Generating entanglement of photon-number states with coherent light via
  cross-Kerr nonlinearity
Generating entanglement of photon-number states with
coherent light via cross-Kerr nonlinearity
Zhi-Ming Zhang∗, Jian Yang, and Yafei Yu
Laboratory of Photonic Information Technology,
School of Information and Photoelectronics,
South China Normal University, Guangzhou 510006, China
November 25, 2018
Abstract
We propose a scheme for generating entangled states of light fields. This scheme only requires
the cross-Kerr nonlinear interaction between coherent light-beams, followed by a homodyne de-
tection. Therefore, this scheme is within the reach of current technology. We study in detail the
generation of the entangled states between two modes, and that among three modes. In addition
to the Bell states between two modes and the W states among three modes, we find plentiful
new kinds of entangled states. Finally, the scheme can be extend to generate the entangled states
among more than three modes.
PACS: 03.67.Mn; 42.50.Dv; 42.50.Ct
1 Introduction
Entanglement is a characteristic feature of quantum states and has important applications in quantum
science and technology, for example, in quantum computation and quantum information [1]. There
are a lot of schemes for generating various kinds of entanglement, for example, the entanglement
between photons, the entanglement between atoms, the entanglement between trapped ions, and
the entanglement between different kinds of particles (for example, between photons and atoms).
In addition to the entanglement between two parties, there are also entanglement of multiparties.
Among these schemes many use single-photon sources and/or single-photon detectors. Although
there are great progresses in the study on these single-photon devices, how to obtain them is still a
challenging task. In this paper we propose a simple scheme for generating entangled states of light
fields. This scheme only requires the cross-Kerr nonlinear interaction between light fields in coherent
states, followed by a homodyne detection. Therefore, this scheme is within the reach of current
technology.The basic idea of this scheme is shown in Figure 1. Mode a is a bright beam which is in a
coherent state |α〉. Mode b is a weak or bright beam which is also in a coherent state. BS is a 50/50
beam splitter. KM1 and KM2 are Kerr media. HD means homodyne detection [2].
This paper is organized as follows: In section 2 we briefly introduce the cross-Kerr nonlinear inter-
action between two field-modes. In section 3 and section 4 we study the generation of entanglement
between two modes and that among three modes, respectively. Section 5 is a summary.
∗Corresponding author: zmzhang@scnu.edu.cn
http://arxiv.org/abs/0704.1560v1
2 Cross-Kerr nonlinear interaction
First, let us briefly review the cross-Kerr nonlinear interaction between a mode A and a mode B. The
interaction Hamiltonian has the form [3]
HCK = ~Kn̂An̂B, (1)
where n̂A and n̂B are the photon-number operator of mode A and mode B, respectively. The coupling
coefficient K is proportional to the third-order nonlinear susceptibility χ(3). The time-evolution
operator is
U (t) = exp
= exp {−iKn̂An̂Bt} = exp {−iτ n̂An̂B} = U (τ) , (2)
in which τ = Kt = K (l/v), it can be named as the scaled interaction time, or the nonlinear phase
shift. Here l is the length of the Kerr medium and v is the velocity of light in the Kerr medium. The
cross Kerr nonlinearity has following property
U (τ) |n〉B |α〉A = |n〉B
∣αe−inτ
, (3)
here |n〉 and |α〉 are the photon number state and the coherent state, respectively.
3 Entanglement between two modes
Now let us study the generation of the entangled states between two modes. The scheme is shown
in Figure 1. Assume that mode a is in a coherent state |α〉 [4]. Mode b is also in a coherent state
which is divided by the 50/50 beam splitter BS into two beams b1 and b2, and both b1 and b2 are in
coherent state |β〉.
We first consider the case of weak coherent state |β〉. In this case we have
|β〉 ≈ 1√
1 + |β|2
(|0〉+ β |1〉) , (4)
where |0〉 and |1〉 are the vacuum state and one-photon state, respectively. Let mode a interacts with
mode b1 and b2 successively. For simplicity, we assume that both the scaled interaction times are τ ,
thai is, τ1 = K1t1 = τ2 = K2t2 = τ.The interactions change the state as following way
|β〉2 |β〉1 |α〉a →
1 + |β|2
|0〉2 |0〉1 |α〉a + β (|1〉2 |0〉1 + |0〉2 |1〉1)
∣αe−iτ
+ β2 |1〉2 |1〉1
∣αe−i2τ
where the subscripts 1 and 2 denote modes b1 and b2, respectively. We note that the internal product
of coherent states satisfies [4]
αe−inτ |αe−i(n+1)τ
= e−4|α|
2 sin2(τ/2) ≈ e−|α|
2τ2 , (6)
in which we have taken into account the fact that in practice τ is small [3] and therefore sin (τ/2) ≈
τ/2. However, if mode a is bright enough so that |α|2 τ2 ≫ 1, then the two coherent states will
be approximately orthogonal. This condition can be easily satisfied in experiments and in following
discussions we assume that it is satisfied. In this case, a homodyne detection can distinguish different
coherent states [5]. Therefore, when we find that mode a is in the coherent state
∣αe−iτ
, then beam
b1 and beam b2 will be projected into the entangled state
(|1〉2 |0〉1 + |0〉2 |1〉1) , (7)
and the probability for getting this entangled state is 2 |β|2 /
1 + |β|2
.This state is one of Bell
states [1] and a special case of the NOON states [6].
Now let us consider the general situation in which beam b1 and beam b2 are normal coherent states
[4]. In this situation,
|β〉 = exp
|n〉 . (8)
The cross-Kerr interactions transform the state as follows
|β〉2 |β〉1 |α〉a = e
−|β|2
βm+n√
|m〉2 |n〉1 |α〉a
→ e−|β|
βm+n√
|m〉2 |n〉1
αe−i(m+n)τ
. (9)
If the homedyne detection finds mode a in the state
∣αe−i(m+n)τ
∣αe−ikτ
(k = m+n = 1, 2, ...),
then mode b1 and mode b2 will be collapse into the entangled state
n! (k − n)!
|k − n〉2 |n〉1 (k = 1, 2, ...). (10)
Since in this state the sum of photon numbers of the two modes is equal to k, we name this state as
the 2-mode k-photon entangled state. The probability for getting this state is exp(−2 |β|2)2
|β|2k .
The entanglement property of the states expressed by Eq.(10) can be proved by using following
entanglement criteria [7]
b+1 b2
> 〈Nb1Nb2〉 , (11)
where Nb1(Nb2), b1(b2) and b
2 ) are the photon-number operator, the photon annihilation operator
and the photon creation operator of mode b1(b2), respectively. For the states of equation (10), we
can find
b+1 b2
k2 , and 〈Nb1Nb2〉 = 14k (k − 1) . Therefore the entanglement condition (11) is
satisfied, and the states (10) are indeed entangled states. For k = 1, equation (10) reduces to equation
(7), and some other examples of the 2-mode k-photon entangled states are listed below.
(|2〉2 |0〉1 + |0〉2 |2〉1) +
2 |1〉2 |1〉1
(k = 2) (12)
(|3〉2 |0〉1 + |0〉2 |3〉1) +
3 (|2〉2 |1〉1 + |1〉2 |2〉1)
(k = 3) (13)
Equations (12) and (13) are new kinds of entangled states. Equation (12) can be understood as a
superposition of a NOON state (|2〉2 |0〉1 + |0〉2 |2〉1) and a product state |1〉2 |1〉1 ,while equation (13)
can be understood as a superposition of a NOON state (|3〉2 |0〉1 + |0〉2 |3〉1) and a NOON − like
state (|2〉2 |1〉1 + |1〉2 |2〉1). We also note that in the superposition (13) the probability of getting the
state (|2〉2 |1〉1 + |1〉2 |2〉1) is larger than that of getting the state (|3〉2 |0〉1 + |0〉2 |3〉1) .That is, the
photons trend to distribute between the two modes symmetrically. The properties and applications
of these new kinds of entangled states will be studied in the future.
4 Entanglement among three modes
We can extend the scheme above to generate the entanglement among three modes. For this purpose
we modify the scheme from Figure 1 to Figure 2, in which BS1 has the reflection/transmission = 1/2
and BS2 has the reflection/transmission = 1/1, so that the three beams b1, b2 and b3 have the same
strength, and we assume all of them are in the coherent state |β〉 . We let mode a, in a coherent state
|α〉, interacts with modes b1, b2 and b3 successively. And for simplicity, we assume that all of the
scaled interaction times are equal, thai is, τ1 = τ2 = τ3 = τ.
For the situation in which |β〉 is weak and can be expressed as in equation (4), the interactions
transform the states in the following way
|β〉3 |β〉2 |β〉1 |α〉a →
1 + |β|2
{|0〉3 |0〉2 |0〉1 |α〉a
+β (|1〉3 |0〉2 |0〉1 + |0〉3 |1〉2 |0〉1 + |0〉3 |0〉2 |1〉1)
∣αe−iτ
+β2 (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1)
∣αe−i2τ
+β3 |1〉3 |1〉2 |1〉1
∣αe−i3τ
}. (14)
As discussed above, we assume that different coherent states in above equation are approximately
orthogonal, and we can use homodyne detection to distinguish them [5]. If we find that mode a is in
state
∣αe−iτ
then modes b1, b2 and b3 will be projected to the entangled state
(|1〉3 |0〉2 |0〉1 + |0〉3 |1〉2 |0〉1 + |0〉3 |0〉2 |1〉1) , (15)
and the probability for obtaining this state is 3 |β|2 /
1 + |β|2
. On the other hand, If we find that
mode a is in state
∣αe−i2τ
then modes b1, b2 and b3 will be projected to the entangled state
(|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1) , (16)
and the probability for getting this state is 3 |β|4 /
1 + |β|2
. Equations (15) and (16) can be named
as 1-photon W state [8] and 2-photon W state, respectively.
For the general case in which |β〉 is not very weak we use equation (8). In this case the interactions
transform the states as follows:
|β〉3 |β〉2 |β〉1 |α〉a = e
−3|β|2/2
l,m,n
βl+m+n√
l!m!n!
|l〉3 |m〉2 |n〉1 |α〉a
→ e−3|β|
l,m,n
βl+m+n√
l!m!n!
|l〉3 |m〉2 |n〉1
αe−i(l+m+n)τ
. (17)
If we find that mode a is in the state
∣αe−i(l+m+n)τ
∣αe−ikτ
(k = l+m+n = 1, 2, ...), then
modes b1, b2 and b3 will be projected to the entangled state
(k −m− n)!m!n!
|k −m− n〉3 |m〉2 |n〉1 (k = 1, 2, ...). (18)
We name this state as the 3-mode k-photon entangled state. The probability for getting this state
is exp(−3 |β|2)3
|β|2k .The entanglement property of the states of Eq.(18) can be proved by using
following entanglement criteria [7]
b+1 b2
> 〈Nb1Nb2〉 and
b+2 b3
> 〈Nb2Nb3〉 . (19)
For the states (18), we can find
b+1 b2
b+2 b3
k2, and 〈Nb1Nb2〉 = 〈Nb2Nb3〉 = 19k (k − 1) .Therefore
the entanglement condition (19) is satisfied, and the states (18) are indeed entangled states of three
modes. For k = 1, equation (18) reduces to equation (15), and some other examples of the 3-mode
k-photon entangled state are as follows:
{(|2〉3 |0〉2 |0〉1 + |0〉3 |2〉2 |0〉1 + |0〉3 |0〉2 |2〉1)
2 (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1)} (k = 2) , (20)
{(|3〉3 |0〉2 |0〉1 + |0〉3 |3〉2 |0〉1 + |0〉3 |0〉2 |3〉1)
3 (|2〉3 |1〉2 |0〉1 + |2〉3 |0〉2 |1〉1 + |1〉3 |2〉2 |0〉1 + |1〉3 |0〉2 |2〉1 + |0〉3 |2〉2 |1〉1 + |0〉3 |1〉2 |2〉1)
6 |1〉3 |1〉2 |1〉1} (k = 3) . (21)
Equation (20) is a superposition of two 2-photon W states. While equation (21) is a superposition
of a 3-photon W state (the first line), a product state (the third line), and a state (the second line)
which can be expressed as
|1〉j |0〉k + |0〉j |1〉k
+ |1〉i
|2〉j |0〉k + |0〉j |2〉k
+ |0〉i
|2〉j |1〉k + |1〉j |2〉k
, (22)
where the subscripts i = 1,or 2, or 3,and j,k are the other two, respectively. We also note that in
the superposition (20) the probability of getting the state (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1)
is larger than that of getting the state (|2〉3 |0〉2 |0〉1 + |0〉3 |2〉2 |0〉1 + |0〉3 |0〉2 |2〉1). This shows again
that the photons trend to distribute among different modes symmetrically.
5 Summary
In summary, we have proposed a scheme for generating entangled states of light fields. This scheme
has following advantages: First, the scheme only involves the cross-Kerr nonlinear interaction between
coherent light-beams, followed by a homodyne detection. It is not necessary that the cross-Kerr
nonlinearity is very large, as long as the coherent light is bright enough. Therefore, this scheme is
within the reach of current technology. Second, in addition to the Bell states between two modes and
the W states among three modes, plentiful new kinds of entangled states can be generated with this
scheme. We also found that in the generated entangled states, the photons have a trend to distribute
among different modes symmetrically. Finally, we would like to point out that the scheme can be
extend to generate the entangled states among more than three modes.
Acknowledgement This work was supported by the National Natural Science Foundation of
China under grant nos 60578055 and 10404007.
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Figure captions
Figure 1. Scheme for generating entanglement between two modes. KM:cross-Kerr medium; BS:
beam splitter; M: mirror, HD: homodyne detection.
Figure 2. Scheme for generating entanglement among three modes. KM:cross-Kerr medium; BS:
beam splitter; M: mirror, HD: homodyne detection.
	Introduction
	Cross-Kerr nonlinear interaction
	Entanglement between two modes
	Entanglement among three modes
	Summary
