Download Mean spin direction and spin squeezing in superpositions of spin

DOI: 10.2478/s11534-007-0029-2
Research article
CEJP 5(3) 2007 367–376
Mean spin direction and spin squeezing in
superpositions 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. China
2
Zhejiang Institute of Modern Physics,
Department of Physics, Zhejiang University,
HangZhou 310027, P.R. China
3
School of Science,
Lanzhou University of Technology,
Lanzhou 730050, P. R. China
4
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 coherent
states (SCS) | ± μ, and superpositions of |μ and |μ∗ with a relative phase. We find that the
azimuthal angle exhibits a π transition for both states when we vary the relative phase. The spin
squeezing 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 direction
PACS (2006): 03.65.Ud, 42.50.Dv
Some investigations have partially focused on squeezing in spin systems [1–17] in recent 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 information 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]
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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]
N (ΔSn⊥ )2
ξ =
,
|S ◦ n|2
2
(1)
where thesubscript 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 ξ 2 < 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],
2 −j
|μ = (1 + |μ| )
2j 1/2
2j
n=0
n
μn |n.
(3)
It has been already normalized and the parameter μ is complex. The SCS satisfies the
equation [16]
S+ |μ = μ−1 N |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 superposition states also show other non-classical properties including anti-bunching effects and
sub-Poissionian distribution. There are many studies of Schrödinger states such as superpositions of bosonic states |α and |α∗ [25], superpositions of states |α and | − α [26],
or superpositions of arbitrary bosonic states |α and |β.
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In this study, we consider the following superpositions of two SCSs |μ and |ν with
relative phase γ
(5)
|μ, ν, γ =A(|μ + eiγ |ν),
where A is a normalization factor which is given by
1
A = [2 + eiγ μ|ν + e−iγ ν|μ]− 2
(6)
with the overlap being
(1 + μ∗ ν)2j
.
[(1 + |μ|2)(1 + |ν|2 )]j
Two special cases of the above superposition state are given by
μ|ν =
(7)
|μ, γ1 = A1 (|μ + eiγ | − μ),
(8)
|μ, γ2 = A2 (|μ + e |μ ),
(9)
iγ
∗
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 perpendicular 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 ⎟ ⎜ sin θ cos φ sin θ sin φ cos θ ⎟
⎟
⎜ ⎟ ⎜
⎟,
⎜ n ⎟ = ⎜ − sin φ
(10)
cos
φ
0
⎟
⎜ 2⎟ ⎜
⎠
⎝ ⎠ ⎝
− cos θ cos φ − cos θ sin φ sin θ
n3
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−1 e−iγ ν|N |μ + ν −1 eiγ 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)
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4
N=4
3.5
3
φ
2.5
2
1.5
1
0.5
N=5
0
1
2
3
γ
4
5
6
4
5
6
0.7
0.6
N=4
0.5
N=5
θ
0.4
0.3
0.2
0.1
0
0
1
2
3
γ
Fig. 1 Azimuthal angle and the polar angle of the mean spin direction versus γ for state
1. Parameter μ = 3 + 4i.
9
8
N=4
7
5
2
ξ ,L
6
N=8
4
N=8
3
2
N=4
1
0
0
1
2
3
γ
4
5
6
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
2j(1 + μ∗ ν)2j−1 μ∗ ν
∂F (λ) .
μ|N |ν =
=
∂λ λ=1 [(1 + |μ|2)(1 + |ν|2 )]j
(15)
Let us first consider the state |μ, γ1 (8), a superposition state of two SCSs | ± μ.
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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, γ1 =
−2iA2 μ−1 μ|N | − μsin 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
⎧
⎪
Sx ⎨ arccos
if Sy > 0,
|S| sin(θ)
φ=
(19)
⎪
Sx ⎩ 2π − arccos
if Sy ≤ 0.
|S| sin(θ)
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 ξ 2 , the following variance must be calculated [16]
(ΔSn⊥ )2 = Sn2 2 + Sn2 3 − (Sn2 2 − Sn2 3 )2 + [Sn2 , Sn3 ]+ 2
(20)
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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, γ2 [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, γ2
= 2A22 [(μ−1 )μ|N |μ
+(μ∗−1 eiγ 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(p̂/x̂) = arctan(a/a),
(24)
where x̂, p̂, and a are the position, momentum, and annhilation operators, respectively.
We have used the fact that a = √12 (x̂ + ip̂) to get the above equation. We may write the
bosonic counterpart of |μ, γ1 and |μ, γ2 as
|α, γ1 = A1 (|α + eiγ | − α),
|α, γ2 = A2 (|α + eiγ |α∗ ),
(25)
(26)
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3.5
3
N=4
N=5
2.5
φ
2
1.5
1
0.5
0
0
1
2
3
γ
4
5
6
1
N=4
0.8
θ
0.6
0.4
N=5
0.2
0
0
1
2
3
γ
4
5
6
Fig. 3 Azimuthal angle and the polar angle of the mean spin direction versus γ for state
2. Parameter μ = 3 + 4i.
9
N=4
8
ξ2,L
7
6
5
4
N=8
N=8
3
2
N=4
1
0
0
1
2
3
γ
4
5
6
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
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1 α, γ|a|α, γ1
2 α, γ|a|α, γ2
= −2iαα| − αA21 sin γ,
= A22 (α + α∗ + eiγ α∗ α|α∗
+ e−iγ αα∗ |α)
(28)
For the coherent state, relevant overlaps are given by
2
α| − α = e−2|α| , 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, γ1 ,
and tan ϕ = 0 for state |α, γ1 .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 university (NCET). Specialized Research Fund for the Doctoral Program of Higher Education
(SRFDP) with grant No. 20050335087.
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