Spin Noise Spectroscopy in K and Rb

Nick Krueger & Robin Heinonen

Advisors: Paul Crowell & Gordon Steckein

Introduction

Spin noise spectroscopy (SNS) was shown to be a valid method for determining the atomic properties of Rb and K atoms. SNS is a novel, effectively non-perturbative probe of the atomic systems. Quantitative and qualitative comparisons are made to quantities from literature. Future development of this technique could include applications to ultra-cold gases and semiconductor systems.

Theory

The fluctuation-dissipation theorem guarantees that we may determine the response of a system to an external perturbation from the system's statistical fluctuations at equilibrium [6,7]. In our experiment, a laser detuned sufficiently from an optical transition is used to probe a warm gas of alkali atoms at equilibrium. The laser detuning ensures that no absorption occurs and that the laser is a non-perturbative probe [6,7]. A magnetic field B, small enough that the Zeeman energies are at most comparable to the hyperfine splitting energies, is applied transverse to the direction of laser propagation. Statistical fluctuations in the net magnetization precess about this magnetic field as superpositions of Zeeman states, and as a result coherence between Zeeman sublevels occurs in the projection of the precessing magnetization noise in the direction of laser propagation [7]. The precession also shifts the equilibrium above the low-frequency regime where environmental noise dominates [7]. The component of magnetization fluctuations in the direction of laser propagation causes Faraday rotation fluctuations δφ, proportional to the magnetization fluctuations, in the laser polarization, which are measurable to a high degree of precision [5]. (Recall that the Faraday effect is a small change in the polarization angle of light effected by the presence of a magnetic field.)

The coherences between Zeeman sublevels results in peaks in the noise spectrum at frequencies corresponding to Zeeman energy differences. At low field, the peak Ω satisfies

where g_F is the effective g-factor. As the field increases, the nuclear and electron spin decouple and the peak splits. The Zeeman energies now are given by the Breit-Rabi formula:

where Δ_hf is the hyperfine splitting energy at zero field, I is the nuclear spin, g_I is the nuclear g-factor, g_I is the electronic g-factor, F is the total spin, and the +/- is determined by F=I+/-1/2.

For transitions with Δ_hf=0, the peaks will occur at, to quadratic order in the magnetic field,

For transitions with Δ_hf=1, which we see only for potassium since the hyperfine energy is much higher for rubidium, the peaks occur at, to leading order,

providing an especially accurate method for finding the hyperfine splitting.

The noise power at a peak corresponding to a transition from Zeeman state i to state </j> can be shown [6] to be proportional to the square of the spin matrix element for the transition:

where A depends on the optical transition, the isotope, and properties of the laser beam, the components of σ are the Pauli spin matrices, and n is a unit vector in the direction of laser propagation. These matrix elements are in general complicated functions of the magnetic field, but may be computed by diagonalizing the quadratic Zeeman hamiltonian.

Apparatus

An open cavity adjustable wavelength laser is detuned from any optical transition in K or Rb. The laser is polarized with a linear polarizer and sent through a warm vapor or Rb or K. A pair of Helmholtz coils provides the out of plane magnetic field. After passing through the cell a polarized beam splitter is used to separate the linear polarized light into the LH- and RH- circular polarized light components. These components are incident on a balanced photo bridge which converts the difference in power of each diode into a voltage which is then sent through filters and amplifiers and finally to a digitizer. This digitized voltage has a Fourier spectrum applied thus obtaining the spin noise in the frequency domain.

Results

A and C show spin noise in the frequency domain. Error bars contribute from the standard deviation of flat areas of noise. B and D show the location of peaks predicted the by Breit-Rabi formula. Peak colors correspond to the number and color from A and C. Error bars in indistinguishable in both. Note that a linear background (low field Zeeman energy called Larmor frequency here) was removed to emphasize splitting.

Expected and measured values of various constants determined through fitting of the experimental data. "Zeeman" means the results were found from fitting of the low field Zeeman energy and "Hyperfine" implies the fitting was done on the higher energy (and frequency) hyperfine lines. The hyperfine energy was to large to observe directly for Rb.

Potassium spin noise data similar to the Rb. This includes the hyperfine energy for the two isotopes of K as well.

Potassium-39 integrated peak areas versus field. The theory curves are provided from the spin matrix elements discussed above. Only qualitative comparison is made due to an inability to determine the beam size or accurate detuning energy.

Conclusion

In conclusion we have presented the relevant spin noise and atomic theory for a determination of various constants to values from literature. Good quantitative and qualitative comparison are made.

References

[1] Happer, W. ``Optical pumping." Rev. Mod. Phys. 44 169--249 (1972).

[2] Kastler, A. ``Optical methods for studying Hertzian resonances." Science 158, 214--221 (1967).

[3] Itano WM, Bergquist JC, Bollinger JJ, Gilligan JM, Heinzen DJ, Moore FL, Raizen MG, and Wineland DJ. "Quantum projection noise: Population fluctuations in two-level systems." Phys. Rev. A 47, 3554--3570 (1993).

[4] Sorensen JL, Hald J, and Polzik ES. "Quantum noise of an atomic spin polarization measurement." Phys. Rev. A 80, 3487--3490 (1998).

[5] Happer W and Mathur BS. "Off--resonant light as a probe of optically pumped alkali vapors." Phys. Rev. Lett. 18, 577--580 (1967).

[6] Mihaila B, Crooker SA, Rickel DG, Blagoev K, Littlewood PB, and Smith DL. "Quantitative study of spin noise spectroscopy in a classical gas of ⁴¹K atoms." Phys. Rev. A 74, 043819 (2006).

[7] Crooker SA, Rickel DG, Balatsky AV, and Smith DL. "Spectroscopy of spontaneous spin noise as a probe of spin dynamics and magnetic resonance." Nature 431, 49--52 (2004).

[8] Steck D. "Rubidium 87 D Line Data." 25 Sept. 2001. http://steck.us/alkalidata/rubidium87numbers.1.6.pdf

[9] Steck D. "Rubidium 85 D Line Data." 20 Sept. 2013. http://steck.us/alkalidata/rubidium85numbers.pdf

[10] Tieke TG. "Properties of Potassium." Feb. 2010. http://staff.science.uva.nl/~tgtiecke/PotassiumProperties.pdf

Acknowledgments

We thank Paul Crowell and Gordon Stecklein for their invaluable mentorship and assistance on this project.