The possibility of representing the quantum states of a harmonic oscillator not on the whole alpha-plane but on its one dimensional manifolds is considered. It is shown that a simple Gaussian distribution along a straight line describes a quadrature squeezed state while a similar Gaussian distribution along a circle leads to the amplitude squeezed state. The connection between the one dimensional representations and the usual Glauber representation is discussed

Adam, P.

Foldesi, I.

Janszky, J.

Vinogradov, An. V.

English

NASA Technical Reports Server

N94-10570
ONE DIMENSIONAL REPRESENTATIONS IN
QUANTUM OPTICS
J. Janszky, P. Adam, I. FSldesi
Research Laloratory /or Crystal Physics
PO Boz 18_, H.l$Ot Budapest, Hsn]ary
An. V. Vinogradov
Leledev Inatit,_te o/ Pkpiee, Moscow, Rt##id
Abstract
The possibility to represent the quantum states of a harmonic oscillator not on the whole
a.plane but on its one dimensional manifolds is considered. It is shown that a simple Gaus-
sian distribution along a straight line describes a quadrature glueesed state while a similar
Gaussian distribution along a circle leads to the amplitude squeesed state. The connec-
tion between the one dimensional representations and the usual Clauber representation is
discussed.
1 Introduction
There are several widely used representations to describe a state of a quantum oscillator in the
Hilbert space. The most natural one is the expansion of the state into the number state
(3O
I_>= _ _.l'>.
n=O
A nosher well known possibility is the coherent state representation [1,2]
1), >= 1jrf )'(a*)exp(-{o [2/2)is > d2o, ¢12a = ¢l(R¢o)d(Imo),
(1)
(_)
/{o*) being an analyticalhnction of o*. Here the state is represented by a superposition of
nonorthogonal coherent states all over the complex o-plane.
As already Glauber pointed out, there is an infinite number of ways of expanding any state
in terms of coherent states due to the overcompleteness of the latter states
I/>= / Io> (3)
here the expansion function G(a,a*) may be a rather general function of a and a*. Being
confined to some given class of functions the uncertainty in finding G(o,o*) can be reduced. In
55
https://ntrs.nasa.gov/search.jsp?R=19940006115 2020-06-16T21:09:36+00:00Z
this paper we shall deal with such representations that correspond to kern functions G(o,o*)
leadingto integration over a one dimensional manifold of the o-plane in Eq. (3). The possibility
to represent any state on a subspace of the complex plane comes from Cahill's theorem on
overcompleteness [3,4[. We shall show that such nonclassical states as the quadrature and
amplitude squeezed states can be represented very naturally by superposition of coherent states
along a straight line or along a circle in the o-plane correspondingly.
2 Representation along a straight line
The most simple states emerging from superposition of coherent states are the even [ x, + >
and the odd Ix,- > states
t_,* >-- _+(I• > ÷ I-* >), (4)
Ix,- _= c-fix >- I-_), (5)
where [ x > is a usual coherent state with real eigenvahe of the annihilation operator a I x >=
x [ x >. It is remarkable that the even state [ x, + > being a superposition of two classical
states is squeer.ed [6]
1 x2/[1 ÷ exp(2.r2)] (6)(_'2)2= _-
where _1 and a2 are the Hermitian quadratures of the annihilation operator.
The squeezingcan be further enhanced by adding the vacuum state to I x, + >
Ix,p >= _v(I• > +p Io> + I-: >). (7)
This way one can achieve a variance (_a2) 2 = 0.0661 instead of 0.111 for [ x, + > or 0.26 for
the vacuum state. Superposing more and more even states to it one can get even more squeesing
I/>= _/(_) I_ > 4_. (s)
--00
In fact for any positive even function .f(x), but for the/(x) = 6(:r) describing the vacuum,
the state defined by Eq. {8) is squeezed. Amoet important particular case is the Gaussian
superposition function [6,6]
describing the usual squeezed vacuum state with uncertainties of the quadratures
(9)
(_1) 2= (I+72)/4, (A.2)2 = 1f4(1+72). (lo)
56
Similardistributions can be constructed not only along the real axis but along any straight
line. For example the squeezed coherent state with coherent signal c_and squeezing parameter
= r exp(i0) can be written in the form [7,8]
[o,£ >- Tf(x'°'£)l°+exp(iS/2)x>dx' (11)
--00
/(x,a,[)=cexp(-x2/72-i6x), _=Im[o*exp(iO/2)], 7= c_f_-l., (12)
As the Gaussian superposition of coherent states of Eq. (8,0) was a. useful generalisation of
the e_en states of Eqs. (4,7) analogously one can build an odd state [ 7,1 > resembling Eq. (5)
17,1>= T G(x,7,1) [.v>dx, (13)
--(30
G(x,7,1) = clzexp(-ar2172), c 1 = _ (1 + )3/j..
The mean photon number and the uncertainty of the quadrature a2 in this state are
<7,1[ata [7,] >- 1÷ 33'4
4(1 + 7Z) '
and
(14)
(15)
3 (16)(Aa2)2 - 4(1 + 72)"
We can see that the state [ 7,1 > coincides with the one photon state [ 1 > in the limit 7 = 0
and with increasing 7 at < 7,1 [ ata [ 7,1 >= 2, 7 = V_ it becomes squeezed.
Similarlyone can define states [ 7, a > withx n instead of • in their weight function G(z,7, n).
Superpositions of such states leading to Hermite polynomial weight functions are rather remark-
able
--00
An(x) = _fv_l(2gn!)Hn(-_)exp(-x2/2),
The states [ An > are orthonormali,.ed (< An [ Am >= _nm), satisfy the relation
alhm>=_llAn+l>+_iAn-I >,,
and correspondingly
(17)
(18)
(10)
• 57
<An[.[A.>=O, <Anla21A.>= _(2.+1), <Anlatalh.>=_
The projection operator constructed from the Hermite states
oo
t'h-- _ Ih.><h.I,
n=O
is a unity operator both for the coherent and photon number states
< x IPh It >=< x It >= exp[-(x- _)2/2],
5n+1 (20)
(21)
(22)
<. le A Im >= ,5.m, (23)
which shows that any state can be represented by them. For example one can expand a [ f >=
f_°°oo.f(x) lx > tx state into the IAn > states
oo
11 >= E I. IA. >, (24)
fl-O
where
In = / T ]{x)ba(Y)exPI-(x-Y)2/2]dxdy"
-oo
(26)
3 Representation along a circle
Let us now consider a state emerging from superposition of coherent states with the same
amplitude j a [= R i. e. we choose only those coherent states which lie on the same circle in
the a-plane [6].
i F, R >= exp ,u22,/2) f F(÷)[ R exp(i÷) > d_.
If the radius of the cir.cle is chosen big enough so that Eq. (2} can be replaced by
(2s)
F(÷) = R exp (i_) [
la I< R
I1 >= I_ [ /(_*)exp(-I_ t2/2)[_ > _%, (27)
_rla (<R
then we can find connections between the distribution function F(¢) and Olauber's weight
function f(a*)
/(o*) exp (-I °_ 12)d2or, z = Rexp(i0) (28)
2--Or
58
and
f(o') = / F(_)_p(o*_)a_. (20)
We note that if one knows the time behaviour of the annihilation operator a(t) the analytic
expansion function f(or*, t) can be found from the expression [9]
/(o*,t) = /Fvx(,,t)exp(-1¢ 12-qo*)/If fF¢x(,,t)exp(-I_ 12), (30)
where X(_, t) is the normally ordered characteristic function (p being the density operator)
X(_,t) = Trlpexp(_,t(t))exp(-q*,(t))l. (31)
Using Eq. (2S) we tlnd for the n-photon state and the coherent state correspondingly
F(÷,m) = V_R -n exp(-im÷), (32)
F(¢,a) = zexp(-la [2/2), [a l< R. (33)
Z--O
According to Eq. (32) we can obtain the coefficients of the n-photon representation ¢n of Eq.
(1) if we know the distribution function F(¢)
c. = RnFn/V_,
where Fn are Fourier, coefficients of F(O)
(84)
OO
F(_)= _ exp(-in÷)Fn. (35)
n=0
An interesting state is the state with Oaussian distribution function ] w >
F(÷,w) = cu exp(-i$# - ._¢2). (86)
In case of extremely large u it describes the usual coherent state while in the opposite limit
it is the n-photon state (n = 6). Between these states it will be an amplitude squeezed banana
state. Graphically it can be understood if one visualizes how with decreasing u the muffin-like
coherent state going through a squeezed crescent-like state deforms along the circle into the
donut-like n umber state.
It is also worth mentioning that the Gaussian superposition of coherent states along an arc
are not only describe amplitude squeezing [10,11 t but they are also approximate number-phase
intelligent states [10] associated with the Peg,g-Barnett phase operator [12].
Remarkable feature of this state is the complete analogy with the usual quadrature squeezed
state discussed in the previous Section, as the Gaussian arc distribution is amplitude squeezed
while the Gaussian straight line distribution is quadrature squeezed. Moreover, as the even
59
superposition of two coherent states from Eq. (4) can be derived by truncation from the straight
line Gaussian state of Eqs. (6,0) so Schleich's superposition state I13]
>= ca÷(Iaei÷/2 > ÷ Ioe-;÷/2>), (37)
similarlycan be considered as a truncated arc Gaussian state of Eqs. (26,36).
A physical example, the so called phonon squeeging [14], where an arc distribut.ed state
occurs is the Franck-Condon transition induced by short coherent light pulse in a molecule
[6,14,15]. It is worth mentioning that using Eq. (28) one can to some extent purposefully shape
the molecular vibrational state by special choice of the characteristics of the exiting light pulse.
For example we showed that by appropriate linear chirp the vibrational state can be turned
in the c_ -plane while using nonlinear chirp the amplitude squeezed vibrational state can be
deformed into a typical quadrature squeezed form.
4 Acknowledgments
This work was supported by the National Re_rch Foundation of Hungary under contra_t 1444.
References
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Inc., New York, 1968).
[3] K. E. Cahill, Phys. Rev. 188, B1866 (1968).
[4] A. Vourdas, Phys. Rev. A 41, 1683 (1090).
[8] J. Janszky and An. V. Vinogradov, Phys. Rev. Lett. 64, 2771 (1900).
[6] V. Busek and P. L. Knight, Optics. Comm. 81,331 (1901), V. Busek, A. Vidiella-
Barranco and P. L. Knight, Phys. Rev. A 48, 6870 (1002).
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[8] J. Janszky, P. Adam and An. V. Vinograd_, Phys. Rev. Lett. 68, 3816 (1002).
[O]P. Adam and J. Janssky, Phys. Lett. A 149, 67 (1090).
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[11] J. Jansr.ky, P. Adam, M. Bertolottiand C. Sibilia, Quantum Opt. 4, 163 (1092).
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6O


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