Analyzing atomic noise with a consumer sound card

Carsten H. H. Schulte, Georg M. Muller, Hauke Horn, Jens Hubner, and Michael Oestreicha)

Institut fur Festkorperphysik, Leibniz Universitat Hannover, Appelstraße 2, D-30167 Hannover, Germany

(Received 8 March 2011; accepted 2 November 2011)

We discuss an advanced undergraduate project on spin noise spectroscopy of atomic rubidium

vapor. The spin noise is digitized using a consumer sound card and analyzed by a fast Fourier

transform. Students gain competence in digital data acquisition and processing, and the idea of

analyzing a noise signal is emphasized. VC 2012 American Association of Physics Teachers.

[DOI: 10.1119/1.3663275]

I. INTRODUCTION

In 1905 Einstein proposed a fundamental relation betweennoise and dynamics in terms of the stochastic Brownianmotion of a particle suspended in a liquid.1 The universalityof this relation is described by the fluctuation-dissipation the-orem.2 This theorem links, for example, the magnetizationfluctuations in an equilibrium system to its magnetic suscep-tibility. Analogously, the time correlations of the fluctuationsof the magnetization reveals the underlying spin dynamicsunder an infinitesimal external perturbation.3 Hence, the fullspin dynamics can be determined by investigating the fluctu-ations and time correlations of the system if the experimentalprobe creates only an infinitesimal perturbation.

Spin noise spectroscopy records the spin fluctuations viathe optical Faraday effect, which enables such a perturbation-free mapping of the spin noise onto the rotation of the linearlight polarization of a probe laser whose energy is slightlydetuned from an appropriate optical transition.4,5 Spin noisespectroscopy is used in atomic optics as a non-destructivemeasurement of the macroscopic magnetization,6 and enableslight-matter or matter-matter entanglement.7–9 In semiconduc-tor physics the technique yields a perturbation-free probe ofelectron spin dynamics.10–12

An undergraduate project on spin noise spectroscopyintroduces students to atomic physics (energy levels, Lorentzoscillator model, and the relation between dispersion and dis-sipation, that is, the Kramers-Kronig relation), statisticalphysics (fluctuations from the mean, correlation functionsand their relation via Fourier transform to the spectral powerdensity function), different types of physical and measure-ment noise (classical 1=f noise, photon shot noise, projectionnoise, and digitizer noise), and the basics of optical experi-ments (laser and polarization optics). Students also becomefamiliar with concepts such as probing with off-resonantlight13 and non-destructive measurement techniques.

In this paper, we discuss an implementation of spin noisespectroscopy that utilizes a consumer sound card for analyzingthe noise. The usually most expensive part of the experimentis replaced by an inexpensive consumer product. Our experi-ence is that the required programming expertise correspondsto the skills of most undergraduate physics students. The real-ization of this project can be a valuable resource for studentsin subsequent laboratory work because digital data acquisitionand processing is of much importance in many experiments.

II. BASICS OF SPIN NOISE SPECTROSCOPY

Figure 1(a) depicts the basic setup. Spin noise spectroscopymeasures the fluctuating macroscopic spin component of thesystem of electrons or atoms along the direction of light

propagation sz by off-resonant Faraday rotation. To this end,linearly polarized laser light with its energy slightly detunedfrom the optical transition of interest is sent through thesample. The rotation of the light polarization is measured by aWollaston prism oriented 45� with respect to the originalpolarization. The prism projects the linearly polarized laserlight onto theþ 45� and� 45� linear polarization states, whichenables the sensitive measurement of small rotations of thelinear polarization with a pair of balanced photodiodes. Thedifference of the electrical signal of the balanced photodiodesis digitized and spectrally analyzed.

Typical noise spectra are depicted in Fig. 1(b). In additionto the actual spin noise, the measured noise signal also con-tains random photon shot noise and electrical backgroundnoise. The subtraction of a reference curve without the spinnoise yields the pure spin noise signal.

A. Spin noise

The magnetization of a system in thermal equilibrium isintrinsically subject to fluctuations, and the random deviationsfrom the mean dephase after a finite time.14 Also the projec-tion to basis states by the measurement introduces projectionnoise,15 even for the case of infinitely long spin coherence.(Projection noise results from the inherent uncertainty in themeasurement of a superposition state.) It becomes important ifa spin polarization transverse to the direction of light propaga-tion is induced by optical pumping by a second laser.7 In theexperiments discussed below, the dephasing of the incompleterandom cancellation of up and down spins at thermalequilibrium yields the measured spin noise. The mean devia-tion of the magnetization at thermal equilibrium is given fornoninteracting spins by the binomial distribution and is pro-portional to 1=

ffiffiffiffi

Np

, where N is the number of probed spins.16

Accordingly, spin noise spectroscopy is an especially interest-ing tool for small spin systems such as semiconductornanostructures.11,17

B. Below band-gap Faraday rotation

The spin noise is mapped in the experiment onto the orien-tation of the linear polarization of the probe light via the Far-aday effect. For rubidium vapor the spin polarization of the5S valence electrons is probed. To this end, the D1 transitionto 52P1=2 (794.98 nm) or the D2 transition to 52P3=2 (780.24nm) is virtually excited by detuning the probe laser from thecorresponding resonance (see Fig. 2 for the energy level dia-gram of rubidium). A random spin polarization in the 5Slevel introduces a different absorption of left (rþ) and right(r�) circularly polarized probe light. This difference in ab-sorbance yields a circular birefringence (Dn¼ nþ� n�) via

240 Am. J. Phys. 80 (3), March 2012 http://aapt.org/ajp VC 2012 American Association of Physics Teachers 240

the Kramers-Kronig relation and results in a rotation of line-arly polarized probe light by the angle h / Dn / sz. Here,nþ and n� are the real part of the refractive indices for rþ

and r� light, z is the direction of laser propagation, and sz isthe random spin polarization along z. The rotation of the lin-ear polarization by the electron spin polarization is known asthe Faraday effect.18 The Kramers-Kronig relations revealthat the dispersive (real) part of the refractive index, whichdetermines the strength of the Faraday rotation, decreasesmore weakly with detuning from the optical resonance thanthe dissipative (imaginary) part, which describes energyabsorption in the sample. This effect corresponds to theeveryday experience that glass is transparent in the opticalregime but shows significant dispersion, which is the basisfor refraction. Correspondingly, the Faraday effect allowsoff-resonant probing of the electron spin at negligible energytransfer to the sample and enables experimental access to thespin dynamics in thermal equilibrium. Operation close toresonance with non-negligible energy transfer is also possi-ble and yields higher signal amplitudes, but the systemis driven more out of equilibrium, and thus the spin lifetimedecreases.19 Faster dynamics require a higher detectionbandwidth. Therefore, a detuned laser is preferred becausethe detection bandwidth is limited by the sound card.

C. Spectral analysis

The resulting polarization is converted into an electricalsignal by means of a polarizing beam splitter and a pair ofbalanced photodiodes. Besides the actual spin noise, the sig-nal from the balanced receiver also contains photon shotnoise of the probe laser, some electrical noise introduced bythe electronic components, and classical laser noise thatscales with the inverse frequency and is therefore referred toas 1=f noise. Although 1=f noise contributes only at very lowfrequencies, photon shot noise becomes evident as frequencyindependent noise. In contrast, the spectral shape of the spinnoise is determined by the underlying spin dynamics. Thepower spectral density of the spin noise S 2pfð Þ / F sz tð Þf gj j2(F denotes the Fourier transform) is proportional to the realpart of the Fourier transform of the spin autocorrelation func-tion. This link between the autocorrelation function and thepower spectral density is universal and is known as theWiener-Chintchin theorem.20,21 The spin noise spectrum,which is the power spectral density, is given by a Lorentziancurve centered at zero frequency with the full width at halfmaximum given by 1=(pT2) for an exponential spin decayhsz 0ð Þsz tð Þi / exp �t=T2ð Þ, where T2 is the spin dephasingtime.22 Applying a magnetic field B transverse to the direc-tion of light propagation induces a Larmor precession of thespin polarization with the Larmor frequency xL. The Larmorprecession is equivalent to a modulation of the detected spinnoise. In other words, the spin correlation function in thepresence of a magnetic field in the x-direction is given byhsz 0ð Þsz tð Þi / cos xLtð Þexp �t=T2ð Þ. Accordingly, the Lor-entzian spin noise curve is centered at the angular frequencyxL. This modulation allows for separation of the spin noisefrom classical 1=f noise which is both centered at zero fre-quency in the absence of a magnetic field. The Larmor fre-quency is given by xL ¼ g�lBB�h�1, where lB is the Bohrmagneton, g* is the effective electron Lande g-factor, and �his the reduced Planck constant. The effective Lande factor g*

can significantly differ from the free electron value g0 � 2.This deviation originates from spin-orbit and=or hyperfinecoupling. For atomic rubidium vapor, the coupling with thenuclear spin yields g*¼ gF¼ 1=2 for the isotope 87 Rb andg*¼ gF¼ 1=3 for 85 Rb (see Fig. 2).

D. Spin dephasing

The spin information in the probe laser volume decreasesdue to the spatial motion of the atoms within the cell. Theresulting finite interaction time of light and atomic spinsyields a broadening of the spin noise curve in accordancewith the Fourier uncertainty principle.23 The helium buffergas inside the cell reduces the mean free path of the rubidiumatoms and lowers the efficiency of this time-of-flight broad-ening which still dominates spin dephasing in the experi-ment. Due to inter-atomic collisions, the atomic motion isdiffusive and accordingly a 1=w2 dependence of the meas-ured spin dephasing rate on the laser beam waist w isobserved.24 Inter-atomic collisions do not contribute to spindephasing. Collisions among the rubidium atoms conservethe total spin of the ensemble, and no spin can be transferredto the helium atoms due to the noble gas configuration. Incontrast, collisions with the internal walls of the cell are animportant source of depolarization of the atomic spin.

In semiconductors the electron spin couples via hyperfineor spin-orbit interaction much more strongly to its

Fig. 1. (Color online) (a) Off-resonant linearly polarized laser light is trans-

mitted through a rubidium gas cell. The direction of the linear light polariza-

tion acquires a rotation due to the random spin polarization in the sample

based on the magneto-optical Faraday effect. The resulting polarization

noise is converted to electrical noise which is digitized and spectrally ana-

lyzed on a computer using a fast Fourier transform. (b) The black curve

shows a noise spectrum including contributions from spin noise, shot noise,

electrical noise, and classical 1=f noise. Subtracting a reference spectrum

containing only background noise yields the pure spin noise signal. The

spikes in the spectrum slightly below 80 kHz result from electromagnetic in-

terference with unknown sources. The noise power above 90 kHz drops due

to the limited bandwidth of the sound card.

241 Am. J. Phys., Vol. 80, No. 3, March 2012 Schulte et. al. 241

environment. Accordingly, spin dephasing times are signifi-cantly reduced and the collision-free time evolution of aconduction-band electron is usually not spin-conserving.A brief discussion on this topic is given in Ref. 12.

III. EXPERIMENTAL DETAILS

A. Probe laser

We use a laser diode with an external resonator in the Lit-trow configuration25 for probing the optical D2 transition inrubidium. The back facet of the laser diode and a diffractiongrating form the laser cavity. The first-order diffracted lightis sent back into the laser diode while the zeroth-order lightis the specular reflection out of the cavity. Rotating the dif-fraction grating enables the tuning of the laser wavelengthclose to the narrow optical transition which lies within thegain spectrum of the laser diode. The reduction of the freespectral range of the laser due to the external cavity enablesstable operation in a single laser mode without mode jumps.The optical frequency drifts in the kilohertz to megahertzrange because it is not additionally stabilized, for example,by locking it to an external reference. The drift is not rele-vant to the experiment due to the large detuning from theatomic resonance. The Littrow configuration allows a sweep-ing of the laser wavelength over the rubidium absorptionspectrum and is described in more detail in Ref. 25. Using afree running laser with a relatively broad laser line around awavelength of 780 nm that is tuned by changing the diodetemperature is sufficient to achieve a proof of principle real-ization of this experiment. A variety of AlGaAs-based laserdiodes with the corresponding wavelength are commerciallyavailable. A low-cost battery-based laser diode driver kit is asufficiently stable current source.

B. The rubidium cell

In this work, we use a quartz cell with purified 87 Rb andhelium buffer gas at a pressure of 1 mbar. The buffer gasreduces the mean free path of the rubidium and increases themeasured effective spin dephasing times by decreasingthe efficiency of the time-of-flight broadening. However, anyrubidium reference cell can be used for this experiment.The cell is heated to around 450 K, and the correspondingrubidium vapor density can be calculated by the Clausius-Clapeyron relation and is approximately 2� 1014cm�3 (seeRef. 26).

C. Terrestrial and applied magnetic fields

A magnetic field is applied for modulation of the spinnoise with the Larmor frequency. Because the Earth’s mag-netic field yields a Larmor frequency of several hundred kHz(the Earth’s magnetic field is � 49.2 lT in Hannover), themagnetic field of the Earth needs to be compensated in allthree directions by three pairs of Helmholtz coils. Magneticfields of this strength can be measured with high accuracy bya low-cost fluxgate magnetometer (Forster probe). Magneticfields that correspond to Larmor frequencies greater than thedetection bandwidth of the setup enable the recording of ref-erence noise spectra which do not contain any spin noisecontribution. These reference spectra can be subtracted fromthe noise spectra to remove the photon shot noise back-ground [see Fig. 1(b)].

D. Photodetection

A commercial balanced photoreceiver based on siliconphotodiodes with a relatively high bandwidth of 80 MHz

Fig. 2. Energy states of 87 Rb and 85 Rb.26 The different hyperfine splitting for the two isotopes results in different effective g -factors g*¼ gF¼ 1=2 (87 Rb)

and g*¼ gF¼ 1=3 (85Rb). The hyperfine splitting is negligible for the experiment.

242 Am. J. Phys., Vol. 80, No. 3, March 2012 Schulte et. al. 242

was used. This high bandwidth is not needed for the experi-ment, and even a hand-made balanced photodetector usingcommercial silicon photodiodes can be utilized. A secondbalanced receiver can be used to measure the absorption ofthe rubidium cell. This extension of the setup is especiallyhelpful if the laser energy is continuously swept and enablesrecording of the rubidium absorption spectrum and calibra-tion of the laser energy via the sharp absorption lines.

E. Spectral analysis

We employed a hand-made fast Fourier transform (FFT)spectrum analyzer for this experiment. An FFT spectrum an-alyzer consists of an analog-to-digital (A=D) converter andan implementation of the FFT algorithm to convert the timeseries into the frequency domain. We use a consumer PCIsound card for A=D conversion (see the Appendix fordetails). The idea of analyzing the spin noise with a soundcard is appealing to students and results in a significant costreduction of the experiment. The actual FFT analysis can beeasily realized by modules in the LABVIEW programmingenvironment or the free GNU scientific library.27 We use asound card with a sampling rate of 192 kHz and a resolutionof 24 bit. According to the Nyquist-Shannon theorem,28,29

the sampling rate corresponds to a detection bandwidth of 96kHz. The high dynamic resolution (bit depth) of the utilizeddigitizer enables FFT spectrum analysis without any influ-ence of noise due to binning.30 In our case a FFT containing4096 points gives sufficient frequency resolution. We aver-age 100 of these spectra before the applied magnetic field isswitched off to acquire a reference noise spectrum. Switch-ing off the applied magnetic field shifts the spin noise out ofour detection window due to the uncompensated terrestrialmagnetic field. Measurement times of around 20 min areused including some idle time for switching the magneticfield and computation. A sweeping spectrum analyzer canalso be used for spectral analysis and shows decent detectionsensitivity for the experiment. However, at high bandwidths,the FFT spectrum analysis combines the advantages of moreefficient data averaging and lower cost.12,31 The higher sen-sitivity results from the fact that the FFT spectrum analysisuses the complete data stream for averaging, while a usualsweeping spectrum analyzer acquires at a particular timeonly the frequency component of the signal at the corre-sponding frequency of the internal oscillator.

IV. EXPERIMENTAL RESULTS

Figure 1(b) depicts typical noise curves. The pure spinnoise spectrum is obtained by subtracting a reference noisespectrum which contains no spin noise. The ratio of peakspin noise density and the white photon shot noise back-ground is much larger than one, indicating a high signalstrength, which facilitates the implementation of the experi-ment. The laser wavelength is adjusted by first tuning it tothe D2 absorption line. For coarse tuning a low-resolutionspectrometer can be used. The fine tuning is best achievedwith an infrared viewer or a CCD camera because the reso-nant excitation of the D2 transition results in a strong fluores-cence of the cell. In the second step the laser is slightlydetuned from resonance because the spin dephasing ratesunder resonant excitation are increased due to the finite life-time of the excited states. Figure 3 demonstrates the modula-tion of the spin noise with the Larmor frequency by applying

a transverse magnetic field. A detailed evaluation of the spinnoise spectra by fitting the spectra with two Lorentziancurves reveals residual 85Rb in the reference cell as shown inFig. 4. The ratio of the two frequencies of 2=3 correspondsto the ratio of the two g-factors. The extracted peak width inFig. 4 yields a spin dephasing time of 36 ls. With a calcu-lated laser beam waist of around 20 lm, this dephasing timeis consistent with time-of-flight broadening due to transienteffects. Increasing the beam waist increases the spin dephas-ing time and reduces the width of the spin noise curves. Theheight of the spin noise peak and, consequently, the signal-to-noise ratio is roughly independent of the probe laser beamwaist, because the integrated spin noise power also scaleswith the reciprocal laser beam area.5

V. DISCUSSION

We have discussed the use of spin noise spectroscopy onrubidium vapor as a project at the advanced undergraduate

Fig. 3. Spin noise spectra at different effective magnetic fields. The spin

noise is centered at the Larmor frequency which is proportional to the mag-

netic field. For clarity, the spectra are shifted vertically by 45 mV2=Hz each.

Fig. 4. (Color online) A detailed fit of the experimental spectra reveals re-

sidual 85Rb content in the cell. The presence of 85Rb is verified by a fit con-

taining the sum of two Lorentzian curves. The peak width corresponds to a

dephasing time of 36 ls.

243 Am. J. Phys., Vol. 80, No. 3, March 2012 Schulte et. al. 243

level in which atomic noise is demonstrated with compara-tively simple equipment. The project has been successfullyrealized in Hannover as an undergraduate practical project.The students gain understanding of a variety of differentphysical aspects. They learn, for example, how to tune semi-conductor laser diodes by varying the temperature, the cur-rent, or the external cavity. They also gain experience withatomic energy states and optical transitions. In addition, theyunderstand the optical Faraday effect, which is important formany experiments and applications. The students also learnabout important concepts such as correlation functions andtheir relation to noise spectra. Moreover, the students imple-ment efficient digital data acquisition and processing.

APPENDIX: METHODS

We used a AlGaAs-quantum-well laser diode with ARcoating and a center wavelength of 770 nm manufactured byeagleyard Photonics GmbH.32 Alternatively, a non-ARcoated laser diode may be used in the cavity, in which casethe laser diode current and the grating angle must be used totune the laser to the required wavelength.33 In our experi-ments the optimum detuning was found by searching for themaximum rubidium photoluminescence intensity caused byabsorption, followed by a slight detuning (the rubidium cellwas still glowing). For these observations an infrared vieweror a cheap CCD camera can be used, because the emittedlight of the D2 transition at 780 nm is not easily visible withthe naked eye. A webcam is suitable for this purpose. Thelaser power at the rubidium cell was typically 5–20 mW.

The rubidium cells were purchased from Triad Technol-ogy and Toptica.34,35 By increasing the size of the laserfocus, the time-of-flight broadening is reduced and the use ofrubidium cells without buffer is possible. Using a cell with anatural isotope ratio and a collimated laser beam of 2.9 mmdiameter, a spin noise peak of 33 kHz width was measured.Because most of the noise spectrum is generated within theRayleigh range around the focus, possible broadening effectsdue to magnetic field inhomogeneities become more signifi-cant when a larger focal spot, that is, a longer Rayleighrange, is used. The effects of the laser focus size on the spinnoise power are discussed in Ref. 5.

The balanced photo receiver with silicon photo diodes waspurchased from New Focus.36 The balanced receiver’s signalwas amplified with a voltage amplifier made by Femto37 toexploit the complete input range of the sound card. As men-tioned, a hand-made balanced photo receiver is easily built andshows satisfying performance. An amplifier could also be builtinto the balanced receiver, but is not essential due to the factthat the sound card worked with a resolution of 24 bits. Weused a Trace Alpha PCI sound card produced by Marian.38

a)Author to whom correspondence should be addressed. Electronic mail:

oest@nano.uni-hannover.de1A. Einstein, “Uber die von der molekularkinetischen Theorie der Warme

geforderte Bewegung von in ruhenden Flussigkeiten suspendierten

Teilchen,” Ann. Phys. 17, 549–560 (1905).2R. Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys. 29 (1),

255–284 (1966).3S. Kos and P. Littlewood, “Statistical physics: Hear the noise,” Nature

(London) 431, 29 (2004).4E. B. Aleksandrov and V. S. Zapasskii, “Magnetic resonance in the

Faraday-rotation noise spectrum,” Sov. Phys. JETP 54 (1), 64–67 (1981).5S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D. Smith, “Spectroscopy

of spontaneous spin noise as a probe of spin dynamics and magnetic reso-

nance,” Nature (London) 431, 49–52 (2004).

6A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P.

Bigelow, “Quantum nondemolition measurements of collective atomic

spin,” Phys. Rev. A 60 (3), 2346–2350 (1999).7B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived

entanglement of two macroscopic objects,” Nature (London) 413 (6854),

400–403 (2001).8B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik,

“Experimental demonstration of quantum memory for light,” Nature (Lon-

don) 432 (7016), 482–486 (2004).9K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface

between light and atomic ensembles,” Rev. Mod. Phys. 82 (2), 1041–1093

(2010).10M. Oestreich, M. Romer, R. J. Haug, and D. Hagele, “Spin noise

spectroscopy in GaAs,” Phys. Rev. Lett. 95 (21), 216603-1–4

(2005).11G. M. Muller, M. Romer, D. Schuh, W. Wegscheider, J. Hubner, and

M. Oestreich, “Spin noise spectroscopy in GaAs (110) quantum wells:

Access to intrinsic spin lifetimes and equilibrium electron dynamics,”

Phys. Rev. Lett. 101 (20), 206601-1–4 (2008).12G. M. Muller, M. Oestreich, M. Romer, and J. Hubner, “Semiconductor

spin noise spectroscopy: Fundamentals, accomplishments, and

challenges,” Physica E 43 (2), 569–587 (2010).13W. Happer and B. S. Mathur, “Off-resonant light as a probe of opti-

cally pumped alkali vapors,” Phys. Rev. Lett. 18 (15), 577–580

(1967).14F. Bloch, “Nuclear induction,” Phys. Rev. 70 (7–8), 460–474 (1946).15W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen,

F. L. Moore, M. G. Raizen, and D. J. Wineland, “Quantum projection

noise: Population fluctuations in two-level systems,” Phys. Rev. A 47 (5),

3554–3570 (1993).16A. Papoulis, Probability, Random Variables, and Stochastic Processes,

2nd ed. (McGraw-Hill, New York, 1991).17S. A. Crooker, J. Brandt, C. Sandfort, A. Greilich, D. R. Yakovlev,

D. Reuter, A. D. Wieck, and M. Bayer, “Spin noise of electrons and holes

in self-assembled quantum dots,” Phys. Rev. Lett. 104 (3), 036601-1–4

(2010).18M. Faraday, “On the magnetic affection of light,” Philos. Mag. 29,

153–156 (1846).19W. Chalupczak and R. M. Godun, “Near-resonance spin-noise

spectroscopy,” Phys. Rev. A 83 (3), 032512-1–7 (2011).20N. Wiener, “Generalized harmonic analysis,” Acta Math. 55 (1), 117–258

(1930).21A. Chintchin, “Korrelationstheorie der stationaren stochastischen

Prozesse,” Math. Ann. 109 (1), 604–615 (1934).22M. Braun and J. Konig, “Faraday-rotation fluctuation spectroscopy with

static and oscillating magnetic fields,” Phys. Rev. B 75 (8), 085310-1–6

(2007).23D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recov-

ery,” SIAM J. Appl. Math. 49 (3), 906–931 (1989).24G. E. Katsoprinakis, A. T. Dellis, and I. K. Kominis, “Measurement of

transverse spin-relaxation rates in a rubidium vapor by use of spin-noise

spectroscopy,” Phys. Rev. A 75 (4), 042502-1–6 (2007).25L. Ricci, M. Weidemuller, T. Esslinger, A. Hemmerich, C. Zimmermann,

V. Vuletic, W. Konig, and T. W. Hansch, “A compact grating-stabilized

diode laser system for atomic physics,” Opt. Commun. 117 (5–6),

541–549 (1995).26D. A. Steck, “Rubidium 87 D line data,” revision 2.1.4 (Theoretical Divi-

sion (T-8), MS B285, Los Alamos National Laboratory, Los Alamos,

2010).27M. Frigo and S. G. Johnson, “FFTW 3.3,” <www.fftw.org/>.28H. Nyquist, “Certain topics in telegraph transmission theory,” Trans.

AIEE 47, 617–644 (1928).29C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37,

10–21 (1949).30G. M. Muller, M. Romer, J. Hubner, and M. Oestreich, “Efficient data

averaging for spin noise spectroscopy in semiconductors,” Appl. Phys.

Lett. 97 (19), 192109-1–3 (2010).31M. Romer, J. Hubner, and M. Oestreich, “Spin noise spectroscopy in semi-

conductors,” Rev. Sci. Instrum. 78 (10), 103903-1–6 (2007).32Ridge waveguide laser with AR-coating from Eagleyard Photonics

GmbH, Rudower Chaussee 29 (IGZ), D-12490 Berlin, Germany, <www.

eagleyard.com/>.33M. Gilowski, C. Schubert, M. Zaiser, W. Herr, T. Wubbena, T. Wendrich,

T. Muller, E. Rasel, and W. Ertmer, “Narrow bandwidth interference

244 Am. J. Phys., Vol. 80, No. 3, March 2012 Schulte et. al. 244

filter-stabilized diode laser systems for the manipulation of neutral atoms,”

Opt. Commun. 280, 443–447 (2007).34Wavelength reference cells from Triad Technology, 640 South Sunset St.,

Longmont, CO 80501, <www.triadtechno.com/>.35Spectroscopy cells from Toptica Photonics AG, Lochhamer Schlag 19,

D-82166 Graefelfing (Munich), <www.toptica.com/>.

3680 MHz balanced photoreceiver 1807 from New Focus, 2584 Junction

Ave., San Jose, CA 95134-1902, <www.newfocus.com/>.37DHPVA-200 variable gain 200 MHz wideband voltage amplifier from Femto

Messtechnik GmbH, Klosterstr. 64, D-10179 Berlin, <www.femto.de/>.38Trace Alpha from Marian GmbH, Berggartenstrasse 12, D-04155 Leipzig,

<www.marian.de/en/>.

245 Am. J. Phys., Vol. 80, No. 3, March 2012 Schulte et. al. 245