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Nanoscale Microscopy using Electromagnetically Induced Transparency
N. A. Proite and D. D. Yavuz
Department of Physics, University of Wisconsin, Madison, WI, 53706
Introduction
Electromagnetically Induced
Transparency
Over the last two decades, nanoscale optical microscopy has been the subject of
intense research [1]. While it is well known that the Abbe Limit physically constrains
how well a beam of light can resolve a sample, several techniques are being
investigated and employed which work around this limit:
J. E. Thomas and collaborators employ a strong position dependent
Stark Shift focused into a cold atom gas to localize probed atoms [2-4].
Their method utilizes an intense, off-resonant light field which shifts the
energies of the atoms. The magnitude of the shift is determined by the
position of the atom in the light field’s intensity profile. This method is
experimentally verified to break the diffraction limit [3].
Electromagnetically Induced Transparency (EIT) is a quantum interference scheme
which eliminates absorption on a resonant probe beam propagating through a Raman
medium. This technique, in its most simple form, calls for two near-resonant laser
beams, a probe beam and a coupling beam [Fig. 2]. Under certain conditions, the
beams reduce the imaginary part of the complex linear susceptibility to zero, and
absorption on the probe beam therefore vanishes [7].
We propose to focus a laser beam very tightly into an ultracold atomic cloud and
use a particular nonlinear interaction, specifically the dark state of
Electromagnetically Induced Transparency, to localize the atomic excitation to a
sub-wavelength spot.
Intensity Distribution
(a)
440 nm
4

50 nm
3
Γ : spontaneou s decay rate from 3
p
c
[1] Stelzer, Nature 417, 806-807 (2002).
[2] J. E. Thomas, Optics Letters 14, 21 (1989).
[3] Gardner, Marable, Welch, and Thomas, PRL 70, 22 (1993)
[4] Stokes, Schnurr, Gardner, Marable, Welch and Thomas, PRL 67, 15 (1991)
[5] E. Paspalakis and P. L. Knight, Phys. Rev. A 63 065802 (2001)
[6] Qamar, Zhu and Zubairy, Phys. Rev. Lett. 61 063806 (2000)
coupling
laser
probe
laser
atomic
medium
scanning
lens
2
Figure 2: A three state Λ system. States |1 and |3 are coupled with a “probe” beam, and states |1 and |2 are
coupled with a “coupling” beam. Under the correct conditions, state |3 is rendered invisible to the system and
absorption on the probe beam is eliminated.
The following Hamiltonian describes the interaction of the two beams with the threestate Λ system:
Ωp 
|  dark 
 *c
| c |  |  p |
2
2
1
Figure 3: A probe and coupling laser are focused into a gas of ultracold mediums which contains a nanoscale
object. Before applying the fluorescence laser, we first prepare all atoms in state |1, and then adiabatically
transfer them to state |2. Only atoms in a tightly confined volume will transfer to |2 and therefore be excited by
the fluorescence laser.
We initialize the sample in state |1 and then, using a counterintuitive pulse sequence,
we adiabatically transfer the populations from |1 to a superposition of states |1 and |2.
The key idea is that the population of state |2 will sensitively and nonlinearly depend on
the intensity profile of the probe and coupling light fields, as noted in Eq. (3).
E pμ

(1)
Eμ
Ωc  c

| 2
 *p
| 1 
| c |  |  p |
2
2
| 2  0 | 3
(2)
| 
dark
| p |
2
2
|  p | | c |
2
(3)
2
From state |2, we use standard resonance fluorescence techniques to excite the atoms
to state |4 and collect photons emitted by spontaneous decay (Fig 3). The analytical
consequence stemming from Eq. (3) is shown in Figs. 1 and 4. We utilize a
counterintuitive pulse sequence in which the coupling field is first turned on slowly,
followed by the probe field. We then smoothly turn these fields off simultaneously in
order to preserve the populations of |1 and |2 during fluorescence.
[7] S. E. Harris, Phys. Today 50, No. 7, 36 (1997).
Numerical Simulation
Numerical Simulation
51.1 nm
0.5
1
-200 -100
(b)
50
100 200
x (nm)
100
0.25
0
c /
p /
0.5
0
(a)
0.25
0
These simulations are analytical solutions that assume the system is in the darkstate of Eq. (1).
0
1
100 200
-4
x (nm)
0
x (microns)
Figure 4: Localization of atomic excitation to spots much smaller than the diffraction limit. Plot (a) shows the
population transfer, |ψ|2|2, as a function of the transverse coordinates, x. Plot (b) shows the spatial distribution of
probe and coupling beams. These distributions are calculated numerically by taking into account vectorial
properties of light near a focus. Plot (c) is a zoom in on plot (a) detailing the spatial structure of the excitation. The
excitation has a FWHM of 51.1 nm, which is about 16 times shorter than the wavelength of the excitation beams.
50 nm
FWHM
-2
0
2
4
time (s)
Figure 6: We recalculate Fig. 4 (c) numerically. Plot (a) compares the analytical curve (solid line) with the numerical
curve (points). Rather than assuming the system is in the dark state, we now consider finite Gaussian pulses,
resulting in a nonvanishing population in |3 of the dark state. Plot (b) shows the pulse shape that is applied at each
point in (a) to produce the numerical curve; dashed line is the coupling beam, solid line is the probe beam. The
atoms are probed at +2 μs, represented by the tick mark.
Considering the results of Fig. 6, we numerically simulate additional non-ideal experimental
parameters to test our system’s sensitivity to error. We find that the scheme presented here
is robust against both time and intensity fluctuations [Fig. 7].
0.5
population transfer
Results from these simulations suggest a factor of 16 improvement in the atomic
excitation waist versus the wavelength of the light. We can arbitrarily improve the
spatial resolution by manipulating the parameters of Eq. (4) until we are limited by
Heisenberg’s position-momentum uncertainty [2].
-1
excited state fraction
|2||2
Using light fields saturated absorption locked to the 5S1/2-5P3/2 transition, λ=780nm,
the coupling and probe beams will originate from the same master oscillator and
then split by 6.8GHz, corresponding to the ground state hyperfine splitting. The
spontaneous decay rate from the excited state is Γ = 2π x 6.06 MHz. We simulate
the results shown here assuming the peak values of the Rabi Frequencies are
Ωp,peak = Γ/2 and Ωc,peak = 100Γ.
0
We proceed by considering general parameters to understand the scanning speed that
this method demands. A first realizable experiment of this microscopy scheme may
include a nanotube mounted in an ultracold atomic cloud, similar to Fig. 3. The cloud can
be obtained through standard Magneto-Optical Trap and Far-Off Resonant Dipole Trap
techniques. If our volume of excitation is (50 nm)3 ≈ 10-16 cm3 in an ultracold atomic
cloud with density N = 1014 cm-3 [8], then we would expect to require ~100 scans per
point, with each scan requiring about 10ms to obtain good signal to noise ratios [9-11].
Thus each point requires ~1 second integration time.
Both the scanning speed and resolution requirements can be improved with more intense
coupling and probe light fields. We report that developments in producing high-intensity
fields appropriate for a system similar to this has recently been demonstrated [12].
[8] R. Newell, J. Sebby, and T. G. Walker, Opt. Lett. 28, 1266 (2003).
[9] N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, Nature (London) 411, 1024 (2001).
[10] D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, Phys.
Rev. Lett. 93, 150501 (2004).
[11] D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman,
Phys. Rev. Lett. 96, 063001 (2006).
[12] B. E. Unks, N. A. Proite, and D. D. Yavuz, Rev. Sci. Inst., submitted.
We proposed a novel use of EIT which shows promise for sub-wavelength
imaging. Our simulations show that two, on-resonant laser beams focused to a
diffraction limited spot produces an excitation that is a factor of nine tighter than
the optical waist.
Possible experimental issues include mechanical noise and the heating due to
van der Waals interactions at close range (r < 10 nm). Heating by up to 20 mK
can be expected, but this problem is addressed with correspondingly faster Rabi
Frequencies on the probe and coupling beams [12].
0
-200 -100
 coupling, peak
(4)
(b)
intensity
0
 probe, peak
Conclusions
Fig. (4) assumes that the system is in the dark state. We next discuss temporal evolution.
We repeat the results of Fig. 4 (c) numerically by using a counterintuitive pulse sequence in
which first the coupling field, and then the probe field, is smoothly turned on. We assume a
the probe field is a finite Gaussian pulse-shape in time [Fig. 6(b)]. When we compare the
results of this lab situation to the ideal situation, we find that our system’s resolution ability is
nearly unaffected.
population transfer
0.25
-1
After numerically solving for the intensity distribution of the light, we simply evaluate
Eq. (3) at every point of interest [Figs. 1 and 4]. We choose the well studied
experimental medium of 87Rb for our atomic gas. The ground state, 5S1/2 is an ideal
system suited for our purpose. We use the hyperfine splitting, |F=1 and |F=2 for
states |1 and |2 in Fig. 2. We take the excited state |3 in Fig. 2. to be 5P3/2 |F=1.
population transfer
|2||2
We are concerned with focusing light down to a diffraction limited spot. It is well
known that the Gaussian approximation breaks down where waist ≈ λ. For these
simulations, we turn to a formalism developed by Richards and Wolf to numerically
find the intensity profiles by considering the vectorial nature of light [13]. We
assume that λ = 780nm and that we focus the beams from a semi-aperture angle of
α = 60°.
population transfer
(c)
0.5 (a)
Based on Eq. (3), we find the waist of the atomic excitation in the atom cloud is directly
proportional to the intensity waist of the light fields. Noting Eqn. (4), the constant of
proportion depends directly on the strength of the coupling field and probe fields in the
limit that Ωp,peak << Ωc,peak.
waist excitation  waist light *
probe and
coupling
lasers
|1
An eigenstate of this Hamiltonian is termed the “dark state,” defined as:
Figure 1: Two laser beams (λ = 780nm) are focused into an ultracold cloud of atoms. Plot (a) shows is the
diffraction limited focal spot calculated using the vectorial properties of light. Plot (b) shows the tightly confined
excitation fraction. The excitation is a factor of 16 smaller than the light wavelength. The key idea is to utilize
the dark state of Electromagnetically Induced Transparency with focused beams to resolve better than the
diffraction limit.
fluorescence
laser
|2
0
p 
 0


H  0
0
c 
 p * c *  / 2 


At the center of the region of interest, the Rabi Frequency of the probe field is at its
maximum and the Rabi Frequency while the coupling field is at a minimum.
nano-scale
object
Ωp : Rabi Frequency of probe beam
Ωc : Rabi Frequency of coupling beam
Atomic Excitation
(b)
We proceed with an experimental schematic for applying EIT to achieve sub-wavelength
optical microscopy. We propose to focus probe and coupling lasers along a common
axis to a tight diffraction-limited spot in a cloud of ultracold atoms. The cold atom
sample is obtained using standard Magneto-Optical Trap (MOT) and dipole trap
techniques. An nanoscale object placed mounted in the atomic cloud could then be
imaged [Fig 3].
|3
P. L. Knight and colleagues propose a three-level Λ-system interacting
with a probe laser field and a classical standing-wave coupling field [5].
The proposed scheme localizes atoms to the sub-wavelength regime.
Zubairy and co-workers propose several standing-wave schemes to
localize the positions of atoms. One such scheme involves utilizing a
standing wave and detecting photon emission from spontaneous decay.
The frequency of these photons contains position information, as the
standing wave shifts the levels in a position dependent manner [6].
Nanoscale Microscopy Schematic
We note several important advantages of our system over prior work [1-6]. This
system would ideally be employed in the common and well-studied system of Rb87. A key advantage is that we require only two resonant fields, both of which
may be inexpensively provided by a single, common master-oscillator [12]. A
strong off-resonant field is not required [2-4]. Additionally, our system shows
great robustness to common experimental errors such as intensity and timing
fluctuations.
Future work includes further theoretical calculations to carefully test problems
due to heating by the sample. Additionally, we plan to consider this system as a
possible single atom / single photon source. Experimental verification of the
results shown here is realizable with present-day instruments.
0.25
Acknowledgements
0
Figure 5: Localization of the atomic excitation in the focal plane. The excitation waist is much smaller than the
waist of the diffraction limited beams.
[13] B Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959).
-200 -100
0
100 200
x (nm)
Figure 7: We recalculate Fig. 4 (c) numerically. The solid line represents the numerical simulation of Fig. 5.
We assume 10% intensity fluctuation at each point (solid points) and +/- 1μs temporal fluctuation in the
pulse timings (open points).
We would like to thank Thad Walker for helpful discussions as well as an
anonymous referee for important suggestions. This work was supported by a
start-up grant from the Department of Physics at University of WisconsinMadison.