61), and their associated experimental
regimes.

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1.Pump-signals partition
We first consider the separability of the sole pump mode from all signal modes. We define u1and v1as
u1=
n
?
i=1
αi
α1(Qi+ Q−i) +2
x
n
?
i=1
αi
α1Qp
(65)
v1=
n
?
i=1
(Pi+ P−i) − xPp,(66)
with the real parameter x > 0. Based on Eq. (64), the separability of mode p implies
S1= V (u1) + V (v1) ? 2(|h−ng−n+ ··· + h−1g−1+ h1g1+ ··· + hngn| + |hpgp|)
? 2(2|h1g1+ ··· + hngn| + |hpgp|)
? 2(|2
α1
i=1
xα1
n
?
(67)
(68)
n
?
αi| + |
2
n
?
i=1
αi(−x)|)(69)
?
8
α1
i=1
αi.(70)
Here, and in the following, we make the assumption that all classical amplitudes {αi}i are equal, for the sake of
simplicity. This doesn’t lessen the generality of our treatment but makes numerical evaluations easier. Under this
assumption, we get
S1? 8n.(71)
Figure 3 displays the maximum violation of the above inequality versus n and σ, for optimized values of the arbitrary
weight x. As can be seen, there always exist values of (n,σ) for which S1−8n is negative, which proves the inseparability
of the pump mode from the signal modes. Unsurprisingly, entangling a larger number of pairs (n) requires a higher
pump to threshold power ratio (σ). However, a large pump power (σ) does degrade the inseparability, which might
just be due to the increasing depletion of the intracavity pump field.
a1
a2
a−1
a−2
a−n
an
ap
FIG. 3. Left, sketch of the mode partition studied. Center, plot of the optimum values of x = xopt which give maximum
violation S1(x) − 8n of the vLF inequality, at a given pump to threshold power ratio σ and a given number n of mode pairs
inside the cavity. Right, plot of the maximum vLF inequality violation S1(xopt) − 8n, versus σ and n. We took the particular
case Ω = 0 and αi = αj ∀i,j.

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2. Partition of one (aj,a−j) EPR pair
We now study the inseparability of an entangled pair (aj,a−j) from the rest of the signals and the pump. For such
a partition, we define
u2= Qj++
n
?
i?=j
αi
αjQi++
2
xαj
n
?
i
αiQp
(72)
v2=
n
?
i?=j
(αi
αjPj+− Pi+),(73)
and the vLF inequality is
S2= V (u2) + V (v2) ? 2(|2hjgj| + |2h1g1+ ··· + 2hngn+ hpgp|)
??????
?
αj
i?=j
? 8(n − 1).
(74)
? 2

21
n
?
αj
n
?
i?=j
αi


??????
+ 2
??????
−2
αj
n
?
i?=
αi+ 0
??????
(75)
8
αi
(76)
(77)
Figure 4 shows the violation of this inequality for a broad range of parameters. It also demonstrates the necessity of
applying larger input pump intensity when considering more pairs inside the cavity in order to generate inseparability
between all pairs, again unsurprisingly.
aj
a−j
a1
a2
a−1
a−2
a−n
an
ap
FIG. 4. Left, sketch of the mode partition studied. Center, plot of the optimum values of x = xopt which give maximum
violation S2(x) − 8(n − 1) of the vLF inequality, at a given pump to threshold power ratio σ and a given number n of mode
pairs inside the cavity. Right, plot of the maximum vLF inequality violation S2(xopt) − 8(n − 1), versus σ and n. We took the
particular case Ω = 0 and αi = αj ∀i,j.
3.Pair set partition
We next turn to partitions {(a1,b1)...(ak,bk)}{(ak+1,bk+1)...(an,bn)}. The operators are
u3=
k
?
k
?
i=1
Qi++
n
?
j=k+1
αj
α1Qj++
2
xα1
n
?
l
αlQp
(78)
v3=
i=1
n
?
j?=i
(Pi+−αi
αjPj+), (79)

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10
and the vLF inequality is
S3= V (u3) + V (v3) ? 2(|2h1g1+ ··· + 2hkgk| + |2hk+1gk+1+ ··· + 2hngn+ hpgp|)
? 8k(n − k).
(80)
(81)
Assuming 1 ? k < n, then it is straightforward to show that 8(n − 1) ? 8k(n − k) ? 2n2. As a consequence, the vLF
inequality S3is automatically violated when vLF inequality S2is, and does not need to be considered separately.
4. Single signal mode partition
Finally, we consider partitions of a single signal mode, or several such, all belonging to different EPR pairs (i,−i).
Because the signal mode aiis highly entangled to the mode a−i—and to all other signals by virtue of the preceding—
the separability test is simple in the present case. It is straightforward to show that the necessary vLF inequality for
the partition {a1,...,ak}{a−1,...,a−k,(ak+1,a−(k+1)),...,(an,a−n))} ,
S4=V

Qj− Q−j+
? 2(|h1g1+ ··· + hkgk| + |h−1g−1+ ...h−kg−k+ 2hk+1gk+1+ ...2hngn|)
?
? 4(n − k),
?
i?=j
αi
αj(Qi− Q−i)

+ V

?
i?=j
Pi+ P−i−αi
αj(Pj+ P−j)


? 2| −
i?=1
αi
α1
+
k
?
i=2
αk
α1| + 2|
?
i?=1
α−i
α1
+
k
?
i=2
α−k
α1
+ 0|
(82)
is always violated in the presence of single EPR pair entanglement. This is because the left hand side term of Eq. (82)
contains EPR nullifiers [37], a.k.a. EPR entanglement witnesses, whose squeezed variances tend toward zero. The
inseparability of any other form of partitions on modes when modes a1and a−1are placed in different partitions,
can be examined by inequalities similar to S4and with nonzero boundaries. Such inequalities are always violated.
Therefore, if EPR entanglement is present (the checking of which is a staple of the experimental calibration of a
regular two-mode squeezer), then the violation of both S1 and S2 is a necessary and sufficient condition to mode
inseparability for all possible partitions in the optical frequency comb of a single OPO.
C. Entanglement between pairs without considering the pump
Here we ask the question of the possibility of multipartite entanglement between twins without considering the
pump field. For that, we rewrite inequality S2without the pump quadratures:
S
?
2= V (u
?
2) + V (v
?
2) ? 8(n − 1)(83)
with
u
?
2= Qj++
n
?
i?=j
αi
αjQi+
(84)
v
?
2=
n
?
i?=j
αi
αjPj+− Pi+.(85)
In Fig. 5, we plot the violation of this inequality. As can be seen by comparing with Fig. 4, Inequality S
slightly larger σ to be violated, compared to S2, for small values of n. It shows we need to pump harder (and get
closer to total depletion) in order to see pure entanglement between twin pairs. The arguments of subsections and
IIIB3 and IIIB4 may be reused here to complete the inseparability proof.
?
2requires

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11
FIG. 5. Plot of the vLF inequality violation S
mode pairs inside the cavity, for Ω = 0 and αi = αj, ∀i,j.
?
2− 8(n − 1), versus σ, the pump to threshold power ratio, and n, the number of
IV. CONCLUSION
We showed that a single OPO operating well above threshold can generate multipartite entanglement in its optical
frequency comb. We verified the multipartite nature of the entanglement by evaluating the van Loock-Furusawa
separability criterion on all possible Qmode partitions. We showed that all of these inequalities can be violated, for
an arbitrary large number of pairs n, simply by increasing the input pump power higher above threshold.
While the presence of multipartite entanglement in such a simple system is a remarkable feature, it is important to
keep in mind that the exact type of entanglement that is produced here (GHZ, W, cluster) is difficult to determine
and is not likely to be useful for quantum computing. Indeed, previous work has shown that multipartite cluster-state
generation, which was experimentally demonstrated in a single OPO below threshold [10], should actually fail above
threshold [28]. However, the multipartite entanglement that we discovered in the simple OPO above threshold could
still be useful for quantum communication applications.
This work was supported by U.S. National Science Foundation grants No. PHY-0855632 and No. PHY-0960047.
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