The general theory of atomic angular momentum states is used to derive the Wigner distribution function for atomic angular momentum number states, coherent states, and squeezed states. These Wigner functions W(theta,phi) are represented as a pseudo-probability distribution in spherical coordinates theta and phi on the surface of a sphere of radius the square root of j(j +1) where j is the total angular momentum

Agarwal, G. S.

Dowling, Jonathan P.

Schleich, Wolfgang P.

English

NASA Technical Reports Server

N94-10603
WIGNER FUNCTIONS FOR NONCLASSICAL STATES
OF A COLLECTION OF TWO-_ ATOMS
G. S. Agarwal
School of Physics, University of Hyderabad
Hyderabad 500 134, India
and
Moa-Planch-Institut fl_r Quantenoptik, Postfach 1513.D so4 Ge,many
Jonathan P. Dowling
Weapons Sciences Dirsctomte, AMSMI.RD-WS.ST
Research, Development, and Engineering Center
?.7. S. Army Missile Command, Redstone Arsenal, Alabama 35898.5248, USA
Wolfgang P. Schleich
Abteilung f_r Theoretische Physik HI, Universitdt Ulm
Oberer Eselsberg W.7900, Ulm, Germany
AbsWact
The general theory of atomic angular momentum statos is used to derive the
Wigner distribution function for atomic angular momentum number states, co-
herent states, and squeezed states. These Wigner functions W(O,_) are
represented as a pseudo-probability distribution in spherical coordinates 0 and 9 on
the surface of a sphere of radius _ffO+l) where j is the total angular momentum.
1 Introduction
The phase space description of electromagnetic fields has had great success in
leading to an understanding of the relationship between semiclassical and quantum
theories of light. It was Sudarshan [1] who proved the optical equivalence theorem, i.e.,
he derived the relationship between the quantifies measured by a photodetector and the
mean values of the corresponding operators. He showed that the function appearing in
the diagonal coherent state representation, that is calculated from the density matrix,
provides a link between the semiclassica] and quantum descriptions. This function,
now denoted by P(cO, is generally singular for nonclassical states [2]. In such cases the
Wigner function [3,4] has proved to be especially attractive as an alternative. The
Wigner function has also proved to be quite useful in discussing related topics [5] such
as the photon number distribution and the phase operator distribution. In these
problems, the concept of the area of overlap in phase space has been especially useful.
The nonclassical characteristics of the atomic systems, particularly a collection of
two-level atoms, has been a subject of much investigation [6,7]. Much of the work has
concentrated on the direct calculation of the variances in the atomic operators such as
Jx, J+, and J - Very little has been done on the relationship between the nonclassical
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aspects and the phase space distributions for atomic operators. For general angular
momentum systems, Arecchi, et al. [8] introduced the analog of the diagonal coherent
state representation
p:I P_.,_)I-,_><_,_1.in._ d_,
where / a,_) represents the atomic coherent state
" 2j "_ j÷m j-ffio,> C'm:l' C":l
(1)
(2)
and where/jm) is the eigeustate of j2 and Jz" The parameters u and fl correspond to
8 and p except that a is measured off the south pole. The coherent state obtains the
minimum of the angular momentum uncertainty relation ('Aj2_)('AJ2y,) _ I(Jz,)/2/4,
where x;y; and z" form an orthogonal coordinate system with z'in the _" direction with
angular coordinates (a,_). The coherent state is just a rotation of the ground Fock state
IJ,-J) away from the south pole. Arecchi and co-workers discussed the utility of the
function P(a,_)in atomic problems, and Scully and co-workers have discussed the
. 1
Wigner function for spm-_ particles [9]. Using the general theory of multipole
operators [10], Agarwa] [11] introduced the Wigner function for systems of arbitrary
angular momentum. To arrive at this distribution, we first expand the atomic angular
momentum operators as
G = GKQTKQ,
where T/_Q is the multipole operator defined by
TKQ =,.=_jm_. (-1)J'In_ m Q ' [
-- _j
(3)
jm) Gin'[, (4)
where [_nOm'] is the usual Wigner 3j symbol. The expansion coefficients
in Eq. (3)_---"are obtained from the orthogonality of the multipole operators, namely
GKQ= Tr (GT;Q) . (5)
The Wigner function associated with G is then defined by [11]
2j ÷K
W(O,,) = _Q=__ YKQ(O,q_)GKQ ,
(6)
where YKQ are the usual spherical harmonics. Note that
330
TrG = _/W(e,cp) sineded_ ffi 1, (7)
a general property desired of any distribution function. Note further that if two
operators G (1) and G (2) are represented respectively by the Wigner functions W (1)
and W (2), then
Tr (G xG2) = f W(1)(e,_) W(2)(0,_) sine de d_, (8)
a defining property of the Wigner di_'cribution. In fact thue two features, Eqs. (7) and
(8), can be used to derive the form, Eq. (6), of the Wigner f_mctio_ Thus, unlike the P
function, all expectation values can be obtained in terms of the Wigner functions alone.
In this paper we shall consider the structure of the Wigner function associated
with the important states like (i) Fock states /j,m), (ii) coherent states/a,_), and
(iii) squeezed states/_,m) associated with a collection of two-level atoms interacting
with a squeezed photon bath. We examine how the quantum character of the state is
reflected in the properties of the Wigner function.
2 Angular Momentum States I jm)
We first obtain the Wigner function for the state/jm). The density matrix can be
written in the form
p ffi [jm) (jm[ . (9)
Upon using Eqs. (4) through (6), that are used in defining the Wigner function W, we
find that
iK (10)WJm(6'cP) " YKO(eJP)(-I_'_ _ m 0 m
As expected Wjm is independent of 9-
This function is plotted in Fig. 1 as a function of 0 c (0,z) and _oe (-_9 for jr5 and
mffiO, -I, ... -5. We plot the distribution both as planar and spherical surfaces. If we
suppose that/jm) is an orbital angular momentum state, then quantum-mechanically
we would expect the angular momentum vector of length _ _ to be oriented inside
a sphere of radius _ _ such that its z component is m)_ where m = - 5, ..., 5. This
situation is depicted in Fig. 2 [12]. The Wigner function W(O, cp), when integrated over
the domain of spherical angle, 0 e (0,z) and q e (-_,z), contributes the most positive
probability at precisely these locations in 0. At these 0 values there is always one peak
on the "wavy sea" that is not cancelled by a trough and so contributes a large amount of
probability. In Fig. 1 we plot the function W(_ _0)as a two-dimensional surface, and also
the normalized function W= W/'O_-_ in spherical coordinates so that the oscillations
can be viewed as variations in the surface of a sphere of radius one.
331
(e_ _d)
332
(h)
fit)
{I)
333
FIG. 1. Here we plot for 8 • (0,_ and _ e C-x,x), the normalized Wigner
function _Fo©k = WFock/_) , where wFock(o,q_) is given by Eq. (I0). The
angular momentum Fock states represented here are ljm)=/5,m) where m = O,
t-l'he-2, --3, -4, -_. When integrated over e and _, the Wigner function contributesmost positive probability precisely at the 16ca_-ons Where the angular
momentum vector for/jm) of length _ _ has z component m_ (see Fig. 2).
These contributions occur where the dominant positive crest of the Wigaer
function -- the peak that is not cancelled by any troughs -- contributes. To b_
out all the features of W(O,_) we plotit first as a two-dimensional surface
function of" 8 and _ in (a), (c), (e), (g), (i), and (k). This method of presentation
brings out the scale of the local pos/tive and negative variations of W with
respect to the plane _8; _) • 0. Then in Co), (d), (f), (h), (j), and (1), we take a global
view by plotting _,_8, _) = W{O,_)/_ on a sphere of radius one.
Z
m=0
m=-I
m=-2
m=-3
m=-4
mffi-5
h
FIG. 2.' Here we show a schematic diagram of the angular momentum
vector for the Fock states inside a sphere of radius _ _. The vectors all have
length _ _ but z component m_. These vector locations correspond to the
maximal contributions from the Wigner functions shown in Fig. 1. In
particular, the Wigner function always has an uncance]]ed dominant peak at
precisely these locations in the angle 0.
334
$ Atomic Coherent State io_fl)
We next consider the Wigner function for the atomic coherent state, Eq. (2),
p = l a,_) (a,_SI . (11)
Using Eqs. (2), (4), and (S), the coemcionts of the operator G for the density matrix,
Eq. (11), are found to be
] - 2j .I/2 2j -I/2
mm--J
(12)
The WiSher function wc°herent(_ 9) is then g/yen by Eq. (6) and is plotted in Fig. 3 for
a = ,8 = x/4, recalling that a is measured at the south pole. (Again we have
normalized W = W/'_j_).) The coherent state appears u a positive perturbation on
the surface of a unit sphere. It is a Gaussian-llke distribution located on the sphere's
surface at e = 3z/4, qp= g/4; the "Wigner toothache" state. It is just a rotation of the
ground Fock state _ner function from section 2. The Gaussian shape is analogous to
that found for the WiSher distribution for coherent states of the sin_e mode radiation
field.
(b)
FIG. 3. Here we plot the Wigner distribution Wcoherent/_+'_ for the
coherent state/a,D), Eq. (2). We choose the parameters o_=x/4 that correspond
to a Gaussian distribution localized at 0 = 3g/4, V'= 1r/4. This distribution is
qualitatively similar to that of the coherent state for photons. Again we present a
two-dimensional surface view (a) and a spherical coordinate perspective (b).
335
4 Atomic Squeezed State I_m)
We finally consider the state [12,13] ofthe angular momentum system defined by
where cA m is the normalization constant. For m ffi -6, this state -- generated by a non-
Hermitian operator -- describes the behavior of a collection of two-level atoms
interacting with a squeezed coherent photon state. In the state, Eq. (13), the x
quadrature, i.e., Jz is squeezed as per Eq. (13),
where _" is defined by
20
• -- tanh (2 I_l). (15)
This relation implies that ((AJx) 2) < /(Jz)[/2 that shows a suppression of x noise, AJ=,
in uncertainty relation ((AJz) 2) ((_Jy)2) • /(Jz)/2/4 at the _ex_pense of the y fluctuations,
AJy. Thus the states of Eq. (13) can be considered as..statable candidates for sque_e_
states of the general angular momentum system, besides, Agarwal aria run txoJ na
shown that the states, Eq. (13), are the eigenstates of the operator
(J- cosh/_'/ + J+ einh/_/)/__g'_" with the eigenvalue m, and that these states
are the analog of the two photon coherent states [2] for photons. Note further that
Eq. (13) can be written in terms of the elements of the rotation operator coefficients
)
dram, (g/2) via the relationship
where we define
djmp(v__'2) = C(J+m)! (j-m)! (j+p)V.(j...p)!)l/2
2 j
(16)
÷J (_l)q
Z (i-p-q)! qt (q+p-m)'t(j+m-.q)t "
q--j
(17)
Upon using Eq,. (13), (16), and (17) for the squeezed state, and _s. (-4)and (5) for the
definition of the Wigner function, we find the coefficients of the squeezed density
operator G to be
J J
m--j m',.-j
m'pI[.. v :_
÷,°--,,° J
(18)
where we have also introduced the value of the normalization constant. The Wigner
distribution W ulueezed (a_) obtained from Eq. (6), using Eqs. (17) and (18), is plotted in
336
Fig. 4 for j - 5 and p - -5. We take the squeezing parameter O equal to --2.13xi0-5
which, in the Agarwal and Puri system of two-level atoms interacting with a squeezed
photon bath, corresponds to a mean photon occupation number of
. sinh2 (/arctanh (e2e)}-" 50 corresponding-- to _- 2.65. The plot is again normalized
I"- /
so that the elongated Gauasian of the squeezed state appears as a "Wigner banana"
draped across the surface of sphere of radius one at the south pole. (To see this, one
must take the surface in Fig. 4a and mentally map it onto a sphere of radius one, as in
Fig. 41).) Notice that the localization of the state is squeezed in the g direction at the
expense of knowledge about the y location. Agarwal and Purl [13] have shown how the
atomic states Eq. (13) can be produced ifthe collect/on of two-level atoms interacts with a
broad band squeezed bath and if one concentrates only on the steady-state solution for
the collective system. The parameter _"characterizes the squeezed bath with average
photon number equal to sinh 2_'.
W_L t_
'ii
|
(s)
Co)
FIG. 4. Here we plot the Wigner function for a squeezed ,nmdAr momen .turn
state/_', -5) defined by Eq. (13). The function wsquxsed(0,9) is computed umng
Eqs. (6), (17), and (18) for a squeezing parameter of O- -2.13 x 10 "_ corresponding
to a mean occupation number of n- 50. In (a) we plot the function as a surface
W(O, q_) as before. We have normalized the variation in the surface in spherical
coordinates to a sphere of radius _ in (b) so that the elongated Gaussian
appears here as a "Wigner banana" draped across the surface of the sphere of
radius one at the south pole. Notice that the squeezed state is more localized in
the x" direction than the coherent state, Fig. 3, at the expense of decreased
localization or increased noise in the y'direction.
337
5 Summary and Conclusions
In summary the Wigner distribution for a genera] angular momentum state has
been derived and given explicitly for a Fork state, a coherent state, and a squeezed state.
Represented as a pseudo-probability distribution on the sphere of radius one, the
Wigner function is plotted for these three situations. These plots enable us to
understand the nonclassical nature of the states of a collection of identical two-level
atoms since the collection is described by the addition of the spin operators for each
atom.
Acknowledgements
Both G. S. A. and J. P. D. thank Professor H. Walther for his hospitality and
support at the Max-Planck-Institut fttr _mntenoptik where part of this work was done.
G. S. A. also acknowledges travel support under National Science Foundation grant
number _ 9100685, and one of us (J. P. D.) would like to acknowledge the National
Research Council for financial support.
References
[1]
[2]
E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963); see also, It. J. Glauber, Phys. Rev.
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H. P. Yuen, Phys. Rev. A I$, 2226 (1976); for many important properties of the P
function, see also R. J. Glauber in Ref. I.
[3]
[4)
[5]
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[7)
[s)
[9)
E. P. WiSher, Phys. Rev. 40, 749 (1932).
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338
[II] G. S. Agarwal, Phys. Rev. A 24, 2889 (1981).
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339



Wigner functions for nonclassical states of a collection of two-level atoms

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