We outline an analytical framework for the treatment of radial Rydberg wave
packets produced by short laser pulses in the absence of external electric and
magnetic fields. Wave packets of this type are localized in the radial
coordinates and have p-state angular distributions. We argue that they can be
described by a particular analytical class of squeezed states, called radial
squeezed states. For hydrogenic Rydberg atoms, we discuss the time evolution of
the corresponding hydrogenic radial squeezed states. They are found to undergo
decoherence and collapse, followed by fractional and full revivals. We also
present their uncertainty product and uncertainty ratio as functions of time.
Our results show that hydrogenic radial squeezed states provide a suitable
analytical description of hydrogenic Rydberg atoms excited by short-pulsed
laser fields.Comment: published in Physical Review 

A. ten Wolde

D. Delande

D. R. Meacher

D. S. McAnally

G. Alber

I. Sh. Averbukh

J. A. Yeazell

J. Hare

J. Mostowski

J. Parker

J.-C. Gay

L. S. Brown

M. M. Nieto

M. Nauenberg

R. G. Hulet

Robert Bluhm

V. A. Kostelecký

V. Alan Kostelecký

V. P. Gutschick

W.-M. Zhang

Z. Dačić Gaeta

English

arXiv

Crossref

Radial squeezed states and Rydberg wave packets

We outline an analytical framework for the treatment of radial Rydberg wave packets produced by short laser pulses in the absence of external electric and magnetic fields. Wave packets of this type are localized in the radial coordinates and have p-state angular distributions. We argue that they can be described by a particular analytical class of squeezed states, called radial squeezed states. For hydrogenic Rydberg atoms, we discuss the time evolution of the corresponding hydrogenic radial squeezed states. They are found to undergo decoherence and collapse, followed by fractional and full revivals. We also present their uncertainty product and uncertainty ratio as functions of time. Our results show that hydrogenic radial squeezed states provide a suitable analytical description of hydrogenic Rydberg atoms excited by short-pulsed laser fields

Bluhm, R.

Kostelecký, V.A.

IUScholarWorks

PHYSICAL REVIEW A VOLUME 48, NUMBER 6 DECEMBER 1993
Radial squeezed states and Rydberg wave packets
Robert Bluhm' and V. Alan Kostelecky
'Physics Department, Colby College, 8'aterUille, Maine 04901
Physics Department, Indiana Uniuersity, Bloomington, Indiana 47405
(Received 19 August 1993)
We outline an analytical framework for the treatment of radial Rydberg wave packets produced by
short laser pulses in the absence of external electric and magnetic fields. Wave packets of this type are
localized in the radial coordinates and have p-state angular distributions. We argue that they can be de-
scribed by a particular analytical class of squeezed states, called radial squeezed states. For hydrogenic
Rydberg atoms, we discuss the time evolution of the corresponding hydrogenic radial squeezed states.
They are found to undergo decoherence and collapse, followed by fractional and full revivals. We also
present their uncertainty product and uncertainty ratio as functions of time. Our results show that hy-
drogenic radial squeezed states provide a suitable analytical description of hydrogenic Rydberg atoms
excited by short-pulsed laser fields.
PACS number(s): 32.80.—t, 03.65.Ge, 11.30.Pb
Excitations of Rydberg atoms by short-pulsed laser
fields produce wave packets that are localized in the radi-
al coordinates [1,2]. These wave packets initially oscillate
with the classical Keplerian period between inner and
outer apsidal points corresponding to those of the classi-
cal Keplerian orbit [3]. They subsequently collapse and
then eventually revive almost to their original shape [4].
In the interval between collapse and full revival, subsidi-
ary wave packets form. These are known as fractional re-
vivals, and their orbital periods are rational fractions of
the classical period [5].
Depending on the excitation process and whether
external fields are present, several different electronic or-
bital geometries are possible for the quantum-mechanical
motion. For single-photon excitations without external
fields, the radial motion is localized but the angular dis-
tribution is that of a p state. For multiphoton excita-
tions, the angular distribution can be that of a d or higher
state. If an external electric field is present at the time of
excitation, a parabolic wave packet is formed. This ex-
hibits beats in the angular momentum [6]. If instead the
atomic excitation is in the presence of a strong radio-
frequency field or crossed electric and magnetic fields,
circular atoms may be produced [7]. These are Rydberg
atoms with high angular momentum that are localized in
the angular coordinates.
Since Rydberg wave packets are localized and exhibit
some features of the classical motion, a description in-
volving some type of coherent state [8) would seem ap-
propriate. This possibility has been realized in a number
of cases for circular and elliptical quantum geometries
[9,10), suitable for characterizing the motion of a hydro-
genic Rydberg atom excited by a short laser pulse in the
presence of external fields. These coherent states are ei-
ther a superposition of angular-momentum eigenstates or
eigenstates with a large value for the angular momentum.
However, when a wave packet is produced by single-
photon excitation in the absence of external fields, a @-
state eigenfunction with I = 1 results.
In this Rapid Communication, we show that a particu-
lar kind of analytical squeezed state, called a hydrogenic
radial squeezed state, can be used to describe aspects of a
wave packet generated by exciting a hydrogenic Rydberg
atom with a short laser pulse in the absence of external
fields. Like coherent states, squeezed states [11] for a
given system are quantum wave functions minimizing an
operator uncertainty relation. However, for squeezed
states, the ratio of the operator uncertainties typically has
a different value than that of the ground state. This
means that the squeezed-state uncertainty product and
ratio display a distinct time dependence.
Direct attempts to obtain analytical squeezed states in
the variables r and p, meet with a variety of obstacles.
Instead, we have adopted an indirect method suggested in
Ref. [10], in which a change of variables is made from r
and p, to a new set, R and P, chosen to have certain
properties of harmonic-oscillator variables and hence
more amenable to analysis. A complete derivation of the
analytical form of a general class of radial squeezed
states, also valid for Rydberg atoms other than hydrogen
(in particular the alkali-metal atoms used in experiments),
is somewhat lengthy and is given elsewhere [12]. We re-
strict ourselves here to summarizing the results for hy-
drogenic p-state excitations. In what follows, we work in
atomic units with A =e =m, = 1.
The effective radial potential for a hydrogenic Rydberg
atom with angular momentum I = 1 is
V,rr(r) =(1 2r)/2r . For this cas—e, it turns out that the
appropriate new quantum operator R replacing the usual
r is R —= (2 —r) I2r, while P =p„remains unchanged. The
associated commutation relation is [R,P]= ir, lead-—
ing to the uncertainty relation b,R b,P ~
—,
' ( r ) . We
take the hydrogenic radial squeezed states as the wave
functions minimizing this uncertainty relation. They are
given by
g(r)=Nr e 'e
where a, yo, y, are three real parameters and X is a nor-
malization constant [13].
1050-2947/93/48(6)/4047(4)/$06. 00 48 R4047 1993 The American Physical Society
ROBERT BLUHM AND V. ALAN KOSTELECKY
t =-, TiC
5000
L
10000
7 (a.u. )
('~)
15000 5000 10000
v (a.u. )
(b)
15000
I
5000 10000
~ (a.u. )
(c}
15000 5000 10000
v (a.u. )
(cj)
15000
t = 4Tcg
5000 10000
~ (a.u. )
(e)
15000 5000 10000
~ (a.u, )
15000
t .
q t„„+—.1T
5000 10000
~ (a.u. )
(g)
15000 5000 10000
~ (a.u. )
(h)
15000
v+ Tc~
the radial
~~nnhAI'P&NRiR. ~
5000 10000
~ (a.u. )
150005000 010000 15000
~ (a.u. )
(j)(1)
n with n =85. The unnormalized radial probability ens' y ' pe ' ' ' de sit is lotted as a function ofRdi1 q d t t ofhydog
t =0 (b) t= Tcl (C) t =T 1, (d) t = cl e cl ecoordinate r in a.u. at times (a)
t =2.6 nsec.2 Tcl~ (&) t rev~ V rev cl~cl p) t —t + T, where Tcl =93 3 Psec and trev
III) ~ ~ If 1llf ll ~ ~ I II ~ ' ll II
RADIAL SQUEEZED STATES AND RYDBERG WAVE PACKETS
We fix the three parameters a, yo, y, for a particular
hydrogenic radial squeezed state at time t =0 by match-
ing to the corresponding hydrogenic Rydberg wave pack-
et at its point of closest-to-minimum uncertainty. This
occurs at r =r,„„represe nti ng the outer apsidal point [1].
In the present case, we assume that the laser excites a
range of energy states with principal quantum numbers
centered on the value n and that contributions from con-
tinuum states are negligible. We take the outer apsidal
point to be r,„,=n +n+n —2. This point is near the
outer classical apsidal point given by r, =n (1+e), where
the eccentricity e is e=+1—1/n . We perform the
match by imposing the three conditions
(p„)=0, (r ) =r,„„(H) =E (2)
Here, E„=—1/2n is the energy corresponding to n
Roughly, these conditions mean that the hydrogenic radi-
al squeezed state at t =0 is located at the outer turning
point, has no radial velocity, and has specified energy.
The conditions (2) determine the three parameters o., yo,
y, in terms of n and l=1. Our full three-dimensional
wave packet at t =0 is then given by I'&o(8, $)g 7 &(r).
The time evolution of hydrogenic Rydberg wave pack-
ets has been previously studied [5]. After its formation, a
packet is expected to execute radial oscillations with
periodicity equal to the classical orbital period
T,& =2mn . Significant self-interference due to the gradu-
al decoherence of the packet is anticipated at a time
t;„,-nT, ]/36n, where 6n measures the range of dom-
inant principal quantum numbers in the initial packet.
Later, the packet should reconstitute itself almost to its
original shape. This full revival is expected to occur at a
time t„,= —,' nT, &. It should persist for several orbits, per-
forming oscillations with period T„. Between t;„, andt„, there should be times t, when the packet is gathered
into r spatially separated pieces, oscillating with wave-
function periodicity T„=T,]/r. Among the values of t„
are the ones we discuss below at t„=t„,/r.
The analytical form of the hydrogenic radial squeezed
states permits relatively simple expressions to be found
for the time evolution. As these are somewhat lengthy,
we do not provide them here [12]. Instead, we illustrate
key features of the time evolution of the hydrogenic radi-
al squeezed states by focusing on a particular example:
radial p-state excitations of hydrogen centered on n =85.
The associated classical orbit is one of high eccentricity,
e = 1, and the period is T„=93.3 psec. The outer apsidal
point is r,„,=2n =14450 a.u. Together with the condi-
tions (2), this fixes the parameters of the radial squeezed
state as a = 168.225, yo =0.011 746 5, and y & =0.
Figure 1 shows the radial probability density
f(r)=r g 7,(r)I for this radial squeezed state at vari-7
ous times. The initial (r =0) form of the state is displayed
in Fig. 1(a). The packet is located around r,„,. Its
smooth, peaked shape rejects a small uncertainty prod-
uct Arhp, =0.5015. The initial motion of the radial
squeezed state is towards the inner turning point, near
the origin.
Figure 1(b) shows the packet halfway through its first
orbit, t =
—,'T„. The oscillatory nature of the probability
distribution reAects the quantum nature of the packet
near the core. Intuitively, the electron moves faster there
and so has greater momentum, which in turn is rejected
in the greater local curvature of the wave function. The
uncertainty product there is Ar Ap„= 59.5.
Figures 1(c) and 1(d) show the radial squeezed state
after one and two full orbits, at t = T,& and t =2T,&, re-
spectively. The packet is evidently repeatedly returning
to the outer turning point, but after each successive orbit
more decoherence appears and the quantum nature of the
object becomes more apparent. This is rejected in the
uncertainty product, which grows from Arhp„=1. 6 at
t = T ] to Ere, =8.0 at t =2T &.
At the time t;„„which is near -4T,& in this example, it
is no longer possible to attribute a single peak to the radi-
al probability distribution. This situation is shown in Fig.
1(e). The remnants of the original peak are still visible
but there are many oscillations and other subsidiary
peaks present. The decoherence of the packet is reAected
in the uncertainty product, which at Ere, =45.4 is
comparable to that of the highly quantum distribution in
Fig. 1(b) appearing at t = —,' T„.
At the time t„=
—,
' t„„afractional revival is expected to
appear. The corresponding distribution for the radial
squeezed state is shown in Fig. 1(f). Three spatially
separated packets are indeed visible. Similarly, a frac-
tional revival containing two spatially separated packets
appears at t2—-—,'t„,. Its forms are shown at that time
and at half a classical period later, at t =
—,'t„,+ —,' T,&, in
Figs. 1(g) and 1(h), respectively. These figures demon-
70
,)0
A7 AP„40
(B,.11.)
10
0
0 0.0;) 0. 1 0. 1:) 0. '7
t ( 77 s 0'(')
(t)
0. 7;) 0.:3 0.:35
1.4
0 2
0
0 0.0 )
I
0, 1 0. [.") 0. '~
f. ( 7 7 .s ( ' ('.)
(]))
0 0.3 0.:3 )
FIG. 2. (a) The uncertainty product ArAp, as a function of
time in nanoseconds for a radial squeezed state of hydrogen
with n =85. (b) The ratio of the uncertainties Ar/Ap„ in units
of 10 a.u. as a function of time in nanoseconds for a radial
squeezed state of hydrogen with n =85.
R4050 ROBERT BLUHM AND V. ALAN KOSTELECKY
strate that the periodicity of the distribution in time is
indeed
—,
' T,&, in agreement with expectation.
Figures 1(i) and 1(j) display the radial squeezed state
near the revival time, t =t„,=2.6 g.sec, and one classical
period later. As expected, the distribution is once again
similar to that of a single peak located at the outer turn-
ing point, evolving with the classical periodicity.
Figure 2(a) shows the uncertainty product b, rbp„ for
the radial squeezed states as a function of time. Again,
this plot is for hydrogen with n =85. The smallest value
of the uncertainty product in r and p, is close to —,' at the
initialization time t =0, when by construction the uncer-
tainty product in R and P is minimized. The cyclic
behavior of the uncertainty product as a function of time,
expected for a squeezed state, is revealed. The classical
orbital periodicity is again manifest during the first few
orbits, until the wave function begins to decohere. The
product hr bp„ then remains large until the revivals form.
Figure 2(b) displays the uncertainty ratio hr/bp„as a
function of time for the same hydrogenic radial squeezed
state. This provides a measure of the relative amount of
squeezing in the two coordinates. Overall, it is apparent
that the squeezing in p, is greater than that in r by five or
six orders of magnitude. In fact, initially
b, r /bp„= 1.2 X 10 . Halfway through the first orbit, at
t=
—,'Td, the relative squeezing reaches its minimum of
b,r/bp„=S. OX10 . The uncertainty ratio continues to
change with time, but remains large so that p„remains
squeezed.
In summary, the characteristic features of the hydro-
genic radial squeezed states are similar to those of Ryd-
berg wave packets in atoms excited by a short laser pulse
without external fields. Further details of our analysis,
including in particular the generalization of our analyti-
cal framework via supersymmetry-based quantum defect
theory [14] to the case of alkali-metal atoms, is presented
elsewhere [12].
We enjoyed conversations with Charlie Conover and
Duncan Tate. R.B. thanks Colby College for a Science
Division Grant. Part of this work was performed while
V.A.K. was visiting the Aspen Center for Physics.
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[13]In Ref. [10],attention is given to states of the form (1) but
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atoms, but not to three-dimensional p-state excitations for
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Radial Squeezed States and Rydberg Wave Packets

Radial Squeezed States and Rydberg Wave Packets

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