Cooperative Lamb Shift in an Atomic Vapor Layer of Nanometer Thickness
ABSTRACT We present an experimental measurement of the cooperative Lamb shift and the Lorentz shift using a nanothickness atomic vapor layer with tunable thickness and atomic density. The cooperative Lamb shift arises due to the exchange of virtual photons between identical atoms. The interference between the forward and backward propagating virtual fields is confirmed by the thickness dependence of the shift, which has a spatial frequency equal to twice that of the optical field. The demonstration of cooperative interactions in an easily scalable system opens the door to a new domain for nonlinear optics.
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Article: The Theory of Electrons
Bulletin of The American Mathematical Society  BULL AMER MATH SOC. 01/1911; 17(1911).  [Show abstract] [Hide abstract]
ABSTRACT: The book consists of nine chapters. In the first one, Planck’s law is formulated, Einstein’s coefficients are defined, and the principle of action of a laser is described. The second chapter is devoted to the quantummechanical description of atom radiation interaction. The Einstein coefficients are calculated, the optical Bloch equations are obtained, and the connection between them and the rate equations are investigated. In the third one, the author describes the statistical analysis of chaotic light sources, different types of interferometers, and the interference patterns that they give in chaotic light. The field quantization is the subject of the fourth chapter. The quantization of electromagnetic field, the interaction of the quantized field with atoms, and the second quantization of the atomic Hamiltonian are analyzed there. The fifth chapter is devoted to singlemode quantum optics, while the sixth one describes multimode and continuousmode quantum optics. The seventh chapter deals with attenuation and amplification of a light beam, actions frequently needed in experiments. The eighth one is devoted to resonance fluorescence and light scattering. Elastic Rayleigh scattering and inelastic Raman scattering are described in this chapter. An introduction to nonlinear quantum optics is given in the ninth chapter. The book is written as a graduatelevel textbook, and over 100 problems help to reinforce the understanding of the material presented.01/2000; Oxford Science Publications.
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Cooperative Lamb Shift in an Atomic Vapor Layer of Nanometer Thickness
J. Keaveney,1A. Sargsyan,2U. Krohn,1I.G. Hughes,1D. Sarkisyan,2and C.S. Adams1,*
1Department of Physics, Rochester Building, Durham University, South Road, Durham DH1 3LE, United Kingdom
2Institute for Physical Research, National Academy of Sciences—Ashtarak 2, 0203, Armenia
(Received 25 January 2012; published 23 April 2012)
We present an experimental measurement of the cooperative Lamb shift and the Lorentz shift using a
nanothickness atomic vapor layer with tunable thickness and atomic density. The cooperative Lamb shift
arises due to the exchange of virtual photons between identical atoms. The interference between the
forward and backward propagating virtual fields is confirmed by the thickness dependence of the shift,
which has a spatial frequency equal to twice that of the optical field. The demonstration of cooperative
interactions in an easily scalable system opens the door to a new domain for nonlinear optics.
DOI: 10.1103/PhysRevLett.108.173601PACS numbers: 42.50.Nn, 32.30.?r, 32.70.Jz
One of the more surprising aspects of quantum electro
dynamics is that virtual processes give rise to real phe
nomena. For example, the Lamb shift [1] arises from a
modification of the transition frequency of an atom due to
the emissionand reabsorption ofvirtual photons.Similarly,
in cavity quantum electrodynamics [2–4] the reflection of
the virtual field by a mirror modifies the absorptive and
emissive properties of the atom. In a cooperative process
such as superradiance, the lightmatter interaction is modi
fied by the proximity of identical emitters. The dispersive
counterpart of superradiance is known as the cooperative
Lamb shift [5] (also sometimes referred to as the collective
or N atom Lamb shift [6]). The cooperative Lamb shift
and the cooperative decay rate (i.e., super or subradiance)
arise from the real and imaginary parts of the dipoledipole
interaction, respectively.
Although superradiance has been investigated exten
sively [7], experimental studies of the cooperative Lamb
shift are scarce. Evidence for the shift is restricted to two
particular cases, involving threephoton excitation in the
limit of the thickness ‘ being much larger than the tran
sition wavelength ? in an atomic gas [8], and xray scat
tering from Fe layers in a planar cavity [9], demonstrating
the fundamental link between the cooperative shift and
superradiance. However, the full thickness dependence of
the shift in a planar geometry with ‘ < ? predicted four
decades ago [5] has not been observed.
Here we present experimental measurements of the
cooperative Lamb shift in a nanothickness vapor layer of
Rb atoms as a function of both density and vapor thickness.
The atoms are confined in a cell between two super
polished sapphire plates. Similar nanothickness vapors
have been studied extensively over the last two decades,
see, e.g., [10–15]. We extend this work to the high density
regime where dipoledipole interactions dominate. In
addition, by incorporating the effects of dipoledipole
interactions into a sophisticated model of the absorption
spectra we are able to extract the thickness dependence
of the resonant shift and thereby verify that the spatial
frequency of the cooperative Lamb shift is equal to twice
that of the light field [5]. We thus confirm the fundamental
mechanism of the cooperative Lamb shift as the exchange
of virtual photons.
The underlying mechanism of light scattering is the
interference between the incident field and the local field
produced by induced oscillatory dipoles. In a medium with
N two level dipoles per unit volume, the dipolar field is
proportional to the susceptibility which, for a weak field, is
given by the steady state solution to the optical Bloch
equations (see, e.g., [16]),
?0@
where d is the transition dipole moment, ?geis the decay
rate of the coherence between the ground and excited
states, and ? is the detuning from resonance. The res
ponse of an individual dipole is described in terms of the
polarizability,
4??0@
In a dense medium, the field produced by the dipoles
modifies the optical response of each individual dipole.
This modified response is found by adding the incident
field to the dipolar field, Eloc¼ E þ P=3?0, where Elocis
known as the Lorentz local field [17]. The susceptibility
determines the bulk response P ¼ ?0?E, whereas the
polarizabilitydetermines
4??0N?pEloc. Substituting for E and P we find a relation
between the macroscopic variable ? and the single dipole
parameter ?pwhich is referred to as the LorentzLorenz
(LL) law [17]:
? ¼ ?Nd2
? þ i?ge
;
(1)
?p¼
?
4?N¼ ?
1d2
? þ i?ge
:
(2)
thelocalresponse
P ¼
? ¼
4?N?p
1 ?4
3?N?p
:
(3)
Substituting for ?pwe find
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? ¼ ?
Nd2=?0@
? þ i?geþ Nd2=3?0@;
(4)
and hence the first order correction due to dipoledipole
interactions is a shift in the resonance frequency known as
the Lorentz shift
?LL¼ ?Nd2
3?0@:
(5)
However, as discussed by Stephen [18] and Friedberg,
Hartmann, and Manaasah [5], the pairwise dipoledipole
interaction also contains a radiation term. The complete
pair potential for two dipoles Vddhas the form
Vdd¼ ?½ð1 ? ikrÞð3cos2? ? 1Þ þ ðkrÞ2sin2??eikr;
(6)
where ? ¼ ?3@?=4ðkrÞ3, r, and ? are their separation and
relative angle, respectively, and ? is the natural linewidth
of the dipole transition with wave vector k ¼ 2?=?. The
real and imaginary parts of Vddgive rise to a level splitting
and a modification of the spontaneous lifetime (super
radiance or subradiance), respectively [5,18–20]. While
these effects have been demonstrated in experiments on
two ions [21] and two molecules [22], a key advantage of
atomic vapors is that one can easily vary the number of
atoms and their mean spacing hri. By changing the
temperature of the vapor between 20?C and 350?C, one
can smoothly vary the number density over 7 orders of
magnitude. In doing so, we move between two regimes:
Nk?3? 1, hri > ? where dipoledipole interactions are
negligible, and Nk?3? 100, hri ? ?=30 where dipole
dipole interactions dominate.
For more than two dipoles, the cooperative N atom
shift and decay rate are given by a sum of the pairwise
dipoledipole interaction, Eq. (6), for all pairs. For the
relatively simple case of an ensemble of dipoles confined
within a thin plane of thickness ‘, the sum produces a
shift [5],
?dd¼ ?j?LLj þ3
4j?LLj
?
1 ?sin2k‘
2k‘
?
;
(7)
where the first term is the Lorentz shift and the second term
is the cooperative Lamb shift. There are two remarkable
features of Eq. (7). First, the cooperative Lamb shift is a
shift to higher energy. One can understand the opposite
sign of the Lorentz shift and the cooperative Lamb shift
from the pairwise potential, Eq. (6). For a thin slab where
all the dipoles lie in the plane, all the dipoles oscillate in
phase such that the dipoledipole interaction reduces to the
static case, which, after averaging over all angles, gives an
attractive interaction resulting in the Lorentz shift to lower
energy. As one moves out of the plane in the propagation
direction, the relative phase of the dipoles changes and at a
separation of ?=4 the second dipole reradiates a field that is
? out of phase with the source dipole. This switches the
sign of the interaction giving rise to the cooperative Lamb
shift to higher energies. The second interesting property of
the shift is that it depends on twice the propagation phase
k‘ which arises due to the reradiation by the second dipole
[5]. Finally, we note that while superradiance requires
excitation of the medium, the cooperative Lamb shift can
be observed in the limit of weak excitation where there is
negligible population of the excited state. Consequently,
preparation of the medium is not required to observe the
shift.
It is important to note that the shift ?ddapplies to a static
medium. For a gaseous ensemble, atomic motion leads to
collisions that also contribute a density dependent shift
?coland broadening ?selfof the resonance lines (see [23]
and references therein), and thus the total shift for a
thermal ensemble becomes
?tot¼ ?ddþ ?col:
(8)
While evidence for density dependent shifts has been
observed in experiments on selective reflection [24], it is
important to measure ?totas a function of the thickness of
the medium to separate the thickness independent colli
sional shift ?col[5] from the thickness dependent coopera
tive Lamb shift. Below, we present experimental data that
allow that distinction to be made for the first time.
To measure the cooperative Lamb shift, we use a
gaseous layer of Rb confined in a vapor cell with thickness
‘ < ?. The cell is shown in Fig. 1(a), and consists of a Rb
reservoir and a window region, where the Newton’s rings
indicate the variation in the cell thickness from 30 nm at
the center to 2 ?m near the bottom of the photograph.
The wedgeshaped thickness profile arises due to the slight
curvature of one of the windows (radius of curvature
FIG. 1 (color online).
mental data. (a) The Newton’s rings interference pattern can be
observed on the windows of the cell due to the curvature of one
of the windows, with a radius of curvature >100 m. At the
center of these rings the cell has its minimum thickness ‘ ?
30 nm, which increases to ?2 ?m near the stem at the bottom of
the photo. (b) Transmission spectra for layer thickness ‘ ¼
90 nm for measured Rb temperatures of 190?C (black),
207?C (light blue), 250?C (blue), 265?C (dark green),
280?C (green), 290?C (orange), and 305?C (red). The shift
can be seen clearly as the density is increased.
Photograph of the nanocell and experi
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R > 100 m). The local thickness at the position of the
probe laseris measuredatan operationaltemperatureusing
an interferometric method outlined in Ref. [25]. The local
surface roughness measured over an area of 1 mm2is less
than 3 nm, for any part of the window, and the focus of the
beam is ? 1 mm2. The reservoir can be heated almost
independently of the windows and its temperature deter
mines the Rb number density, while the windows are kept
>50?C hotter to prevent condensation of the Rb vapor. By
changing the temperature of the vapor between 20?C
and 350?C, we can vary the atomic density between the
regimes Nk?3? 1 where dipoledipole interactions are
negligible and Nk?3? 100, where dipoledipole inter
actions dominate.
To determine the optical response of the medium, we
record transmission spectra as a narrowband laser is
scanned across the D2 resonance in Rb at 780 nm. The
light is reduced to a power P ? 100 nW and focused to a
30 ?m spot size inside the cell, leading to a local vapor
thickness variation due to the wedgeshaped profile of less
than 3 nm. The accuracy in determining the cell thickness
is therefore limited by the surface flatness of the windows.
Though the intensity of the light is greater than the con
ventional saturation intensity (Isat? 1:7 mW=cm2for the
Rb D2 line), the extremely small thickness of the cell
means that optical pumping is strongly suppressed. The
transmission is recorded on a photodiode, and a reference
cell and FabryPerot interferometer are used to calibrate
the laser frequency. Example experimental spectra for a
thickness of ‘ ¼ 90 nm are shown in Fig. 1(b), where the
shift is clearly visible. The shift is extracted by fitting the
observed spectra to a comprehensive model of the absolute
transmission, based on a MarquardtLevenberg method
(see, e.g., Ref. [26]). The model includes the effect of
collisional broadening and has been shown to predict the
absolute absorption of the Rb vapor to better than 0.5%
[23,27]. To this we add the effects of Dicke narrowing [10],
where the Doppler effect is partially suppressed as a result
of theshortlengthscale,cavityeffects[13],since thecellis
a lowfinesse etalon (with finesse F ? 1), and a single
parameter which accounts for a frequency shift of the
whole spectrum.
Figure 2 shows experimental data and the theoretical fit
for two cases. Panel (a) shows a relatively large vapor
thickness (‘ ¼ ?=2 ¼ 390 nm) with a low atomic density
where dipoledipole interactions are negligible (Nk?3?
0:07), and highlights the effects of Dicke narrowing.
Clearly visible are the individual excited state hyperfine
components that are normally masked by Doppler broad
ening in a conventional cmthickness cell. In stark contrast,
panel (b) shows the spectrum obtained in the dipoledipole
dominated regime (Nk?3? 50) for a thickness ‘ ¼
90 nm. The individual hyperfine transitions are no longer
resolved and there is a clear shift of the resonance to a
lower frequency. To illustrate this, we also plot the
theoretical prediction with the line shift removed.
From fitting the data, the collisional broadening is found
to be ?self¼
thicknesses greater than ?=4 in agreement with previous
work (see [23] and references therein). For thicknesses
shorter than ?=4, we observe additional broadening that
requires furtherinvestigation.
van der Waals shift due to atomsurface interactions, which
we extract by fixing the density and varying the cell
thickness (see also [15]); but even for the smallest thick
ness (90 nm) this is more than an order of magnitude
smaller that the cooperative Lamb shift.
By comparing the experimental data with the theoretical
prediction, we extract the line shift as a function of number
density and the thickness of the medium. In Fig. 3 we show
the measured shift as a function of number density for two
thicknesses, ‘ ¼ 90 and 250 nm. Hyperfine splitting gives
rise to a different effective dipole for each transition in the
spectrum, which at low densities shifts independently.
However, in the high density regime (N > 1016cm?3)
dipoledipole interactions dominate the line shape and
hyperfine splitting becomes negligible. We can then treat
the line as a single s1=2! p3=2transition which shifts
linearly with the density, as shown in Fig. 3. We fit the
gradient of the linear region to obtain the coefficient of the
shift, and repeat these measurements for 13 thicknesses up
to 600 nm. For thicknesses greater than 600 nm, the high
optical depth of the sample impairs the resolution of the
line shift. Measurements for a thickness of 420 nm do not
fit to Eq. (8) and have been excluded from the plot. This is
thought to be due to the wellknown energy pooling
process where 420 nm light is generated by a frequency
up conversion of 780 nm light in a dense Rb vapor [28].
ffiffiffi
8
p
?N?k?3¼ 2?ð1:0 ? 10?7ÞN Hz cm3for
Wealso observea
FIG. 2 (color online).
theory. Transmission spectrum as a function of linear detuning
for thickness (a) ‘ ¼ 390 nm, Nk?3? 0:1 (T ¼ 130?C) and
(b) ‘ ¼ 90 nm, Nk?3? 50 (T ¼ 305?C). The black line is
experimental data, while the solid green and dashed red lines
are the fits to the model outlined in the main text. The dotdashed
red line in panel (b) is the theory without the line shift included.
The residuals show the difference between experiment and
theory. Zero on the detuning axis represents the weighted line
center of the D2 line.
Transmission spectra—experiment and
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In Fig. 4 we plot the gradient of the line shift as a
function of cell thickness. For the Rb D2 resonance,
?LL=N ¼ ?2??k?3, where we have used the relationship
between the dipole moment for the s1=2! p3=2transit
ion and the spontaneous decay rate, d ¼
1jerjLg¼ 0i (see Ref. [27]). We extract the collisional
shift by comparing the data to Eq. (8) with ?colthe only
free parameter. The amplitude and period of the oscillatory
part are fully constrained by Eq. (7). We find the collisional
shift to be ?col=2? ¼ ð?0:25 ? 0:01Þ ? 10?7Hzcm3,
similar to previous measurements on potassium vapor
[24]. In this high density limit, the collisional shift is also
independent of hyperfine splitting. The solid line is the
prediction of Eq. (7), and the agreement between the
measured shifts and the theoretical prediction is remark
able (the reduced ?2for the data is 1.7). As well as
measuring the thickness dependence of the cooperative
ffiffiffiffiffiffiffiffi
2=3
p
hLe¼
Lamb shift, our data also provide a determination of the
Lorentz shift which can only be measured in the limit of
zero thickness. An important advance on previous studies
[8] is that the results clearly show the oscillations in the
shift versus the thickness which arises due to the relative
phase of the reradiated dipolar field.
The demonstration of the cooperative Lamb shift and
coherentdipoledipole interactions inmediawiththickness
??=4 opens the door to a new domain for quantum optics,
analogous to the strong dipoledipole nonlinearity in
blockaded Rydberg systems [29,30]. As the cooperative
Lamb shift depends on the degree of excitation [5], exotic
nonlinear effects such as mirrorless bistability [31,32] are
now accessible experimentally. In addition, given the
fundamental link between the cooperative Lamb shift and
superradiance, subquarterwave thickness vapors offer an
attractive system to study superradiance in the small
volume limit. Finally, we note that the measured coopera
tive Lamb shift is the average dipoledipole interaction
for a homogeneous gas which contains both positive and
negative contributions. It could therefore be enhanced by
eliminating directions that contribute with the undesired
sign, for example, by patterning the distribution of dipoles.
These topics will form the focus of future research.
We would like to thank M.P.A. Jones for stimulating
discussions. We acknowledge financial support from
EPSRC and Durham University.
*c.s.adams@durham.ac.uk
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