When a lithium atom in a Rydberg state (n  80) is exposed to a short, intense microwave pulse we find that substantial population is left in extremely highly excited states (n . 120), in spite of the fact that the microwave field amplitude is more than 40 times larger than required to classically ionize these states

Gallagher, T. F.

Griffith, W. M.

Noel, Michael W.

English

Scholarship;  Research;  and Creative Work at Bryn Mawr College

Bryn Mawr College
Scholarship, Research, and Creative Work at Bryn Mawr
College
Physics Faculty Research and Scholarship Physics
1999
Population Trapping in Extremely Highly Excited
States in Microwave Ionization
Michael W. Noel
Bryn Mawr College, mnoel@brynmawr.edu
W. M. Griffith
T. F. Gallagher
Let us know how access to this document benefits you.
Follow this and additional works at: http://repository.brynmawr.edu/physics_pubs
Part of the Physics Commons
This paper is posted at Scholarship, Research, and Creative Work at Bryn Mawr College. http://repository.brynmawr.edu/physics_pubs/76
For more information, please contact repository@brynmawr.edu.
Custom Citation
Michael W. Noel, W. M. Griffith, and T. F. Gallagher, "Population Trapping in Extremely Highly Excited States in Microwave
Ionization," Phys. Rev. Lett. 83, 1747 (1999).
VOLUME 83, NUMBER 9 P HY S I CA L REV I EW LE T T ER S 30 AUGUST 1999
Population Trapping in Extremely Highly Excited States in Microwave Ionization
Michael W. Noel, W.M. Griffith, and T. F. Gallagher
Department of Physics, University of Virginia, Charlottesville, Virginia 22901
(Received 18 December 1998)
When a lithium atom in a Rydberg state (n ! 80) is exposed to a short, intense microwave pulse
we find that substantial population is left in extremely highly excited states (n . 120), in spite of the
fact that the microwave field amplitude is more than 40 times larger than required to classically ionize
these states.
PACS numbers: 32.80.Rm, 42.50.Hz
The interaction between an atom and a strong oscil-
lating field has revealed a multitude of interesting phe-
nomena, often involving the ionization of the atom [1].
Among the most fascinating are those cases in which
ionization does not occur, in spite of its appearing to
be very probable at first glance. Several immediately
relevant examples occur in the microwave ionization of
Rydberg atoms. Classical stabilization of Rydberg atoms
against microwave ionization occurs when the microwave
frequency or one of its harmonics matches the classical
Kepler frequency of the Rydberg atom [2–4]. Of particu-
lar interest for this work, it was observed long ago in clas-
sical calculations that the Rydberg atom could be left in
very high lying states by a microwave pulse [5–7]. More
recently there have been extensive calculations of the fi-
nal state distributions subsequent to a microwave pulse
[8–11]. At low microwave fields the final state popu-
lation is found to fall off roughly exponentially with the
change in n of the final state from the initial state, and
experiments have confirmed these predictions [8,9]. Here
n is the principal quantum number. At fields high enough
to produce ionization, the calculated distributions of final
states are continuous from the initially populated state to
the continuum, with structure due to resonant single pho-
ton transitions [11]. To date, however, no experiments
have been carried out to test these predictions.
Here we report the analysis of the final state distribution
of lithium Rydberg atoms exposed to intense, short mi-
crowave pulses. We find the rather surprising result that
when n ! 80 lithium atoms are exposed to 0.5 ns long mi-
crowave pulses .35% of the population is transferred to
states of n . 120, with no apparent population in states
of n , 120, contrary to the calculations, in spite of the
fact that the microwave field amplitude exceeds the classi-
cal ionization field of n ! 120 by a factor of 40. As the
pulse is made longer the number of n . 120 atoms sur-
viving decreases, but it is still !3% with a 10 ns pulse.
Recent measurements of Rydberg atoms kicked by a se-
ries of unipolar pulses also show atoms left in very high n
states, again in disagreement with calculations [12]. Us-
ing the fact that there can be no transfer of energy from
the field to the electron when it is far from the ion core and
accounting for the state redistribution that occurs when the
electron comes near the core we have developed a simple
picture that reproduces the observed population transfer to
what have been termed extremely highly excited states [5].
The general scheme for this experiment was to excite
lithium atoms to a high n Rydberg state, allow them
to interact with a short microwave pulse, and measure
the state distribution of the atoms that remained bound.
Lithium atoms, from an effusive source, entered a piece of
WR90 waveguide through a small hole in one of its sides.
Three nanosecond dye laser pulses entered the waveguide
through a hole on the opposite side and excited the
atoms to a Rydberg state through the 2s ! 2p, 2p ! 3s,
and 3s ! np transitions. The final laser was polarized
orthogonal to the microwave field so as to excite jmj ! 1
states where m is the magnetic quantum number. With
the laser bandwidth of 4 GHz, we could selectively excite
Rydberg states beyond n ! 100.
The microwave pulse was formed by mixing the
11.5 GHz continuous wave output of a sweep oscillator
with a fast rising edge (!100 ps) voltage step in one
mixer and a fast falling edge (!100 ps) voltage step in a
second mixer. The relative delay between the steps could
be precisely adjusted to give the desired pulse width.
A subsequent mixing stage with a 1 to 100 ns pulse
(depending on the pulse length set with the fast steps)
served to further suppress the wings of this fast pulse.
The microwave source was phase locked to the pulse
envelope, which allowed us to monitor the microwave
electric field, not just the field envelope. After two stages
of preamplification we passed the pulse through a fast mi-
crowave switch which was gated open for a time slightly
longer than the pulse. Finally, the pulse was amplified in
a gated traveling wave tube amplifier and passed through
a voltage controlled attenuator with which we could
vary its power over 40 dB. The pulse was coupled into
and out of the short piece of waveguide containing the
prepared Rydberg atoms after which it was attenuated
and monitored with a sampling oscilloscope. We cali-
brated the power using a Hewlett Packard 432A power
meter and estimate a microwave field uncertainty of 10%.
After the microwave interaction a negative field ioniza-
tion pulse was applied to a septum below the atoms in the
waveguide. The field ionized electrons passed through a
0031-9007"99"83(9)"1747(4)$15.00 © 1999 The American Physical Society 1747
VOLUME 83, NUMBER 9 P HY S I CA L REV I EW LE T T ER S 30 AUGUST 1999
small hole in the top of the waveguide and were collected
with a dual microchannel plate detector. By adjusting a
bias voltage on the septum to minimize excitation of high
ns states we reduced the static electric field to less than
50 mV"cm, which can ionize states with n . 280.
With the laser tuned to the desired Rydberg state
we repetitively scanned the microwave amplitude over
many laser shots as we downloaded the time resolved
electron signals recorded on a digital oscilloscope on each
shot. The time resolved electron signal revealed the state
distribution of the atoms which remained bound after the
microwave interaction. Electrons left in high n states
were ionized early during the slowly rising field ionization
pulse and arrived early in time, whereas those of lower n
required higher fields and consequently arrived later in
time. We calibrated the arrival time versus n by directly
exciting known states, and we can distinguish states up
to about n ! 130, above which the electrons all arrive
at the detector at the same time. Electrons which were
ionized directly by the microwave pulse dispersed before
the arrival of the field ionization pulse, so they were not
registered at the detector.
In Fig. 1 we show a typical set of data collected
according to the above prescription for an initial state of
n ! 80 interacting with a 0.5 ns (5 cycle) pulse. The
Kepler period is given by 2pn3 in atomic units, and
for n ! 80 it is 78 ps. The primary classical resonance,
where the Kepler period is equal to the microwave period,
FIG. 1. Bound state distribution as a function of field strength
for an atom initially prepared in the 80p state interacting with
a 0.5 ns pulse. In (a) electron signal is mapped to a gray scale
with black corresponding to large signal and white to no signal.
The horizontal axis, which experimentally is the detection time
of an electron after being ionized by the slowly rising field
ionization pulse, has been calibrated according to n. The labels
are placed at the peak in the arrival time of the electron signal
for a given n. Bound states above n ! 130 are not resolved
and we label them collectively with “limit.” (b) Electron signal
integrated over all n (light line) and electron signal integrated
from 120 , n , ` (bold line). The three horizontal lines in
(b) are the 10%, 50%, and 90% ionization fields from top to
bottom, respectively.
occurs at n ! 83. The duration of the short 0.5 ns
pulse matches the Kepler period at n ! 150. The state
distribution as a function of field amplitude is plotted in
Fig. 1(a) by mapping the electron signal to a gray scale
with black corresponding to a large signal and white to no
signal. Figure 1(b) shows the electron signal integrated
over all n, the total remaining atom signal, as well as the
integrated state population for 120 , n , ` as functions
of field amplitude. Finally in Fig. 2 we show several
lines out of the full data set in Fig. 1(a). These are the
state distributions at microwave amplitudes of 2 V"cm
(no ionization), 9.5 V"cm (10% ionization), and 48 V"cm
(90% ionization).
In both Figs. 1 and 2 it is apparent that when a
significant fraction of the atoms is ionized, a surprisingly
large fraction of the remaining bound population has
been transferred to high n states (as much as 35%).
Furthermore, some population remains bound in these
high n states for fields as high as 60 V"cm, which is more
than 40 times the field necessary to classically ionize an
atom of n ! 120.
In an effort to explain the remarkable accumulation
of population in these high n states, even in very
strong microwave fields, we consider the interaction of
a classical Rydberg atom with the microwave field. If
the electron is far from the core it simply oscillates in
the field as if it were a free electron. The amplitude of
the oscillations for a 60 V"cm field is 0.2 mm, much less
than the size of the orbit for n . 120. In contrast, when
the electron returns to the core it can gain energy from
or lose it to the oscillating field, depending on the phase
of the field. Since the orbit time for n . 150 is longer
than our microwave pulse, if such states are formed in
the first cycle of the pulse, the field is off by the time
the electron returns to the core, and the electron remains
bound in these high n states.
With a 9.5 ns (110 cycle) pulse there is still some
excitation of high n states, however, the amount of
population transferred to these states is much smaller.
As shown by Fig. 3, the maximum transfer for an atom
prepared in the 80p state is about 3%. In this case
65758595limit
principle quantum number, n
0
1
ele
ctr
on
 si
gn
al 
(a
rb
itr
ar
y u
nit
s)
FIG. 2. State distributions from Fig. 1 for field amplitudes of
2 V"cm (no ionization, light line), 9.5 V"cm (10% ionization,
bold line), and 48 V"cm (90% ionization, dotted line).
1748
VOLUME 83, NUMBER 9 P HY S I CA L REV I EW LE T T ER S 30 AUGUST 1999
FIG. 3. (a) State distribution as a function of field strength for
an atom initially prepared in the 80p state interacting with a
9.5 ns pulse. (b) Electron signal integrated over all n (light
line) and electron signal integrated from 120 , n , ` (bold
line). For clarity the bold line in (b) has been expanded by a
factor of 3.
the Kepler period does not exceed the pulse width until
n . 400, and a wave packet at n ! 280, the highest
lying bound state possible in our experiment, passes the
core 3 times during the microwave pulse. Each time the
electron passes the core there is some probability that it
will be ionized by the microwave field, so with 110 field
cycles it is not surprising that few bound atoms are left.
As might be expected, with a 100 ns (1150 cycle) pulse
we did not see any excitation to high n states for any of
the initial states used (n ! 47 to n ! 94). In this case
the Kepler period does not exceed the pulse width until
n . 870 and at n ! 280 the electron passed the core
30 times during the microwave pulse.
The dynamic nature of the above classical description
can also be incorporated into a simple model for the
final state distributions. In this model we consider the
microwave interaction one cycle at a time in the following
iterative procedure. The effect of the first cycle is to
spread the population from its initial state over an energy
band of states. The next cycle spreads the population
of each of these states in an identical way for n , n1
and does nothing for states with n . n1, where n1 is the
principle quantum number at which the Kepler period is
equal to the microwave period. On the third field cycle
atoms put into states with n1 , n , n2, by the first cycle,
where n2 is the principal quantum number for which the
Kepler period is twice the microwave period, as well as
atoms put into states with n , n1 by the second field
cycle, are spread over the energy band. This process is
repeated for each field cycle, with those atoms that have
returned to or remained near the ion core being spread
over the energy band.
The evolution of the state distribution during this itera-
tion process is shown in Fig. 4 for a five cycle pulse in-
teracting with an atom initially in n ! 80. The function
60708090limit
principle quantum number, n
po
pu
lat
ion
 (a
rb
itra
ry
 u
nit
s)
0
1
FIG. 4. Model of the state distributions for the interaction
between a five cycle pulse and an n ! 80 Rydberg atom. We
show distributions after the first (light line), third (dotted line),
and fifth (bold line) cycles.
used to spread population was a Gaussian whose width
was adjusted so that, after five iterations, 10% of the fi-
nal distribution had positive energy for comparison to the
10% ionization trace in Fig. 2. For low n states the mi-
crowave field oscillates slowly compared to the classical
period, and ionization proceeds via field ionization rather
than through the diffusion process seen when the time
scales are comparable. In order to account for this ef-
fect the Gaussian was cut off sharply at n ! 58, where
the microwave period is 3 times the Kepler period. This
model contains three essential parameters, the width of the
distribution with which we spread population and the low
and high n cutoffs for the interaction, all of which have
clear physical origins. The gross features of the final dis-
tribution, the peak at high n, and the broad tail to lower
n, are relatively insensitive to changes in these parameters
of the model, and it is evident that the model captures the
essential features of the data shown in Fig. 2.
Finally, we have studied the excitation to the high
n states as a function of the initial state populated.
In Fig. 5(a) we show, using a gray scale, the popu-
lation of high n states (120 , n , `) as a function
of field amplitude for the series of initial states from
n ! 50 to n ! 95. Figure 5(b) is composed of lines
out of Fig. 5(a) for selected initial states. In Fig. 5(a)
we also plot the 10% microwave ionization fields for
lithium, which exhibit a resonant structure similar to that
seen in hydrogen [3]. It is evident that the excitation
to very high n states occurs only for fields sufficient
to produce some ionization, which is not surprising since
producing these high n states is not so different from pro-
ducing free electrons. In Fig. 5(a) we have marked the
positions at which the microwave frequency is equal to or
a subharmonic of the classical Kepler frequency (1"2pn3
in atomic units). It is evident that the production of the
very highly excited states is maximized near the funda-
mental resonance, but exactly why is not apparent.
In conclusion, when Rydberg atoms are exposed
to intense microwave pulses a remarkable amount of
population is left in states with n . 120 in spite of the
1749
VOLUME 83, NUMBER 9 P HY S I CA L REV I EW LE T T ER S 30 AUGUST 1999
FIG. 5. Redistribution to states of 120 , n , ` from various
initial states. In (a) we represent the high n populations with
a gray scale for initial states ranging from n ! 50 to n ! 95.
The line in (a) is the 10% ionization field for the different
initial states and the arrows mark the points at which the
microwave frequency is equal to, one-half, and one-third the
classical frequency. In (b) we show the population of high
n states as a function of field amplitude for initial states of
n ! 60 (dotted line), n ! 70 (dashed line), n ! 80 (bold line),
and n ! 90 (thin line).
fact that the microwave amplitude is 40 times larger than
the classical ionization field for these states. The excita-
tion to these weakly bound states has an onset matching
the microwave ionization threshold field and is most
pronounced for a microwave frequency near the Kepler
frequency. Although unexpected, this behavior can be
explained by a simple model based on the likelihood of
the Rydberg electron’s absorbing energy from the field at
different points in a classical orbit.
This work was supported by the Air Force Office of
Scientific Research and the National Science Founda-
tion. We thank R. R. Jones and P.M. Koch for useful
discussions.
[1] Atoms in Intense Laser Fields, Advances in Atomic,
Molecular, and Optical Physics, edited by M. Gavrila
(Academic, San Diego, 1992), Suppl. 1.
[2] K.A.H. van Leeuwen, G.V. Oppen, S. Renwick, J. B.
Bowlin, P.M. Koch, R. V. Jensen, O. Rath, D. Richards,
and J. G. Leopold, Phys. Rev. Lett. 55, 2231 (1985).
[3] P.M. Koch and K.A.H. van Leeuwen, Phys. Rep. 255,
289 (1995).
[4] M.M. Sanders and R.V. Jensen, Am. J. Phys. 64, 21
(1996).
[5] J. G. Leopold and I. C. Percival, Phys. Rev. Lett. 41, 944
(1978).
[6] J. G. Leopold and I. C. Percival, J. Phys. B 12, 709 (1979).
[7] D.A. Jones, J. G. Leopold, and I. C. Percival, J. Phys. B
13, 31 (1980).
[8] J. N. Bardsley, B. Sundaram, L.A. Pinnaduwage, and J. A.
Bayfield, Phys. Rev. Lett. 56, 1007 (1986).
[9] R. Blumel, A. Buchleitner, R. Graham, L. Sirko,
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1750


Population Trapping in Extremely Highly Excited States in Microwave Ionization

Scholarship, Research, and Creative Work at Bryn Mawr College | Bryn Mawr College Research

Bryn Mawr CollegeScholarship, Research, and Creative Work at Bryn MawrCollegePhysics Faculty Research and Scholarship Physics1999Population Trapping in Extremely Highly ExcitedStates in Microwave IonizationMichael W. NoelBryn Mawr College, mnoel@brynmawr.eduW. M. GriffithT. F. GallagherLet us know how access to this document benefits you.Follow this and additional works at: http://repository.brynmawr.edu/physics_pubsPart of the Physics CommonsThis paper is posted at Scholarship, Research, and Creative Work at Bryn Mawr College. http://repository.brynmawr.edu/physics_pubs/76For more information, please contact repository@brynmawr.edu.Custom CitationMichael W. Noel, W. M. Griffith, and T. F. Gallagher, "Population Trapping in Extremely Highly Excited States in MicrowaveIonization," Phys. Rev. Lett. 83, 1747 (1999).VOLUME 83, NUMBER 9 P HY S I CA L REV I EW LE T T ER S 30 AUGUST 1999Population Trapping in Extremely Highly Excited States in Microwave IonizationMichael W. Noel, W.M. Griffith, and T. F. GallagherDepartment of Physics, University of Virginia, Charlottesville, Virginia 22901(Received 18 December 1998)When a lithium atom in a Rydberg state (n ! 80) is exposed to a short, intense microwave pulsewe find that substantial population is left in extremely highly excited states (n . 120), in spite of thefact that the microwave field amplitude is more than 40 times larger than required to classically ionizethese states.PACS numbers: 32.80.Rm, 42.50.HzThe interaction between an atom and a strong oscil-lating field has revealed a multitude of interesting phe-nomena, often involving the ionization of the atom [1].Among the most fascinating are those cases in whichionization does not occur, in spite of its appearing tobe very probable at first glance. Several immediatelyrelevant examples occur in the microwave ionization ofRydberg atoms. Classical stabilization of Rydberg atomsagainst microwave ionization occurs when the microwavefrequency or one of its harmonics matches the classicalKepler frequency of the Rydberg atom [2–4]. Of particu-lar interest for this work, it was observed long ago in clas-sical calculations that the Rydberg atom could be left invery high lying states by a microwave pulse [5–7]. Morerecently there have been extensive calculations of the fi-nal state distributions subsequent to a microwave pulse[8–11]. At low microwave fields the final state popu-lation is found to fall off roughly exponentially with thechange in n of the final state from the initial state, andexperiments have confirmed these predictions [8,9]. Heren is the principal quantum number. At fields high enoughto produce ionization, the calculated distributions of finalstates are continuous from the initially populated state tothe continuum, with structure due to resonant single pho-ton transitions [11]. To date, however, no experimentshave been carried out to test these predictions.Here we report the analysis of the final state distributionof lithium Rydberg atoms exposed to intense, short mi-crowave pulses. We find the rather surprising result thatwhen n ! 80 lithium atoms are exposed to 0.5 ns long mi-crowave pulses .35% of the population is transferred tostates of n . 120, with no apparent population in statesof n , 120, contrary to the calculations, in spite of thefact that the microwave field amplitude exceeds the classi-cal ionization field of n ! 120 by a factor of 40. As thepulse is made longer the number of n . 120 atoms sur-viving decreases, but it is still !3% with a 10 ns pulse.Recent measurements of Rydberg atoms kicked by a se-ries of unipolar pulses also show atoms left in very high nstates, again in disagreement with calculations [12]. Us-ing the fact that there can be no transfer of energy fromthe field to the electron when it is far from the ion core andaccounting for the state redistribution that occurs when theelectron comes near the core we have developed a simplepicture that reproduces the observed population transfer towhat have been termed extremely highly excited states [5].The general scheme for this experiment was to excitelithium atoms to a high n Rydberg state, allow themto interact with a short microwave pulse, and measurethe state distribution of the atoms that remained bound.Lithium atoms, from an effusive source, entered a piece ofWR90 waveguide through a small hole in one of its sides.Three nanosecond dye laser pulses entered the waveguidethrough a hole on the opposite side and excited theatoms to a Rydberg state through the 2s ! 2p, 2p ! 3s,and 3s ! np transitions. The final laser was polarizedorthogonal to the microwave field so as to excite jmj ! 1states where m is the magnetic quantum number. Withthe laser bandwidth of 4 GHz, we could selectively exciteRydberg states beyond n ! 100.The microwave pulse was formed by mixing the11.5 GHz continuous wave output of a sweep oscillatorwith a fast rising edge (!100 ps) voltage step in onemixer and a fast falling edge (!100 ps) voltage step in asecond mixer. The relative delay between the steps couldbe precisely adjusted to give the desired pulse width.A subsequent mixing stage with a 1 to 100 ns pulse(depending on the pulse length set with the fast steps)served to further suppress the wings of this fast pulse.The microwave source was phase locked to the pulseenvelope, which allowed us to monitor the microwaveelectric field, not just the field envelope. After two stagesof preamplification we passed the pulse through a fast mi-crowave switch which was gated open for a time slightlylonger than the pulse. Finally, the pulse was amplified ina gated traveling wave tube amplifier and passed througha voltage controlled attenuator with which we couldvary its power over 40 dB. The pulse was coupled intoand out of the short piece of waveguide containing theprepared Rydberg atoms after which it was attenuatedand monitored with a sampling oscilloscope. We cali-brated the power using a Hewlett Packard 432A powermeter and estimate a microwave field uncertainty of 10%.After the microwave interaction a negative field ioniza-tion pulse was applied to a septum below the atoms in thewaveguide. The field ionized electrons passed through a0031-9007"99"83(9)"1747(4)$15.00 © 1999 The American Physical Society 1747VOLUME 83, NUMBER 9 P HY S I CA L REV I EW LE T T ER S 30 AUGUST 1999small hole in the top of the waveguide and were collectedwith a dual microchannel plate detector. By adjusting abias voltage on the septum to minimize excitation of highns states we reduced the static electric field to less than50 mV"cm, which can ionize states with n . 280.With the laser tuned to the desired Rydberg statewe repetitively scanned the microwave amplitude overmany laser shots as we downloaded the time resolvedelectron signals recorded on a digital oscilloscope on eachshot. The time resolved electron signal revealed the statedistribution of the atoms which remained bound after themicrowave interaction. Electrons left in high n stateswere ionized early during the slowly rising field ionizationpulse and arrived early in time, whereas those of lower nrequired higher fields and consequently arrived later intime. We calibrated the arrival time versus n by directlyexciting known states, and we can distinguish states upto about n ! 130, above which the electrons all arriveat the detector at the same time. Electrons which wereionized directly by the microwave pulse dispersed beforethe arrival of the field ionization pulse, so they were notregistered at the detector.In Fig. 1 we show a typical set of data collectedaccording to the above prescription for an initial state ofn ! 80 interacting with a 0.5 ns (5 cycle) pulse. TheKepler period is given by 2pn3 in atomic units, andfor n ! 80 it is 78 ps. The primary classical resonance,where the Kepler period is equal to the microwave period,FIG. 1. Bound state distribution as a function of field strengthfor an atom initially prepared in the 80p state interacting witha 0.5 ns pulse. In (a) electron signal is mapped to a gray scalewith black corresponding to large signal and white to no signal.The horizontal axis, which experimentally is the detection timeof an electron after being ionized by the slowly rising fieldionization pulse, has been calibrated according to n. The labelsare placed at the peak in the arrival time of the electron signalfor a given n. Bound states above n ! 130 are not resolvedand we label them collectively with “limit.” (b) Electron signalintegrated over all n (light line) and electron signal integratedfrom 120 , n , ` (bold line). The three horizontal lines in(b) are the 10%, 50%, and 90% ionization fields from top tobottom, respectively.occurs at n ! 83. The duration of the short 0.5 nspulse matches the Kepler period at n ! 150. The statedistribution as a function of field amplitude is plotted inFig. 1(a) by mapping the electron signal to a gray scalewith black corresponding to a large signal and white to nosignal. Figure 1(b) shows the electron signal integratedover all n, the total remaining atom signal, as well as theintegrated state population for 120 , n , ` as functionsof field amplitude. Finally in Fig. 2 we show severallines out of the full data set in Fig. 1(a). These are thestate distributions at microwave amplitudes of 2 V"cm(no ionization), 9.5 V"cm (10% ionization), and 48 V"cm(90% ionization).In both Figs. 1 and 2 it is apparent that when asignificant fraction of the atoms is ionized, a surprisinglylarge fraction of the remaining bound population hasbeen transferred to high n states (as much as 35%).Furthermore, some population remains bound in thesehigh n states for fields as high as 60 V"cm, which is morethan 40 times the field necessary to classically ionize anatom of n ! 120.In an effort to explain the remarkable accumulationof population in these high n states, even in verystrong microwave fields, we consider the interaction ofa classical Rydberg atom with the microwave field. Ifthe electron is far from the core it simply oscillates inthe field as if it were a free electron. The amplitude ofthe oscillations for a 60 V"cm field is 0.2 mm, much lessthan the size of the orbit for n . 120. In contrast, whenthe electron returns to the core it can gain energy fromor lose it to the oscillating field, depending on the phaseof the field. Since the orbit time for n . 150 is longerthan our microwave pulse, if such states are formed inthe first cycle of the pulse, the field is off by the timethe electron returns to the core, and the electron remainsbound in these high n states.With a 9.5 ns (110 cycle) pulse there is still someexcitation of high n states, however, the amount ofpopulation transferred to these states is much smaller.As shown by Fig. 3, the maximum transfer for an atomprepared in the 80p state is about 3%. In this case65758595limitprinciple quantum number, n01electron signal (arbitrary units)FIG. 2. State distributions from Fig. 1 for field amplitudes of2 V"cm (no ionization, light line), 9.5 V"cm (10% ionization,bold line), and 48 V"cm (90% ionization, dotted line).1748VOLUME 83, NUMBER 9 P HY S I CA L REV I EW LE T T ER S 30 AUGUST 1999FIG. 3. (a) State distribution as a function of field strength foran atom initially prepared in the 80p state interacting with a9.5 ns pulse. (b) Electron signal integrated over all n (lightline) and electron signal integrated from 120 , n , ` (boldline). For clarity the bold line in (b) has been expanded by afactor of 3.the Kepler period does not exceed the pulse width untiln . 400, and a wave packet at n ! 280, the highestlying bound state possible in our experiment, passes thecore 3 times during the microwave pulse. Each time theelectron passes the core there is some probability that itwill be ionized by the microwave field, so with 110 fieldcycles it is not surprising that few bound atoms are left.As might be expected, with a 100 ns (1150 cycle) pulsewe did not see any excitation to high n states for any ofthe initial states used (n ! 47 to n ! 94). In this casethe Kepler period does not exceed the pulse width untiln . 870 and at n ! 280 the electron passed the core30 times during the microwave pulse.The dynamic nature of the above classical descriptioncan also be incorporated into a simple model for thefinal state distributions. In this model we consider themicrowave interaction one cycle at a time in the followingiterative procedure. The effect of the first cycle is tospread the population from its initial state over an energyband of states. The next cycle spreads the populationof each of these states in an identical way for n , n1and does nothing for states with n . n1, where n1 is theprinciple quantum number at which the Kepler period isequal to the microwave period. On the third field cycleatoms put into states with n1 , n , n2, by the first cycle,where n2 is the principal quantum number for which theKepler period is twice the microwave period, as well asatoms put into states with n , n1 by the second fieldcycle, are spread over the energy band. This process isrepeated for each field cycle, with those atoms that havereturned to or remained near the ion core being spreadover the energy band.The evolution of the state distribution during this itera-tion process is shown in Fig. 4 for a five cycle pulse in-teracting with an atom initially in n ! 80. The function60708090limitprinciple quantum number, npopulation (arbitrary units)01FIG. 4. Model of the state distributions for the interactionbetween a five cycle pulse and an n ! 80 Rydberg atom. Weshow distributions after the first (light line), third (dotted line),and fifth (bold line) cycles.used to spread population was a Gaussian whose widthwas adjusted so that, after five iterations, 10% of the fi-nal distribution had positive energy for comparison to the10% ionization trace in Fig. 2. For low n states the mi-crowave field oscillates slowly compared to the classicalperiod, and ionization proceeds via field ionization ratherthan through the diffusion process seen when the timescales are comparable. In order to account for this ef-fect the Gaussian was cut off sharply at n ! 58, wherethe microwave period is 3 times the Kepler period. Thismodel contains three essential parameters, the width of thedistribution with which we spread population and the lowand high n cutoffs for the interaction, all of which haveclear physical origins. The gross features of the final dis-tribution, the peak at high n, and the broad tail to lowern, are relatively insensitive to changes in these parametersof the model, and it is evident that the model captures theessential features of the data shown in Fig. 2.Finally, we have studied the excitation to the highn states as a function of the initial state populated.In Fig. 5(a) we show, using a gray scale, the popu-lation of high n states (120 , n , `) as a functionof field amplitude for the series of initial states fromn ! 50 to n ! 95. Figure 5(b) is composed of linesout of Fig. 5(a) for selected initial states. In Fig. 5(a)we also plot the 10% microwave ionization fields forlithium, which exhibit a resonant structure similar to thatseen in hydrogen [3]. It is evident that the excitationto very high n states occurs only for fields sufficientto produce some ionization, which is not surprising sinceproducing these high n states is not so different from pro-ducing free electrons. In Fig. 5(a) we have marked thepositions at which the microwave frequency is equal to ora subharmonic of the classical Kepler frequency (1"2pn3in atomic units). It is evident that the production of thevery highly excited states is maximized near the funda-mental resonance, but exactly why is not apparent.In conclusion, when Rydberg atoms are exposedto intense microwave pulses a remarkable amount ofpopulation is left in states with n . 120 in spite of the1749VOLUME 83, NUMBER 9 P HY S I CA L REV I EW LE T T ER S 30 AUGUST 1999FIG. 5. Redistribution to states of 120 , n , ` from variousinitial states. In (a) we represent the high n populations witha gray scale for initial states ranging from n ! 50 to n ! 95.The line in (a) is the 10% ionization field for the differentinitial states and the arrows mark the points at which themicrowave frequency is equal to, one-half, and one-third theclassical frequency. In (b) we show the population of highn states as a function of field amplitude for initial states ofn ! 60 (dotted line), n ! 70 (dashed line), n ! 80 (bold line),and n ! 90 (thin line).fact that the microwave amplitude is 40 times larger thanthe classical ionization field for these states. The excita-tion to these weakly bound states has an onset matchingthe microwave ionization threshold field and is mostpronounced for a microwave frequency near the Keplerfrequency. Although unexpected, this behavior can beexplained by a simple model based on the likelihood ofthe Rydberg electron’s absorbing energy from the field atdifferent points in a classical orbit.This work was supported by the Air Force Office ofScientific Research and the National Science Founda-tion. We thank R. R. Jones and P.M. Koch for usefuldiscussions.[1] Atoms in Intense Laser Fields, Advances in Atomic,Molecular, and Optical Physics, edited by M. Gavrila(Academic, San Diego, 1992), Suppl. 1.[2] K.A.H. van Leeuwen, G.V. Oppen, S. Renwick, J. B.Bowlin, P.M. Koch, R. V. Jensen, O. Rath, D. Richards,and J. G. Leopold, Phys. Rev. Lett. 55, 2231 (1985).[3] P.M. Koch and K.A.H. van Leeuwen, Phys. Rep. 255,289 (1995).[4] M.M. Sanders and R.V. Jensen, Am. J. Phys. 64, 21(1996).[5] J. G. Leopold and I. C. Percival, Phys. Rev. Lett. 41, 944(1978).[6] J. G. Leopold and I. C. Percival, J. Phys. B 12, 709 (1979).[7] D.A. Jones, J. G. Leopold, and I. C. Percival, J. Phys. B13, 31 (1980).[8] J. N. Bardsley, B. Sundaram, L.A. Pinnaduwage, and J. A.Bayfield, Phys. Rev. Lett. 56, 1007 (1986).[9] R. Blumel, A. Buchleitner, R. Graham, L. Sirko,U. Smilansky, and H. Walther, Phys. Rev. A 44, 4521(1991).[10] G. Casati, I. Guarneri, and D. L. Shepelyansky, Phys. Rev.A 36, 3501 (1987).[11] R.V. Jensen, S.M. Susskind, and M.M. Sanders, Phys.Rep. 201, 1 (1991).[12] M. T. Frey, F. B. Dunning, C. O. Reinhold, S. Yoshida,and J. Burgdörfer, Phys. Rev. A 59, 1434 (1999).1750

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Population Trapping in Extremely Highly Excited States in Microwave Ionization

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