On the quantum polarization and entanglement of superpositions of two two-mode coherent states

On the quantum polarization and entanglement of superpositions of two two-mode coherent states

Optics Communications 281 (2008) 6034–6039 Contents lists available at ScienceDirect Optics Communications journal homepage: www.elsevier.com/locate...

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Optics Communications 281 (2008) 6034–6039

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

On the quantum polarization and entanglement of superpositions of two two-mode coherent states João Batista Rosa Silva a,b,*, Rubens Viana Ramos a a b

Department of Teleinformatic Engineering, Federal University of Ceará, 60455-760, C.P. 6007, Fortaleza-Ce, Brazil University of Fortaleza, Telecommunications Engineering, 60811-905, Fortaleza-Ce, Brazil

a r t i c l e

i n f o

Article history: Received 2 July 2008 Received in revised form 1 September 2008 Accepted 2 September 2008

a b s t r a c t This work discusses the entanglement and quantum polarization of superpositions of two-mode coherent states of the types |w1i = N1(|a, bi + |b, ai) and |w2i = N2(|a, ai + |a, ai). We use the concurrence to measure their entanglements and the quantum Stokes parameters and the Q function in order to analyze their polarization and degree of polarization. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Quantum polarization is an important property that has been extensively used in quantum information processing. Regarding coherent states, their quantum polarization has similarities with polarization of classical light: the average values of the Stokes parameters for coherent states coincide with the classical values of the Stokes parameters. However, the quantum Stokes paramS2 and ^ S3 do not commute, that is, it is not possible to eters ^ S1 ; ^ know the values of any two of them simultaneously without uncertainties. This fact has been used in [1] in order to create a continuous variable quantum key distribution system. Basically, polarization of coherent states has been used for quantum key distribution [1–4]. On the other hand, the use of superpositions of coherent states for quantum computing purposes has been considered in [5–8]. However, from the best of our knowledge, the quantum polarization of superposition of coherent states has not been investigated. In this direction, here we use the quantum Stokes parameters and the Q function in order to analyze the quantum polarization of superposition of coherent states of the type jwi ¼ Nðja; bi þ je; kiÞ (N is the normalization constant), as well we use the concurrence to measure their entanglements. This work is outlined as follows: in Section 2 a review of quantum polarization is presented; in Section 3 the quantum polarization of superpositions of coherent states is analyzed; in Section 4 the entanglement is calculated; Finally, Section 5 brings the conclusions.

2. Review of quantum polarization The quantum Stokes operators are defined as [9–11]

^S0 ¼ a ^H þ a ^V ; ^yH a ^yV a y y ^S1 ¼ a ^H  a ^V ; ^H a ^V a y y ^S2 ¼ a ^V þ a ^H ; ^ a ^ a H

ð1Þ ð2Þ ð3Þ

V

^S3 ¼ iða ^H  a ^V Þ; ^yV a ^yH a ½S^j ; ^Sk  ¼ i2ejkm ^Sm ; ½^S0 ; ^Sj  ¼ 0 for j; k; m 2 f1; 2; 3g:

^yH ða ^H Þ and a ^yV ða ^V Þ are, respectively, the creation (anniIn (1)–(4) a hilation) operators of the horizontal and vertical modes. The average values of the quantum Stokes parameters of a coherent state are equal to its classical Stokes parameters values. However, since there is a variance on the Stokes parameters values, the polarization is not well defined. The mean values and the variances of the Stokes parameters of the two-mode coherent state |a, bi are given by:

ja; bi ¼

1 X

e

jaj2 2

n¼0

an

pffiffiffiffiffi jniH  n!

h^S1 i ¼ jaj2  jbj2 ;

1 X

k jbj2 b e 2 pffiffiffiffi jkiV k! k¼0

h^S2 i ¼ a b þ ab ;

ð7Þ h^S22 i ¼ ða bÞ2 þ ðab Þ2 þ jaj2 þ jbj2 þ 2jaj2 jbj2 ;

2

V 2 ¼ jaj þ jbj

0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.09.046

ð6Þ

h^S21 i ¼ ðjaj2  jbj2 Þ2 þ jaj2 þ jbj2 ;

V 1 ¼ jaj2 þ jbj2

2

* Corresponding author. Address: Department of Teleinformatic Engineering, Federal University of Ceará, 60455-760, C.P. 6007, Fortaleza-Ce, Brazil. E-mail addresses: [email protected] (J.B.R. Silva), [email protected] (R.V. Ramos).

ð4Þ ð5Þ

ð8Þ

h^S3 i ¼ iðab  a bÞ; h^S2 i ¼ ða bÞ2  ðab Þ2 þ jaj2 þ jbj2 þ 2jaj2 jbj2 ; 3

V 3 ¼ jaj2 þ jbj2

ð9Þ

J.B.R. Silva, R.V. Ramos / Optics Communications 281 (2008) 6034–6039

V j ¼ h^S2j i  h^Sj i2 ¼ ha; bj^S2j ja; bi  ha; bj^Sj ja; bi2 ; j 2 f0; 1; 2; 3g

ð10Þ

The larger the optical power the larger are the variances of all three parameters. In order to apply a phase shift / between the horizontal and vertical modes of |a, bi, resulting in |aeiu/2, beiu/2i, the operator used is Cð/Þ ¼ expði/^ S1 =2Þ. On the other hand, if the goal is to apply a geometric rotation of h in the polarization, then the operator to be used is RðhÞ ¼ expðih^ S3 Þ. Thus, R(h)|a, bi = |b sinh + acosh, b cosh  a sinhi. A light beam can be considered unpolarized if its observables do not change after an application of a geometric rotation and/or a phase shift between the components. These conditions are mathematically described by [12]:

½q; ^S3  ¼ ½q; ^S1  ¼ 0;

ð11Þ

where q is the density matrix of the light quantum state. The general form of the quantum state of an unpolarized light is [13,14]



X

pn

n

n 1 X jkijn  kihkjhn  kj; n þ 1 k¼0

ð12Þ

where pn is the total photon number probability distribution. Concerning the quantum degree of polarization, some measures have been proposed [15,16] and here we consider the measure based on the Q function because it is always nonnegative for all states, being a truly probability distribution obtained by projection on the SU(2) coherent states, which can be regarded as the states with minimum polarization fluctuations. The Q function is defined as [16,17]:

jn; h; /i ¼

  n  X n 1=2 m

m¼0

sin

6035

As an illustration, using (13), (14) we plotted in Fig. 1 the Q functions for the states |a = 2, b = 0i, |a = 0, b = 2i, |a = 21/2, b = ± 212i, |a = 21/2, b = ± i21/2i. In this figure, x = Q(h,/)sin(h)cos(/), y = Q(h,/)sin(h)sin(/) and z = Q(h,/)cos(h). One can also analyze the polarization using the adimensional quadrature operators that represent the electric field in the xy plane (similar to position operator of the harmonic oscillator) [16]:

^x ¼

 1 ^H þ a ^yH ; a 2

^¼ y

 1 ^V þ a ^yV a 2

ð20Þ

^ðy ^Þ, with eigenstates |xi(|yi) form an The eigenvalues x(y) of x unbounded continuous set. There are probability distributions ^. For coherent states, the probaassociated with measuring ^ x and y bility density at the point x(y), px =|hx|ai|2(py = |hy|bi|2) is a Gaussian distribution having mean equal to the real part of the field amplitude, Re(a) (Re(b)) [19]. Using numerical simulations, we show the amplitude probability distributions for the quantum states |a = 2, b = 0i and |a = 0, b = 2i, in Fig. 2, and |a = 21/2, b = ± i21/2i in Fig. 3 [16,20]. From the numerical data we obtained hxi  2 and hyi  0 for |a = 2, b = 0i and hxi  0 and hyi  2 for |a = 0, b = 2i. Similarly, we obtained for both |a = 21/2, b = i21/2i and |a = 21/2, b = i21/2i, hxi  1.41 and hyi  0 (since the real part of b is zero). At least, the figures we obtained are in a good agreement with the expres2 2 sion ð2=pÞe2ðReðaÞxÞ e2ðReðbÞyÞ .

 nm  m h h cosð Þ eim/ jmijn  mi 2 2 ð13Þ

1 X nþ1 Q ðh; /Þ ¼ hn; h; /jqjn; h; /i 4p n¼0 2 Z 2p Z p  1 Q ðh; /Þ  sinðhÞdhd/; D ¼ 4p 4p 0 0 D P¼ ; 0 6 P  1; 1þD

ð14Þ ð15Þ ð16Þ

where |min  mi are the product of photon number states in the corresponding horizontal and vertical modes, and h and / are, respectively, the polar and azimuthal angles on the Poincaré sphere. In (15) 1/(4p) is the Q function of the unpolarized light whose quantum state is given by (12). For the two-mode coherent state |a, bi=||a| exp(i/a), |b| exp (i/b)i the Q function is [18]

Q ðh; /Þ ¼ eðjaj

2

þjbj2 Þ

ð1 þ zÞez =4p;

Fig. 1. Graphs of Q functions of the states ja ¼ 2; b ¼ 0i, (1) |a = 0, b = 2i (2), |a = 21/2, b = 21/2i, (3) |a = 21/2, b = - 21/2i, (4) |a = 21/2, b = i21/2i (5) and |a = 21/2, b = -i21/2i (6).

ð17Þ

        2 h h z ¼ jaj cos cos ð/a þ /Þ þ jbj sin cos /b 2 2         2 h h þ jaj cos sin ð/a þ /Þ þ jbj sin sin /b : 2 2

ð18Þ

Using (15)–(18) the quantum degree of polarization of the state |a, bi can be calculated analytically

P ¼1

4ðjaj2 þ jbj2 Þ 2

2

1 þ 2ðjaj þ jbj Þ½1 þ ðjaj2 þ jbj2 Þ  e2ðjaj

2

þjbj2 Þ

ð19Þ

As can be noted in (19), any two two-mode coherent states having the same optical power will have the same quantum degree of polarization. When |a|2+|b|2  1, (19) can be approximated by P  1  2/(|a|2 + |b|2) showing that the larger the optical power the larger is the quantum degree of polarization.

Fig. 2. Amplitude probability distributions for the quantum states | a = 2, b = 0i (1) and | a = 0, b = 2i.

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n h^S22 i ¼ jNj2 jaj2 þ jbj2 þ jej2 þ jkj2 þ 2ðjaj2 jbj2 þ jej2 jkj2 Þ þ ða bÞ2 þðab Þ2 þ ðe kÞ2 þ ðek Þ2 þ ½a e þ b k þ ae þ bk o þða k þ b eÞ2 þ ðe b þ ak Þ2 d

ð25Þ

n h^S23 i ¼ jNj2 jaj2 þ jbj2 þ jej2 þ jkj2 þ 2ðjaj2 jbj2 þ jej2 jkj2 Þ  ða bÞ2 ðab Þ2 

ðe kÞ2  ðek Þ2 þ ½a e þ b k þ ae þ bk o ða k  b eÞ2  ðe b  ak Þ2 d

ð26Þ

h i where d ¼ exp a e þ b k  ðjaj2 þ jbj2 þ jej2 þ jkj2 Þ=2 . On the other hand, the Q function of the state jwi ¼ Nðja; bi þ je; kiÞ is given by

Qðh; /Þ ¼

jNj2 4p

( eðjaj

2

þjbj2 Þ

þe

3. Quantum polarization of superposition of two two-mode coherent states In this section we consider the quantum Stokes parameters and the quantum degree of polarization of quantum states composed by the superposition of two bimodal coherent states jwi ¼ Nðja; bi þ je; kiÞ, where jNj2 ¼ f2 þ ½f þ f  exp½ðjaj2 þ jbj2 þ jej2 þ jkj2 Þ=2g1 and f ¼ expða e þ b kÞ. The averages of the quantum Stokes parameters and of their squared values, for |wi are:

h^S1 i ¼ jNj2 fðjaj2  jbj2 Þ þ ðjej2  jkj2 Þ þ ½ða e  b kÞ þ ðae  bk Þdg

ð21Þ

h^S2 i ¼ jNj2 fða b þ ab Þ þ ðe k þ ek Þ þ ½ða k þ eb Þ

2

þjkj2 Þ

ð1 þ z2 Þez2



jaj2 þ jbj2 þ jej2 þ jkj2



Fig. 3. Amplitude probability distributions for the states | a = 21/2, b = ± i21/2i.

ð1 þ z1 Þez1 þ eðjej

2

) z12

2Re½ð1 þ z12 Þe 

        2 h h cos ð/a þ /Þ þ jbj sin cos /b z1 ¼ jaj cos 2 2         2 h h sin ð/a þ /Þ þ jbj sin sin /b þ jaj cos 2 2  2     h h z2 ¼ jej cos cos ð/e þ /Þ þ jkj sin cos ð/k Þ 2 2  2     h h sin ð/e þ /Þ þ jkj sin sin ð/k Þ þ jej cos 2 2     h jð/a /e Þ h jð/b /k Þ 2 þ jbkkj sin e e 2 2   jakkjejð/a /k þ/Þ þ jbkejejð/b /e /Þ sinðhÞ: þ 2

ð27Þ

ð28Þ

ð29Þ

z12 ¼ jakej cos2

ð30Þ

ð23Þ

In this work, we are going to be concerned only with the following particular cases of |wi:|w1i = N1(|a, bi + |b, ai) and |w2i = N2(|a, ai + |a, ai) as well the particular case of |w1i, |w3i = N3(|a, 0i + |0, h

i1=2 h

i1=2 2 2 2 ai), where N1 ¼ 2 1 þ e2abðjaj þjbj Þ ; N 2 ¼ 2 1 þ e4jaj

n h^S21 i ¼ jNj2 jaj2 þ jbj2 þ jej2 þ jkj2 þ ðjaj2  jbj2 Þ2 þ ðjej2  jkj2 Þ2 o þ½a e þ b k þ ae þ bk þ ða e  b kÞ2 þ ðae  bk Þ2 d ð24Þ

and N3=N1(b=0). Hereafter we will consider only the case where a and b are real values. From (20)–(25), the averages and variances of the quantum Stokes parameters of the states and |w1i, |w2i and |w3i are:





þ ðe b þ ak Þdg h^S3 i ¼ ijNj2 fðab  a bÞ þ ðek  e kÞ þ ½ðeb  a kÞ þ ðak  be Þdg

ð22Þ

8 2 > h^S1 i ¼ 0; V 1 ¼ 2N 21 ½a2 þ b2 þ 2abeðabÞ þ ða2  b2 Þ2 ; > > < n o 2 2 2 2 2 jw1 i h^S2 i ¼ 2N 21 ½2ab þ ða2 þ b2 ÞeðabÞ ; V 2 ¼ 2N 21 2abð2ab þ eðabÞ Þ þ ða2 þ b2 Þ½1 þ ða2 þ b2 ÞeðabÞ   2N 21 ½2ab þ ða2 þ b2 ÞeðabÞ  ; > > > 2 : 2 V 3 ¼ 2N 21 fa2 þ b2 þ ½2ab  ða2  b2 Þ eðabÞ g; h^S3 i ¼ 0; 8 2 ^ > V 1 ¼ 4N 22 a2 ð1  e4a Þ; > < hS1 i ¼ 0; jw2 i h^S2 i ¼ 4N 22 a2 ð1 þ e4a2 Þ; V 2 ¼ 4N 22 a2 f2a2 ð1 þ e4a2 Þ þ ð1  e4a2 Þ½1  4N 22 a2 ð1  e4a2 Þg; > > 2 : ^ hS3 i ¼ 0; V 3 ¼ 4N 22 a2 ð1  e4a Þ; 8 > h^S1 i ¼ 0; V 1 ¼ 2N 23 a2 ð1 þ a2 Þ; > > > 2 2 2 < ^ hS2 i ¼ 2N 23 a2 ea ; V 2 ¼ 2N 23 a2 ½1 þ a2 ð1  2N 23 ea Þea ; jw3 i 2 > > h^S3 i ¼ 0; V 3 ¼ 2N 23 a2 ð1  a2 ea Þ; > > :

ð31Þ ð32Þ

ð33Þ

J.B.R. Silva, R.V. Ramos / Optics Communications 281 (2008) 6034–6039

One can observe in (30)–(32) that hS1i and hS3i are equal to zero for the three states. This happens because the average optical powers in horizontal and vertical polarizations are equal for all of them and only real values of a and b are being considered. Hence, |w1i, |w2i and |w3i correspond to a diagonal polarization, |±p/4i. An interesting behavior caused by the superposition can be seen at the variances of the Stokes parameters. For states of the type |a, bi, the variances of all Stokes parameters are equal and they vary proportionally to the optical power, as can be seen in (7)–(9). Hence, any two two-mode coherent states having the same optical power will have the same variances of the Stokes parameters. On the other hand, for the superpositions of coherent, two states having the same optical power, in general, will not have the same variances of the Stokes parameters. Furthermore, the variances of the Stokes parameters increase in a non-linear way when optical power increases. These facts can be seen in Fig. 4 for V3 of |w1i, where d = tan1(b/a). Now, using (27)–(30) we obtain the following Q functions for |w1i,|w2i and |w3i:

6037

Fig. 5. Q functions of the states | w1i (1), | w2i (2) and | w3i (3) having total mean photon number equal to 4.

8 2 2 eða þb Þ N 21 > > Q 1 ðh; /Þ ¼ fð1 þ z1 Þez1 þ ð1 þ z2 Þez2 þ 2ð1 þ z12 Þez12 g; > 4p > > < 2 h 2 2 2 h ð34Þ jw1 i z1 ¼ a cos ð2Þ þ b sin ð2Þ þ ab sinðhÞ cosð/Þ; > 2 > z ¼ b cos2 ðhÞ þ a2 sin2 ðhÞ þ ab sinðhÞ cosð/Þ; > 2 > 2 2 > : z12 ¼ ab½1 þ sinðhÞ cosð/Þ; ( 2 e2a N22 z1 z12 jw i Q 2 ðh; /Þ ¼ 2p fð1 þ z1 Þe þ ð1 þ z12 Þe g; ð35Þ 2

jw3 i

8 <

z1 ¼ a2 ½1 þ sinðhÞ cosð/Þ; Q 3 ðh; /Þ ¼

2 ea N 23

4p

: z ¼ a2 cos2 h; 1 2

z12 ¼ a2 ½1 þ sinðhÞ cosð/Þ;

fð1 þ z1 Þez1 þ ð1 þ z2 Þez2 þ 2Re½ð1 þ z12 Þez12 g; j/ 2  z2 ¼ a2 sin 2h ; z12 ¼ a2 e2 sinðhÞ: ð36Þ

As an illustration, it is shown in Fig. 5 the Q functions of the states |w1i, |w2i and |w3i, having all of them total mean photon number equal to 4. Finally, using (33)–(35) in (15), (16), we calculated the quantum degree of polarization of the states jw1 i:ðb=a ¼ 1=8; 1=4; 1=2Þ; jw2 i and |w3i and we compared them to the quantum degree of polarization of the state | r, ri, 2r2 = a2 + b2. The results are shown in Fig. 6 for states | r, ri(1) and | w1i(2) and in Fig. 7 for states | r, ri(1), | w2i(2) and | w3i(3). From Figs. 6 and 7, is straightforward to note that, the larger the mean photon number the larger is the quantum degree of polarization. Furthermore, for the conditions specified, the state | r, ri has

Fig. 4. Variances of S3 (V3), versus a2+b2 and d=tan-1(b/a) for |w1i.

Fig. 6. Quantum degree of polarization, P, versus optical power (a2+b2) for states | r, ri(1) and |w1i(2).

Fig. 7. Quantum degree of polarization, P, versus optical power (2 a2) for states |r, ri(1), |w2i(2) and |w3i(3).

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the upper bound for the quantum degree of polarization, P(|r, ri) P P(|wni) for n = 1, 2, 3. At last, in Fig. 7 the quantum degree of polarization of the state |w3i is always lower than the others because of the vacuum component. 4. Quantum entanglement of superposition of two two-mode coherent states Quantum entanglement is, in fact, the most interesting quantum property that, when applied in communication and computation, allows powerful ways of information sending and processing. Hence, to quantify the amount of entanglement present in a quantum state is a crucial task. Several works have been done in this direction and there are several measures reliable as, for instance, those discussed in Refs. [21–24]. Here, we are going to use the concurrence, which is defined as [24] C(q) = max{0, k1  k2  k3  k4}, where the ki’s are the square roots of the eigenvalues of q(ryry) q*(ryry) put in nonincreasing order, ry is the Pauli spin matrix, and asterisk stands for complex conjugation. The amount of quantum entanglement of the state jwi ¼ Nðja; bi þ je; kiÞ calculated using the concurrence is [25,26]



qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  jhajeij2 Þð1  jhkjbij2 Þ=ð1 þ RefhajeihbjkigÞ

ð37Þ

In particular, the concurrences of |w1i, |w2i and |w3i are, respectively:

C1 ¼

1  ejabj

2

1 þ ejabj

2

;

C2 ¼

1  e4jaj

2

1 þ e4jaj

2

;

C3 ¼

1  ejaj

2

1 þ ejaj

2

Fig. 9. Concurrence of |w4iC(u2)R(h)C(u1) | w1i (Eq. (40)) versus (rotator’s angle) for u1 e {0, p/8, p/4} and | a - b | 2 = 4.

ð38Þ

:

For the states |w2i, and |w3i, the larger |a|2 the larger are their entanglements. For the state |w1i, the larger the difference |a  b|2 the larger is the entanglement. This obvious since for a = b, |w1i reduces to |a, ai that has none entanglement. In Fig. 8 it is shown the entanglement of |w1i versus a2 + b2 and d = tan1(b/a). As expected, the entanglement is zero when d = p/4 (that implies a = b) and it grows when d tends to p/2 or 0 and when a2 + b2 grows. Now, let us assume the state |w1i passes by a compensator-rotator-compensator device, C(/2)R(h)C(/1) The quantum state at the output and its entanglement are given by

jw4 i ¼ N1 jb sinðhÞejð/2 /1 Þ=2 þ aejð/2 þ/1 Þ=2 cosðhÞijbejð/2 þ/1 Þ=2 cosðhÞ aejð/2 /1 Þ=2 sinðhÞi þ ja sinðhÞejð/2 /1 Þ=2 þ bejð/2 þ/1 Þ=2 cosðhÞi jaejð/2 þ/1 Þ=2 cosðhÞ  bejð/2 /1 Þ=2 sinðhÞi ; ð39Þ

C4 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 þ e2jabj  2ejabj cos h½ja  bj2 sinð2hÞ cosð/1 Þ 2

1 þ ejabj

ð40Þ

Fig. 9 shows the change in the entanglement according to variations in /1 and h, for |a  b|2 = 4. The compensator does not change the entanglement since it is a local operation. On the other hand, regarding the entanglement, the polarization rotator plays the same role as a beam splitter for entanglement in spatial modes. Hence, it can create or destroy entanglement. In particular the rotation of p/4 applied in the state N1(|a, bi + |b, ai) destroys completely its entanglement: R(p/ 4)N1(|a, bi + |b, ai) = N1(|(a-b)/21/2i + (b  a)/21/2i)(b + a)/21/2i. 5. Conclusions We have presented explicit formulas for the mean values and the variances of the quantum Stokes parameters, as well the Q functions of superposition of two two-mode coherent states. The results showed that the superposition has an interesting influence in the variances of the Stokes parameters. In general, different states having the same optical power will have different variances values. Furthermore, the variances increase in a non-linear way when the optical power increases. These two characteristic do not happen when two-mode coherent states (without superposition) are considered. Moreover, for the cases studied, the degree of polarization still increases monotonically with the total mean photon number, as happens with normal two-mode coherent states. Concerning the entanglement, as expected, it was shown that it can be controlled by a polarization rotator. This is similar to the entanglement creation and destruction caused by beam splitters in superposition of spatial two-mode coherent states Nab(|a, bi12 + |b, ai12). Acknowledgments Useful discussions with Dr. Jonas Söderholm are gratefully acknowledged. References

2

Fig. 8. Concurrence of the state |w1iN1(|a, bi + |b, ai) versus a +b {r = tan -1(b/a)}.

2

and

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