Mean spin direction and spin squeezing in superpositions of spin coherent states

DOI: 10.2478/s11534-007-0029-2Research article

CEJP 5(3) 2007 367–376

Mean spin direction and spin squeezing insuperpositions of spin coherent states

Dong Yan1,3, Xiaoguang Wang2, Lijun Song1,4∗, Zhanguo Zong1

1 Institute of Applied Physics,Changchun University,Changchun 130022, P.R. China2 Zhejiang Institute of Modern Physics,Department of Physics, Zhejiang University,HangZhou 310027, P.R. China3 School of Science,Lanzhou University of Technology,Lanzhou 730050, P. R. China4 School of Science,Changchun University of Science and Technology,Changchun 130022, P.R. China

Received 15 January 2007; accepted 8 May 2007

Abstract: We consider the mean spin direction (MSD) of superpositions of two spin coherentstates (SCS) | ± μ〉, and superpositions of |μ〉 and |μ∗〉 with a relative phase. We find that theazimuthal angle exhibits a π transition for both states when we vary the relative phase. The spinsqueezing of the states, and the bosonic counterpart of the mean spin direction are also discussed.c© Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.

Keywords: spin squeezing, mean spin directionPACS (2006): 03.65.Ud, 42.50.Dv

Some investigations have partially focused on squeezing in spin systems [1–17] in re-

cent years due to its practical applications e.g., in the fields of high-precision atomic

clocks [4] and interferometers [3]. Also, it was found that the spin squeezing is closely

related to quantum entanglement [17–22], which plays an important role in quantum infor-

mation and quantum computation. For an ensemble of two-level system, spin squeezing

implies entanglement. At the same time, it was shown that a single spin-like system

with S greater than 1/2 can also manifest entanglement [19–21], moreover, the single-

∗ E-mail:[email protected]

368 D. Yan / Central European Journal of Physics 5(3) 2007 367–376

spin squeezing is exactly the same as the spin entanglement, while spin-coherent states

are always unentangled [22]. Because quantum entanglement is still puzzling, especially

many-body entanglement, one can use spin squeezing to study entanglement.

The concept of the squeezing in spin systems was clarified by Kitagawa and Ueda in

1993 [3] who also described the basic notion of spin squeezed states, which associated both

quantum correlations and uncertainty relationships. By now, there are several definitions

to measure spin squeezing [3, 4, 18]. Here, we use the definition given by Wineland et

al. [4]

ξ2 =N (ΔSn⊥)2

|〈S ◦ n〉|2 , (1)

where the subscript n⊥ refers to an axis perpendicular to the mean spin direction

n = 〈S〉/√〈S〉 ◦ 〈S〉, where the minimal value of the variance (ΔS)2 is obtained. The

integer N = 2j, Sn⊥ = S ◦ n⊥, and j is the spin number. The inequality ξ2 < 1 indicates

that the system is spin squeezed.

In this paper, we study the MSD and spin squeezing in a general superpositions of

two spin coherent states (SCSs) [24], the study of the MSD is important at least for

the following reasons. First, we regard the spin as squeezed only if the variance of one

spin component normal to the mean spin vector is smaller than the standard quantum

limit of S/2 [3], so we must confirm the MSD firstly in the studies of the spin squeezing.

Secondly, the MSD has a direct measurable classical counterpart and it is important from

the experimental point of view. Thirdly, the MSD can reveal the underlying chaos in the

quantum kicked top model. Finally, the concurrence (measure of entanglement) can be

directly expressed in terms of the length of the mean spin vector [22].

We work in the (2j + 1)-dimensional angular momentum Hilbert space {|j, m〉; m =

−j, ..., +j} and for convenience define the ‘number’ operator N = Sz + j and the number

state |n〉 ≡ |j,−j + n〉. They satisfy

N|n〉 = n|n〉. (2)

The SCS is defined in this Hilbert space and is given by [24],

|μ〉 = (1 + |μ|2)−j

2j∑

n=0

(2j

n

)1/2

μn|n〉. (3)

It has been already normalized and the parameter μ is complex. The SCS satisfies the

equation [16]

S+|μ〉 = μ−1N|μ〉, (4)

where S± = Sx±iSy. This equation will be useful for the discussions below.

Squeezing can be obtained by superpositions of two coherent states. These superpo-

sition states also show other non-classical properties including anti-bunching effects and

sub-Poissionian distribution. There are many studies of Schrodinger states such as super-

positions of bosonic states |α〉 and |α∗〉 [25], superpositions of states |α〉 and | − α〉 [26],

or superpositions of arbitrary bosonic states |α〉 and |β〉.

D. Yan / Central European Journal of Physics 5(3) 2007 367–376 369

In this study, we consider the following superpositions of two SCSs |μ〉 and |ν〉 with

relative phase γ

|μ, ν, γ〉 =A(|μ〉 + eiγ |ν〉), (5)

where A is a normalization factor which is given by

A = [2 + eiγ〈μ|ν〉 + e−iγ〈ν|μ〉]− 12 (6)

with the overlap being

〈μ|ν〉 =(1 + μ∗ν)2j

[(1 + |μ|2)(1 + |ν|2)]j . (7)

Two special cases of the above superposition state are given by

|μ, γ〉1 = A1(|μ〉 + eiγ | − μ〉), (8)

|μ, γ〉2 = A2(|μ〉 + eiγ |μ∗〉), (9)

The former state with γ = 0 (π) is just the even (odd) coherent state and the latter with

with γ = 0 (π) is the real (imaginary) coherent state. Even and odd SCSs [17], and real

and imaginary SCSs [16] have been investigated.

In the following, we denote the MSD by n1 and the other two directions perpendic-

ular to it are denoted by n2 and n3, respectively. Without loss of generality, spherical

coordinates are adopted, and the directions can be expressed as⎛

⎜⎜⎜⎜⎝

n1

n2

n3

⎞

⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎝

sin θ cos φ sin θ sin φ cos θ

− sin φ cos φ 0

− cos θ cos φ − cos θ sin φ sin θ

⎞

⎟⎟⎟⎟⎠

, (10)

where θ and φ are polar angle and azimuth angle, respectively. They are determined by

θ = arccos(〈Sz〉/√〈Sx〉2 + 〈Sy〉2 + 〈Sz〉2), (11)

φ = arctan(〈Sy〉/〈Sx〉) = arctan(�〈S+〉/〈S+〉). (12)

Once we know the expectation values 〈Sα〉 (α ∈ {x, y, z}) or 〈S+〉 and 〈Sz〉, the MSD are

completely determined.

For the superposition state (5), by using Eq. (4), the expectation of S+ is obtained as

〈μ, ν, γ|S+|μ, ν, γ〉= A2(μ−1〈μ|N |μ〉+ ν−1〈ν|N |ν〉

+μ−1e−iγ〈ν|N |μ〉+ ν−1eiγ〈μ|N |ν〉). (13)

Hence, in order to determine the expectation value, it is sufficient to know the the quantity

〈μ|N |ν〉,which can be computed by using the generating function method. From Eq. (3),

the generating function F (λ) = 〈μ|λN |ν〉 is given by

F (λ) =(1 + λμ∗ν)2j

[(1 + |μ|2)(1 + |ν|2)]j . (14)


0 1 2 3 4 5 60.5

1

1.5

2

2.5

3

3.5

4

γ

φ

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

γ

θ

N=4

N=5

N=4

N=5

Fig. 1 Azimuthal angle and the polar angle of the mean spin direction versus γ for state

1. Parameter μ = 3 + 4i.

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

γ

ξ2 , L

N=4

N=8

N=4

N=8

Fig. 2 Length of mean spin and the squeezing parameter versus γ for state 1. Parameter

μ = 0.2 + 0.3i.

Thus, the quantity 〈μ|N |ν〉 is obtained as

〈μ|N |ν〉 =∂F (λ)

∂λ

∣∣∣λ=1

=2j(1 + μ∗ν)2j−1μ∗ν

[(1 + |μ|2)(1 + |ν|2)]j . (15)

Let us first consider the state |μ, γ〉1(8), a superposition state of two SCSs | ± μ〉.


Immediately, from Eq. (15), we find

〈μ|N |μ〉 = 〈−μ|N | − μ〉, 〈μ|N | − μ〉 = 〈−μ|N |μ〉. (16)

By using Eq. (16), the expectation value of S+ (13) reduces to

1〈μ, γ|S+|μ, γ〉1= −2iA2μ−1〈μ|N | − μ〉sin γ. (17)

Then, from Eq. (11), the azimuthal angle φ can be determined by

tan φ =(μ)

�(μ). (18)

Thus, the azimuth angle can take two values, arctan((μ)/�(μ)) and

π + arctan((μ)/�(μ)). Alternatively, the angle φ can be given by

φ =

⎧⎪⎨

⎪⎩

arccos(

〈Sx〉|S| sin(θ)

)if 〈Sy〉 > 0,

2π − arccos(

〈Sx〉|S| sin(θ)

)if 〈Sy〉 ≤ 0.

(19)

The above expression is valid for θ �= 0, π. For θ = 0, π, the mean spin is along the ±z

direction, and one possible choice of φ can be φ = 0, π.

To illustrate the above finding, we plot in Fig. 1, the azimuth angle and the polar

angle versus the relative phase γ for state 1. Dependent on γ, the angle takes only two

values, and they differ by π, implying that the MSD is always in a same plane. The

azimuth angle displays a π transition at γ = 0, π, which corresponds to the even and odd

SCSs. These two states have a fixed parity. At γ = π, for even N, the phase jumps down,

while for odd N, it jumps up.

The polar angle first increases to a maximum, and then decreases to zero when the

azimuth angle varies from 0 to π. It is obvious that the polar angle is symmetric with

respect to the point γ = π. The MSD is always in a same plane, so it behaves as follows

when we vary γ from 0 to 2π. First, the mean spin points to the z direction, and then

moves to one side of z axis with a fixed azumuthal angle, until the polar angle reaches

its maximum. After this, the MSD moves back to z axis. When γ increases from π, the

MSD leaves z axis again with a fixed azumuthal angle which differs from the former one

by π, it moves to another side of z axis. After the polar angle reaches the same maximum

value as the former, it returns back to z axis, and a cycle is completed.

Having studied the MSD, we now consider the length of mean spin and compare it

with spin squeezing parameter. The length of mean spin is just

L =√〈Sx〉2 + 〈Sy〉2 + 〈Sz〉2.

For determining the parameter ξ2, the following variance must be calculated [16]

(ΔSn⊥)2 = 〈S2n2

+ S2n3〉 −

√(〈S2

n2− S2

n3〉)2 + 〈[Sn2 , Sn3]+〉2 (20)


with

Sn2 = − Sx sin φ + Sy cos φ

Sn3 = − Sx cos θ cos φ − Sy cos θ sin φ + Sz sin θ. (21)

From the variance and the length of the mean spin, the squeezing parameter can be

readily obtained.

In Fig. 2, we plot the length of mean spin and the squeezing parameter versus γ. We

observe that the maximum spin length as well as strongest squeezing occurs at γ = 0.

The minimum length occurs at γ = π, and here, there is no squeezing. The larger spin

length and the smaller variance are of course good for spin squeezing.

In a similar manner, we study the state |μ, γ〉2 [Eq. (9)], a superposition of two SCSs

|μ〉 and |μ∗〉. From Eq. (15), one finds

〈μ|N |μ〉 = 〈μ∗|N |μ∗〉,〈μ|N |μ∗〉 = (〈μ∗|N |μ〉)∗.

By using the above equation, Eq. (13) reduces to

2〈μ, γ|S+|μ, γ〉2 = 2A22[(μ−1)〈μ|N |μ〉

+(μ∗−1eiγ〈μ|N |μ∗〉)], (22)

which is real. Therefore, from Eq. (11), we have

tan(φ) = 0, (23)

implying that the azimuthal angle can only take two values 0 and π.

In Fig. 3, we give the numerical results of the azimuthal angle φ and the polar angle

of the mean spin direction versus γ for state 2. Similar to the case of state 1, there

are two π transitions of the azimuthal angle, but the transitions do not occur at the

points γ = 0, π. At the transition points, the polar angle becomes zero. Comparing with

Fig. 1, there is no well symmetry, and the behavior of the polar angle is more irregular.

We also plot the length of mean spin and the squeezing parameter versus γ in Fig. 4.

The behavior is similar to Fig. 2 except that the minima of the length (maximum of the

squeezing parameter) occurs at some point of γ �= π.

Finally, we consider the bosonic counterpart of the MSD, and explore if its behavior

is similar to the MSD. For the bosonic system, in two-dimensional phase space, the polar

angle is defined as

ϕ = arctan(〈p〉/〈x〉) = arctan(�〈a〉/〈a〉), (24)

where x, p, and a are the position, momentum, and annhilation operators, respectively.

We have used the fact that a = 1√2(x + ip) to get the above equation. We may write the

bosonic counterpart of |μ, γ〉1 and |μ, γ〉2 as

|α, γ〉1 = A1(|α〉 + eiγ | − α〉), (25)

|α, γ〉2 = A2(|α〉 + eiγ |α∗〉), (26)


0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

γ

φ

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

γ

θ

N=4 N=5

N=4

N=5

Fig. 3 Azimuthal angle and the polar angle of the mean spin direction versus γ for state

2. Parameter μ = 3 + 4i.

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

γ

ξ2 ,L

N=4

N=8

N=8

N=4

Fig. 4 Length of mean spin and the squeezing parameter versus γ for state 2. Parameter

μ = 0.2 + 0.3i.

where |α〉 is the bosonic coherent state satisfying

a|α〉 = α|α〉. (27)

By using the above equation, one finds


1〈α, γ|a|α, γ〉1 = −2iα〈α| − α〉A21 sin γ,

2〈α, γ|a|α, γ〉2 = A22(α + α∗ + eiγα∗〈α|α∗〉

+ e−iγα〈α∗|α〉) (28)

For the coherent state, relevant overlaps are given by

〈α| − α〉 = e−2|α|2, 〈α|α∗〉 = e−|α|2+α∗2. (29)

and hence from Eqs. (24), (28), and (29), we obtain tan ϕ = −α1/α2, for state |α, γ〉1,and tan ϕ = 0 for state |α, γ〉1.These results exhibit a well similarity with the azimuthal

angle given by Eqs. (24), (28) for spin states, namely, the polar angle can only take two

possible values when we vary the relative phase.

In conclusion, we have studied the MSD of the superpositions of two coherent states

| ± μ〉, and the superpositions of coherent states |μ〉 and |μ∗〉. Interestingly, we find that

the azimuthal angle displays a π transition for both states when we vary the relative

phase γ from 0 to 2π. At the transition points, the polar angle becomes zero, and the

MSD is in the z direction. Because an spin S is squeezed only if one of the components

normal to the mean spin vector has a variance smaller than S/2 [3],the present work in

the MSD is best suited for detecting and studying spin squeezing.

Acknowledgments

This work is supported by NSFC with grant Nos. 10405019 and 90503003; NFRPC

with grant No. 2006CB921206; Program for new century excellent talents in univer-

sity (NCET). Specialized Research Fund for the Doctoral Program of Higher Education

(SRFDP) with grant No. 20050335087.

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Mean spin direction and spin squeezing in superpositions of spin coherent states

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