1. Nanoscale Microscopy using Electromagnetically Induced Transparency N. A. Proite and D. D. Yavuz Department of Physics, University of Wisconsin, Madison, WI, 53706 Introduction 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: Electromagnetically Induced Transparency Nanoscale Microscopy Schematic 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]. 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. 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]. [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) Numerical Simulation Conclusions Acknowledgements 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 Wisconsin- Madison. 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. 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. 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 three- state Λ system: An eigenstate of this Hamiltonian is termed the “dark state,” defined as: (2) [7] S. E. Harris, Phys. Today 50, No. 7, 36 (1997). 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). (3) 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°. [13] B Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959). 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 87 Rb for our atomic gas. The ground state, 5S 1/2 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 5P 3/2 |F=1 . 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. 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. x (nm) population transfer time (  s) intensity 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. (a)(b) 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 cm 3 in an ultracold atomic cloud with density N = 10 14 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. 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]. 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]. 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. 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). 440 nm (a) 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]. 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]. Intensity Distribution Atomic Excitation |1  |2  |3   pp cc (1) 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. 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 Ω p,peak << Ω c,peak. (4) [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. 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]. Using light fields saturated absorption locked to the 5S 1/2 -5P 3/2 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 Ω p,peak = Γ/2 and Ω c,peak = 100Γ. 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 Rb- 87. 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. 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]. These simulations are analytical solutions that assume the system is in the dark- state of Eq. (1). 50 nm FWHM 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. excited state fraction |  2|  | 2 (b) 50 nm 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. |  2|  | 2