Cross-correlation spin noise spectroscopy of heterogeneous interacting spin systems

Interacting multi-component spin systems are ubiquitous in nature and in the laboratory. As such, investigations of inter-species spin interactions are of vital importance. Traditionally, they are studied by experimental methods that are necessarily perturbative: e.g., by intentionally polarizing or depolarizing one spin species while detecting the response of the other(s). Here, we describe and demonstrate an alternative approach based on multi-probe spin noise spectroscopy, which can reveal inter-species spin interactions - under conditions of strict thermal equilibrium - by detecting and cross-correlating the stochastic fluctuation signals exhibited by each of the constituent spin species. Specifically, we consider a two-component spin ensemble that interacts via exchange coupling, and we determine cross-correlations between their intrinsic spin fluctuations. The model is experimentally confirmed using “two-color” optical spin noise spectroscopy on a mixture of interacting Rb and Cs vapors. Noise correlations directly reveal the presence of inter-species spin exchange, without ever perturbing the system away from thermal equilibrium. These non-invasive and noise-based techniques should be generally applicable to any heterogeneous spin system in which the fluctuations of the constituent components are detectable.

T here are numerous natural and engineered systems in which interactions between ''spins of different kind'' lead to the emergence of new and interesting physics. Notable examples include the interaction between electron spins from different Bloch bands that gives rise to heavy-fermion behavior and Kondo-lattice effects in correlated-electron materials 1,2 , the decoherence of solid-state electronic spin qubits by a nuclear spin bath [3][4][5][6] , carrier-mediated ferromagnetism in diluted magnetic semiconductors 7,8 , and spin-exchange pumping of noble gas nuclei for enhanced medical imaging 9 . Measurements of inter-species spin interactions are therefore essential to our understanding of many magnetic and spin phenomena.
Where possible, inter-species spin interactions are generally studied by well-developed perturbative experimental methods that, for example, selectively polarize or depolarize one spin species while separately monitoring the influence on the other(s) 10 . However, interaction cross-sections often depend strongly and non-linearly on the non-equilibrium spin polarizations that are experimentally induced 11 . For many spin systems, it may therefore be desirable to measure spin interactions under conditions as close to thermal equilibrium as possible.
As an alternative to conventional perturbation-based techniques for measuring spin and magnetization dynamics, methods for optical spin noise spectroscopy [12][13][14] have been recently developed in which electron and hole spin dynamics are revealed solely via the passive detection of their intrinsic and random spin fluctuations in thermal equilibriumi.e., without any polarization, excitation, or pumping. To date, spin noise spectroscopy has been applied to many different single species of spins, such as specific alkali atoms 12,[15][16][17][18] , itinerant electron spins in semiconductors [19][20][21][22] , and localized hole spins in quantum dot ensembles 6,23 . These studies have shown that dynamic properties of spin ensembles such as g-factors, relaxation rates and decoherence times are measurable simply by ''listening'' (typically via optical Faraday rotation) to the system's intrinsic and random spin noise -an approach ensured by the fluctuation-dissipation theorem.
Based on these developments, here we explore whether spin interactions between different spin ensembles can also be directly revealed and studied -under conditions of strict thermal equilibrium -through their stochastic spin fluctuations alone. We envision a type of experiment shown schematically in Fig. 1, wherein two spin species A and B in thermal equilibrium interact, e.g., by direct spin exchange as depicted 24,25 . If the intrinsic spin fluctuations from species A and B can be independently detected, then signatures of spin interactions should appear in the cross-correlation of these two spin noise signals. For example, anti-correlations can be expected in the presence of purely spin-exchange coupling, since interactions causing positive spin fluctuations in one species will be accompanied by corresponding negative fluctuations in the other species.
In this paper we show that multi-probe spin noise spectroscopy can indeed reveal inter-species spin-spin interactions by detecting the system's intrinsic stochastic spin fluctuations. We develop a theory for such cross-correlations in heterogeneous spin systems at thermodynamic equilibrium. In particular, we prove a universal sum rule (a ''no-go theorem'') that imposes restrictions on such cross-correlators. These results are directly compared to a proofof-principle experimental study of a well-understood interacting spin system (a mixture of warm Rb and Cs vapors) by applying a new type of ''two-color'' spin noise spectroscopy 26 , and excellent agreement is found. Thus, we introduce a framework for both theoretical and experimental exploration of a broad class of heterogeneous interacting spin systems by detecting their spin fluctuations in equilibrium.

Results
Experimental setup. To most easily introduce the notion of spin noise spectroscopy and to describe how spin fluctuations are detected and correlated, we first describe the experiment and its results. Figure 2(a) depicts the setup. A 20 mm long glass cell containing both Rb and Cs metal (and 100 Torr of Ar buffer gas) is heated to ,140uC, giving a classical alkali vapor with Rb and Cs particle densities of about 0.6 and 1.4 3 10 14 cm 23 respectively. To independently probe the intrinsic spin fluctuations in both species, we perform spin noise spectroscopy 12 using two linearly-polarized probe lasers with wavelengths l Rb , 795.0 nm and l Cs , 894.6 nm that are, respectively, tuned close to (but not on) the fundamental D1 ( 2 S 1/2 -2 P 1/2 ) optical transitions of Rb and Cs. In fact, l Rb and l Cs are intentionally detuned below their respective D1 transitions (anywhere from 30-100 GHz), to enable detection of the Rb and Cs spin polarization via optical Faraday rotation 27 . This large detuning significantly exceeds the Doppler or pressure broadening or hyperfine splitting of the D1 transitions, ensuring that the probe lasers do not pump or excite the atoms to leading order. Moreover, the large detuning of the probe lasers compared to the pressure broadening of the D1 lines due to the buffer gas (,10 GHz) greatly simplifies the analysis of the data because we can ignore the hyperfine sub-structure of the D1 transition and can effectively consider the Rb and Cs atoms as having simple spin-1/2 magnetic ground states 28 .
The random spin fluctuations of the Rb and Cs valence electrons along theẑ direction -S Rb,z (t) and S Cs,z (t) -are detected by the optical Faraday rotation (FR) fluctuations h Rb (t) and h Cs (t) that they impart on the detuned probe lasers. This detection scheme is made possible by the optical selection rules in alkali atoms, and because FR depends not on absorption constants but rather on the right-and left-circularly polarized indices of refraction of the alkali vapors (h / n R 2 n L ), which are dispersive and decay slowly (inversely) with large laser detuning 27 . The two detuned lasers can therefore be regarded as passive, non-perturbing probes of the Rb and Cs vapor's intrinsic spin fluctuations 12,17,18,29 . The FR sensitivity of the lasers to spin fluctuations scales inversely with (and can be adjusted via) the laser detuning from the D1 line.
The two probe lasers are first combined in a single-mode optical fiber to ensure optimal spatial overlap before being focused through the vapor cell (,50 mm diameter beam waist), after which they are separated by a dichroic beamsplitter. FR fluctuations h Rb (t) and h Cs (t) are measured by separate balanced photodiode pairs. The fluctuating output voltages V a (t) ( / h a (t), where a 5 Rb, Cs) are continuously digitized and processed in real time. Specifically, we compute the frequency spectrum of the spin noise power density for each species a, which is equivalent to the Fourier transform of the spin-spin correlator: Importantly, we also compute the real part of the cross-correlation spectrum between the Rb and Cs spin fluctuations: which has not been considered previously for spin noise studies but which, as shown below, contains specific information about interspecies spin coupling and interactions. Note that for clarity, the subscript 'z' was omitted from all the spin projections S z (t) in Eqs. (1) and (2). Finally, three orthogonal pairs of large Helmholtz coils are used to cancel out ambient magnetic fields and to apply small (0-5 G) uniform magnetic fields B x along the transversex direction. This forces the spin fluctuations S z (t) to precess, thereby shifting the measured spin noise to higher (Larmor) frequencies. The entire setup is constructed of nonmagnetic materials; and magnetic field variations across the 20 mm path length of the probe beams are less than 10 22 G.
Spin noise power spectra. Figure 2(b) shows the power spectra of the detected spin noise from the Rb and Cs spins at B x 5 0 [P Rb (v) and P Cs (v); blue and red curves respectively]. The detuning and intensity of the two probe beams were independently adjusted to give approximately equal Rb and Cs spin noise power as measured by the detectors. At B x 5 0 these spin noise peaks are centered at zero frequency and they exhibit approximately Lorentzian lineshapes, indicating exponentially-decaying spin correlations. In principle the linewidth of the spin noise is inversely proportional to the intrinsic relaxation time t s of the spin species. However, t s is very long in alkali atoms, and so here the observed noise linewidths are additionally broadened in part by the transit-time broadening of the atoms diffusing across the small ,25 mm radius of the focused probe beams (this timescale is of order 10 ms), and also by the fact that comparatively small detunings of the probe lasers were used here in order to obtain large noise signals (l Rb and l Cs were detuned by 28 GHz and 56 GHz from their D1 transitions, respectively). At smaller detunings and/or larger laser intensities, some residual amount of optical pumping will occur which reduces the effective spin lifetimes and broadens the observed spin noise, as was observed and noted in previous spin noise studies of alkali vapors 17,30 and also in solid-state systems 20 .
Noise cross-correlation spectra. Figure 2(c) shows the corresponding and simultaneously-measured noise cross-correlation spectrum between the two species, P cr (v). Crucially, P cr (v) is not zero, indicating that interspecies spin interactions do appear in -and are measurable through -intrinsic spin fluctuations alone. P cr (v) exhibits a very narrow peak centered at zero frequency, revealing positive correlations between Rb and Cs spin fluctuations at small frequencies. In addition, P cr (v) also exhibits a broader negative feature at larger frequencies, revealing anti-correlations between Rb and Cs spins at these frequencies. Importantly, P cr (v) can be fit extremely well by the difference of two Lorentzians with equal area (i.e., P cr has zero total integrated area), the origin and significance of which are discussed below. We note that qualitatively similar results are also obtained when using much larger laser detunings, smaller laser intensities, and/or larger beam waists, all of which give narrower noise linewidths [as shown below, e.g., in Fig. 2 (f) and Fig. 3].
To confirm these cross-correlation signals, Figs. 2 (d,e) show similar measurements acquired when one of the probe laser wavelengths is tuned above (rather than below) its corresponding D1 transition. While the noise power spectra for the individual vapors are unaffected as expected (because noise power scales as the square of FR; see Eq. (1)), P cr (v) inverts sign because the FR induced by a polarized ground-state alkali spin is an odd function of wavelength about the D1 transition. That is, a given spin fluctuation will induce a positive-or negative-going FR fluctuation depending on whether the probe laser is red-or blue-detuned from the D1 transition (see inset diagrams). Importantly, it was also verified that cross-correlation signals exist only when the two probe lasers are spatially overlapped in the vapor mixture. Figures 2 (f,g) show that P cr (v) 5 0 when the two probe beams were spatially separated in the vapor, such that they probed Rb and Cs populations that were not directly interacting. Note that in Figs. 2 (f,g), l Rb and l Cs were detuned by 70 GHz and 101 GHz from their D1 transitions, which gives narrower noise linewidths.
Magnetic field dependence. Figures 3(a-c) show the measured spin noise power spectra from Rb and Cs at different values of B x . With increasing B x , the noise peaks shift to higher frequency (due to precession) at different rates in accord with their g-factors (1/3 and 1/4, respectively). At 0.2 G the Rb and Cs spin noise peaks are still largely overlapped, while at 1.2 G they are mostly separated. The higher-frequency spin noise peak from the less abundant 87 Rb isotope (g 5 1/2) is also visible 12 . Importantly, Figs. 3(d-f) show that the corresponding cross-correlator P cr (v) also shifts to higher frequencies, diminishes in amplitude, and develops a more complex structure. At larger B x when the Rb and Cs spin noise peaks no longer overlap at all, P cr (v) disappears entirely (not shown). Finally, Figs. 3(g-i) show P cr (v) calculated from the theoretical model that is developed immediately below.
Sum rule for noise cross-correlators. In order to model and understand these experiments, we develop and apply a theory for interpreting cross-correlations between spin fluctuations in a twocomponent spin-1 2 ensemble. We introduce vectors S A and S B whose components are the total (unnormalized) spin polarization along the x, y, z-axes of type A and B spin in the observation volume. Given numbers N Az" and N Az# of " and # spins of type A, then spin noise power (a.u.) S Az 5 (N Az" 2 N Az# )/2. We now formulate a useful sum rule, which is valid irrespective of further model details: No-Go Theorem: At thermodynamic equilibrium and in the limit of large spin temperature, the integral of the cross-correlator over frequency is zero.
Proof: The integral of P cr (v) over v, defined in (2), gives a delta function in time, which is removed by integration over time to produce a cross-correlator at equal time t 5 0: where curly brackets are the anti-commutator. The equilibrium spin density matrix at large temperature is proportional to a unit matrix. The trace of its product with a traceless operator, such as S Az (t)S Bz (t), is zero. Q.E.D.
The no-go theorem shows that, in the limit of large temperatures where k B T greatly exceeds the energy splittings of available spin states, useful information about interactions between different spin species is contained only in the functional form of P cr (v). In this limit, it rules out strategies for inferring inter-species spin interactions that are based only on measurements of integrated spin noise power (e.g., Ref. 31). For the case of warm alkali vapors, where k B T .25 meV but Zeeman and hyperfine energies do not exceed ,10 meV, the no-go sum rule applies with high accuracy. Hence, it is a valuable tool to test the validity of our theoretical and experimental results.
Finally, we note that the large temperature limit does not mean that the noise power spectrum is trivial. When the temperature scale exceeds dynamic energy scales of the system, all microstates are equally probable. However, this restricts only the distribution of the sizes of spin fluctuations, as it would be revealed by ''one-time'' measurements of the spin density. In contrast, the two-time spin correlator, as measured with our setup, reveals the dynamics of such fluctuations, which is described by the intrinsic spin Hamiltonian. The no-go theorem only introduces a thermodynamic constraint on a single parameter that is the area under the spectral curve. Moreover, this theorem will not hold for ultra-cold atomic gases and many condensed matter systems in which exchange interaction energies exceed the temperature scale. In such situations, the area of P cr (v) can reveal intrinsic static cross-correlations in the ground state. Another possibility to avoid the restrictions of the no-go sum rule is to consider spin noise in systems that are not in strict thermodynamic equilibrium 32 .
Theoretical model of spin noise cross-correlators. To model spin interactions and the essential role of spin fluctuations, we first assume that species A and B each have an intrinsic net spin relaxation process with rate c A and c B per particle (due to, e.g., interactions with cell walls, buffer gas, etc.). To define these rates more quantitatively, we will assume that if species A has polarization S Az then, on average, this polarization changes by dS Az 5 2c A dtS Az during a small time interval dt (and similarly we define a relaxation rate c B for species B). In addition, spin-exchange interactions between A and B spins lead to the total-spin-conserving co-flip processes with a rate c AB per pair of atoms of different kind. On average, each atom of type A can interact with N B /2 atoms of opposite species, and each atom of type B can exchange spin polarization with N A /2 of atoms of type A, where 1/2 factor follows from the fact that an atom with (say) spin up can exchange spin only with an atom with spin down. With such a definition of kinetic rates, we arrive at the standard evolution equations describing the dynamics of the average spin polarization with spin exchange kinetics: However, in order to study spin fluctuations near thermodynamic equilibrium, Eqs. (4) must be amended to include stochastic fluctuations. It will be convenient to introduce the vector S 5 (S A , S B ) T , which has components that we will treat as variables having stochastic dynamics. The latter can be written in the form of the following multivariate Langevin equation: whereR is the relaxation matrix with elements R ab ij , i, j 5 x, y, z and a, b 5 A, B, that can be read directly from Eq. (4): Here we assume that the magnetic field is applied along the x-axis and we introduce the bar operator: A~B and B~A. We discuss in the Methods section that, due to the large number of spins in the observation region (N A,B ?1) the noise source j can be considered as a white Gaussian noise, i.e. its components have correlators with some coefficients G ab ij that can be considered as elements of the correlation matrixĜ. It is the property of the multivariate Langevin equation that in a statistical equilibrium steady state, matricesĜ and R are related by (see, e.g. Eq. (4.4.51) in Ref. 33)  whereŝ is the stationary covariance matrix whose elements are the equal-time correlators of the components of the vector S: In a general stationary but nonequilibrium stochastic process, Eq. (8) cannot be used to determineĜ because the matrixŝ, itself, has to be determined by solving Eq. (5). However, at thermodynamic equilibrium the form of the matrixĜ is uniquely determined by the form of the relaxation matrixR and by the condition that the probabilities of a system's microstates are given by the Boltzmann-Gibbs distribution. This result is generally known as the fluctuation-dissipation theorem.
In our case (the large temperature limit) all spin microstates have equal probabilities. In particular, each spin of each atom can be found with equal probability to have spin projection along the z-axis equal to either 11/2 or 21/2, independently of states of other spins at the same moment of time. Since the variance of the sum of independent variables is the sum of individual variances, the covariance matrix elements are where the Kronecker symbol d ab accounts for the fact that spins of different species can have polarization 61/2 independently of each other, and d ij reflects the fact that the free energy of the system is isotropic. The factor 1/4 in (9) accounts for the fact that the variance of the polarization of a single spin-1/2 is equal to 1/4. Substituting (6) and (9) into (8) we determine elements of the correlation matrix: To obtain the cross-correlator, we define S v ð Þ: where i, j 5 x, y, z; a, b 5 A, B; and1 is a unit matrix. The crosscorrelator in Eq. (2) is then given by: A compact expression can be obtained by assuming identical values for the relaxation rates: c A 5 c B ; c 1 (this assumption is not unreasonable for our Rb/Cs mixture).
where  Figs. 3(d-f). The calculations used N A 5 N B , which is a reasonable approximation here because l Rb and l Cs were adjusted in the experiment to give equal spin noise power.

Discussion
We now discuss three different limits of Eq. (13).
(i) First, at zero applied magnetic field, we find which is simply the difference of two equal-area Lorentzians with widths c 1 and c 2 . Here Q 5 N A N B / (N A 1 N B ).
(i) In Fig. 4(a) we show the experimentally measured P cr (v) at B x 5 0. In good agreement with Eq. (15) and the ''no-go'' theorem, P cr (v) has zero total area, being well fit by the difference of two equal-area Lorentzians (dashed lines). Figure 4(b) shows the extracted c 1,2 as a function of the total vapor density n Rb 1 n Cs , which is tuned with the cell temperature. Here we recall that c 1 describes the relaxation of the total spin S Az 1 S Bz . The spin exchange rate is therefore characterized by the difference c 2 -c 1 . As is typical for alkali vapors, we find that the relaxation rate of the total spin is much smaller than the exchange rate, since the latter conserves the total spin. Now we can interpret the negative part of P cr (v) (15) as emerging from the expected anti-correlations induced by fast spin coflips between Rb and Cs atoms. On the other hand, the positive-valued peak in (15) and in the data is due to the fact that, at fast co-flip rate, the total spin polarization is equally observed by both beams at longer time scales (i.e., the total spin relaxation is 'shared' between the interacting Rb and Cs atoms), which corresponds to positive cross-correlations. (ii) Next, we consider the limit c 1 = g A {g B ð Þ B x =c 2 , which is close to the case measured in Figs. 3(b,e). Here, one can disregard effects of c 1 and obtain where which indicates that the spectrum shifts to an effective (weighted average) Larmor frequency V L , while the positivevalued peak is broadened by the magnetic field (an effective total spin relaxation rate), in agreement with Figs. 3(e,h). (iii) In the limit of a large magnetic field, Eq. (13) predicts that the cross-correlator P cr (v) vanishes, also in agreement with experimental observation.
Thus, we see that the theoretical model is confirmed by the experimental data and therefore captures the essential physics of spin fluctuations and their correlations. We note, however, that a more rigorous and quantitative description of the observed noise power and cross-correlations should include not only the presence of all different isotopes but also the coupled dynamics of their nuclear and electronic spinsi.e., the fact that alkali atoms actually have a nontrivial magnetic ground state due to hyperfine splitting -which would lead to multiple correlated resonances even within the same atomic species. On the other hand, our experimental results using widely-detuned lasers apparently validate the simple twocomponent approximation. This can be explained by the presence of fast intra-species spin-exchange interactions that smear the physics related to presence of multiple resonances from the same atomic species. We also note that, in order to achieve a better theoretical precision, it is straightforward to amend our approach to include multiple intra-species resonances by extending the set of equations (5). In summary we have shown, both experimentally and theoretically, that cross-correlations between the stochastic spin fluctuations of different species do reveal specific information about spin interactions. Crucially, these interactions can be detected using unperturbed spin ensembles under conditions of strict thermal equilibrium. Such non-invasive characterization techniques may find future applications in metrology, e.g. to reveal the physics that limits the efficiency of various magnetometers 18,24 . We also envision applications of this technique to mixtures of ultra-cold atomic gases and condensates 35,36 , which are sensitive to the probe interference 37 . Studies of cross-correlations of spin noise in solid state physics, e.g. in multiple Bloch bands and in new layered materials as well as in artificial semiconductor nanostructures, represent additional and asyet-unexplored avenues for applications of cross-correlation studies and two-color spin noise spectroscopies.

Methods
Microscopic derivation of the correlation matrix. Consider a time interval dt much smaller than the ensemble average spin relaxation time but sufficiently large for many random spin flips to happen. The existence of such a time scale is guaranteed by the presence of the large number of atoms in the observation region (N A,B ?1). In order to include stochastic fluctuations, we will interpret the kinetic rate, e.g. c A , in terms of the probability, p A , for an arbitrary spin of type A to flip per unit of time. The definition of this probability should be consistent with our definition of c A in terms of the average relaxation rate. Suppose at time t there are N A" (t) atoms with spin 11/2 and N A# (t) with spin 21/2. Since each atom can change its spin by the amount dS Az 5 61, after a time dt the change of the total polarization, on average, will be Comparing this with the definition of the relaxation rate, i.e. with equation AEdS Az (t)ae/ dt 5 2c A S Az (t), we find that p A 5 c A /2. Consider now the variance of the total polarization change due to this process: var(dS Az ) ; AE(dS Az (t)) 2 ae 2 AEdS Az (t)ae 2 . If we had only one atom, the variance of spin polarization change would be just p A (dS Az ) 2 5 c A dt/2. The variance of the total change of polarization of N A atoms is the sum of variances of all independent processes that contribute to it, i.e.
Since the interval dt is much smaller than the spin relaxation time, the number of atoms that experience a spin flip during dt is much smaller than the total number of atoms. Hence, it is unlikely for any given spin to produce more than one flip in two consecutive time intervals of size dt. In turn, this means that spin fluctuations in nearby time intervals are produced essentially by different atoms and can be considered statistically independent. Hence, on a much larger time scale of the spin relaxation, one can assume that spin fluctuations are produced by a white noise, i.e. dS Az /dt 5 2c A S Az 1 g Az (t), where AEg Az (t)ae 5 0, AEg Az (t)g Az (t9)ae 5 rd(t 2 t9), and where the coefficient r can be obtained by comparing the correlator of the variable ð Þdt' with Eq. (18). This leads us to AEg Az (t)g Az (t9)ae 5 c A N A d(t 2 t9)/2. Finally, to include fluctuations along different axes, we note that because of the spherical symmetry the equal-time correlator is invariant of coordinate rotation, and such fluctuations must be considered uncorrelated. To describe this situation, we can introduce a vector spin fluctuation source g A such that the total spin polarization vector S A changes as dS A /dt , g A (t), where AEg Ai (t)g Aj (t9)ae 5 d ij c A N A d (t 2 t9)/2; i, j 5 x, y, z. Stochastic spin flips of atoms of type B are treated similarly, leading to a noise source g B with AEg Bi (t)g Bj (t9)ae 5 d ij c B N B d(t 2 t9)/2. Obviously, also AEg A (t)g B (t9)ae 5 0. Next, consider spin-exchange between A and B spins. During time interval dt, each atom of type A can experience co-flip with one of approximately N B /2 atoms of type B, where the factor 1/2 appears because, on average, only half of the opposite species atoms can participate in the exchange interaction with a given atom.
Each of the N A N B /2 allowed elementary co-flip processes can be considered independent and having a small probability, p AB , to happen per unit of time. We defined the rate c AB so that the average change in S Az (t) during short time interval dt due to co-flip processes is then given, up to the linear order in spin polarization, by AEdS Az (t)ae 5 2c AB dt[S Az (t)N B 2 S Bz (t)N A ]/2. To make the definition of p AB consistent with this assumption, we should assume that p AB 5 c AB /2.
If we had only one such a possible process, i.e. between only one atom A and one atom B, then a spin of atom A would have a chance to change by 61 and simultaneously spin of atom B would change by +1. The variance of such a process would correspond to AE(dS Az ) 2 ae 5 AE(dS Bz ) 2 ae 5 (61) 2 p AB dt 5 p AB dt, and AE(dS Az )(dS Bz )ae 5 2p AB dt. Summing over all independent processes, we find that the variance of total spin polarization fluctuation is given by  Combining the intrinsic spin dynamics with random inter-species co-flip processes, and introducing external magnetic fields, we find: where the noise sources are correlated as where i, j 5 x, y, z and a, b 5 A, B. Eqs. (21)- (22) reproduce the elements of the correlation matrix (10) with j A~gA zg AB , j B~gB {g AB : Here we emphasize that the same noise source g AB appears in both equations (19) and (20) with opposite signs. This guarantees the conservation of the total spin at co-flip events. We note that a related model was explored in Ref. 31 in a context different from two-color spin noise spectroscopy. However, instead of a single exchange noise g AB , two independent noise sources were introduced in Eqs.  www.nature.com/scientificreports SCIENTIFIC REPORTS | 5 : 9573 | DOI: 10.1038/srep09573 The corresponding system of stochastic equations leads to essentially different predictions and results inconsistent with the no-go theorem, and hence cannot be applied to our case.