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Uncertainty relations for a q-deformed coherent spin state
Physics Letters A 376 (2011) 14–18
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Uncertainty relations for a q-deformed coherent spin state M. Reboiro a,b , O. Civitarese a,∗ a b
Department of Physics, University of La Plata, c.c. 67 1900, La Plata, Argentina Faculty of Engineering, University of La Plata, La Plata, Argentina
a r t i c l e
i n f o
Article history: Received 7 July 2011 Received in revised form 5 October 2011 Accepted 20 October 2011 Available online 26 October 2011 Communicated by P.R. Holland Keywords: Squeezing Intelligent spin states Coherent states Quantum algebras
a b s t r a c t A Coherent Spin State (CSS) is defined as an eigenstate of the spin component in the direction specified by angles (θ0 , φ0 ). This state satisfies minimum uncertainty relation, with uncertainties equally distributed on any two orthogonal components normal to the direction of the total spin vector S. Starting from this concept, we apply the notion of CSS to quantum groups and discuss the properties of q-deformed CSS and the associated uncertainty relations. We show that these states behave as Intelligent Spin States (ISS) on two orthogonal components normal to the direction of the mean value of the spin operator. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The concept of entanglement in quantum many body systems have received significant attention in different contexts [1, 2]. Among them, and because of the experimental consequences of it, the study of spin squeezed atoms and ions could play a major role in the improvement of high precision measurements, like those required in the implementation of quantum computing devices. In recent experiments, spin squeezed states have been detected [3–7]. The authors of [4] have reported a direct experimental demonstration of interferometric phase precision beyond the standard quantum limit in a nonlinear Ramsey interferometer. They have reported the observation of coherent squeezed states in a Bose–Einstein condensate of rubidium atoms (Ru37 ). In their experiment, the nonlinear atomic beam splitter follows the “one-axis twisting” scheme proposed by M. Kitagawa and M. Ueda [8]. From the theoretical point of view, the concept of squeezing is closely related to the analysis of Heisenberg Uncertainty Relations. Given a physical system, one may be interested in the minimization of the fluctuation of an observable at the expense of the increment of the fluctuation of the conjugate variable. The definition of a parameter associated to the concept of squeezing offers several possibilities, as discussed in Refs. [8–10]. Some innovative views in the field have been advanced in [11–14]. Particularly, the authors of [11] have introduced a definition of the spin-squeezing parameter which obeys the additional condition of su(2) invariance.
*
Corresponding author. E-mail addresses: reboiro@fisica.unlp.edu.ar (M. Reboiro), osvaldo.civitarese@fisica.unlp.edu.ar (O. Civitarese). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.10.028
In [14], it has been analyzed an angular performance measurement for which the angle fluctuation can be properly quantified and which provides simple bounds for the conjugate variable. The authors of [14] have searched for intelligent states by minimizing uncertainty relations. They referred to these states as circular squeezed states. Among the first references in the literature to intelligent states there is the paper of C. Argone and co-workers [15]. Following [15], a considerable amount of work was devoted to the study of both the properties of intelligent spin states [16–18] as well as to the construction of such states [19–23]. As an example, the authors of [19] have considered the steady states generated by a collection of two-level atoms interacting with a broadband squeezed radiation field. They have shown that these states behave as Intelligent Spin States (ISS), that is: states which minimize Heisenberg Uncertainty Relations for the SU (2) spin operators. In [20] polynomial intelligent states have been constructed. The construction of intelligent states as coupled su(2) coherent states is addressed in [21]. Recently, the theory of intelligent states for the su(1, 1) algebra have been developed in [22], and the relation between intelligent su(2) states and squeezing have been discussed in [23]. The search for minimum uncertainty relations and intelligent states is a matter of interest also for quantum groups and Hopf algebras [24]. The construction of minimum angular uncertainty states for quantum group SUq (2) have been discussed in [25]. The authors of [25] have constructed squeezed angular momentum states for the SUq (2) in terms of Wigner C-functions. In this Letter, we shall follow the work of [8] and study angular momentum suq (2) uncertainty relations, for a q-deformed coherent spin state [26]. Deformed coherent states have been constructed as a natural extension of the notion of coherent states
M. Reboiro, O. Civitarese / Physics Letters A 376 (2011) 14–18
associated to Lie groups. These states have attracted attention in different branches of physics. As an example, they have been used to describe a large class of quantum systems characterized by different potentials [27]. In the spirit of [28] we may think of the q-deformed scheme as a trade-off mechanism which allow us to absorb, algebraically, model degrees of freedom in favor of effective ones. In [29] it has been shown that sub-Poissonian and superPoissonian photon number statistics of real lasers can be modelled by introducing q-coherent deformed states, by a proper choice of the deformation parameter. Recently, in [30], it was analyzed the generation of entanglement using a coherent suq (2) spin state transmitted through a beam splitter. From a mathematical point of view, properties of coherent states on SUq (2) homogeneous space have been studied in [31]. In this work, we shall look at the fluctuations of the angular momentum in the direction perpendicular to the mean value of the total spin, and show that these states behave as intelligent spin states. We have followed arguments, concerning the analysis of spin fluctuations, which are similar to the ones presented in [8]. The Letter is organized as follows. The formalism is presented, and applied to the analytically solvable case with total spin S = 1, in Section 2. In Section 3 we show the results of the calculations for a general case (S = 10). The conclusions are drawn in Section 4. 2. Formalism In this section we shall introduce the essentials of the formalism. Section 2.1 is devoted to present a brief review of the concept of intelligent spin state. In Section 2.2 we study the behavior of the uncertainty relations for a q-coherent suq (2) state, and we present some analytical results for the case of spin S = 1.
where ζ12 and ζ22 relate the deviations of the 1- and 2-component of the spin with the expectation value of the 3-component. From Eq. (5), they are written
ζ12 = |α |, ζ22 =
1
|α |
(7)
,
and obviously they fulfill the equality ζ12 ζ22 = 1. In fact, the parameters ζ12 and ζ22 may be taken as analogous to the spin-squeezing ones of [8]. Notice that in Ref. [8], the squeezing parameters are given as ratios of the standard deviations of the normal spin components respect to the mean value of the total spin, and that they are su(2) invariant [11,23]. This requirement is not generally fulfilled by quantities like ζ12 and ζ22 of Eq. (7). Therefore, we shall take them as representatives of relative fluctuations of the 1- and 2-components of the spin, respect to the 3-component. For a more general discussion of spin squeezing the reader is kindly referred to [11,23]. We shall return to this point later on (see Section 2.2). The value S 3 can be computed easily and it reads
1
S 3 = N 2S M α c (θ)2M 2 ( S − M )! s(θ)2k+1 ( S + M + k + 1)! × c (θ) ( S + M )! k!(k + 1)!( S − M − k − 1)! k
+ c (θ)−1
An intelligent spin state is a state which satisfies the limiting value of the Heisenberg Uncertainty Relations, that is
2 2M N− S M α = c (θ)
(1)
(S i = 1, 2, 3 are the components of the spin operator). The condition (1) is achieved if and only if
(i) (ii)
S 1 − S 1 , S 2 − S 2
= 0,
S 1 − S 1 |Ψ = iα S 2 − S 2 |Ψ ,
(2)
with α ∈ R. This is equivalent to find the eigenvectors of the nonHermitian operator S α = S 1 − iα S 2 , namely
S α | w = w | w .
(3)
It is well known [15] that intelligent spin states can be constructed as π
|Ψ = | S M α = N S M α eθ S 3 e−i 2 S 2 | S M ,
(8) with
4
( S − M + 1)! s(θ)2k−1 ( S + M + k − 1)! ( S + M − 1)! k!(k − 1)!( S − M − k + 1)! k
2.1. Intelligent spin states
2 1 2 S 1 2 S 2 = S 3 ,
15
( S − M )! s(θ)2k ( S + M + k)! , ( S + M )! k!2 ( S − M − k)!
(9)
k
and
c (θ) = cosh(θ), s(θ) = sinh(θ).
(10)
2.2. Coherent q-deformed states In this subsection we study the uncertainty relations of a suq (2) coherent state. The generators of the suq (2) algebra, S + , S − , S z , obey the relations
[S z , S±] = ±S±,
[ S + , S − ] = [2S z ]q ,
(11)
with
(4)
sin( zx)
α with θ = 12 ln( 11− +α ) (−1 < α < 1). Intelligent spin states satisfy
[x]q =
S 1 = α S 3 , 2 1 S3 2 , S2 =
and where q = e iz . As in the case of the su(2) algebra, the irreducible representation of the suq (2) are indexed by S. The orthonormal basis of representation is denoted as | S M , with M = − S , − S + 1, . . . , S. The generators of the suq (2) act on this basis following the rules
1
2
2
α
(5)
thus, one may define the parameters
ζ12
=
(12)
S z | S M = M | S M ,
2 2 S 1
, | S 3 | 2 2 S 2 , ζ22 = | S 3 |
sin( z)
S±|S M = (6)
[ S ∓ M ]q [ S ± M + 1]q | S M ± 1.
(13)
In a given S-representation of the suq (2) algebra, q-deformed CSS states, |η, are defined by [25,26,30]
16
M. Reboiro, O. Civitarese / Physics Letters A 376 (2011) 14–18
η S+
|η = N eq
|S −S
1/2 2S
2S
ηk
=N
k
k =0
| S − S + k,
(14)
q
where eq is the q-exponential function
eqx =
xk , [k]q !
(15)
k
and
x k
= q
[x]q ! , [x − k]q ![k]q !
(16)
is the q-binomial coefficient [24]. The normalization constant N is given by
N=
1
(17)
,
(1 + |η|2 )q2S
1 + |η|2
2S q
=
2S
|η|2k
k =0
2S k
.
(18)
q
Some mathematical aspects related to the q-coherent spin state (14) can be found in [30]. For the components x and y of the spin defined as
Sx = Sy =
S+ + S− 2
2 1 2 S x 2 S y [ S x , S y ] . 4
=
1 + |η|2
2
,
(25)
η = − tan(θ0 /2)e−iφ0 .
(26)
q
The mean value of the spin components can be calculated straightforwardly
S x = −[2]q Re(η)
S z = − (20)
Following the work of M. Kitagawa and M. Ueda [8], we shall look for the mean value of the spin operator, S, for the state of Eq. (14). We shall define a new system of axes, such that the z direction coincides with the direction of the mean value of the total spin S; that is the direction determined by the unitary vector {˘nx , n˘ y , n˘ z }, with
S ˘ = . |S|
n z
1 + |η|2
(1 + |η|2 )q2
2 1 2 S x 2 S y = [ S x , S y ] . 4
|[ S x , S y ]| 2 2 S y
|[ S x , S y ]|
(22)
, .
1 + |η|2
(1 + |η|2 )q2
1 − |η|4
(1 + |η|2 )q2
, (27)
,
S 2x =
[2]q 4
[2]q
S 2y =
4
S 2z =
+ +
| η |2 [2]q − 2 cos(2φ0 ) , 2 2 4 (1 + |η| )q
[2]q
| η |2 [2]q + 2 cos(2φ0 ) , 4 (1 + |η|2 )q2
[2]q
1 + |η|4
(1 + |η|2 )q2
(28)
,
and
In this respect, it can be said that a q-deformed coherent spin state behaves as an intelligent spin state. For z = 0, the usual su(2) algebra is recovered. In this limit, the coherent spin state satisfies the minimum uncertainty relationship, with uncertainties equally distributed over any two orthogonal components normal to the vector defined by S. For z = 0, 2 S x and 2 S y are, in general, not equal and fulfill the uncertainty relation (22), thus the q-coherent spin state resembles the state of a correlated system. In view of the previous results we shall adopt, as indicators of the relative fluctuations, the quantities
22 S x
,
while
(21)
Respect to this new system
ζ y2 =
N
−1
(19)
,
(24)
with
S y = −[2]q Im(η)
the uncertainty relation respect to the state (14) reads
ζx2 =
|η = N |1 − 1 + η [2]q |10 + η2 |11 ,
,
S+ − S− 2i
2.2.1. Some analytical results for a S = 1 q-deformed coherent spin state Let us consider a system with total spin S = 1. The coherent q-deformed state of Eq. (14) reads
and
with the usual q-binomial expansion
These definitions are consistent with the property (22). However, due to the fact that in constructing the new system of orthogonal axes we have not performed an SUq (2)-rotational transformation, in general the commutation relations between the rotated components of the spin do not coincide with the commutation relations valid in the original system where the inequality (20) holds. Next, we shall verified the validity of the previous statements.
(23)
{ S x , S y } = [2]q
1 − |η|2 { S x , S z } = −[2]q |η| cos(φ0 ), (1 + |η|2 )q2
1 − |η|2 { S y , S z } = −[2]q |η| sin(φ0 ). (1 + |η|2 )q2
| η |2 sin(2φ0 ), (1 + |η|2 )q2
(29)
Thus, as an example, for the case with φ0 = π , the mean value of S is given by
S = 2
cos( z) sin(θ0 )˘ı + cos(θ0 )k˘ 2 + (cos( z) − 1) sin(θ0 )2
,
(30)
and we shall take the vector n˘ z in the direction of the mean value of the total spin, that is S. We shall fix x by searching for the minima in the parameter (23). The new set orthogonal system of axes is then defined by the vectors
M. Reboiro, O. Civitarese / Physics Letters A 376 (2011) 14–18
Fig. 1. Parameters, ζ y2 q and ζx2 q of Eq. (23), as a function of the deformation parameter, z. The figure shows the results obtained for a q-coherent spin state. For this calculation we have adopted the values S = 10, θ0 = π /3 and φ0 = 0.
17
Fig. 2. Mean value of the spin operator S z , S z , as a function of the deformation parameter z, for the q-coherent state of Fig. 1.
cos( z) sin(θ0 )˘ı + cos(θ0 )k˘ n˘ z = , cos(θ0 )2 + cos( z)2 sin(θ0 )2
− cos(θ0 )˘ı + cos( z) sin(θ0 )k˘
n˘ x =
cos(θ0 )2 + cos( z)2 sin(θ0 )2
,
n˘ y = j˘.
(31)
For this new set of axes one obtains
1
2 S x =
2
cos(θ0 )2 + cos( z)2 sin(θ0 )2 [ S x , S y ] , 1
2
S y =
2 cos(θ0
)2
+ cos( z)2 sin(θ0 )2
[ S x , S y ] ,
(32)
with
[ S x , S y ] =
2 cos( z) 2 + (cos( z) − 1) sin(θ0 )2 cos(θ0 )2 + cos( z) sin(θ0 )2
×
cos(θ0 )2 + cos( z)2 sin(θ0 )2
.
(33)
Clearly, the components of the spin along this new set of axes verified Eq. (22), as it can be seen from the above results. The parameters (23), in this case, take the values
ζx2 q =
cos(θ0 )2 + cos( z)2 sin(θ0 )2 ,
ζ y2 q =
1 cos(θ0
)2
+ cos( z)2 sin(θ0 )2
.
(34)
In the following section we present some numerical results which illustrate these concepts for a general case. 3. Results and discussion In this section we shall present the results of the calculations which we have performed to test, numerically, the validity of the correspondence described in the previous section (spin S = 1 case). We shall take an arbitrary larger value of the spin, in order to get larger values of the spin components and their fluctuations. Fig. 1 shows the values of the parameters of Eq. (23), as a function of the deformed parameter z. We have chosen a q-deformed
Fig. 3. The behavior of the mean value of the S z component of the spin, S z , as a function of θ0 of the initial q-coherent spin state, is shown in the figure. We have considered a system with total spin S = 10, and an initial state with z = 0.08 and φ0 = 0.
CSS with total spin S = 10. The orientation angles of the state are θ0 = π /3 and φ0 = 0. These values are arbitrary ones, a choice which is not affecting the validity of our conclusions, as we shall see later on. As it can be seen from Fig. 1, as ζ y2 q increases ζx2 q decreases, while the product ζ y2 q ζx2 q is constant and it equals unity. Thus, the q-deformed CSS behaves as an intelligent spin state. Figs. 2 and 3 show the dependence of S z with both the deformation parameter z and the orientation angle θ0 , respectively, for the same total spin (S = 10) and azimuthal angle φ0 = 0. The curves are indeed smooth functions of z and θ0 . In order to show that the features displayed in Fig. 1 remain, regardless the particular choice of the orientation angle θ0 , we have calculated the same parameters (23) for a set of values of θ0 . Indeed, the curves displayed in
18
M. Reboiro, O. Civitarese / Physics Letters A 376 (2011) 14–18
4. Conclusions In this Letter we have reported on the realization of q-deformed coherent spin states as intelligent spin states. We have demonstrated it by performing a calculation of the uncertainty relations, both analytically (for the S = 1 case) and numerically (for the S = 10 case). Work is in progress concerning the application of this type of states to mesoscopic systems [32,33], as for instance molecular spin-clusters which are paradigmatic cases to study the cross-over between quantum and classical behavior. Acknowledgement This work was partially supported by the National Research Council of Argentine (CONICET). References
Fig. 4. ζi2 , as a function of the orientation angle θ0 of the initial q-coherent spinstate. The adopted parameters are those given in the captions of Fig. 3.
Fig. 5. Mean value of the spin operator S 3 , S 3 of Eq. (8), as a function of α . The figure displays the results corresponding to an intelligent spin state with S = 10. The curves from top to bottom correspond to M = 0, M = ±1, . . . , M = ±10, respectively.
Fig. 4 show that the product ζ y2 q ζx2 q equals unity, as it is the case of the results shown in Fig. 1. We can now turn the attention to the same set of parameters, when calculated from Eq. (7), that is for an intelligent spin state. Fig. 5 shows the values of S 3 , for different values of α , for a state with total spin S = 10. The correspondence with the results obtained with a q-deformed CSS can be established by taken equal values of S 3 and S z q . Thus, for instance, if one takes S 3 ≈ S z q ≈ −7, one gets ζ12 = 0.6, corresponding to α = 0.6 (| S = 10 M = −9 α = 0.6), and ζx2 q ≈ 0.6 for z ≈ 0.1035. The present result may indicate that q-deformed coherent spin states can be regarded as intelligent spin states, when the orientation of the components of the spin are properly defined.
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Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Uncertainty relations for a q-deformed coherent spin state M. Reboiro a,b , O. Civitarese a,∗ a b
Department of Physics, University of La Plata, c.c. 67 1900, La Plata, Argentina Faculty of Engineering, University of La Plata, La Plata, Argentina
a r t i c l e
i n f o
Article history: Received 7 July 2011 Received in revised form 5 October 2011 Accepted 20 October 2011 Available online 26 October 2011 Communicated by P.R. Holland Keywords: Squeezing Intelligent spin states Coherent states Quantum algebras
a b s t r a c t A Coherent Spin State (CSS) is defined as an eigenstate of the spin component in the direction specified by angles (θ0 , φ0 ). This state satisfies minimum uncertainty relation, with uncertainties equally distributed on any two orthogonal components normal to the direction of the total spin vector S. Starting from this concept, we apply the notion of CSS to quantum groups and discuss the properties of q-deformed CSS and the associated uncertainty relations. We show that these states behave as Intelligent Spin States (ISS) on two orthogonal components normal to the direction of the mean value of the spin operator. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The concept of entanglement in quantum many body systems have received significant attention in different contexts [1, 2]. Among them, and because of the experimental consequences of it, the study of spin squeezed atoms and ions could play a major role in the improvement of high precision measurements, like those required in the implementation of quantum computing devices. In recent experiments, spin squeezed states have been detected [3–7]. The authors of [4] have reported a direct experimental demonstration of interferometric phase precision beyond the standard quantum limit in a nonlinear Ramsey interferometer. They have reported the observation of coherent squeezed states in a Bose–Einstein condensate of rubidium atoms (Ru37 ). In their experiment, the nonlinear atomic beam splitter follows the “one-axis twisting” scheme proposed by M. Kitagawa and M. Ueda [8]. From the theoretical point of view, the concept of squeezing is closely related to the analysis of Heisenberg Uncertainty Relations. Given a physical system, one may be interested in the minimization of the fluctuation of an observable at the expense of the increment of the fluctuation of the conjugate variable. The definition of a parameter associated to the concept of squeezing offers several possibilities, as discussed in Refs. [8–10]. Some innovative views in the field have been advanced in [11–14]. Particularly, the authors of [11] have introduced a definition of the spin-squeezing parameter which obeys the additional condition of su(2) invariance.
*
Corresponding author. E-mail addresses: reboiro@fisica.unlp.edu.ar (M. Reboiro), osvaldo.civitarese@fisica.unlp.edu.ar (O. Civitarese). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.10.028
In [14], it has been analyzed an angular performance measurement for which the angle fluctuation can be properly quantified and which provides simple bounds for the conjugate variable. The authors of [14] have searched for intelligent states by minimizing uncertainty relations. They referred to these states as circular squeezed states. Among the first references in the literature to intelligent states there is the paper of C. Argone and co-workers [15]. Following [15], a considerable amount of work was devoted to the study of both the properties of intelligent spin states [16–18] as well as to the construction of such states [19–23]. As an example, the authors of [19] have considered the steady states generated by a collection of two-level atoms interacting with a broadband squeezed radiation field. They have shown that these states behave as Intelligent Spin States (ISS), that is: states which minimize Heisenberg Uncertainty Relations for the SU (2) spin operators. In [20] polynomial intelligent states have been constructed. The construction of intelligent states as coupled su(2) coherent states is addressed in [21]. Recently, the theory of intelligent states for the su(1, 1) algebra have been developed in [22], and the relation between intelligent su(2) states and squeezing have been discussed in [23]. The search for minimum uncertainty relations and intelligent states is a matter of interest also for quantum groups and Hopf algebras [24]. The construction of minimum angular uncertainty states for quantum group SUq (2) have been discussed in [25]. The authors of [25] have constructed squeezed angular momentum states for the SUq (2) in terms of Wigner C-functions. In this Letter, we shall follow the work of [8] and study angular momentum suq (2) uncertainty relations, for a q-deformed coherent spin state [26]. Deformed coherent states have been constructed as a natural extension of the notion of coherent states
M. Reboiro, O. Civitarese / Physics Letters A 376 (2011) 14–18
associated to Lie groups. These states have attracted attention in different branches of physics. As an example, they have been used to describe a large class of quantum systems characterized by different potentials [27]. In the spirit of [28] we may think of the q-deformed scheme as a trade-off mechanism which allow us to absorb, algebraically, model degrees of freedom in favor of effective ones. In [29] it has been shown that sub-Poissonian and superPoissonian photon number statistics of real lasers can be modelled by introducing q-coherent deformed states, by a proper choice of the deformation parameter. Recently, in [30], it was analyzed the generation of entanglement using a coherent suq (2) spin state transmitted through a beam splitter. From a mathematical point of view, properties of coherent states on SUq (2) homogeneous space have been studied in [31]. In this work, we shall look at the fluctuations of the angular momentum in the direction perpendicular to the mean value of the total spin, and show that these states behave as intelligent spin states. We have followed arguments, concerning the analysis of spin fluctuations, which are similar to the ones presented in [8]. The Letter is organized as follows. The formalism is presented, and applied to the analytically solvable case with total spin S = 1, in Section 2. In Section 3 we show the results of the calculations for a general case (S = 10). The conclusions are drawn in Section 4. 2. Formalism In this section we shall introduce the essentials of the formalism. Section 2.1 is devoted to present a brief review of the concept of intelligent spin state. In Section 2.2 we study the behavior of the uncertainty relations for a q-coherent suq (2) state, and we present some analytical results for the case of spin S = 1.
where ζ12 and ζ22 relate the deviations of the 1- and 2-component of the spin with the expectation value of the 3-component. From Eq. (5), they are written
ζ12 = |α |, ζ22 =
1
|α |
(7)
,
and obviously they fulfill the equality ζ12 ζ22 = 1. In fact, the parameters ζ12 and ζ22 may be taken as analogous to the spin-squeezing ones of [8]. Notice that in Ref. [8], the squeezing parameters are given as ratios of the standard deviations of the normal spin components respect to the mean value of the total spin, and that they are su(2) invariant [11,23]. This requirement is not generally fulfilled by quantities like ζ12 and ζ22 of Eq. (7). Therefore, we shall take them as representatives of relative fluctuations of the 1- and 2-components of the spin, respect to the 3-component. For a more general discussion of spin squeezing the reader is kindly referred to [11,23]. We shall return to this point later on (see Section 2.2). The value S 3 can be computed easily and it reads
1
S 3 = N 2S M α c (θ)2M 2 ( S − M )! s(θ)2k+1 ( S + M + k + 1)! × c (θ) ( S + M )! k!(k + 1)!( S − M − k − 1)! k
+ c (θ)−1
An intelligent spin state is a state which satisfies the limiting value of the Heisenberg Uncertainty Relations, that is
2 2M N− S M α = c (θ)
(1)
(S i = 1, 2, 3 are the components of the spin operator). The condition (1) is achieved if and only if
(i) (ii)
S 1 − S 1 , S 2 − S 2
= 0,
S 1 − S 1 |Ψ = iα S 2 − S 2 |Ψ ,
(2)
with α ∈ R. This is equivalent to find the eigenvectors of the nonHermitian operator S α = S 1 − iα S 2 , namely
S α | w = w | w .
(3)
It is well known [15] that intelligent spin states can be constructed as π
|Ψ = | S M α = N S M α eθ S 3 e−i 2 S 2 | S M ,
(8) with
4
( S − M + 1)! s(θ)2k−1 ( S + M + k − 1)! ( S + M − 1)! k!(k − 1)!( S − M − k + 1)! k
2.1. Intelligent spin states
2 1 2 S 1 2 S 2 = S 3 ,
15
( S − M )! s(θ)2k ( S + M + k)! , ( S + M )! k!2 ( S − M − k)!
(9)
k
and
c (θ) = cosh(θ), s(θ) = sinh(θ).
(10)
2.2. Coherent q-deformed states In this subsection we study the uncertainty relations of a suq (2) coherent state. The generators of the suq (2) algebra, S + , S − , S z , obey the relations
[S z , S±] = ±S±,
[ S + , S − ] = [2S z ]q ,
(11)
with
(4)
sin( zx)
α with θ = 12 ln( 11− +α ) (−1 < α < 1). Intelligent spin states satisfy
[x]q =
S 1 = α S 3 , 2 1 S3 2 , S2 =
and where q = e iz . As in the case of the su(2) algebra, the irreducible representation of the suq (2) are indexed by S. The orthonormal basis of representation is denoted as | S M , with M = − S , − S + 1, . . . , S. The generators of the suq (2) act on this basis following the rules
1
2
2
α
(5)
thus, one may define the parameters
ζ12
=
(12)
S z | S M = M | S M ,
2 2 S 1
, | S 3 | 2 2 S 2 , ζ22 = | S 3 |
sin( z)
S±|S M = (6)
[ S ∓ M ]q [ S ± M + 1]q | S M ± 1.
(13)
In a given S-representation of the suq (2) algebra, q-deformed CSS states, |η, are defined by [25,26,30]
16
M. Reboiro, O. Civitarese / Physics Letters A 376 (2011) 14–18
η S+
|η = N eq
|S −S
1/2 2S
2S
ηk
=N
k
k =0
| S − S + k,
(14)
q
where eq is the q-exponential function
eqx =
xk , [k]q !
(15)
k
and
x k
= q
[x]q ! , [x − k]q ![k]q !
(16)
is the q-binomial coefficient [24]. The normalization constant N is given by
N=
1
(17)
,
(1 + |η|2 )q2S
1 + |η|2
2S q
=
2S
|η|2k
k =0
2S k
.
(18)
q
Some mathematical aspects related to the q-coherent spin state (14) can be found in [30]. For the components x and y of the spin defined as
Sx = Sy =
S+ + S− 2
2 1 2 S x 2 S y [ S x , S y ] . 4
=
1 + |η|2
2
,
(25)
η = − tan(θ0 /2)e−iφ0 .
(26)
q
The mean value of the spin components can be calculated straightforwardly
S x = −[2]q Re(η)
S z = − (20)
Following the work of M. Kitagawa and M. Ueda [8], we shall look for the mean value of the spin operator, S, for the state of Eq. (14). We shall define a new system of axes, such that the z direction coincides with the direction of the mean value of the total spin S; that is the direction determined by the unitary vector {˘nx , n˘ y , n˘ z }, with
S ˘ = . |S|
n z
1 + |η|2
(1 + |η|2 )q2
2 1 2 S x 2 S y = [ S x , S y ] . 4
|[ S x , S y ]| 2 2 S y
|[ S x , S y ]|
(22)
, .
1 + |η|2
(1 + |η|2 )q2
1 − |η|4
(1 + |η|2 )q2
, (27)
,
S 2x =
[2]q 4
[2]q
S 2y =
4
S 2z =
+ +
| η |2 [2]q − 2 cos(2φ0 ) , 2 2 4 (1 + |η| )q
[2]q
| η |2 [2]q + 2 cos(2φ0 ) , 4 (1 + |η|2 )q2
[2]q
1 + |η|4
(1 + |η|2 )q2
(28)
,
and
In this respect, it can be said that a q-deformed coherent spin state behaves as an intelligent spin state. For z = 0, the usual su(2) algebra is recovered. In this limit, the coherent spin state satisfies the minimum uncertainty relationship, with uncertainties equally distributed over any two orthogonal components normal to the vector defined by S. For z = 0, 2 S x and 2 S y are, in general, not equal and fulfill the uncertainty relation (22), thus the q-coherent spin state resembles the state of a correlated system. In view of the previous results we shall adopt, as indicators of the relative fluctuations, the quantities
22 S x
,
while
(21)
Respect to this new system
ζ y2 =
N
−1
(19)
,
(24)
with
S y = −[2]q Im(η)
the uncertainty relation respect to the state (14) reads
ζx2 =
|η = N |1 − 1 + η [2]q |10 + η2 |11 ,
,
S+ − S− 2i
2.2.1. Some analytical results for a S = 1 q-deformed coherent spin state Let us consider a system with total spin S = 1. The coherent q-deformed state of Eq. (14) reads
and
with the usual q-binomial expansion
These definitions are consistent with the property (22). However, due to the fact that in constructing the new system of orthogonal axes we have not performed an SUq (2)-rotational transformation, in general the commutation relations between the rotated components of the spin do not coincide with the commutation relations valid in the original system where the inequality (20) holds. Next, we shall verified the validity of the previous statements.
(23)
{ S x , S y } = [2]q
1 − |η|2 { S x , S z } = −[2]q |η| cos(φ0 ), (1 + |η|2 )q2
1 − |η|2 { S y , S z } = −[2]q |η| sin(φ0 ). (1 + |η|2 )q2
| η |2 sin(2φ0 ), (1 + |η|2 )q2
(29)
Thus, as an example, for the case with φ0 = π , the mean value of S is given by
S = 2
cos( z) sin(θ0 )˘ı + cos(θ0 )k˘ 2 + (cos( z) − 1) sin(θ0 )2
,
(30)
and we shall take the vector n˘ z in the direction of the mean value of the total spin, that is S. We shall fix x by searching for the minima in the parameter (23). The new set orthogonal system of axes is then defined by the vectors
M. Reboiro, O. Civitarese / Physics Letters A 376 (2011) 14–18
Fig. 1. Parameters, ζ y2 q and ζx2 q of Eq. (23), as a function of the deformation parameter, z. The figure shows the results obtained for a q-coherent spin state. For this calculation we have adopted the values S = 10, θ0 = π /3 and φ0 = 0.
17
Fig. 2. Mean value of the spin operator S z , S z , as a function of the deformation parameter z, for the q-coherent state of Fig. 1.
cos( z) sin(θ0 )˘ı + cos(θ0 )k˘ n˘ z = , cos(θ0 )2 + cos( z)2 sin(θ0 )2
− cos(θ0 )˘ı + cos( z) sin(θ0 )k˘
n˘ x =
cos(θ0 )2 + cos( z)2 sin(θ0 )2
,
n˘ y = j˘.
(31)
For this new set of axes one obtains
1
2 S x =
2
cos(θ0 )2 + cos( z)2 sin(θ0 )2 [ S x , S y ] , 1
2
S y =
2 cos(θ0
)2
+ cos( z)2 sin(θ0 )2
[ S x , S y ] ,
(32)
with
[ S x , S y ] =
2 cos( z) 2 + (cos( z) − 1) sin(θ0 )2 cos(θ0 )2 + cos( z) sin(θ0 )2
×
cos(θ0 )2 + cos( z)2 sin(θ0 )2
.
(33)
Clearly, the components of the spin along this new set of axes verified Eq. (22), as it can be seen from the above results. The parameters (23), in this case, take the values
ζx2 q =
cos(θ0 )2 + cos( z)2 sin(θ0 )2 ,
ζ y2 q =
1 cos(θ0
)2
+ cos( z)2 sin(θ0 )2
.
(34)
In the following section we present some numerical results which illustrate these concepts for a general case. 3. Results and discussion In this section we shall present the results of the calculations which we have performed to test, numerically, the validity of the correspondence described in the previous section (spin S = 1 case). We shall take an arbitrary larger value of the spin, in order to get larger values of the spin components and their fluctuations. Fig. 1 shows the values of the parameters of Eq. (23), as a function of the deformed parameter z. We have chosen a q-deformed
Fig. 3. The behavior of the mean value of the S z component of the spin, S z , as a function of θ0 of the initial q-coherent spin state, is shown in the figure. We have considered a system with total spin S = 10, and an initial state with z = 0.08 and φ0 = 0.
CSS with total spin S = 10. The orientation angles of the state are θ0 = π /3 and φ0 = 0. These values are arbitrary ones, a choice which is not affecting the validity of our conclusions, as we shall see later on. As it can be seen from Fig. 1, as ζ y2 q increases ζx2 q decreases, while the product ζ y2 q ζx2 q is constant and it equals unity. Thus, the q-deformed CSS behaves as an intelligent spin state. Figs. 2 and 3 show the dependence of S z with both the deformation parameter z and the orientation angle θ0 , respectively, for the same total spin (S = 10) and azimuthal angle φ0 = 0. The curves are indeed smooth functions of z and θ0 . In order to show that the features displayed in Fig. 1 remain, regardless the particular choice of the orientation angle θ0 , we have calculated the same parameters (23) for a set of values of θ0 . Indeed, the curves displayed in
18
M. Reboiro, O. Civitarese / Physics Letters A 376 (2011) 14–18
4. Conclusions In this Letter we have reported on the realization of q-deformed coherent spin states as intelligent spin states. We have demonstrated it by performing a calculation of the uncertainty relations, both analytically (for the S = 1 case) and numerically (for the S = 10 case). Work is in progress concerning the application of this type of states to mesoscopic systems [32,33], as for instance molecular spin-clusters which are paradigmatic cases to study the cross-over between quantum and classical behavior. Acknowledgement This work was partially supported by the National Research Council of Argentine (CONICET). References
Fig. 4. ζi2 , as a function of the orientation angle θ0 of the initial q-coherent spinstate. The adopted parameters are those given in the captions of Fig. 3.
Fig. 5. Mean value of the spin operator S 3 , S 3 of Eq. (8), as a function of α . The figure displays the results corresponding to an intelligent spin state with S = 10. The curves from top to bottom correspond to M = 0, M = ±1, . . . , M = ±10, respectively.
Fig. 4 show that the product ζ y2 q ζx2 q equals unity, as it is the case of the results shown in Fig. 1. We can now turn the attention to the same set of parameters, when calculated from Eq. (7), that is for an intelligent spin state. Fig. 5 shows the values of S 3 , for different values of α , for a state with total spin S = 10. The correspondence with the results obtained with a q-deformed CSS can be established by taken equal values of S 3 and S z q . Thus, for instance, if one takes S 3 ≈ S z q ≈ −7, one gets ζ12 = 0.6, corresponding to α = 0.6 (| S = 10 M = −9 α = 0.6), and ζx2 q ≈ 0.6 for z ≈ 0.1035. The present result may indicate that q-deformed coherent spin states can be regarded as intelligent spin states, when the orientation of the components of the spin are properly defined.
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