Spin-noise correlations and spin-noise exchange driven by low-field spin-exchange collisions
aa r X i v : . [ phy s i c s . a t o m - ph ] S e p Spin-noise correlations and spin-noise exchange driven by low-field spin-exchangecollisions
A. T. Dellis , M. Loulakis and I. K. Kominis ∗ Department of Physics, University of Crete, 71103 Heraklion, Greece School of Applied Mathematical and Physical Sciences,National Technical University of Athens, 15780 Athens, Greece
The physics of spin exchange collisions have fueled several discoveries in fundamental physics andnumerous applications in medical imaging and nuclear magnetic resonance. We here report on theexperimental observation and theoretical justification of spin-noise exchange, the transfer of spin-noise from one atomic species to another. The signature of spin-noise exchange is an increase of thetotal spin-noise power at low magnetic fields, on the order of 1 mG, where the two-species spin-noiseresonances overlap. The underlying physical mechanism is the two-species spin-noise correlationinduced by spin-exchange collisions.
PACS numbers: 42.50.Lc, 03.65.Yz, 05.30.-d, 07.55.Ge
I. INTRODUCTION
The Pauli exchange interaction, of fundamental im-portance for understanding the structure of matter, alsounderlies spin-dependent atomic collisions [1, 2]. Spin-exchange collisions in atomic vapors have fueled a widerange of scientific investigations, ranging from enhancedNMR signals and new MRI techniques [3–5] to nuclearscattering experiments sensitive to the nuclear or nucleonspin structure [6]. Many of the aforementioned phenom-ena rely on the spin-exchange transfer of large spin po-larizations from one atomic species to another.We here extend spin exchange into a deeper layer ofcollective spin degrees of freedom, namely we demon-strate the transfer of quantum spin fluctuations fromone atomic species to another, a phenomenon we termspin-noise exchange. Quantum fluctuations and their in-terspecies transfer are central to emerging technologiesof quantum information, like quantum memories usingatomic spin or pseudo-spin ensembles [7, 8]. Spin noise[9], in particular, determines the quantum limits to theprecision of atomic vapor clocks [10] and the sensitiv-ity of atomic magnetometers [11–14], the most recent ofwhich utilize several spin species [15]. The fundamentalunderstanding of spin-noise exchange could have furtherrepercussions, from noise-energy harvesting in spintronicdevices [16], to novel spin-dependent phenomena in in-tergalactic hydrogen gas [17]. A similar effect to the onedescribed herein was observed with solid-state nuclearspins [18, 19], but the transfer of nuclear spin fluctua-tions was evoked with externally applied magnetic fields.In our case the transfer is spontaneous and driven byincessant atomic spin-exchange collisions.Spin-exchange collisions are central to optical pump-ing of atomic vapors [20]. Even without externally ma-nipulating atoms with light, i.e. leaving them in an un- ∗ Electronic address: [email protected] polarized equilibrium state, spin-exchange collisions leadto continuous spin fluctuations around the average valueof zero. Such spontaneous spin noise has been recentlydemonstrated [21–27] to be a versatile spectroscopic toolin atomic and condensed matter physics. In particu-lar, spin noise in a rubidium vapor was measured [21]at a magnetic field of several Gauss, allowing the spin-noise resonances of Rb and Rb (occurring approx-imately in the ratio 3:1 in rubidium of natural abun-dance) to be clearly distinguished. This is so since therespective gyromagnetic ratios are g = 466 kHz / G and g = 700 kHz / G, whereas the resonance line width wason the order of 10 kHz.The total area under the spectral distribution of spin-noise power is the total spin variance, intuitively ex-pected to be constant, i.e. independent of the magneticfield at which the measurement is performed, or equiva-lently, independent of where along the frequency axis thetwo spin resonances are positioned.We will here demonstrate experimentally and provetheoretically that the total spin-noise power of a two-species spin ensemble, like Rb - Rb, exhibits acounter-intuitive dependence on the applied magneticfield. This is the experimental signature of spin-noise ex-change, which is observable when the two atomic specieshave overlapping spin-noise resonances. For the reso-nance width in our measurement, of about 1 kHz, thisoverlap happens at magnetic fields on the order of 1 mG.In Section II and III we will describe the experimen-tal measurement and data/error analysis, respectively,while in Section IV the observed effect is explained the-oretically based on spin-noise correlations that build upat low magnetic fields due to spin-exchange collisions.
II. MEASUREMENT
The experimental scheme is shown in Fig.1, and is sim-ilar to previous studies of spin noise using a dispersivelaser-atom interaction [9, 21, 28–30]. An off-resonant
PD1 PD2
DiodeLaser
Oven
Rubidium Cell(s)
MagneticField Coils
Linear Polarizer
PolarizingBeam Splitter
Magnetic Shields yz Current Source B z x = Rb = Rb probe laserOne cell with rubidium of natural abundance Two cells back-to-back with isotopically enriched rubidium OR + SpectrumAnalyzer
PD=photodiode
FIG. 1: (Color online). Experimental schematic of the spinnoise measurement. For the actual measurement we used a10 cm long cell with Rb of natural abundance, while for aconsistency-check we used two 5 cm cells back-to-back, eachhaving isotopically enriched Rb. The temperature was mea-sured with a thermocouple placed at the oven’s center, read-ing 112 ◦ C. The temperature inferred from the collisionalline width of the spin noise resonance was 100 ◦ C, and it wasthe corresponding Rb density that we used in the theoreticalprediction. The laser power and detuning from the D2 linewere 3.3 mW and 43 GHz, respectively, while the pressure-broadened optical linewidth at 100 torr of nitrogen is about4 GHz. The magnetic field was set by a computer-controlledswitch at either the desired value or a much larger value push-ing spin noise out of the detector’s bandwidth, enabling a fastsubtraction of the background spectrum (no spin noise) fromthe spin-noise spectrum. The balanced polarimeter outputwas fed into a spectrum analyzer, and the spectra were aver-aged at the computer. laser illuminates a magnetically shielded rubidium vaporcell. A balanced polarimeter measures the Faraday rota-tion angle fluctuations of an initially linearly polarizedand far-detuned laser. These fluctuations result fromthe fluctuating transverse spin, simultaneously precess-ing about a dc magnetic field transverse to the laserpropagation direction. As well known, at high laser de-tunings δ the Faraday rotation angle scales as θ ∝ /δ [31]. Since the measured rotation signal is proportionalto θ and to the laser power, both the laser wavelengthand laser power were monitored and their fluctuations ordrifts were less than 1% and hence negligible. Typical S p i n N o i s e P o w e r S pe c t r a l D en s i t y ( a r b . un i t s ) B = 6.6 mG B = 11.3 mG B = 49.3 mG Rb Rb (a) I n t eg r a t ed S p i n N o i s e P o w e r ( a r b . un i t s ) Frequency (kHz) (b)
10 20 30 40 500
Magnetic Field B (mG)
One cell with rubidium of natural abundanceTwo cells back-to-back with isotopically enriched rubidiumTheory
FIG. 2: (Color online). (a) Measured spin noise spectra forthree different magnetic fields. Upper graphs are from a dataset with the experiment cell (C1) containing Rb of natural iso-topic abundance (ratio of peak heights about 3:1), and lowergraphs are from the two back-to-back cells (C2), each enrichedwith one of the two Rb isotopes (ratio of peak heights about1:1). (b) Integrated spin noise power (ISNP) for C1 (red cir-cles) and C2 (blue squares). The former were normalized bytheir ISNP at B = 50 mG, while the latter were normalized bytheir average value. The red solid lines are the theoretical pre-diction S ( B ) /S (50 mG) of Eq. (7) with no free parameters,but with different (by 20%) values of the magnetic gradientas input to the theory. spin-noise spectra at various magnetic fields are shown inFig.2a. They exhibit two peaks centered at the Larmorfrequencies of Rb and Rb. The spin-noise spectraat different magnetic fields are integrated, and the totalspin-noise power is plotted in Fig.2b. Interestingly, thetotal spin-noise power increases at low magnetic fieldswhere the two magnetic resonance lines overlap. Thisnoise increase is the experimental signature of spin-noiseexchange.A consistency check was done to ensure the experi-ment’s and analysis’ ability to detect an actual change inspin-noise power. Instead of using a cell with rubidiumof natural abundance, we performed the same measure-ment with two cells placed back-to-back, each enriched S p i n N o i s e P o w e r S pe c t r a l D en s i t y ( a r b . un i t s ) Frequency (kHz)
10 20 30 40 (b)(a) I n t eg r a t ed S p i n N o i s e P o w e r ( a r b . un i t s ) Magnetic Resonance Line Width (kHz)
Linear fit
P (mW) PS D ( p V / H z ) r m s (d)(c) Linear fit
Frequency (kHz)
10 20 30 40
FIG. 3: (Color online). (a) Spin-noise spectrum and background. (b) The noisy blue line is the subtraction of the two spectrashown in (a) and constitutes a run, while the black line is the average of 50 runs and constitutes a set. (c) The offset of themeasured power spectra scales linearly with laser power, demonstrating a photon shot-noise limited measurement. (d) Totalintegrated spin-noise power at different temperatures. For the integral we used the Rb spin-noise resonance at high enoughmagnetic fields so that there is no overlap with the Rb resonance. The resonance line width is proportional to atom numberprobed by the laser. We have corrected for the other small contributions to line broadening and for the different average laserpower in the cell at different temperatures. Spin-noise signals scale as √ atom number, hence spin-noise power scales linearlywith the line width. by one of the two rubidium isotopes. In this case therecannot be any inter-species spin-noise transfer, and thetotal spin-noise power is expected to be independent ofthe magnetic field, which is the case as shown in Fig.2b. III. DATA AND ERROR ANALYSIS
The integrated spin-noise power (ISNP) data of Fig.2bwere obtained in the following way. A time series of thepolarimeter output was fed into a differential amplifier,the output of which was acquired by the spectrum ana-lyzer (SA) having a measurement bandwidth of 50 kHzand a resolution bandwidth of 62.5 Hz. The correspond-ing measurement time is 16 ms. Sequentially, we mea-sured the background by applying a large magnetic fieldto shift the spin noise way out of the 50 kHz bandwidthof the SA (Fig.3a). The background spectrum was thensubtracted from the spin noise spectrum. A run consistsof 100 averages of such subtracted spectra, and a dataset consists of the average of 50 runs (Fig.3b).The offset in the spectra of Fig.3a is determined byphoton shot noise (PSN), verified by the offset’s lineardependence [32] on laser power, depicted in Fig.3c. Asusual in noise-measurements, we also verified the linear scaling of the total spin-noise power with atom number,shown in Fig.3d.For every magnetic field we measured three data setsboth with the experiment cell and the two back-to-backcells. The ISNP in each set was calculated by fittingthe spin-noise spectra with a Lorentzian lineshape, tak-ing into account the negative frequency folding for thelow-magnetic field spectra. The results of all sets werethen averaged and presented in Fig.2b. An example ofspin noise data with the fit for a relatively high magneticfield is shown in Fig. 4a, whereas Figs. 4b and 4c showthe data and fit for the two lowest magnetic field points.To avoid contamination of the lowest magnetic field data( B = 3 mG) by the 1/f noise tail we start fitting the dataat 1.9 kHz, as the 1/f noise tail disappears into the PSNbackground at 1.5 kHz (Fig. 4d). This fit cut-off over-estimates the true ISNP and needs to be corrected for.To estimate the correction we produce numerical datawith the same signal-to-noise ratio as the real data andfit them starting from various cut-off frequencies. TheISNP of the numerical data is known, and the extractedfit correction is shown in Fig. 4e.For the higher magnetic fields we both fit the data withLorentzians, and independently we numerically integratethe data to find ISNP. Both methods give perfectly con- B = 3 mG
Frequency (kHz)
B = 4 mG (b) (c)
B = 21 mG -0.2 S p i n N o i s e P o w e r S pe c t r a l D en s i t y ( a r b . un i t s ) Frequency (kHz) (a)
Frequency (kHz)
Frequency (kHz) S p i n N o i s e P o w e r S pe c t r a l D en s i t y ( a r b . un i t s ) F i t C o rr e c t i on ( % ) B = 3 mG
Fit Cut-off Frequency (kHz) center of spin noise resonance at B=3 mGcut-off frequency for the fit
B = 8.4 mG (d) (e)
FIG. 4: (Color online) Spin noise spectrum with Lorentzian fit for (a) a high magnetic field, and (b,c) the lowest two magneticfields. (d) The 1/f noise tail falls to the photon-shot-noise level at 1.5 kHz. (e) To avoid contamination of the B =3 mG data bythe 1/f noise tail we start fitting the data from the fit cut-off frequency 1.9 kHz and on (short-dashed line). The long-dashedline at 1.5 kHz shows the line center frequency. To estimate the fit error due to the fit cut-off we generate numerical data withthe same signal-to-noise ratio as the real data, fit them and compare with the known ISNP. For the B =4 mG data this error isnegligible, since the peak of the resonance, being just higher than the cut-off of 1.9 kHz as shown in (c), is included in the fit. sistent results. With the latter method we also estimatethe ISNP error from the statistical distribution of theISNPs of 50 runs. IV. THEORETICAL EXPLANATION
The theoretical explanation of the observed phe-nomenon follows by considering the detailed spin dynam-ics of a coupled spin ensemble. The three physical mech-anisms driving single-species spin noise are (i) dampingof the transverse spin, (ii) transverse spin fluctuationsand (iii) Larmor precession. Processes (i) and (ii) areboth driven by atomic collisions, as also understood bythe fluctuation-dissipation theorem [28]. They involve (a)alkali-alkali spin exchange collisions and (b) alkali-alkaliand alkali-buffer gas spin destruction collisions. Type-(b) collisions have a negligible cross-section compared tothe spin-exchange cross section [1] σ se = 2 × − cm ,hence only type-(a) collisions will be considered. Inthe coupled double-species system there is an additionalphenomenon: spin exchange collisions between differentatoms. These are a sink of spin coherence for one atomand a source of spin polarization for the other. All ofthe above phenomena are compactly described by thecoupled Bloch equations for the transverse spin polariza- tions P j ≡ ˆx h P j,x i + ˆy h P j,y i of Rb ( j = 1) and Rb( j = 2): d P = dt (cid:2) P × ω − γ se ( P − P ) − γ P (cid:3) + d ξ (1) d P = dt (cid:2) P × ω − γ se ( P − P ) − γ P (cid:3) + d ξ (2)where ω i = ˆz ω i = ˆz g i B are the Larmor frequencies of thetwo Rb isotopes in the magnetic field B = B ˆz . Similarequations, albeit for different binary mixtures and unre-lated to spin-noise, have been used elsewhere [33, 34]. A. Relaxation rates and noise terms
Spin exchange collisions transfer spin polarization fromspecies j to i at a rate γ ijse = σ se vn j , where n and n arethe respective number densities and v the rms average rel-ative velocity of the colliding atoms. The transverse spinrelaxation rate of atom j other than due to spin exchangewith different-species atoms is given by γ j and consists of(i) spin-exchange with same-species atoms, γ jjse = σ se vn j and (ii) magnetic field gradient, γ j, ∇ B . The total spinrelaxation rate of atom j will then be Γ j = Γ + γ j, ∇ B ,where Γ = γ se + γ se = γ se + γ se = σ se v ( n + n ).From the fits of the noise peaks, and considering that γ , ∇ B = ( g /g ) γ , ∇ B [35] it was found that for the 10cm rubidium cell Γ = 2 π ×
800 Hz, γ , ∇ B = 2 π ×
300 Hzand γ , ∇ B = 2 π ×
700 Hz. For the two-cell measure-ment we found Γ ≈ Γ ≈ Γ = 2 π ×
800 Hz, consistentwith the fact that in this case the gradient relaxation isnegligible since it scales with the 4 th power of cell dimen-sion and the isotopic cells were 5 cm long each). Thereare two small additional relaxation sources common toboth atoms: (i) the transit time through the probe laser,and (ii) probe laser power broadening. The former canbe safely neglected. The latter is only 5% of the to-tal linewidth. Finally, d ξ j ( j = 1 ,
2) are independentGaussian white noise processes with zero mean and vari-ance Γ dt/N j [36], where N j is the total atom number ofspecies- j probed by the laser. B. Integrated spin-noise power
Introducing the 2-element column-vector π =( π π ) T , with π j = P j,x + iP j,y , the Bloch equations (1)and (2) can be compactly written as d π = − dt A · π + Ξ · d W (3)where the decay matrix is A = (cid:18) Γ + iω − γ se − γ se Γ + iω (cid:19) , (4)and Ξ is the diagonal 2 × jj = p /N j and j = 1 ,
2. The noise vector d W =( dW dW ) T describes two independent complex Gaus-sian processes, dW and dW , having zero mean and vari-ance dt [37]. The total spin σ y probed by the laser is thesum of the y -component of all rubidium atom spins in-side the probe laser beam, σ y = P Nm =1 s m,y , which canbe written as σ y = Im { n π + n π } . The total spin-noise power S ( B ) as a function of the magnetic field B can be computed as S ( B ) = 1 T Z T dtσ y ( t ) = 12 T Z T dt | σ ( t ) | = 12 T Z T dt | n π ( t ) + n π ( t ) | . Since the averaging time T is much longer than the spinrelaxation time, ergodicity of the Ornstein-Uhlenbeckprocess π ensures that the preceding long time averagecan be computed as an expectation under its equilib-rium distribution. Now, π is a two-dimensional complexGaussian process. Its equilibrium distribution has mean0, while the covariance matrix Σ with Σ ij = E (cid:2) π i π ∗ j (cid:3) for i = 1 , (cid:0) cf [38] equation (4.4.51) (cid:1) asthe unique self-adjoint solution to the matrix equation A Σ + Σ A † = ΞΞ † . Solving the system of linear equations we findΣ ii = ΓΓ i N i (cid:18) γ se γ se Q (cid:19) for i ∈ { , } (5)and Σ = Σ ∗ = Γ p γ se γ se Q √ N N (cid:18) i ∆ ω Γ + Γ (cid:19) (6)where ∆ ω = ω − ω and Q = Γ Γ " (cid:18) ∆ ω Γ + Γ (cid:19) − γ se γ se . Hence, S ( B ) = E (cid:2) | n π + n π | (cid:3) = n T Σ n , where n T = ( n , n ), and finally we get S ( B ) S ( ∞ ) = 11 − γ se γ se Γ Γ f ( B ) h n γ se + n γ se n Γ + n Γ f ( B ) i , (7)where 1 f ( B ) = 1 + 4 γ se γ se (Γ + Γ ) (cid:16) BB (cid:17) . (8)Here we have defined B = 4 γ se γ se / ( g − g ) . For ourexperimental parameters B ≈ . γ , ∇ B = γ , ∇ B = 0, the field B signifies a transition from a high-field regime B ≫ B where the eigenvalues γ = Γ + i ( ω + ω ) / ± ω − ω p B /B − A describe two independent spin precessions at ω and ω and decaying at a rate Γ, to a low-field regime B ≪ B where the spin-exchange coupling forces the atoms toprecess together at ( ω + ω ) /
2, the precession havingtwo decay rates Re { γ } = Γ ± p γ se γ se [39–41]. In thisexperiment the lowest field used is just about B and thistransition of the decay rates γ is not observable. Further,in the absence of magnetic gradient the spin-noise powerat zero field takes on the simple form S (0) S ( ∞ ) = r + 4 r + 1 r + r + 1 (9)where r ≡ n /n . The excess spin-noise power is max-imized for r = 1, the maximum being 100%, i.e. thespin-noise power is double at low fields relative to highfields. C. Spin-noise correlations
Towards explaining the observed effect we note thatthe off-diagonal elements of the covariance matrix Σcarry information about the correlation of polarizations P and P . It is E [ P · P ] = E [ Re { π π ∗ } ] = Re { Σ } .We can thus compute the correlation coefficient ρ ( B ) ≡ E [ P · P ] p E [ | P | ] E [ | P | ] = s γ se γ se Γ Γ f ( B ) (10)Again, in the ideal case of no gradient relaxation it is ρ (0) = √ r/ (1 + r ), which is also maximized for r = 1with the maximum being 1/2. Also, ρ ( B ) → B ≫ B . This leads to an intuitive explanation of theobserved phenomenon as an exchange of spin-noise be-tween two atomic species. In the rotating frame of atom i the transverse spin of atom j precesses at the frequency δω = | ω − ω | . If δω ≫ Γ, in other words if the two spinnoise resonances are far apart, the spin polarization ofatom j seen in the rotating frame of atom i averages outto zero within the spin-exchange time of 1 / Γ. If, however, δω ≤ Γ, then the noise polarization of atom j transferredto i adds up, to some extent coherently due to the non-zero ρ ( B ), to the noise polarization of i . This is dueto the strong polarization-noise correlations produced byspin-exchange. Hence the total spin-noise power is in-creased relative to the case where the two noise powersadd just in quadrature for δω ≫ Γ.To quantify the above discussion, let Π i be the total power of atom- i polarization fluctuations. We can thinkof Π i = Π (0) i + Π ij as consisting of two terms, the noisepower Π (0) i that we would observe if atoms- i were alone,and the transfer of polarization noise from j to i , de-scribed by the term Π ij . Clearly, Π (0) i = Γ / (Γ i n i ). Inview of (5) we find that indeed Π i = Π (0) i + Π ij , withΠ ij = Π j [( γ ijse ) / Γ i ] f ( B ). D. Discussion
For completeness we note the following. (i) In the twohyperfine levels of rubidium the spin precesses in oppo-site directions, corresponding to positive and negativefrequencies. In the measured power spectrum both ap-pear at the same positive frequency. (ii) Spin noise is gen-uine quantum noise produced by atomic collisions. Thelinear scaling of the total spin-noise power with atomnumber (Fig. 3d) does not by itself prove the previ-ous assertion. Instead, the physics of spin-noise gener-ation must be understood. Spin exchange collisions havetwo roles: they damp spin coherence and they generatenoise coherence. As well known [2], atoms can jump fromone hyperfine level to the other during a spin-conservingspin-exchange collision, thereby perturbing their coher-ent spin precession and leading to loss of spin coherence.The same mechanism can generate fluctuations of spincoherence as shown in the example of Fig. 5. In everycollision there are a number of potential final states, theprobability of which is determined by the quantum spin-dependent scattering of the atoms [20], and hence spin-noise bears the fundamental quantum-mechanical unpre- se ~ s s Free precession in the field B z x x x x y y FIG. 5: (Color online). Example of a spin-exchange collisiongenerating spin noise. The spin states | F, m F i of the collidingatoms are written in the x -basis. Two Rb atoms in the F = 3 state and opposite m F collide, and after the collisionone jumps to the F = 2 manifold. The total m F is conserved,however the spin in the F = 2 state precesses in the oppositesense. Under the action of the magnetic field B ˆz the twoatoms will momentarily generate a non-zero contribution tothe spin-noise signal along the y-axis (balanced of course bythe nuclear spins). Subsequent spin-exchange collisions willdamp the transverse spin and so on and forth. dictability. (iii) There is an apparent disagreement be-tween data and theory at intermediate-field data. Themagnetic gradient was found to have a 20% variationwith magnetic field. In Fig. 2b we plot the theoreticalprediction with a constant value for the gradient, but wehaven shown how the theoretical prediction is affected bychanging this constant value within its observed variationrange. Either an unidentified systematic or a statisticaloutlier effect could be responsible for the aforementioneddiscrepancy. To demonstrate the spin-noise effect pre-sented in this work without the added complication ofmagnetic gradients and with better statistics a short cellin the multi-pass arrangement of Romalis and co-workers[42] would be most appropriate. V. CONCLUSIONS
Concluding, we have experimentally demonstrated theinterspecies transfer of spin noise through the spin-exchange coupling of two alkali vapors. This transfer,also seen as a positive correlation of the two-species po-larization noise, manifests itself as a total noise powerincrease at low-magnetic fields, or to put it differently, asthe decrease of the total spin-noise power at high fieldswhere the spin-noise correlation vanishes. Although wedemonstrated the phenomenon using an unpolarized spinstate, the same phenomenon would occur in the coherentspin state of a maximally polarized spin ensemble [43],directly relevant to precision metrology applications.
Acknowledgments
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