## Abstract

Optical magnetic resonance tomography uses optical pumping and the paramagnetic Faraday effect to image spin density distributions in optically thin media. In this paper we present an apparatus that allows to measure the distribution of spin-polarized Cs atoms, which we applied to study the diffusion of Cs in Ne buffer gas by time-resolved 2D-mapping of the evolution of an initial inhomogeneous spin distribution. The diffusion constant *D*
_{0} for Cs in a Ne buffer gas of 1013 mbar is determined as 0.20(1) cm^{2}/s.

©2000 Optical Society of America

## 1 Introduction

Nuclear magnetic resonance tomography is a well-known technique that allows the mapping of spin-densities, relaxation times and flow velocities in dense media by measuring the precession frequencies of nuclear magnetic moments in inhomogeneous magnetic fields. It is best known for its spectacular success in medical science. *Optical Magnetic Resonance Tomography* (OMRT) is a related technique which detects the precessing electronic magnetic moments of paramagnetic atoms manipulated through the interaction with light [1, 3, 4]. Besides producing a macroscopic polarization by optical pumping, light is also used to monitor the precession of the electron spins. In this work we have applied OMRT to study the diffusional motion of spin-polarized cesium atoms in neon buffer gas. In contrast to conventional techniques where low temperatures and/or high magnetic fields are needed to produce (small) population differences of the nuclear magnetic sublevels, optical pumping of cesium vapor with resonant laser radiation may yield population inversions close to 100%. Moreover, the precessing magnetization is detected via the rotation of the polarization vector (paramagnetic rotation) of a near-resonant linearly polarized probe laser beam which traverses the medium perpendicular to the pumping radiation. To get information about the spatial spin distribution a small magnetic field gradient is superposed to the main magnetic field thus encoding the spatial spin distribution into a distribution of Larmor frequencies. Projection reconstruction is used to reconstruct the initial spin density distribution from frequency spectra obtained in gradients with different orientations. The diffusional motion can be monitored by varying the time interval between the spin preparation and the imaging process.

## 2 Experiment

The experiments were performed in a cubic quartz cell of 1 cm3 filled with Ne buffer gas (*p*=135 mbar) and condensed Cs (Fig. 1). The cell is located in the center of a solenoid used to apply the main field and two saddle coil pairs as well as one anti-Helmholtz coil pair for the generation of the field gradients. Radio-frequency is applied by a pair of Helmholtz coils. The cell and all coils are surrounded by a 1.5 mm thick cylindrical *µ*-metal shield (Ø=36 cm, l=60 cm) to reduce stray fields. Optical pumping is done with the beam from a diode laser whose frequency is locked to the F=4 component of the Cs D1-line at 894 nm using a *dichroic atomic vapor lock* [5]. The pumping beam - expanded by means of a telescope - passes through a polarizer and a λ=4-plate and finally traverses the cell parallel to the main magnetic field. The inhomogeneity and position of the beam are controlled by monitoring the light scattered from a screen by means of a CCD camera. The beam is chopped mechanically with a period of 20 ms.

After the pumping beam is blocked by the chopper the longitudinal magnetization *M*_{z}
is transferred into a transverse magnetization *M*
_{+}=*M*_{x}
+*iM*_{y}
by a radio—frequency *π*/2-pulse, after which the transverse polarization starts to precess in the static field B_{0}, producing a free induction decay (FID) of *M*
_{+}.

A second - free running - diode laser produces the linearly polarized probe beam which, after expansion to a diameter of 1.5 cm, intersects the pump beam at right angles within the cell. The probe laser is current-tuned to the frequency of maximum dispersion of the D2 transition without any active stabilization. The power of the probe beam is less than 100
_{µ}
W in order to minimize disturbing effects due to absorption of the probe beam. The paramagnetic rotation of the light polarization produced by the vapor magnetization is detected using an ‘autobalanced’ photodetector (New Focus, model Nirvana) and a polarizing beamsplitter cube orientated at 45° with respect to the incident polarization. The coherent transient signal (FID) is observed as the difference of the photocurrents of two photodiodes at the output ports of the polarizer.

In order to synchronize the radio-frequency pulses (provided by a waveform generator), the strength and orientation of the gradients and the detection of the FID (recorded by an ADC plug-in module), all devices are triggered by a multi-channel pulse generator and controlled by a personal computer. The pulse generator itself is triggered by the mechanical shutter.

After a delay time tdelay following the optical pumping pulse the imaging process is started through the application of the *π*/2-rf-pulse (Fig.2). Gradient fields are switched on at the same time to encode the spin density. The FID is recorded and stored on hard disk for off-line processing. A single tomographic snapshot of the diffusion process is taken by recording 40 FID’s with gradients of identical magnitude and different orientations in a plane perpendicular to B̂_{0}. A *projection reconstruction* (PR) algorithm is used to reconstruct the 2-dimensional spin density distribution [6]. For the tomography of our diffusion-limited gaseous Cs sample the PR algorithm is more suitable than a *Fourier imaging* algorithm.

The space-time evolution of the magnetization can be calculated by solving the diffusion equation including relaxation for the magnetization *M*_{z}
(*x⃗*)

for a given initial distribution of magnetization. *K* is the rate of homogeneous depolarization due to collisions with bu er gas atoms and residual absorption of the probe beam. *D* is the diffusion constant at the actual buffer gas density. As collisions with the cell walls destroy the polarization, the boundary conditions of Eq. (1) are simply that the magnetization *M*_{z}
vanishes at the cell walls. The solutions of Eq. (1) for a cubic cell can hence be expanded in terms of cosine functions with a spatial periodicity of ${n}_{i}\xb7\frac{a}{2}$ where *a* is the cell dimension and *n*_{i}
are integers:

These functions are commonly referred to as *diffusion modes*. Each three-dimensional mode is labeled by a triplet (*l*, *m*, *n*) of integers and decays exponentially with a time constant

In order to simulate the diffusion process we expand the measured initial spin distribution into cubic diffusion modes. As the time development of each mode is known, this allows to compare measured and calculated distributions at later times and to determine the diffusion constant *D*. We determined the spatial resolution of the imaging process and calibrated the magnitude of the gradient fields by recording the magnetization generated by pumping the cell through a calibration mask inserted into the pump beam. Fig. 3 shows an example where the mask consisted of six circular apertures (*Ø*=1 mm, 2.5 mm separation). The tomographic image of the created polarization pattern is amplitude coded. Each initial cylindrically shaped distribution of the magnetization is convoluted with a Gaussian distribution due to diffusion during the FID. From the widths of the peaks we determined the experimental resolution to be 360(20) *µ*m.

## 3 Discussion

In order to measure the diffusion of the spin polarized cesium atoms within the cell, we first prepare a spatially inhomogeneous distribution of spin polarization using a masking aperture of approximately 2 mm clear diameter centered in the pump beam. Stroboscobic pictures of the diffusion process are obtained by varying t_{delay} in steps of 0.5 ms from 0 to 10 ms after the pump pulse. The animation linked to Fig. 4 (left side) shows the measured evolution in a cell with a buffer gas pressure of 135 mbar. Cuts through the distribution are shown in Fig. 6. According to the model outlined above the higher order diffusion modes vanish quickly until only the lowest order mode remains visible; the distribution evolves from an initial Gaussian-shaped distribution to a pure cosine-shaped distribution (lowest order diffusion mode).

In order to extract a quantitative value of the diffusion coffiecient *D* from the measured distributions we proceeded as follows: We first determined the width Δ*x*
_{FWHM} of each measured distribution. The results are shown as dots in Fig. 7. For an open cell (no boundaries) one would expect Δ*x*
_{FWHM} to vary according to

The deviation of the experimental results from this simple law is due to the cell boundaries i.e. to the rapid damping of the higher order diffusion modes. We simulated the actual experimental situation by decomposing numerically the distribution at t_{delay}=0 into diffusion modes (Eq. (2)), calculating the relaxation of each mode according to Eq. (3), reassembling the relaxed modes and finally determining the widths of the relaxed distributions. The values thus obtained are shown as curved line in Fig. 7. Adjustment of the calculated widths was done by a *X*
^{2}-minimizing procedure that varied the parameter *D*. Note that one of the main advantages of our method is the fact that the determination of *x*
_{FWHM} depends only on *D* and is independent of *K*. We obtained a minimal *X*
^{2} value for *D*=1.73(4) cm^{2}=s from data recorded at *T*=51(2) °C. The straight line in Fig. 7 is given by Eq. (4) with this optimal value of *D*. The animations on the right hand sides of Figs. (4,5) show the simulated distributions after the fitting procedure.

It is customary to scale the diffusion constant *D*
_{0} to standard pressure (*p*
_{0}=1013 mbar) and temperature (*T*
_{0}=20°C) via the relation

We determined the Ne buffer gas pressure *p* in our cell in an auxiliary experiment by measuring the pressure shift of the F=3, M=0→F=4, M=0 hyperfine clock transition of the ground state in an optical-microwave double resonance experiment. A small magnetic field (5 *µ*T) was applied in order to lift the Zeeman degeneracy of the ground state sublevels and to isolate the 0-0 transition. The transition-frequency was found to be shifted by 62.0(2) kHz with respect to the clock frequency of 9.19263177 GHz. From the pressure shift measurements described in [8] we conclude that the buffer gas pressure in our cell at *T*=51(2) °C is *p*=135(5) mbar. This gives a reduced diffusion coefficient of

The literature shows a broad range of experimental values for the diffusion constant of cesium in neon buffer gas, ranging from 0.40 cm^{2}/s down to *D*
_{0}=0.10 cm^{2}/s (Measured values up to 1989 are tabulated in [9]). In most of these experiments, the diffusion constant is deduced from the measurement of the longitudinal relaxation time T1 as a function of buffer gas pressure. Diffusion to the walls as well as depolarizing collisions cause the magnetization to vanish. In the analysis of this integral measurement there is a strong correlation between the parameters describing the diffusion and the depolarization and even small (depolarizing) admixtures of paramagnetic impurities to the buffer gas can influence the inferred diffusion constants significantly. The direct space- and time-resolved measurement of the diffusion constant in our experiment is not affected by depolarization processes due to impurities or depolarizing scattering of probe laser radiation.

## 4 Conclusion and outlook

We have developed a novel technique for the quantitative measurement of the diffusion constant of spin polarized alkali atoms in a noble gas. The technique allows to determine the diffusion constant *D* of Cs in Ne with a precision of 2%, while the reduced diffusion constant *D*
_{0} could be determined with precision of 4% only, due to the uncertainty of the buffer gas pressure. Compared to previously used techniques our method is insensitive to uncertainties in the relaxation times. Moreover the diffusion constant D - of dimension (length)^{2}/time - is easily calibrated, the length scale being determined by a calibrating mask and the time scale being given by the delay generating synthesizer used. In the future we plan to apply the technique presented here to samples of Cs atoms embedded in crystalline ^{4}
*He* matrices [10] in order to study their spatial distribution. The strongly suppressed diffusion in these samples and the very long coherence times *T*
^{2}>300 ms should allow to achieve spatial resolutions in the micron range.

## References and links

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