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Phase optimized states via discrete coherent-state superpositions
10 June I996
PHYSICS
LETTERS
A
Physics Letters A 215 (1996) 229-233
Phase optimized states via discrete coherent-state
superpositions
P. Adam, S. Szabo, J. Janszky Research Laboratory Received
12 January
for Crystal Physics, FO. Box 132. H-1502 Budapest, Hungary
1996; revised manuscript received 4 March 1996; accepted Communicated by P.R. Holland
for publication
5 March
1996
Abstract
It is shown that phase optimized quantum states of the electromagnetic field can be constructed by superpositions of a small number of coherent states along the positive real semiaxis in phase space. A systematic method is developed for finding optimal coherent-state superpositions.
Recently, much attention has been devoted to the problem of generating field states that have minimal quantum noise in ameasured physical quantity. Prominent states exhibiting this property are phase optimized states. In such a state the phase noise is minimal for a given mean photon number. Various phase optimized states associated with different measures of phase noise have been investigated so far [ l-51. In previous papers it was shown that several quantum states can be approximated at arbitrary precision by discrete coherent-state superpositions [ 6,7]. This approximation offers novel experimental schemes for generating quantum states. A single-atom [8] and a Ramsey-type atom interference method [ 9,101 have been developed for generating general coherent-state superpositions on a circle in phase space. Generalization of nonlinear optical processes leading to coherentstate superpositions [ 1 I- 141 may result in further experimental arrangements for producing arbitrary superpositions. In this paper we discuss the possibility of constructing phase optimized quantum states via superpositions of a small number of coherent states. Approximating discrete superpositions can be found from the one-
dimensional coherent-state representation of the given state [ 10,151. As such a representation of phase optimized states is not known, we will develop a systematic method for determining the coefficients and the coherent signals of the constituent coherent states. For simplicity we will deal with phase optimized states with zero mean phase. In accordance with this choice let us consider the following discrete coherentstate superpositions along the real positive semiaxis of phase space,
eA.ilXi)* .i=o
19%) =
where Ix~) denotes a coherent state with a positive real coherent signal X,j. In Ref. [I] it was shown that after removing a common complex factor the numberstate expansion of the phase optimized state with zero mean phase contains real coefficients only. To meet this property, the coefficients Ai are chosen to be real. The states I$,,) are required to be normalized,
(I)~ II+!I,,)=
5 k,i=l
0.775.9601/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved P/f SO37S-9601 (96)00230-7
(1)
AkAje-IX’-X~12~2 z 1.
(2)
230
P. Adam et al./Physics
Letters A 215 (1996) 229-233
For the investigation of phase properties of the states I$,,) we will use the Pegg-Burnett formalism [ 161. This formalism is based on a Hermitian phase operator &e1 which exists in a finite dimensional state space, spanned by the number states IO). . .js). Expectation values and variances of physical quantities are to be calculated in the finite space and the infinite limit in the dimension of the space has to be taken after cnumber expressions are obtained. The Hermitian phase operator &,w in the s-dimensional space is defined as
44 = =J+l@,,)(@n,l ,
(3)
nr=a where IO,,,) = (s-t
l)-1/2Cexp(in0,,)ln), n=o
+ -
(5)
sf 1’
andm=O,l,... , s. The value of 00 is arbitrary and defines a particular basis set. For the states I$p) it is convenient to choose the reference phase 00 to be -7r. In the following we omit the index 0 of the phase operator. Using the Fock state expansion of ]ep) (6)
IF0
and Eqs. (3) and (4) the phase variance for the states I+,,) can be easily evaluated, (A&‘) = (&‘) - (6)’
cc(-I)!+’ k>, (k-
The
mean
value
of
1
Q2 m
the
phase
= xi-(p-l)
ifi=
l,...,p
ifi=p,...,2p-
- I, 1.
(8)
The initial number p of the constituent coherent states has to be chosen so large that the final result of the optimization should not depend on it. The algorithm starts at an arbitrary point of the parameter space {r;} and it finds a point with the required mean photon number
k,,j=l
25-m
=fa2+4~-----
~i=Ai+i/Ai
(4)
are the orthonormal phase states. They are the eigenstates of the phase operator & with the eigenvalues
o,,,= 00
of the constituent coherent states in the superposition can be adjusted numerically. The developed algorithm of numerical optimization uses a (2p - 1) -dimensional parameter space {n} whose points are related to the state Ilcl,) in the following way,
operator
(&)
=
((cl,,l~lVQ,J = 0. In order to find the states I+$,) of minimal phase uncertainty, the coefficients Al and coherent signals x,j
=
const.
(91
Then it finds the direction Dp* of the maximal decrease of the phase variance (Aq2) and the direction DN of the maximal increase of the photon number (fi) in the parameter space. Next the algorithm takes a step in the direction which is orthogonal to DN but nearest to the direction DA@ of maximal phase variance decrease. This step in the parameter space leaves the mean photon number (fl) approximately unchanged. To correct the possible small deviation the algorithm finds the nearest point with the required mean photon number using a finite difference Newton method. The process described above is repeated until the phase variance (A$*) reaches the minimal value, i.e., its change is practically negligible. The length of the steps in the parameter space are determined step by step to ensure that (A$2) should always decrease. We note that the runtime of the program is proportional to the square of the required mean photon number and rapidly grows with the number p of the constituent coherent states. The result of the numerical optimization can be seen in Table 1, where the coefficients Aj and the coherent signals X,j of the coherent states in the resulting discrete coherent-state superposition phase optimized states (DPOS) and the corresponding phase and photon number variances are indicated for different mean photon numbers. We find that the number p of the constituent coherent states whose coefficients differ from
P. Adam et al./ Physics Letlers A 215 (19961229-233
231
Table I The coefficients Aj and coherent signals Xj of the coherent states in the DPOS with different mean photon numbers, and the resulting phase and photon number variances (fi)
A@
AN
Al.Xl
A2,xt
A?,X?
s S.? 7 7.2 8 II
0.338557 0.234458 0.226747 0.174856 0.170521 0.155138 0.115924
1.809 2.622 2.713 3.519 3.608 3.966 5.308
0.987,1.777 0.524,1.816 0.497, I .8 I8 0.334.1.830 0.322,1.831 0.281.1.835 0.180,1.840
0.020.2.684 0.553.2.618 0.580,2.644 0.614,2.755 0.611,2.764 0.592.2.796 0.467.2.847
0.008,3.793 0.198.3.574 0.219,3.597 0.296,3.675 0.455,3.783
14
0.092537
5.308
0.129.1.844
0.371.2.879
0.479,3.859
3.2
zero at a fixed mean photon
computing
number.
as the mean energy
precision
This number
depends
gradually
on the
.44.x4
0.60
As,xs
0.001,4.996 0.009,4.725
0.149,4.691 0.289,4.819
0.007,5.701 0.059,5.830
1
increases
a
rises. The DPOS with mean pho-
ton numbers (8) = 5 and (a) = 7 only consist of 2 and 3 coherent states respectively. One can also realize that the difference of coherent signals for two adjacent coherent states is approximately 0.9. It is worth mentioning that this quantity is equal to the distance of two coherent states in a Schriidinger-cat state when the quadrature squeezing is maximal. Fig. 1 shows the coherent signals xj and the coefficients A,j of coherent states in the DPOS for mean photon numbers (fi) = 6.5 (Fig. la) and (A) = 12 (Fig. lb). To demonstrate how perfect DPOS are, we compare them to the mathematically constructed approximating phase optimized states ( AiPOS) of Ref. [ I 1. The definition of these states is
0 00
1 00
2.00
3.00
4.00
5.00
‘k
0.40 ,
030 -I
where Ai is the Airy function, ba M -2.34 is its first zero value, 0 < b < 1 is a shift parameter, parameter a presets the mean photon number, N is a normalization constant. The parameter b has to be optimized
to reach
the possible
least phase
variance.
can be seen in Fig. 2, where the Fock coefficients of the DPOS and the AiPOS are indicated for the mean photon number (fi) = 9. One can see that they almost perfectly coincide. A direct
l&5)
0.10
reference,
states defined
= &&5)lOL
one can choose
quadrature
in the usual way, (11)
4 I
comparison
As another squeezed
Ako20-
0.00
+-
0.00
,
-7
1 .oo
I__. 2.00
3.00
400
500
6.00
‘k Fig. 1. The coefficients Aj and the coherent signals Xj of the COherent states in the DPOS for the mean photon numbers (fi) = 6.5 (a) and (h) = I2 (b).
P. A&m
232
et al./Physics
Letters A 215 (1996) 229-233 Table 2 The phase, photon number and quadrature variances of the AiPOS, the DPOS and the OPSS for different mean photon numbers
0.40
0.30
(fi) =6.5 c,
020
(A) = 9
010
i'
I
000
‘,
AQ,
AN
AX
AY
AiPOS DPOS OPSS AiPOS DPOS OPSS
0.186481 0.186725 0.187 I 19 0.139291 0.139417 0.139909
3.291318 3.295108 3.272385 4.409345 4.413638 4.376340
0.64601 I 0.646538 0.640857 0.736995 0.737648 0.728900
0.387439 0.387090 0.390103 0.339688 0.339399 0.342983
_
----
0
5
10
15
20
25
30
n
Fig. 2. The Fock coefficients c,, of the DPOS (solid line) and the AiPOS (dashed line) for the mean photon number (fi) = 9. The two lines almost perfectly coincide.
where the squeezing 3( [) = exp( @f’
“ 5
1
2c
7
P(7J)
j
- $*ti*>, (12)
and the displacement
operator ‘; 0
= exp(aa+
1
operator
i = sexp(i0),
i>(a)
state
- a!*G).
(13)
Here irt and B are the creation and annihilation operators. The phase properties of quadrature squeezed states were discussed in detail in the literature [ 5,18241. When the uncertainty of the quadrature operator p = (2 - Gt) /2i is squeezed and the coherent signal is positive real one has a phase squeezed state with zero mean phase. For certain measures of phase uncertainty the phase noise of an optima1 phase squeezed state decays proportional to the inverse of the average number of photons [5,25] similarly to phase optimized states. Inspired by these results, we have optimized the squeezing parameters and coherent signals of the phase squeezed states with fixed mean photon numbers to obtain optimal phase squeezed states (OPSS) with minimal phase variance in the Pegg-Barnett formalism. Table 2 shows the phase, photon number and quadrature variances of the AiPOS, the DPOS and the OPSS for different mean photon numbers. One can see that the DPOS only have a one thousandth worse phase variance than the AiPOS, but the OPSS
tr--7-T -77
--T1/2
0
n/2
TT
d Fig. 3. The phase distribution
P( @) for the DPOS with (fi) = 6.5.
perform already a one percent higher phase variance than the AiPOS do. The photon number variance AN of the DPOS is larger than that of the AiPOS, while it is smaller for the OPSS than for the AiPOS. It is interesting to notice that the uncertainty of the quadrature operator P is smallest for the DPOS, while the uncertainty of the quadrature operator 2 = i (d + Gt ) is smallest for the OPSS. To characterize the phase properties of DPOS it is useful to calculate the continuous phase-distribution defined as (14) Fig. 3 shows the function P (0) for the DPOS with (fi) = 6.5. The phase distribution is very narrow, the probability of measuring phase values significantly different from zero is practically zero. Finally we calculate the Wigner quasiprobability
P. A&m et d/Physics
Letters A 215 (19961229-233
233
References
Fig. 4. The topological with (&) =9.
picture of the Wigner function of the DPOS
function of the DPOS, W(a)
= -$eziai2 /
d2/3 (-pl~)(~IP)e2’P*“-B”*’
Fig. 4 shows the topological picture of the Wigner function of the DPOS with (fi) = 9. It can be clearly seen that the Wigner function has a drop-like shape stretched along the positive real semiaxis ensuring small phase variance. In conclusion, we have shown that phase optimized quantum states of the electromagnetic field can be constructed by superpositions of a small number of coherent states. These superpositions can be advantageous for an experimental realization of the states. This work was supported by the National Scientific Research Fund (OTKA) of Hungary, under Contracts Nos. T017386, T014083, and F014139.
[ 1 J G.S. Summy and D.T. Pegg, Opt. Commun. 77 ( 1990) 75. [2] A. Bandilla, H. Paul and H.-H. Ritze, Quantum Opt. 3 (1991) 267. [3] A. Bandilla, Opt. Commun. 94 (1992) 273. 14 1 B. Daeubler, Ch. Miller, H. Risken and L. Schoendorff, Phys. Ser. T 48 (1993) 119. 151 M. Freiberger and W. Schleich, Phys. Rev. A 49 I 1994) 5056. 161 J. Janszky, P Domokos and P Adam, Phys. Rev. A 48 (1992) 213. 171 J. Janszky, P. Domokos, S. Szabo and i? Adam, Phys. Rev. A 51 (1995) 4191. [81 l? Domokos, J. Janszky and P. Adam, Phys. Rev. A SO (1994) 3340. 191 M. Brune, S. Haroche, V. Lefevre, J.M. Raimond and N. Zagury, Phys. Rev. L&t. 65 ( 1990) 976. [lOI S. Szabo, P Adam, J. Janszky and P Domokos, Phys. Rev. A (1996), in print. 1111 B. Yurke and D. Staler. Phys. Rev. Lett. 57 ( 1986) 13. 1121 M. Wolinsky and H.J. Carmichael, Phys. Rev. Lett. 60 (1988) 1836. I131 A. Miranowicz, R. TanaS and S. Kielich, Quantum Opt. 2 (1990) 253. P. Knight and S.J.D. Phoenix, I141 V. Buick, H. Moya-Cessa, Phys. Rev. A 45 ( 1992) 6570. 1151 P Domokos, J. Janszky, P Adam and T. Larsen, Quantum Opt. 6 (1994) 187. [ 161 D.T. Pegg and S.M. Bamett, Europhys. L&t. 6 (1988) 483. [ 17 1 D.T. Pegg and SM. Bamett, Phys. Rev. A 39 (1989) 1665. [ 181 B.C. Sanders, SM. Bamett and P.L. Knight, Opt. Commun. 58 (1986) 290. [ I91 R. Lynch, J. Opt. Sot. Am. B 4 ( 1987) 1723. [201 Hong-Yi Fan and H.R. Zaidi, Opt. Commun. 68 ( 1988) 143. [2l] N. Gronbech-Jensen, P.L. Christiansen and P.S. Ramanujam, J. Opt. Sot. Am. B 6 (1989) 2423. 1221 J.A. Vaccaro and D.T. Pegg, Opt. Commun. 70 ( 1989) S29. [231 Ts. Gantsog and R. TanaS, Phys. Lett. A 152 ( 1991) 251. [24) D. Cohen, Y. Ben-Areyh and A. Mann, Opt. Commun. 94 (1992) 227. 1251 M.J. Collett, Phys. Ser. 48 (1993) 124.
PHYSICS
LETTERS
A
Physics Letters A 215 (1996) 229-233
Phase optimized states via discrete coherent-state
superpositions
P. Adam, S. Szabo, J. Janszky Research Laboratory Received
12 January
for Crystal Physics, FO. Box 132. H-1502 Budapest, Hungary
1996; revised manuscript received 4 March 1996; accepted Communicated by P.R. Holland
for publication
5 March
1996
Abstract
It is shown that phase optimized quantum states of the electromagnetic field can be constructed by superpositions of a small number of coherent states along the positive real semiaxis in phase space. A systematic method is developed for finding optimal coherent-state superpositions.
Recently, much attention has been devoted to the problem of generating field states that have minimal quantum noise in ameasured physical quantity. Prominent states exhibiting this property are phase optimized states. In such a state the phase noise is minimal for a given mean photon number. Various phase optimized states associated with different measures of phase noise have been investigated so far [ l-51. In previous papers it was shown that several quantum states can be approximated at arbitrary precision by discrete coherent-state superpositions [ 6,7]. This approximation offers novel experimental schemes for generating quantum states. A single-atom [8] and a Ramsey-type atom interference method [ 9,101 have been developed for generating general coherent-state superpositions on a circle in phase space. Generalization of nonlinear optical processes leading to coherentstate superpositions [ 1 I- 141 may result in further experimental arrangements for producing arbitrary superpositions. In this paper we discuss the possibility of constructing phase optimized quantum states via superpositions of a small number of coherent states. Approximating discrete superpositions can be found from the one-
dimensional coherent-state representation of the given state [ 10,151. As such a representation of phase optimized states is not known, we will develop a systematic method for determining the coefficients and the coherent signals of the constituent coherent states. For simplicity we will deal with phase optimized states with zero mean phase. In accordance with this choice let us consider the following discrete coherentstate superpositions along the real positive semiaxis of phase space,
eA.ilXi)* .i=o
19%) =
where Ix~) denotes a coherent state with a positive real coherent signal X,j. In Ref. [I] it was shown that after removing a common complex factor the numberstate expansion of the phase optimized state with zero mean phase contains real coefficients only. To meet this property, the coefficients Ai are chosen to be real. The states I$,,) are required to be normalized,
(I)~ II+!I,,)=
5 k,i=l
0.775.9601/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved P/f SO37S-9601 (96)00230-7
(1)
AkAje-IX’-X~12~2 z 1.
(2)
230
P. Adam et al./Physics
Letters A 215 (1996) 229-233
For the investigation of phase properties of the states I$,,) we will use the Pegg-Burnett formalism [ 161. This formalism is based on a Hermitian phase operator &e1 which exists in a finite dimensional state space, spanned by the number states IO). . .js). Expectation values and variances of physical quantities are to be calculated in the finite space and the infinite limit in the dimension of the space has to be taken after cnumber expressions are obtained. The Hermitian phase operator &,w in the s-dimensional space is defined as
44 = =J+l@,,)(@n,l ,
(3)
nr=a where IO,,,) = (s-t
l)-1/2Cexp(in0,,)ln), n=o
+ -
(5)
sf 1’
andm=O,l,... , s. The value of 00 is arbitrary and defines a particular basis set. For the states I$p) it is convenient to choose the reference phase 00 to be -7r. In the following we omit the index 0 of the phase operator. Using the Fock state expansion of ]ep) (6)
IF0
and Eqs. (3) and (4) the phase variance for the states I+,,) can be easily evaluated, (A&‘) = (&‘) - (6)’
cc(-I)!+’ k>, (k-
The
mean
value
of
1
Q2 m
the
phase
= xi-(p-l)
ifi=
l,...,p
ifi=p,...,2p-
- I, 1.
(8)
The initial number p of the constituent coherent states has to be chosen so large that the final result of the optimization should not depend on it. The algorithm starts at an arbitrary point of the parameter space {r;} and it finds a point with the required mean photon number
k,,j=l
25-m
=fa2+4~-----
~i=Ai+i/Ai
(4)
are the orthonormal phase states. They are the eigenstates of the phase operator & with the eigenvalues
o,,,= 00
of the constituent coherent states in the superposition can be adjusted numerically. The developed algorithm of numerical optimization uses a (2p - 1) -dimensional parameter space {n} whose points are related to the state Ilcl,) in the following way,
operator
(&)
=
((cl,,l~lVQ,J = 0. In order to find the states I+$,) of minimal phase uncertainty, the coefficients Al and coherent signals x,j
=
const.
(91
Then it finds the direction Dp* of the maximal decrease of the phase variance (Aq2) and the direction DN of the maximal increase of the photon number (fi) in the parameter space. Next the algorithm takes a step in the direction which is orthogonal to DN but nearest to the direction DA@ of maximal phase variance decrease. This step in the parameter space leaves the mean photon number (fl) approximately unchanged. To correct the possible small deviation the algorithm finds the nearest point with the required mean photon number using a finite difference Newton method. The process described above is repeated until the phase variance (A$*) reaches the minimal value, i.e., its change is practically negligible. The length of the steps in the parameter space are determined step by step to ensure that (A$2) should always decrease. We note that the runtime of the program is proportional to the square of the required mean photon number and rapidly grows with the number p of the constituent coherent states. The result of the numerical optimization can be seen in Table 1, where the coefficients Aj and the coherent signals X,j of the coherent states in the resulting discrete coherent-state superposition phase optimized states (DPOS) and the corresponding phase and photon number variances are indicated for different mean photon numbers. We find that the number p of the constituent coherent states whose coefficients differ from
P. Adam et al./ Physics Letlers A 215 (19961229-233
231
Table I The coefficients Aj and coherent signals Xj of the coherent states in the DPOS with different mean photon numbers, and the resulting phase and photon number variances (fi)
A@
AN
Al.Xl
A2,xt
A?,X?
s S.? 7 7.2 8 II
0.338557 0.234458 0.226747 0.174856 0.170521 0.155138 0.115924
1.809 2.622 2.713 3.519 3.608 3.966 5.308
0.987,1.777 0.524,1.816 0.497, I .8 I8 0.334.1.830 0.322,1.831 0.281.1.835 0.180,1.840
0.020.2.684 0.553.2.618 0.580,2.644 0.614,2.755 0.611,2.764 0.592.2.796 0.467.2.847
0.008,3.793 0.198.3.574 0.219,3.597 0.296,3.675 0.455,3.783
14
0.092537
5.308
0.129.1.844
0.371.2.879
0.479,3.859
3.2
zero at a fixed mean photon
computing
number.
as the mean energy
precision
This number
depends
gradually
on the
.44.x4
0.60
As,xs
0.001,4.996 0.009,4.725
0.149,4.691 0.289,4.819
0.007,5.701 0.059,5.830
1
increases
a
rises. The DPOS with mean pho-
ton numbers (8) = 5 and (a) = 7 only consist of 2 and 3 coherent states respectively. One can also realize that the difference of coherent signals for two adjacent coherent states is approximately 0.9. It is worth mentioning that this quantity is equal to the distance of two coherent states in a Schriidinger-cat state when the quadrature squeezing is maximal. Fig. 1 shows the coherent signals xj and the coefficients A,j of coherent states in the DPOS for mean photon numbers (fi) = 6.5 (Fig. la) and (A) = 12 (Fig. lb). To demonstrate how perfect DPOS are, we compare them to the mathematically constructed approximating phase optimized states ( AiPOS) of Ref. [ I 1. The definition of these states is
0 00
1 00
2.00
3.00
4.00
5.00
‘k
0.40 ,
030 -I
where Ai is the Airy function, ba M -2.34 is its first zero value, 0 < b < 1 is a shift parameter, parameter a presets the mean photon number, N is a normalization constant. The parameter b has to be optimized
to reach
the possible
least phase
variance.
can be seen in Fig. 2, where the Fock coefficients of the DPOS and the AiPOS are indicated for the mean photon number (fi) = 9. One can see that they almost perfectly coincide. A direct
l&5)
0.10
reference,
states defined
= &&5)lOL
one can choose
quadrature
in the usual way, (11)
4 I
comparison
As another squeezed
Ako20-
0.00
+-
0.00
,
-7
1 .oo
I__. 2.00
3.00
400
500
6.00
‘k Fig. 1. The coefficients Aj and the coherent signals Xj of the COherent states in the DPOS for the mean photon numbers (fi) = 6.5 (a) and (h) = I2 (b).
P. A&m
232
et al./Physics
Letters A 215 (1996) 229-233 Table 2 The phase, photon number and quadrature variances of the AiPOS, the DPOS and the OPSS for different mean photon numbers
0.40
0.30
(fi) =6.5 c,
020
(A) = 9
010
i'
I
000
‘,
AQ,
AN
AX
AY
AiPOS DPOS OPSS AiPOS DPOS OPSS
0.186481 0.186725 0.187 I 19 0.139291 0.139417 0.139909
3.291318 3.295108 3.272385 4.409345 4.413638 4.376340
0.64601 I 0.646538 0.640857 0.736995 0.737648 0.728900
0.387439 0.387090 0.390103 0.339688 0.339399 0.342983
_
----
0
5
10
15
20
25
30
n
Fig. 2. The Fock coefficients c,, of the DPOS (solid line) and the AiPOS (dashed line) for the mean photon number (fi) = 9. The two lines almost perfectly coincide.
where the squeezing 3( [) = exp( @f’
“ 5
1
2c
7
P(7J)
j
- $*ti*>, (12)
and the displacement
operator ‘; 0
= exp(aa+
1
operator
i = sexp(i0),
i>(a)
state
- a!*G).
(13)
Here irt and B are the creation and annihilation operators. The phase properties of quadrature squeezed states were discussed in detail in the literature [ 5,18241. When the uncertainty of the quadrature operator p = (2 - Gt) /2i is squeezed and the coherent signal is positive real one has a phase squeezed state with zero mean phase. For certain measures of phase uncertainty the phase noise of an optima1 phase squeezed state decays proportional to the inverse of the average number of photons [5,25] similarly to phase optimized states. Inspired by these results, we have optimized the squeezing parameters and coherent signals of the phase squeezed states with fixed mean photon numbers to obtain optimal phase squeezed states (OPSS) with minimal phase variance in the Pegg-Barnett formalism. Table 2 shows the phase, photon number and quadrature variances of the AiPOS, the DPOS and the OPSS for different mean photon numbers. One can see that the DPOS only have a one thousandth worse phase variance than the AiPOS, but the OPSS
tr--7-T -77
--T1/2
0
n/2
TT
d Fig. 3. The phase distribution
P( @) for the DPOS with (fi) = 6.5.
perform already a one percent higher phase variance than the AiPOS do. The photon number variance AN of the DPOS is larger than that of the AiPOS, while it is smaller for the OPSS than for the AiPOS. It is interesting to notice that the uncertainty of the quadrature operator P is smallest for the DPOS, while the uncertainty of the quadrature operator 2 = i (d + Gt ) is smallest for the OPSS. To characterize the phase properties of DPOS it is useful to calculate the continuous phase-distribution defined as (14) Fig. 3 shows the function P (0) for the DPOS with (fi) = 6.5. The phase distribution is very narrow, the probability of measuring phase values significantly different from zero is practically zero. Finally we calculate the Wigner quasiprobability
P. A&m et d/Physics
Letters A 215 (19961229-233
233
References
Fig. 4. The topological with (&) =9.
picture of the Wigner function of the DPOS
function of the DPOS, W(a)
= -$eziai2 /
d2/3 (-pl~)(~IP)e2’P*“-B”*’
Fig. 4 shows the topological picture of the Wigner function of the DPOS with (fi) = 9. It can be clearly seen that the Wigner function has a drop-like shape stretched along the positive real semiaxis ensuring small phase variance. In conclusion, we have shown that phase optimized quantum states of the electromagnetic field can be constructed by superpositions of a small number of coherent states. These superpositions can be advantageous for an experimental realization of the states. This work was supported by the National Scientific Research Fund (OTKA) of Hungary, under Contracts Nos. T017386, T014083, and F014139.
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