# The effect of diversity on group decision-making

Georgi Karadzhov<EMAIL_ADDRESS>
Department of Computer Science and Technology, University of Cambridge
Cambridge, UK Andreas Vlachos<EMAIL_ADDRESS>
Department of Computer Science and Technology, University of Cambridge
Cambridge, UK Tom Stafford<EMAIL_ADDRESS>
Department of Psychology, University of Sheffield
Sheffield, UK

###### Abstract

We explore different aspects of cognitive diversity and its effect on the
success of group deliberation. To evaluate this, we use 500 dialogues from
small, online groups discussing the Wason Card Selection task - the DeliData
corpus. Leveraging the corpus, we perform quantitative analysis evaluating
three different measures of cognitive diversity. First, we analyse the effect
of group size as a proxy measure for diversity. Second, we evaluate the effect
of the size of the initial idea pool. Finally, we look into the content of the
discussion by analysing discussed solutions, discussion patterns, and how
conversational probing can improve those characteristics.

Despite the reputation of groups for compounding bias, we show that small
groups can, through dialogue, overcome intuitive biases and improve individual
decision-making. Across a large sample and different operationalisations, we
consistently find that greater cognitive diversity is associated with more
successful group deliberation.

Code and data used for the analysis are available in the anonymised
repository: https://anonymous.4open.science/r/cogsci24-FD6D.

Keywords: group deliberation; decision-making; collaboration

## Introduction

Individuals come together to discuss, analyse, and evaluate different points
of view. Such group deliberation is considered an essential tool for tackling
complex tasks. It is a cornerstone of decision-making in domains such as
research, government and business. In academia, we have the peer review
process (?, ?, ?), where scientists discuss and improve scientific reports and
academic studies. In government, collective bargaining, resource allocation
and community planning are driven by committees, task forces, and city
councils, where groups of people try to come up with fair decisions and
policies. In business, decisions are made again in small groups, such as
boards of directors or management teams, responsible for running companies,
budgeting, and hiring. The widespread deployment of group deliberation as a
decision-making strategy makes it important to understand the forces which
support or undermine good deliberation. The current work presents a data-
driven approach to understanding one such key factor: cognitive diversity.

Group deliberation is the dialogue between a group that facilitates the
careful weighing of information, providing adequate speaking opportunities,
and bridging the differences among participants’ diverse ways of speaking and
knowing (?, ?). ? (?) review evidence that, in the right conditions, groups
can outperform even the most successful individual members in their decision-
making. These conditions include common goals among group members and the
exchange of arguments. As well as groups exceeding the capacity of any
individual member, another relevant standard is the decision which would be
reached by simple aggregation or averaging of independent individual
judgements (the so-called “Wisdom of the Crowd” effect, ?, ?). In the study of
? (?) small groups outperform the wisdom of the crowd standard. ? (?) argue
that small groups can be more effective than both individuals of superior
ability, and large crowds. Further, this and other evidence (?, ?, ?) supports
the idea that the exchange of arguments is mechanistically related to
successful group decision-making.

That said, group decision-making is not universally successful. A tradition in
social psychology has emphasised the potential for negative outcomes. A
notable diagnosis is the perils of “Group Think” (?, ?). Potential mechanisms
of failures of group decision-making include social loafing, herding,
deference to others’ preferences — real (?, ?), or imagined (?, ?) —, failure
to share knowledge available to individual group members (?, ?),
underweighting dissenting voices, and amplification of individual cognitive
biases.

We note that the mechanisms supporting the Wisdom of Crowds effect relies on
individuals having independent access to some true information (signal), and
uncorrelated errors (noise). To the extent that individuals share systematic
error, such as a shared cognitive bias, aggregation of individual judgements
will reinforce error, not eliminate it.

To study group decision-making under conditions of bias, we use the Wason Card
Selection Task (?, ?), an iconic task used to study human reason (?, ?). For
our purposes, it has two important qualities. First, with high reliability,
the majority of people get the task wrong, and they fail in a consistent way.
This suggests the presence of a confirmation, or matching, bias which draws
people to an intuitive but incorrect response. This makes the task an
attractive paradigm for investigating how group function may interact with
individual biases. Second, the Wason task has a correct answer which can, once
understood, be relatively easily explained and understood by others. This
underlies the remarkable prior result that while individuals tend to fail on
the Wason Task, small groups tend to succeed (discussed in ?, ?).

### Cognitive diversity in group decisions

An established finding in studies of group decision-making is the potential
benefits of diversity (?, ?, ?, ?). Diversity can be in the form of
differences in the backgrounds, knowledge or cognitive styles that individuals
bring to a group, as well as the different roles they may adopt or be assigned
within a group. In the limit, too extreme diversity can hamper group function
— particularly where it facilitates intragroup conflict or hinders agreement
on common goals — but otherwise, there are good reasons to expect greater
diversity to enhance group decision-making.

Diversity can be improved in a number of ways. One approach is by differing
demographic or professional characteristics (?, ?) of the participants. This
can also be in part achieved by considering groups of larger sizes - i.e. by
having a larger group, there is an increased chance that the group would have
people from different backgrounds and experiences. An alternative way is to
adopt procedures or structures which facilitate expressing and exchanging
diverse ideas (?, ?). Our focus here is on cognitive diversity, which reflects
differences between individuals in knowledge, processes or actions within the
group context and is distinct from (but may stem from) demographic
differences, such as gender or ethnic diversity.

#### The current work, hypotheses.

In this work, we leverage a unique dataset of group decisions, the DeliData
corpus (?, ?). In DeliData, 500 groups provided initial judgments and then
discussed the Wason Card Selection Task, for which full transcripts were
recorded. Individuals then submitted their updated answers. These data allow
us to look, with a high degree of granularity, at the features of groups and
the deliberative processes which generated successful and unsuccessful
deliberation. Furthermore, DeliData contains data indicating participants’
solutions, arguments, and deliberative markers, which would be beneficial for
rich and multifaceted analysis.

Concretely, we look into three aspects of cognitive diversity - (i)
indirectly, via the proxy of group size, (ii) via the distribution of initial
solutions possessed by the group, and (iii) directly, via the content of the
discussions, the solutions considered by the group, as well as deliberation
markers affecting said content.

Given previous research, as well as the three aspects of cognitive diversity,
we will be testing the following hypotheses. First, that larger groups will
demonstrate better performance. This has been shown in other studies (?, ?,
?), but not — to our knowledge — in the Wason Task, nor for other tasks which
elicit strong individual biases (for which groups may function to aggregate
rather than eliminate) as a characteristic that may affect group performance.

Second, we hypothesise that a greater diversity of initial ideas will support
better group performance. Following ? (?) we index this by inspecting the
number of distinct individual solutions submitted by participants prior to
entering the group discussion. Third and finally, that diversity in discussion
content will support better group performance. This was shown for forecasting
by ? (?). We index this based on the distinct solutions mentioned in the
conversation, as well as the transitions between mentioned solutions.

## Data & Methods

### DeliData

# of dialogues | 500
---|---
# of utterances | 14003
AVG # of utterances / dialogue | 28
AVG group size | 3.16
AVG utterance length | 8.59
# of Probing utterances | 1739
Solo Performance | 0.59
Group performance | 0.72
Performance Gain | 0.13
Table 1: Summary statistics for the DeliData corpus

The DeliData corpus is described in ? (?). It comprises 500 small group
discussions of the Wason Task, which vary both in terms of group composition
(2 to 5 participants) and performance (both high and low, and pre- (solo) and
post-discussion (final), performance). Statistics of the DeliData corpus are
presented in Table 1.

To create the dataset, the Wason Card Selection Task was presented in its
traditional form: it is explained that cards have a number on one side and a
letter on the other. Four cards are shown displaying an even number (‘E’), odd
number (‘O’), consonant (‘C’) and vowel (‘V’). The task is to answer the
question ”Which card(s) should you turn to test the rule: All cards with
vowels on one side, have an even number on the other”. The intuitive but wrong
answer (henceforth the Lure) is to turn the even number and vowel (‘EV’). The
correct answer (henceforth the Target) is to turn over the odd number and the
vowel (‘OV’).

? (?) report that 64% of the groups performed better following the discussion
compared to their initial performances. In 44% of the groups who had at least
one correct answer as their final solution, none of the participants had begun
with the solution. This demonstrates the benefits of group discussion and
suggests that more is occurring than mere propagation of the best answers
among the groups.

Moreover, ? (?) provides utterance-level annotations that mark deliberative
and argumentation cues. The full details of the annotation scheme and corpus
are presented in ? (?), however in the context of this work, we are interested
in the phenomenon of conversational probing. Conversational probing is defined
as any utterance that provokes discussion, deliberation or argumentation. This
is further split into three categories - Moderation (i.e. modelling group
dynamics), Reasoning (probing further argumentation), and Solution (i.e.
enquiring about a specific solution).

### Analytic strategy

The Wason Card Selection task has one correct answer, allowing for a direct
and objective measure of performance. In this work, we leverage the fine-
grained scoring from (?, ?), where 0.25 points each are given for i) selecting
either card in the Target (O and V) and (ii) for not turning unnecessary cards
(E and C). Therefore, if the participant submitted a correct solution, their
score would be 1, if they are off by one card - 0.75 and so on. If
participants selected the Lure (EV), their fine-grained score would be 0.5.
Similarly, selecting all cards (CEOV) or none, will have a fine-grained score
of 0.5.

Performance gain is calculated by subtracting the task performance of the solo
solution from the task performance in the final submission. For example, if a
participant’s solo submission was 0.5 and improved to 0.75 after the
discussion, the performance gain would be $0.75-0.5=0.25$. In this work, we
investigate group performance gain which is the averaged performance gain of
each of the participants.

#### Different measures of diversity

First, we investigate group size as a proxy measure of diversity, and can be
compared to a baseline of pre-discussion, solo performance (groups of size 1).
The premise is that more members participating in a dialogue would naturally
increase cognitive diversity. Following, we consider the diversity of initial
ideas, which is calculated by counting the number of unique solo solutions
submitted by the group’s participants. Finally, we analyse the factors
measuring the diversity of discussed solutions and their contribution towards
the performance gain. To this end, we evaluate the following characteristics
of a diverse discussion.

  * •

dialogue length measured by the number of utterances. The premise is that
longer conversations will naturally contain more mentions of solutions, thus
contributing towards a more diverse discussion:

  * •

Number of unique solutions that are mentioned in a discussion. We hypothesise,
that conversations with positive performance gain will contain more mentions
of unique solutions, indicating that participants were actively discussing a
diverse set of possible solutions to the Wason task.

  * •

We analyse how participants discuss solutions. To measure this, we extract the
solutions participants talk about in every utterance by keeping track of Wason
card mentions 111And the two special cases when participants mentioned “none”
or “all” where we consider that their solution is to select none or all of the
cards. Then, we calculate the occurrences of all transitions between solutions
in the form of a from-through-to transition triple. For example, if the
conversation starts with the lure (EV), then transitions to a solution that
turns every card, and then to the correct solution, the transition triplet
would be “EV-CEOV-OV”. Having these transition triplets, we measure the
following three quantities:

    * –

Number of unique transitions

    * –

Number of stuck transitions \- where participants recalled the same solution
within a triplet, for example, EV-EV-EV.

    * –

Number of circular transitions \- where participants mention one solution,
then an interim solution and finally return to the first solution. In our
notation, this would be represented as: OV-CEOV-OV, if participants first
discuss odd number + vowel, then discuss turning in all cards, and then they
return back to odd + vowel. This is a special case of stuck solutions, where
participants may try to “break the cycle” but ultimately return to a previous
solution.

Finally, we analyse the interaction of the measures above and the probing
utterances, as annotated in DeliData (?, ?)

## Results

### Effects of group size on diversity

Figure 1: Average performance gain against group size

As a first step of this analysis, we are hypothesising that group size is a
factor that contributes towards better cognitive diversity and ultimately to
improved decision-making performance (as measured by performance gain).

Considering only the size of the group and the performance gain, we show that
the performance gain of a group increases consistently with group size, as
shown in Figure 1. To confirm the significance of these results, we perform a
1-sample t-test against the baseline performance gain of 0. This yields
p-values of 2.5-8, 3.5-15, 4.1-12, and 2.8-10 for groups of 2, 3, 4, 5
respectively. This indicates that for all group sizes, the performance gain
was significantly above 0. Following, we performed an ANOVA (?, ?) comparison
between groups of sizes 2 to 5. This showed a significant result (p=0.0251),
confirming that group size affects performance gain. Following this, we did a
post-hoc analysis, performing a t-test between different group sizes. The
results (also presented in table 2) of the test show that there is no
significant difference between each of the groups. This is also consistent and
confirms the findings of (?, ?). That said, ? (?) argues that groups (3+
people) perform better than dyads (dialogues). To this end, we also confirm
this claim on the 500 dialogues of DeliData, showing that groups (size$>=$3)
significantly outperform dyads (size=2) with a p-value of 0.03.

Groups | p-value
---|---
2 & 3 | 0.17
3 & 4 | 0.75
4 & 5 | 0.06
2 & 3+ | 0.03
Table 2: Comparison between the effect of group size on performance gain.
Significant values in bold. Figure 2: Comparison between conversational
statistics of two-party dialogues (dyads) and group dialogues (groups)

Many factors can contribute towards the difference between dyads (2
participants) and multi-party dialogues (3 or more participants). First,
larger groups may benefit from an increased idea pool, which we investigate in
detail in the next section - Diversity of initial ideas. Secondly, larger
groups may engage in more diverse conversational strategies, thus
incorporating more ideas and arguments during the group discussion. Figure 2
compares basic conversational statistics between dyads and groups. Dialogues
between two interlocutors have mostly between 10 and 25 utterances, while
group dialogues have a wider range of lengths, with a long tail of dialogues
longer than 50 utterances. Likewise, groups in these dialogues tend to use a
larger vocabulary, as shown on the violin plot of the unique tokens. We
further evaluate conversational patterns and characteristics in section
Diversity of discussed solutions.

### Diversity of initial ideas

Figure 3: Performance gain of groups with high (purple) and low (pink)
diversity of initial submissions. Groups are also split by whether any initial
submissions contained a correct answer (right two) or did not contain any
correct answers (left two).

Herein, we investigate how the diversity of the initial pool of ideas
(indicated by submitted solutions in participants’ solo phase) affects
performance gain. We hypothesise that more diverse initial ideas will provide
a basis for more discussion and a more balanced starting point, ultimately
leading to an improved conversation performance (i.e. increase in performance
gain). Figure 3 shows how initial ideas, in terms of both diversity and the
presence of a correct solution among group members, affect performance gain.
We compare two factors - (i) whether the initial idea pool contained a correct
answer and (ii) the level of diversity (i.e. number of unique solutions) of
the initial idea pool. We show that regardless of whether the initial idea
pool contained a correct answer, high diversity is always beneficial for the
group’s performance gain. Further, we analyse the effects of these two factors
by fitting a linear regression, where performance gain is predicted based on
the level of diversity and whether the initial idea pool contained a correct
answer. The diversity component has a statistically significant contribution
with a p-value of 0.001. On the other hand, the p-value of the component
indicating a correct initial solution is 0.079.

These results show that the diversity of initial ideas is a significant
component of a constructive conversation. However, a reasonable question can
be asked - how does the size of the group confound the diversity of the
initial ideal pool with respect to performance gain? Intuitively, larger
groups would have more ideas, thus the chance of encountering a diverse pool
of ideas is higher. To evaluate this interaction between group size and the
diversity of initial ideas, we predict performance gain using a linear
regression model fitted with the least squares method. We consider three
linear models with the following setups - we predict performance gain based on
(i) group size, (ii) diversity of initial ideas, and (iii) a combination of
(i) and (ii), i.e. group size + diversity of initial ideas. We then compare
these models using ANOVA.

First, we compare models (i) and (iii), testing whether including diversity of
ideas to group size gives us extra predictive power. The ANOVA comparison
estimates that these two models are significantly different, with a p-value of
0.0006. Following, we compare the inverse - models (ii) and (iii), to test
whether including group size to diversity of ideas gives us extra predictive
power. This time, the difference is not significant, with a p-value of 0.81.

This analysis shows that most of the predictive power comes from the diversity
of initial solutions, and including the number of participants does not bring
additional predictive power. Another perspective would be that having a higher
diversity of initial solutions already captures the information of having a
larger group size.

### Diversity of discussed solutions

| Pearson Correlation | p-value
---|---|---
dialogue length | 0.10 | 0.02
unique solutions | 0.28 | 2.8-10
unique transitions | 0.22 | 4.01-7
stuck transitions | 0.02 | 0.67
circular transitions | 0.19 | 8.99-6
Table 3: Results of correlation analysis between different factors
representing the diversity of solutions and performance gain. In bold, we show
the statistically significant results.

Here we use the five indicators outlined in the Analytic strategy subsection -
dialogue length, number of unique solutions, number of unique transitions,
stuck transitions and circular transitions. In order to evaluate these five
indicators and their relationship to performance gain, we perform a
correlation study using (?, ?). The values of the coefficients and the
statistical significance values are presented in Table 3. Out of the five
factors, four are statistically significant in predicting the performance gain
- the dialogue length, the number of unique discussed solutions, the number of
unique transitions, and the number of circular transitions. Further, if we
consider the correlation coefficients, the number of unique solutions (0.28)
is considered the most important feature, followed by the number of unique
transitions (0.22), and the circular transitions (0.19).

These results confirm the hypothesis that the diversity of discussed solutions
correlates with group performance gain. Similarly to the diversity of the
initial ideas, the number of unique solutions discussed is highly indicative
of well-performing groups. Following similar intuition, we argue that this is
both a measure of how diverse the discussion is, as well as it is a proxy for
how open the group is towards new ideas.

In the previous subsection (Diversity of initial ideas), we showed that groups
with a high diversity of initial solutions achieve better performance gain.
Likewise, in this section, we showed that the number of discussed solutions is
also associated with increased performance gain. However, not all groups start
with a diverse set of initial ideas. A reasonable research question would be -
what happens in conversations where groups with low initial diversity achieve
a high number of discussed ideas. To evaluate this, we test the hypothesis
that conversational probing affects dialogues with a low diversity of initial
ideas. The premise is that if participants probe the conversation more often,
this will push the group to discuss novel ideas. Here, we investigate the
correlation between the number of conversational probes and the diversity of
discussed solutions. The results of this analysis are presented in figure 4,
and show that there is a weak correlation between the number of probing
utterances and the increase of unique solutions discuss.

Figure 4: Scatter plot of relation between unique solutions discussed and
number of probing utterances, for conversations with low diversity of initial
ideas.

To further analyse this effect, we take advantage of the more granular probing
annotations in DeliData, namely whether the probing was related to Moderation,
Reasoning behind a solution, or whether a specific Solution should be picked.
To that end, we perform a correlation study to investigate the relationship
between different kinds of probing and the diversity of discussed solutions.
Results of this analysis are shown in Table 4. We see that both probing for
reasons and solutions are a significant contributor towards predicting the
diversity of discussed solutions (p-values $\approx$ 0). The correlation
coefficients show that probing for reasons (0.41) has a higher association
with the diversity of discussed solutions than simply enquiring about
solutions (PearsonR = 0.26). This is an interesting result, showing the
importance of reasoning and also confirming the findings of (?, ?) about the
value of deliberation. Moreover, this gives a basis for future work on
interventional analysis and applications, where conversational probing can
increase cognitive diversity and improve conversational outcomes.

| Pearson Correlation | p-value
---|---|---
Moderation | -0.017 | 0.71
Reasoning | 0.41 | 3.24-21
Solution | 0.26 | 4.21-9
Table 4: Results of correlation analysis of different types of probing and
predicting the number of unique solutions discussed. In bold, we show the
statistically significant results.

## Discussion

Across three different operationalisations, we show that greater diversity is
associated with enhanced benefits of discussion in small groups.

On the chosen task, which has a high risk of individual bias and error, online
discussion groups outperform individuals, and further groups of 3 and above
outperform dyads.

Groups incorporating more diverse individuals inherently have access to
greater cognitive diversity. We find support for this idea more directly by
showing that the diversity of initial solutions was a stronger predictor of
group performance gain than the fact of any individual group member possessing
the right answer.

Finally, at the greatest level of analytic granularity we deploy, we inspect
the content of group discussion and show that groups which discuss a wider
range of potential solutions are also more successful. The circularity and
number of discussed solutions transitions were also predictive of an increase
in performance gain, while stuck transitions were insignificant. Moreover, we
show that conversational probing can increase the diversity of discussed
solutions, even in groups which had a limited idea pool.

### Contributions

This analysis of the DeliData corpus shows that it replicates the finding that
group discussion of the Wason Task can reduce, rather than reinforce, the
individual susceptibility to error on the task. It extends this finding to the
increasingly relevant domain of online discussion among strangers. The finding
is all the more remarkable for the low-commitment 222As in DeliData the
participants were recruited from the crowd-sourcing website MechanicalTurk,
brief, nature of the discussions (typically only 20 - 40 utterances).

By focusing on diversity, we confirm prior findings that the initial pool of
ideas is an important factor in successful group deliberation (?, ?), as is
the exchange of ideas during the discussion (?, ?). We show that the
association of diversity with successful deliberation extents beyond the mere
content and volume of discussion to diversity in structure. Circularity in
discussion is not necessarily negative — it could represent focus and
persistence which we also show is important predictor of performance
improvement.

Additionally, we show how analysis of features of groups and group dialogue,
captured at different levels of description, can be applied to the study of
group decision-making. These approaches are applicable to other domains where
dialogue supporting group decision-making is captured.

### Limitations

Since our study is observational, not experimental, in nature, it is only
possible to note associations, not causation. Ultimately, we suspect that
successful groups generate diverse discussions, as well as vice versa. Our
analysis supports the idea that diversity of membership, ideas and discussion
will promote successful group decision-making, but decisive proof of this idea
awaits testing by intervention studies.

Further, in this study, we investigated the effect of cognitive diversity on
performance gain. By definition, performance gain is affected by the baseline
performance of the group - i.e. if a group has a low starting performance, the
performance of this group can grow more compared to groups that start with
high baseline performance. The vice-versa is also true - if a group starts
with high (or even perfect) solo performance, during the discussion they can
only decrease their performance leading to negative performance gain. We
recognise that this is a limitation of the current study, as we only look at
how much a conversation improved, disregarding the starting points. We
performed a small-scale control experiment to evaluate the effect of this
limitation. First, we consider the dialogues with high baseline performance,
which we define as dialogues where more than 50% of the participants had the
correct answer in their solo submissions. In DeliData, there are 9 dialogues
out of 500 (1.8%) where this is the case, suggesting that this affects a
limited portion of the data. Second, we re-ran all of the analyses in this
work without those 9 dialogues and confirmed that the removal of those does
not affect the statistical significance tests, nor does it change any of the
analyses performed as part of this work.

### Conclusion

Despite the reputation of groups for compounding bias, these findings show
that small groups can, through dialogue, enhance individual decision-making.
Further, we offer a worked example of the potential for fine-grained analysis
of dialogue to illuminate exactly how the benefits of deliberation come about.
Across a large sample and different operationalisations, we consistently find
that greater diversity is associated with more successful group deliberation.

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# Raman scattering model of the spin noise

G. G. Kozlov  1 A. A. Fomin1  M. Yu. Petrov1  I. I. Ryzhov2,1 and V. S.
Zapasskii1  1Spin Optics Laboratory, Saint Petersburg State University, St.
Peterbsurg, 198504 Russia
2Department of Photonics, Saint Petersburg State University, St. Peterbsurg,
198504 Russia<EMAIL_ADDRESS>
###### Abstract

The mechanism of formation of the polarimetric signal observed in the spin
noise spectroscopy (SNS) is analyzed from the viewpoint of the light
scattering theory. A rigorous calculation of the polarimetric signal (Faraday
rotation or ellipticity) recorded in the SNS is presented in the approximation
of single scattering. We show that it is most correctly to consider this noise
as a result of scattering of the probe light beam by fluctuating
susceptibility of the medium. Fluctuations of the gyrotropic (antisymmetric)
part of the susceptibility tensor lead to appearance of the typical for the
SNS Faraday rotation noise at the Larmor frequency. At the same time,
fluctuations of linear anisotropy of the medium (symmetric part of the
susceptibility tensor) give rise to the ellipticity noise of the probe beam
spectrally localized at the double Larmor frequency. The results of the
theoretical analysis well agree with the experimental data on the ellipticity
noise in cesium vapor.

††journal: oe

##  1 Introduction

Spin noise spectroscopy (SNS), for the first time demonstrated on a sodium
vapor [1] back in 1981, has been rapidly developing over the past decade and
has proven to be an efficient tool for studying the energy structure of matter
(see, e. g., reviews [2, 3, 4, 5, 6]). To a large extent, the success of the
SNS was inspired by its applicability for studies of semiconductor systems [7,
8, 9, 10, 11], including nanostructures [12, 13, 14]. Not only spin carrier
Larmor precession in volume ([9, 15] etc.) and low-dimensional media (e. g.
[16, 17]) can be addressed by SNS, but also the nuclear dynamics [18, 19, 20],
light-matter interaction effects [21, 22], spatial properties distribution
[23, 24], and even noise of valley redistribution of electrons [25]. Still the
atomic systems attract the significant attention of researchers [26, 27, 28,
29, 30, 31, 32, 33] as model objects which allow to reveal the fundamental
pecularities of spin noise formation mechanics. In typical SNS experiments
with polarimetric detection of the signal, one measures, in fact, the
magnetization-noise power spectrum of the studied sample associated, in
accordance with the fluctuation-dissipation theorem, with the spectrum of its
magnetic susceptibility [34]. The experiments are usually carried out in a
transverse static magnetic field, so that the magnetization-noise power
spectrum turns out to be localized in the region of the Larmor frequency
$\omega_{L}$of precession of the magnetic moments (spins) and represents, in
fact, EPR (electron paramagnetic resonance) spectrum of the system. The fact
that magnetization noise is observed in SNS as the polarization noise of the
probe laser light and is caused by fluctuations of the optical susceptibility
of the sample shows a similarity of the SNS and Raman spectroscopy. Thus, due
to combination of the features of the EPR and Raman spectroscopy, the SNS
reveals a number of unique features related to nonperturbativity of the SNS
[35, 36], to possibility of tuning the frequency of the probe laser light [37,
38] and its spatial localization [39], to possibility of using multi-beam
configurations [40, 41] and some others.

Interpretation of the SNS experiments is performed, as a rule, using
“magnetic” language with the observed Faraday rotation (FR) noise being
identified with the magnetization noise and the FR power spectrum calculated
using appropriate model. Meanwhile, strictly speaking, the fluctuations of the
scattered probe light observed in the SNS experiments, can be caused not only
by fluctuations of gyrotropy (proportional to the magnetization), but also by
fluctuations of the whole optical susceptibility tensor of the medium [42].

The idea of interpretation of the spin noise signal formation mechanism in
terms of Raman scattering is well known to the audience involved in the SNS
and was suggested shortly after the first experiment of Aleksandrov and
Zapasskii [43]. The aim of our work work is to rigorously develop this idea
and to fill the existing theoretical gap with a consistent description of the
mechanism of formation of the light-polarization noise observed in SNS as a
noise of light scattering.

The work is organized as follows. In the first section, we present a brief
description of a typical SNS setup and derive the general relationship between
the polarimetric signal (FR or ellipticity noise) recorded in SNS and optical
susceptibility tensor of the scattering particles. In the second section, we
calculate, in the framework of a simple atomic model, the optical
susceptibility tensor for the case of nonstationary (superposition) atomic
state. The derived expression is used in the third section for calculating the
SNS polarimetric signal. In this section, we analyze spectral peculiarities of
the noise signal and their dependence on the angle between the direction of
the probe beam polarization plane and magnetic field. In the fourth section,
we analyze, in the framework of the developed model, the mechanism of
formation of the recently discovered ellipticity noise signal at double Larmor
frequency (alignment noise) and compare our results with the experimental data
obtained in [30]. The main results of the paper are summarised in Conclusions.

Throughout the article, polarimetric signals recorded in SNS are considered as
the result of Raman scattering in its simplest form, when the modulation of
the optical susceptibility leads to the appearance of side frequencies in the
scattered field, which is described by classical electrodynamics. This
assumption seems to be justified for the purposes of SNS, since the frequency
shift of the scattered radiation is much less than the temperature (in
frequency units) at which the measurements are made.

##  2 The relationship between SNS polarimetric signal and susceptibility
tensor

Schematic of the experimental setup used for the SNS measurements is shown in
Fig. 1. The polarimetric detector in this setup could detect either FR or
ellipticity noise depending on the type of the waveplate (WP) placed in front
of the polarization beamsplitter (PBS). When detecting the FR noise, the WP
was taken half-wave and was used to balance the differential photodetector.
When detecting the ellipticity noise, the WP was taken quarter-wave, with its
axes aligned at 45∘ with respect to axes of the beamsplitter PBS. In this
arrangement of the polarization elements, the output signal of the
differential photodetector was zero, when the input light was polarized
linearly (regardless of the polarization plane azimuth) and became nonzero
only upon appearance of ellipticity in the input light. The output electric
signal $U$ of the polarimetric detector was fed to a digital hardware Fourier-
transform spectrum analyzer which acquired the polarization noise spectrum of
the probe beam transmitted through the sample. Magnetic field could be aligned
either along or across the probe-light propagation axis (in the Faraday or
Voigt geometries).

Figure 1: Schematic of the experimental setup. When observing fluctuations of
the Faraday rotation, a $\lambda/2$ waveplate (WP) was placed and used to
balance photocurrents of the detector. When observing fluctuations of
ellipticity, a $\lambda/4$ WP was placed with its axes aligned at 45∘ with
respect to the PBS eigen directions.

As was shown in [43], the polarization noise signal observed in SNS can be
interpreted as the result of scattering of the probe beam by the studied
sample comprising any paramagnetic particles (atoms, ions, electrons or either
quasiparticles such as holes and excitons), in what follows they will be
referred to as atoms. We will calculate this signal using the following
simplifying assumptions:

  1. (i)

The electromagnetic field acting upon each atom coincides, with an acceptable
accuracy, with that of the probe beam (approximation of single scattering);

  2. (ii)

Atomic polarization can be calculated in the approximation of linear response;

  3. (iii)

The magnetic field is small enough to Zeeman splitting of the atomic
multiplets $\sim\omega_{L}$ be much smaller than the homogeneous linewidth
$\delta$ of the optical transition and not be resolved in the optical spectrum
($\omega_{L}\ll\delta$).

###  2.1 Polarimetric response of a single atom

In this subsection we consider the case of ellipticity noise – the FR noise
spectrum can be calculated in a similar way [44]). In this case, the quarter-
wave plate of the polarimetric detector (Fig. 1) is aligned with its axes at
$45^{\circ}$ with respect to polarization directions of the polarizing
beamsplitter (PBS), which we assign to be the axes $z$ and $x$. The probe beam
propagation direction is taken for the axis $y$ (see Fig. 1). In this
coordinate system, the magnetic field has only $z$ component (in Voigt
geometry), ${\bf B}=(0,0,B)$, while the probe-beam electric field $\bf{\cal
E}_{0}$ has only the $x$ and $z$ components, ${\bf{\cal E}}_{0}=({\cal
E}_{x0},0,{\cal E}_{z0})$ (Fig. 1).

Let us denote electric field of the probe beam at the input of the
polarimetric detector as ${\bf{\cal E}}=({\cal E}_{x},{\cal E}_{y},{\cal
E}_{z})$. Then, as can be shown by direct calculations, the output signal $U$
of the detector, in this mode of operation, is given by

$U={\frac{1}{T}}\int_{0}^{T}dt\int_{S}dxdz\bigg{[}{\cal E}_{x}(t){\cal
E}_{z}(t+\Delta)-{\cal E}_{x}(t+\Delta){\cal E}_{z}(t)\bigg{]},$ (1)

where $\Delta={\pi/2\omega}$ and $\omega$ is the probe light frequency. The
integration over $dxdz$ in Eq. (1) is performed over photosensitive surfaces
of the photodetectors $S$, which are supposed to be identical. The integration
over $t$ corresponds to averaging over the time interval $T$ that contains
integer number of optical periods and meets the requirement $2\pi/\omega\ll
T\ll 2\pi/\omega_{L}$. We see that, indeed, the signal $U$ appears to be
nonzero only for the elliptically polarized input field ${\bf{\cal E}}_{0}$,
while for any linearly polarized field, the output signal vanishes. It is also
seen from Eq. (1) that any rotation of the polarimetric detector around the
$y$ axis does not affect the output signal $U$, since, for any two vectors
$\bf A$ and $\bf B$, the quantity

$A_{x}B_{z}-A_{z}B_{x}=({\bf A,\beta B}),\hskip
8.53581pt\beta\equiv\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ (2)

does not change under arbitrary rotations in the plane $xz$.

The noise signal observed in SNS experiments can be represented as the sum of
the contributions of individual atoms. Therefore, we will start with
calculating the ellipticity signal $\delta u_{e}$, created by a single atom.
For convenience, we will consider the field $\cal E$ at the input of our
detector as a real part of the complex field ${\bf E}$: ${\cal E}=\mathop{\rm
Re}{\bf E}$. The field ${\bf E}$ can be considered as a sum of the field ${\bf
E}_{0}\equiv{\bf A}_{0}\exp{-\imath\omega t}$ of the probe beam $({\cal
E}_{0}=\mathop{\rm Re}{\bf E}_{0})$ and the field ${\bf E}_{1}$ created by the
atomic dipole $({\cal E}_{1}=\mathop{\rm Re}{\bf E}_{1})$. We use calligraphic
letters to denote the observed (real) fields.

We calculate the ellipticity signal in the approximation linear in the field
${\bf E}_{1}$. Since all the fields are assumed quasi-monochromatic
($\sim\exp{-\imath\omega t}$), the time shift by $\pm\pi/2\omega$ is
equivalent to multiplication by $\mp\imath$. Keeping this in mind, we obtain
from Eq. (1) the following expression for the signal $\delta u_{e}$ [45]

$\delta u_{e}=\hbox{ Im }{\frac{2}{T}}\int_{0}^{T}dt\int_{S}dxdz\bigg{[}{\cal
E}_{0x}{E}_{1z}-{E}_{1x}{\cal E}_{0z}\bigg{]}.$ (3)

This formula includes both complex $(E_{1i})$ and real $({\cal E}_{0i})$
fields. The field ${\bf E}_{1}$ of the atomic dipole can be obtained by
solving the inhomogeneous Helmholtz equation $\Delta{\bf E}_{1}+k^{2}{\bf
E}_{1}=-4\pi k^{2}{\bf P}$, where $k=\omega/c$ ($c$ is the speed of light) and
$\bf P(r)\sim\delta({\bf R-r})$ is the complex polarization created by the
atom with the radius-vector $\bf R$. Solution of this equation can be obtained
using Green’s function of the Helmholtz operator (see, e. g., [44, 39]) and
has the form ${\bf E}_{1}({\bf r})=k^{2}\int d^{3}{\bf r^{\prime}}\exp{\imath
k|{\bf r-r^{\prime}}|}{\bf P(r^{\prime})}/|{\bf r-r^{\prime}}|$. By
substituting this expression into Eq. (3), we have:

$\delta u_{e}={2k^{2}\over T}\hbox{ Im }\int_{0}^{T}dt\int d^{3}{\bf
r^{\prime}}\bigg{[}\Phi_{x}({\bf r^{\prime}})P_{z}({\bf
r^{\prime}})-\Phi_{z}({\bf r^{\prime}})P_{x}({\bf r^{\prime}})\bigg{]},$ (4)

where we introduced the following functions $\Phi_{i}({\bf
r^{\prime}})(i=x,z)$[46]:

$\begin{gathered}\Phi_{i}({\bf r^{\prime}})\equiv\int_{S}dxdz\hskip
2.84526pt{\cal E}_{0i}(x,y,z){\exp{\imath k|{\bf r-r^{\prime}}|}\over|{\bf
r-r^{\prime}}|}\bigg{|}_{r_{y}=L}\equiv\\\ \exp({-\imath\omega
t})\Phi_{i}^{+}({\bf r^{\prime}})+\exp({\imath\omega t})\Phi_{i}^{-}({\bf
r^{\prime}}),\hskip 8.53581pti=x,z,\hskip 8.53581pt{\bf
r}=(x,y,z).\end{gathered}$ (5)

When integrating over the photodetector surface $S$ ($dxdz$) in Eq. (5), we
assumed that $r_{y}=y=L$, where $L$ is the distance from the atom to the
polarimetric detector, which we consider to be large: $L\rightarrow\infty$.
Besides, in Eq. (5) we separate explicitly the components $\Phi_{i}^{\pm}({\bf
r^{\prime}})$ of the function $\Phi_{i}({\bf r^{\prime}})$ proportional to
$\exp({\mp\imath\omega t})$. In the approximation of linear response, atomic
polarization is proportional to the probe wave electric field. Therefore,
${\bf P(r^{\prime})}\sim\exp({-\imath\omega t})$ and, after time-averaging in
Eq. (4), only components $\Phi_{i}^{-}({\bf r^{\prime}})$ survive. We show in
Appendix that (the similar result for the case of Gaussian beam has been
obtained in [39])

$\Phi_{i}^{-}({\bf r^{\prime}})={\imath\pi\over k}A_{0i}^{\ast}({\bf
r^{\prime}}),\quad|{\bf r^{\prime}}|\ll L,\quad i=x,z;$ (6)

where $A_{0i}$ is the $i$-th projection of the amplitude of the probe beam
complex field (remind that ${\bf E}_{0}=(A_{0x},0,A_{0z})\exp({-\imath\omega
t})$ ). The polarization ${\bf P(r^{\prime})}$ created by a single atom
entering Eq. (4) can be presented in the form $P_{i}({\bf
r^{\prime}})=\delta({\bf r^{\prime}-R})\langle d_{i}\rangle\exp({-\imath\omega
t})$ where $\langle d_{i}\rangle$ is the complex amplitude of oscillation of
the $i$-th component of the atomic dipole moment, $\bf R$ is the radius-vector
of the atom. By substituting this expression into Eq. (4) and taking into
account Eq. (6), we obtain, for the ellipticity signal $\delta u_{e}$ created
by a single atom, the following expression:

$\delta u_{e}=2\pi k\hbox{ Re }\bigg{[}A_{0x}^{\ast}({\bf R})\langle
d_{z}\rangle-A_{0z}^{\ast}({\bf R})\langle d_{x}\rangle\bigg{]}.$ (7)

When the polarimetric detector operates in the FR-detection mode (i.e., the
$\lambda/4$ wave plate is replaced by the $\lambda/2$ plate), then a similar
calculation leads to the following expression for the FR noise signal $\delta
u_{r}$ produced by a single atom:

$\delta u_{r}=2\pi k\mathop{\rm
Im}\bigg{[}\cos[2\phi]\bigg{(}A_{0x}^{\ast}({\bf R})\langle d_{x}\rangle-
A_{0z}^{\ast}({\bf R})\langle
d_{z}\rangle\bigg{)}-\sin[2\phi]\bigg{(}A_{0x}^{\ast}({\bf R})\langle
d_{z}\rangle+A_{0z}^{\ast}({\bf R})\langle d_{x}\rangle\bigg{)}\bigg{]}.$ (8)

Here, $\phi$ is the angle between $z$-axis and one of the main directions of
the beamsplitter (in the above calculation of the ellipticity signal, we used
a coordinate system for which $\phi=0$; see [47] for explanation).

Equations (7) and (8) can be simplified under the following conditions. First,
we assume that the quantities $\langle d_{i}\rangle$ entering Eqs. (7) and (8)
can be expressed through the probe light electric field amplitude ${\bf
A}_{0}({\bf R})$ with use of the susceptibility tensor $\alpha$: $\langle
d_{i}\rangle=\alpha_{ik}A_{0k}$. Second, the probe beam is assumed to be
linearly polarized. In this case, $A_{0x}=A_{0}\sin\theta$ and
$A_{0z}=A_{0}\cos\theta$, where $\theta$ is the angle between the probe beam
polarization and $z$ axis. And third, we assume that, in the measurements of
the Faraday-rotation noise, the orientation of the polarimetric detector
(specified by the angle $\phi$ or, which is the same, by the orientation of
the half-wave plate, see [47]) corresponds to conditions of balance with no DC
signal at the output of the detector, i. e., $\phi=\theta+\pi/4$ [48]. When
the above conditions are satisfied, Eqs. (7) and (8) can be rewritten in a
compact scalar form as follows

$\begin{gathered}\delta u\equiv\delta u_{e}+\imath\delta u_{r}=2\pi k({\bf
A}_{0}({\bf R}),\beta\alpha{\bf A}_{0}({\bf R}))=\\\ =\pi k|A_{0}({\bf
R})|^{2}\bigg{[}\alpha_{zx}-\alpha_{xz}-(\alpha_{xz}+\alpha_{zx})\cos
2\theta+(\alpha_{zz}-\alpha_{xx})\sin 2\theta\bigg{]}\end{gathered}$ (9)

with matrix $\beta$ defined by Eq. (2).

##  3 Calculation of the atomic susceptibility

Calculation of the linear atomic susceptibility is performed assuming that it
is related to optical transitions between two (ground and excited) atomic
multiplets [49] with the same total angular momenta $F$. Formation of the
optical response of the atom travelling through the probe beam can be imagined
in the following way. The wavefunction of the atom $\Psi(0)$, at the moment of
its entering the beam (let it be $t=0$) is a random superposition of atomic
eigenfunctions $|1M\rangle$ of the ground multiplet with different
$z$-components $M$ of the angular momentum:
$\Psi(0)=\sum_{M=-F}^{F}C_{M}|1M\rangle=\sum_{M=-F}^{F}|C_{M}|\exp({\imath\beta_{M}})|1M\rangle$,
where $|C_{M}|$ and $\beta_{M}$ are the random amplitude of the atomic state
$|1M\rangle$ and its phase (with $\sum_{M=-F}^{F}|C_{M}|^{2}=1$). When the
magnetic field is nonzero, the ground multiplet exhibits Zeeman splitting, and
the above superposition state appears to be nonstationary (even neglecting the
probe-beam induced perturbation). The appropriate unperturbed density matrix
of the atom $\rho_{0}$ also appears to be time-dependent with nonzero matrix
elements only in the subspace of the states of the ground atomic multiplet:

$\langle
1M|\rho_{0}(t)|1M^{\prime}\rangle=|C_{M}||C_{M^{\prime}}|\exp({\imath[\beta_{M}-\beta_{M^{\prime}}]})\exp({\imath\omega_{1L}[M-M^{\prime}]t}),$
(10)

where $\omega_{1L}$ is the Larmor frequency for the ground-state multiplet. As
seen from Eq. (10), the density matrix oscillates in time at frequencies
integer multiples of the Larmor frequency $\omega_{1L}$. Since the linear
optical susceptibility of the atom is related to its unperturbed density
matrix (this connection will be presented below), it may depend on time at
frequencies $\omega_{1L}[M-M^{\prime}]$. This may, in turn, give rise to
appearance of shifted frequencies $\omega\pm\omega_{1L}|M-M^{\prime}|$ in the
spectrum of the field ${\bf E_{1}}$ scattered by the atom (the effect of Raman
scattering) and can be detected in SNS experiments as the noise of
polarimetric signal spectrally localized in the vicinity of the frequencies
$\omega_{1L}|M-M^{\prime}|$. As shown below, only frequencies with
$|M-M^{\prime}|=0,1,2$ can be observed and, correspondingly, only spectral
features at the frequencies $0$, $\omega_{1L}$, and $2\omega_{1L}$ can arise
in the polarization noise spectra.

Let us pass now to calculation of the linear atomic susceptibility. Here we
present calculations only for the most interesting from the experimental
viewpoint case of Voigt geometry as the case of Faraday geometry can be
analyzed in a very similar way. The matrix of the Hamiltonian of the atom in
the representation of the two (ground and excited) multiplets in frequency
units has the form

$\begin{gathered}H=H_{0}+H_{E},\\\ H_{0}\equiv\Omega\begin{pmatrix}I&0\\\
0&0\end{pmatrix}+\begin{pmatrix}\omega_{2L}J_{z}&0\\\
0&\omega_{1L}J_{z}\end{pmatrix},\\\ H_{E}=\omega_{x}\exp({-\imath\omega
t})\begin{pmatrix}0&J_{x}\\\
J_{x}&0\end{pmatrix}+\omega_{z}\exp({-\imath\omega
t})\begin{pmatrix}0&J_{z}\\\ J_{z}&0\end{pmatrix}\end{gathered}$ (11)

Here $H_{0}$ is the Hamiltonian of undisturbed atom with $\Omega$ being the
spectral distance between ground and excited multiplets. The second term in
the expression for $H_{0}$ describe Zeeman splitting with $\omega_{iL}$ being
Larmor frequencies of $i$-th multiplet ($i=1$ refers to the ground state and
$i=2$ the excited state). Term $H_{E}$ describe the action of monochromatic
$\sim e^{-\imath\omega t}$ electromagnetic field of the probe beam which is
linearly polarised in $ZX$ plain. The quantities $\omega_{x,z}$ correspond to
Rabi frequencies defined by the dipole moment $d$ of optical transition
between the ground and excited multiplets and by the amplitudes $A_{0x,z}$ of
electric field projections of the probe beam at point $\bf R$ where the atom
is located: $\omega_{i}\equiv dA_{0i}({\bf R})/\hbar,i=x,z$. Each ‘element’ of
matrices in Eq. (11) is itself a matrix with dimensions $(2F+1)\times(2F+1)$,
with $J_{z}$ and $J_{x}$ being known matrices of the corresponding projections
of the angular momentum $F$ [50]. The matrices of the operators for the needed
$x$ and $z$ projections of the atomic dipole moment have the form

$d_{x}\equiv d\begin{pmatrix}0&J_{x}\\\ J_{x}&0\end{pmatrix},\quad d_{z}\equiv
d\begin{pmatrix}0&J_{z}\\\ J_{z}&0\end{pmatrix}.$ (12)

The standard procedure of the linear-response theory implies representation of
solution of the equation $\imath\dot{\rho}=[H,\rho]$ for the atomic density
matrix $\rho$ in the form $\rho=\rho_{0}+\rho_{1}+O(H_{E}^{2})$ (here
$\rho_{0}$ is unperturbed atomic density matrix satisfying the equation
$\imath\dot{\rho}_{0}=[H_{0},\rho_{0}]$ and $\rho_{1}$ is linear in the probe
beam field correction satisfying the equation
$\imath\dot{\rho}_{1}=[H_{0},\rho_{1}]+[H_{E},\rho_{0}]$), and computation of
the quantities $\langle d_{i}\rangle$ as $\langle d_{i}\rangle=\mathop{\rm
Sp}\rho_{1}d_{i}$, where $i=x,z$. It leads to the expression $\langle
d_{i}\rangle=\alpha_{ik}A_{0k}({\bf R})$, in which the susceptibility tensor
$\alpha$ has the following elements

$\alpha_{ik}\equiv{d^{2}\over\hbar}\sum_{MM^{\prime}M^{\prime\prime}}{\langle
1M|\rho_{0}|1M^{\prime}\rangle\langle
M^{\prime}|J_{k}|M^{\prime\prime}\rangle\langle
M^{\prime\prime}|J_{i}|M\rangle\over\Delta\omega+\imath\delta+\omega_{2L}M^{\prime\prime}-\omega_{1L}M^{\prime}},\qquad
i,k=x,z.$ (13)

Here, $\Delta\omega\equiv\Omega-\omega$ is the optical detuning,
$\imath\delta$ denotes the homogeneous broadening, $\langle
M|J_{k}|M^{\prime}\rangle$ are the matrix elements of the operator of $k$-th
projection of the angular momentum $F$ [50], and the summation over
$M,M^{\prime}$, and $M^{\prime\prime}$ is performed over $2F+1$ states of the
ground-state multiplet. As has been noted above, the tensor $\alpha$ depends
on time (through the matrix elements $\langle 1M|\rho_{0}|1M^{\prime}\rangle$,
see Eq. (10)), with characteristic frequencies of this dependence
corresponding to spectral features of the noise spectra observed in the SNS.

Since the quantity $\langle M|J_{z}|M^{\prime}\rangle=\delta_{MM^{\prime}}M$
and $\langle M|J_{x}|M^{\prime}\rangle$ is nonzero only for
$|M-M^{\prime}|=1$, it follows from Eq. (13) that the only frequencies
$\omega_{1L}|M-M^{\prime}|$ at which oscillations of the tensor $\alpha$ may
occur are: $\omega_{1L}$ (the elements $\alpha_{xz},\alpha_{zx}$ $\sim\langle
M^{\prime}|J_{x}|M^{\prime\prime}\rangle\langle
M^{\prime\prime}|J_{z}|M\rangle$), $2\omega_{1L}$ (the elements $\alpha_{xx}$
$\sim\langle M^{\prime}|J_{x}|M^{\prime\prime}\rangle\langle
M^{\prime\prime}|J_{x}|M\rangle$) and 0 (the elements
$\alpha_{zz},\alpha_{xx}$ $\sim\langle
M^{\prime}|J_{x}|M^{\prime\prime}\rangle\langle
M^{\prime\prime}|J_{x}|M\rangle$, $\langle
M|J_{z}|M^{\prime\prime}\rangle\langle M^{\prime\prime}|J_{z}|M\rangle)$).

It is seen from Eq. (9) for the complex polarimetric signal $\delta u$ that
the components of this signal at the frequency $2\omega_{L}$ behave as
$\sim\sin 2\theta$ and vanish when the probe beam polarization is parallel or
perpendicular to the magnetic field ($\theta=0,\pi/2$), as it has been
observed in experiment [30].

##  4 Calculation of the polarimetric signal

Note that under assumption that $\omega_{1L}\ll\delta$ the dependence of the
denominator in Eq. (13) on the numbers $M^{\prime\prime}$ and $M^{\prime}$ may
be neglected. Then we obtain the following expression for the matrix of the
atomic susceptibility $\alpha$

$\alpha_{ik}={d^{2}\hbox{ Sp
}\rho_{0}J_{k}J_{i}\over\hbar[\Delta\omega+\imath\delta]}={d^{2}\hbox{ Sp
}\rho_{0}[\\{J_{k}J_{i}\\}+\imath\varepsilon_{kil}J_{l}]\over
2\hbar[\Delta\omega+\imath\delta]},$ (14)

in which we distinguished symmetric ($\sim\\{J_{k}J_{i}\\}\equiv
J_{k}J_{i}+J_{i}J_{k}$) and antisymmetric (gyrotropic, $\sim
J_{k}J_{i}-J_{i}J_{k}=\imath\varepsilon_{kil}J_{l}$) parts (here,
$\varepsilon_{kil}$ is the Levi-Civita tensor). After such a simplification,
the expression for the susceptibility $\alpha$ acquires the form of a quantum
mean value of the tensor observable with the operator $\sim J_{k}J_{i}$ in the
state with the density matrix $\rho_{0}$. The appropriate superpositional
wavefunction $\Psi$ (it contains only the components related to the ground
multiplet) satisfies the Schrödinger equation
$\imath\dot{\Psi}=H_{0}\Psi=\omega_{1L}J_{z}\Psi$ and is defined by the
formula: $\Psi(t)=\exp({-\imath\omega_{1L}J_{z}t})\Psi(0)$. Since the operator
$\exp({-\imath\omega_{1L}J_{z}t})$ is the operator of rotation by the angle
$\omega_{1L}t$ around the $z$ axis [50], the function $\Psi(t)$ represents the
function $\Psi(0)$, rotating around the magnetic field with the angular
frequency $\omega_{1L}$. This rotation is accompanied by ‘rotation’ of the
tensor $\alpha_{ik}\sim\mathop{\rm
Sp}\rho_{0}J_{k}J_{i}=\langle\Psi|J_{k}J_{i}|\Psi\rangle$ [51], and the noise
signal detected in our experiments can be understood as a result of scattering
of the probe beam by a quasi-point anisotropic system rotating with the Larmor
frequency $\omega_{1L}$ around the magnetic field. If we substitute Eq. (14)
into (9), we obtain for the complex polarimetric signal $\delta u$ the
following expression

$\delta u={\pi kd^{2}\over\hbar}|A_{0}({\bf R})|^{2}f(t),$ (15)

where

$f(t)\equiv{\mathop{\rm Sp}\rho_{0}\bigg{[}(J_{x}^{2}-J_{z}^{2})\sin
2\theta+(J_{z}J_{x}+J_{x}J_{z})\cos 2\theta+\imath
J_{y}\bigg{]}\over\Delta\omega+\imath\delta}\equiv f_{e}(t)+\imath f_{r}(t)$

Physical meaning of different contributions in this formula can be determined
by considering behavior of the function $f(t)$ at large detunings
$\Delta\omega\gg\delta$. It can be seen that the first two terms in Eq. (15)
describe fluctuations of symmetric part of the tensor $\alpha$ (fluctuations
of _alignment_) and, being real (at $\Delta\omega\gg\delta$), can be observed
only in the regime of detection of ellipticity (see Eq. (9)).

Since the matrix elements $\langle M|J_{x}^{2}-J_{z}^{2}|M^{\prime}\rangle$
are nonzero only at $|M-M^{\prime}|=0$ and $|M-M^{\prime}|=2$, the
contribution $\sim(J_{x}^{2}-J_{z}^{2})\sin 2\theta$ gives rise to peaks in
the spectra of ellipticity noise at zeroth and double Larmor frequencies
(because the elements $\langle M|J_{x}^{2}-J_{z}^{2}|M^{\prime}\rangle$ enter
the expression for the polarimetric signal Eq. (15) together with the elements
$\langle M^{\prime}|\rho_{0}|M\rangle$ of the density matrix whose time
behaviour is determined by Eq. (10)). In a similar way, one can make sure that
the contribution $\sim(J_{z}J_{x}+J_{x}J_{z})\cos 2\theta$ gives rise to a
peak at the frequency $\omega_{1L}$.

The isotropic term in brackets $\sim\imath\mathop{\rm Sp}\rho_{0}J_{y}$
describes fluctuations of gyrotropy of the atomic system and, being pure
imaginary, is revealed only in the FR noise. Since the matrix elements
$\langle M|J_{y}|M^{\prime}\rangle$ are nonzero only at $|M-M^{\prime}|=1$,
this term provides a feature in the Faraday-rotation noise spectrum only at
the frequency $\omega_{1L}$.

The above consideration was related to the case of Voigt geometry. Similar
results can be obtained for the Faraday geometry. In this case, the expression
for the complex polarimetric signal $\delta u$ differs from Eq. (15) by the
permutation of the operators $J_{z}\rightarrow J_{y}$ and $J_{y}\rightarrow
J_{z}$ (leaving the same expression Eq. (10) for the density matrix
$\rho_{0}$). An analysis similar to the above shows that the polarimetric
noise signal recorded in Faraday geometry will have spectral features only at
zero frequency and at a frequency of $2\omega_{1L}$.

The rigorous calculation of the noise power spectrum ${\cal
N}(\nu)=\int\langle f(0)f(t)\rangle\exp({-\imath\nu t})dt$ observed in SNS
experiments requires the calculation of the correlation function $\langle
f(0)f(t)\rangle$. This calculation is somewhat cumbersome, so, in section 5,
we present the results of such calculation with no details. The quantum-
mechanical correlation functions of the operators entering Eq. (15) were
calculated in [30].

##  5  Discussion

The above simplified consideration shows that observation of spectral feature
at the frequency $2\omega_{1L}$ is possible only in the ellipticity noise
spectrum. Note that experiments [30] mainly support this conclusion. A
consistent calculation shows, however, that when the homogeneous width of the
line $\delta$ is getting much smaller than the Doppler broadening, the
difference between the noise spectra of ellipticity and FR (in terms of the
peak at the double Larmor frequency) becomes not so dramatic. It can be
briefly explained as follows. Consider, e.g., the ellipticity noise spectrum,
which is determined by the Fourier-image of the correlation function
$\langle\delta u_{e}(t)\delta u_{e}(0)\rangle\sim\langle|A_{0}({\bf
R}(t))|^{2}|A_{0}({\bf R}(0))|^{2}f_{e}(t)f_{e}(0)\rangle$ [see Eq. (15)].
Calculation of correlator of the signal $f_{e}(t)$ (15) leads to the following
expression:

$\begin{gathered}\langle
f_{e}(t)f_{e}(0)\rangle\sim{5a^{2}}\cos[\omega_{1L}t]+{d^{2}}\bigg{[}F(F+1)-{3\over
4}\bigg{]}\bigg{[}4\cos^{2}2\theta\cos[\omega_{1L}t]+(3+\cos[2\omega_{1L}t])\sin^{2}2\theta\bigg{]},\\\
d+\imath a\equiv 1/[\Delta\omega+kv_{y}+\imath\delta].\end{gathered}$ (16)

A similar expression was obtained in the theoretical section of work [30] by
solving the equations of motion for the correlation functions. Here, we
omitted nonessential factors and accounted for the Doppler shift $kv_{y}$
($v_{y}$ is the projection of the atomic speed upon the probe beam direction).
The expression for the correlator of the FR signal $\langle
f_{r}(t)f_{r}(0)\rangle$ differs from Eq. (16) by the substitutions
$a\rightarrow d$ and $d\rightarrow a$. Despite the fact that frequency
dependencies of the correlators $\langle f_{e}(t)f_{e}(0)\rangle$ and $\langle
f_{r}(t)f_{r}(0)\rangle$ are different, for both of them it has the form of a
sharp feature with the width $\sim\delta$. For this reason, upon Maxwellian
averaging of the Doppler shift $kv_{y}$, with the width $\sim kv_{T}\gg\delta$
(here, $v_{T}^{2}$ is the mean-square thermal velocity), the above difference
(for $\Delta\omega\sim kv_{T}$) will be of no importance. For the Gaussian
probe beam, calculation leads to the following expression for the correlation
function observed in the SNS (nonessential numerical factors are omitted):

$\begin{split}&K(t)\sim{\sigma\hskip 2.84526ptW^{2}\rho_{c}k^{2}d^{4}\hskip
2.84526pt\over v_{T}^{2}\delta}\exp\bigg{[}-{\Delta\omega^{2}\over
k^{2}v_{T}^{2}}\bigg{]}{\exp({-|t|/T_{2}})\over\sqrt{t^{2}+t_{T}^{2}}}\times\\\
&\times\bigg{\\{}5\cos[\omega_{1L}t]+\bigg{[}F(F+1)-{3\over
4}\bigg{]}\bigg{[}4\cos^{2}2\theta\hskip
2.84526pt\cos[\omega_{1L}t]+(3+\cos[2\omega_{1L}t])\hskip
2.84526pt\sin^{2}2\theta\bigg{]}\bigg{\\}}.\end{split}$ (17)

Here, along with the quantities introduced above, we use: $\sigma$ – atomic
density, $W$ – the probe beam power, $\rho_{c}$ – the beam radius in its
waist, $t_{T}\equiv\rho_{c}/v_{T}$ – the time of flight, and $T_{2}$ – spin
relaxation time. The polarization noise power ${\cal N}(\nu)$ is defined as
${\cal N}(\nu)=\int K(t)\exp({\imath\nu t})dt$. As seen from Eq. (17), at
$T_{2},t_{T}\gg\omega_{1L}^{-1}$, the spectrum of the polarization noise power
of the atomic vapor always reveals peaks at $\nu=0,\omega_{1L},$ and
$2\omega_{1L}$. Angular dependence of amplitudes of these peaks at $F>1/2$ is
controlled by the last term in brackets and $\sim\cos^{2}2\theta$ for the peak
at $\nu=\omega_{1L}$ and $\sim\sin^{2}2\theta$ for the peak at
$\nu=2\omega_{1L}$.

Dependence Eq. (17) of the ellipticity noise spectra on azimuth $\theta$ of
the polarization plane of the probe beam qualitatively agrees with the
experimental data (see Fig. 2, which represents data obtained in our work
[30]), in particular, the amplitude of the peak at the frequency
$2\omega_{1L}$ reaches maximum at $\theta=\pi/4$ and vanishes at $\theta=0$
and $\pi/2$.

The following remark should be made regarding Fig. 2 which shows that at
$\theta=\pi/4$, the amplitude of the peak at the Larmor frequency
$\omega_{1L}$ exceeds the amplitude of the maximum at twice the Larmor
frequency $2\omega_{1L}$. This contradicts formula Eq. (17), according to
which, for $F>1/2$, the amplitude of the peak at the Larmor frequency
$\omega_{1L}$ should be less than that of the peak at doubled Larmor
frequency. We attribute this discrepance to nonlinear effects that took place
in experiments [30] which were carried out under conditions of resonant
probing of an atomic system with a laser beam. These effects require special
consideration, which we plan to present in our future publications.

Figure 2: Ellipticity noise spectra detected at angles $\theta=0^{\circ}$ (a),
$45^{\circ}$ (b) and $90^{\circ}$ (c) between the probe beam polarization
plane and magnetic field direction. The second harmonic of the Larmor
frequency is well pronounced at $\theta=45^{\circ}$ and vanishes at $\theta=0$
and $90^{\circ}$.

##  6 Conclusion

The signals observed in spin noise spectroscopy (SNS) are analyzed from the
view point of scattering theory. Expressions connecting SNS signals with
fluctuations of the optical susceptibility tensor are obtained. This tensor is
calculated in the framework of the simplest atomic model, and it is shown that
the fluctuations of its gyrotropic (antisymmetric) part are proportional to
the magnetization fluctuations and determine the Faraday rotation noise at
Larmor frequency observed in typical SNS experiments. However, the
polarization noise of the scattered optical field is not exhausted by the
noise of its Faraday rotation. It is shown, in this paper, that fluctuations
of the symmetric part of the optical susceptibility tensor (the alignment
noise) give rise to the ellipticity noise of the scattered field spectrally
localized at the double Larmor frequency. Using the developed ideas, we
interpret the experiment [30] where the ellipticity noise at a double Larmor
frequency was observed. Our results are in agreement with those obtained in
[30] by means of symmetry analysis of quantum correlation functions of the
system under study.

Note, in conclusion, that atomic systems showing strong and spectrally narrow
signals of spin noise can be considered as highly convenient model objects of
the spin noise spectroscopy. At the same time, theoretical description of the
spin noise formation, in these systems, especially under conditions of strong
resonant probing, proves to be rather complicated due to complex energy
structure of the dynamically broadened atomic states. The proposed theoretical
model provides the the most consistent basis for accurate description of the
effects of nonlinear spectroscopy of spin noise (in the presence of optical
pumping, optical Stark effect, optically induced spin dephasing, etc.).

The mechanism of the formation of noise polarimetric signals considered in our
article can possibly be applied not only to gas, but also to solid-state
systems, including nano-systems such as quantum wells in microcavities,
quantum wires, etc., noise signals in which were calculated using the
"magnetic language" [52, 53, 54]. In this case, a consistent calculation of
optical scattering can be important for describing diffraction effects and
effects associated with the internal motion of excitations in such systems. A
more detailed discussion of these effects is beyond the scope of this article.
Further development of this approach can make important contribution to
physics of the light-matter interaction.

## Acknowledgements

The authors are grateful to M. M. Glazov for useful discussions.

The work was supported by the RFBR Grant No. 19-52-12054 which is highly
appreciated. The authors acknowledge Saint-Petersburg State University for the
research Grant No. 51125686. Experimental part of the work was performed by A.
A. F. and M. Y. P. under support of the Russian Science Foundation (Grant No.
18-72-00078) and partially using equipment of the SPbU Resource Center
“Nanophotonics”.

## Disclosures

The authors declare no conflicts of interest.

## Appendix

We present here calculation of the function $\Phi_{i}^{-}$ [Eq. (6)]. Let us
introduce the auxiliary functions: ${\bf F}({\bf
r^{\prime}})\equiv\int_{S}dxdz\hskip 2.84526pt{\bf E}_{0}({\bf r})\Gamma({\bf
r-r^{\prime}})\sim\exp({-\imath\omega t})$ and ${\bf P}({\bf
r^{\prime}})\equiv\int_{S}dxdz\hskip 2.84526pt{\bf E}_{0}^{\ast}({\bf
r})\Gamma({\bf r-r^{\prime}})\sim\exp({\imath\omega t})$ (do not confuse with
polarization denoted by the same letter in the main text). Then function
$\Phi_{\alpha}({\bf r^{\prime}})$ Eq. (5) can be represented as
$\Phi_{\alpha}({\bf r^{\prime}})=-2\pi[F_{\alpha}({\bf
r^{\prime}})+P_{\alpha}({\bf r^{\prime}})]$ and therefore
$\Phi^{-}_{\alpha}({\bf r^{\prime}})e^{\imath\omega t}=-2\pi P_{\alpha}({\bf
r^{\prime}})$. To calculate the integral in the above expression for
$P_{\alpha}({\bf r^{\prime}})$, we note that the field ${\bf
E}_{0}^{\ast}({\bf r})$ satisfies the Helmholtz equation, $\Delta{\bf
E}_{0}^{\ast}+k^{2}{\bf E}_{0}^{\ast}=0$ and, therefore, the Kirchhoff formula
[55] can be used:

${\bf E}_{0}^{\ast}({\bf R})=\int_{\partial V}\bigg{[}{\bf E}_{0}^{\ast}({\bf
r}){\partial\Gamma({\bf r-R})\over\partial n}-\Gamma({\bf r-R}){\partial{\bf
E}_{0}^{\ast}({\bf r})\over\partial n}\bigg{]}dS$ (18)

Here, $\Gamma({\bf r})=-{\exp({\imath kr})/4\pi r}$ is the Green’s function of
Helmholtz operator, $\partial V$ is an arbitrary closed surface surrounding
the point $\bf R$, and the symbol $\partial/\partial n$ means the normal
derivative to the surface $\partial V$ [56]. For definiteness, we assume that
the photosensitive surface of photodetectors $S$ is larger than the ’cross
section’ of the probe Gaussian beam, and the photodetectors "intercept" almost
the entire flow of its energy (recall that $S$ lies in the plane $y=L\gg|{\bf
R}|$). In addition, we assume that the size of the surface $S$ (we denote this
size by the symbol $\sqrt{S}$) is much smaller than the distance $L$ from the
scattering particle to the photodetectors: $\sqrt{S}\ll L$. Now we choose the
surface $\partial V$ in the form of a cylinder with its axis parallel to the
axis $y$ and leaning on the right to the surface $S,(y=L)$ and on the left -
to the surface $S^{\prime}$ symmetric with respect to the origin, ($y=-L$)
(see Fig. 3).

Figure 3: See explanations in the text.

Since the probe beam is located inside the cylinder constructed in this way,
the integral in the Kirchhoff formula (18) can be calculated only over the
surfaces $S$ and $S^{\prime}$, because on the side surface of the cylinder the
field of the probe beam is negligible. Both surfaces are perpendicular to the
$y$ axis and have oppositely directed normal vectors. Therefore, on the
surface $S$ the normal derivative is calculated as $\partial/\partial
n\rightarrow\partial/\partial y$, and as $\partial/\partial
n\rightarrow-\partial/\partial y$ on the surface $S^{\prime}$. We calculate
the derivative of the Green’s function $\partial\Gamma({\bf r-R})/\partial n$
on the surface $S$ (that is, for $y=L$):

${\partial\Gamma({\bf r-R})\over\partial n}\bigg{|}_{S}={\partial\Gamma({\bf
r-R})\over\partial y}={\partial\Gamma({\bf r-R})\over\partial|{\bf
r-R}|}{\partial|{\bf r-R}|\over\partial y}=$ (19) $={\partial\Gamma({\bf
r-R})\over\partial|{\bf r-R}|}{y-R_{y}\over|{\bf r-R}|}=$ $=\Gamma({\bf
r-R})\bigg{[}\imath k-\overbrace{{1\over|{\bf r-R}|}}^{\approx L^{-1}\ll
k}\bigg{]}\overbrace{{L-R_{y}\over|{\bf r-R}|}}^{\approx 1}\approx\imath
k\Gamma({\bf r-R})$

The approximations used are justified, since $R,\sqrt{S}\ll L$ ($R\equiv|{\bf
R}|$) and therefore $|L-R_{y}|,|{\bf r-R}|\sim L$. This allows us to neglect
the second term in the expression in square brackets, since $|{\bf
r-R}|^{-1}\sim L^{-1}\ll k=2\pi/\lambda$. In addition, the factor
$[y-R_{y}]/|{\bf r-R}|$ for $r,y\gg R$ is practically equal to 1 on the
surface $S$. The derivative of the Green’s function $\partial\Gamma({\bf
r-R})/\partial n$ on the surface $S^{\prime}$ (i.e. for $y=-L$):

${\partial\Gamma({\bf r-R})\over\partial
n}\bigg{|}_{S^{\prime}}=-{\partial\Gamma({\bf r-R})\over\partial y}=$ (20)
$=-{\partial\Gamma({\bf r-R})\over\partial|{\bf r-R}|}{\partial|{\bf
r-R}|\over\partial y}=-{\partial\Gamma({\bf r-R})\over\partial|{\bf
r-R}|}{y-R_{y}\over|{\bf r-R}|}=$ $=-\Gamma({\bf r-R})\bigg{[}\imath
k-\overbrace{{1\over|{\bf r-R}|}}^{\approx L^{-1}\ll
k}\bigg{]}\overbrace{{-L-R_{y}\over|{\bf r-R}|}}^{\approx-1}\approx\imath
k\Gamma({\bf r-R})$

Thus, we obtain that on both surfaces $S$ and $S^{\prime}$ the following
relation holds

${\partial\Gamma({\bf r-R})\over\partial n}=\imath k\Gamma({\bf r-R})$ (21)

Let us turn now to calculation of the derivative ${\partial{\bf
E}_{0}^{\ast}({\bf r})/\partial n}$ entering Eq. (18). First, we calculate it
on the surface $S$, where $\partial/\partial n=\partial/\partial y$. We now
take into account that the probe beam field ${\bf E}_{0}$ is close to that of
the plane wave $\sim\exp({\imath ky})$ propagating along $y$-axis. For this
reason, we can write the following expression for the considered derivative

${\partial{\bf E}_{0}^{\ast}({\bf r})\over\partial
n}\bigg{|}_{S}={\partial{\bf E}_{0}^{\ast}({\bf r})\over\partial y}=-\imath
k{\bf E}_{0}^{\ast}({\bf r})$ (22)

A similar calculation of the derivative ${\partial{\bf E}_{0}^{\ast}({\bf
r})/\partial n}$ on the surface $S^{\prime}$ leads to the expression

${\partial{\bf E}_{0}^{\ast}({\bf r})\over\partial
n}\bigg{|}_{S^{\prime}}=-{\partial{\bf E}_{0}^{\ast}({\bf r})\over\partial
y}=\imath k{\bf E}_{0}^{\ast}({\bf r})$ (23)

Substituting the obtained relations into formula (18), we obtain

${\bf E}_{0}^{\ast}({\bf R})=\int_{S}\bigg{[}{\bf E}_{0}^{\ast}({\bf
r}){\partial\Gamma({\bf r-R})\over\partial n}-\Gamma({\bf r-R}){\partial{\bf
E}_{0}^{\ast}({\bf r})\over\partial n}\bigg{]}dS+$ (24)
$+\overbrace{\int_{S^{\prime}}\bigg{[}{\bf E}_{0}^{\ast}({\bf
r}){\partial\Gamma({\bf r-R})\over\partial n}-\Gamma({\bf r-R}){\partial{\bf
E}_{0}^{\ast}({\bf r})\over\partial n}\bigg{]}dS}^{=0}=$ $=2\imath
k\int_{S}{\bf E}_{0}^{\ast}({\bf r})\Gamma({\bf r-R})\hskip
2.84526ptdS=2\imath k\hskip 2.84526pt{\bf P}({\bf R})$

Taking into account that $\Phi^{-}_{\alpha}({\bf r^{\prime}})e^{\imath\omega
t}=-2\pi P_{\alpha}({\bf r^{\prime}})$, we have $E_{0\alpha}^{\ast}({\bf
R})=-{\imath k\over\pi}\Phi_{\alpha}^{-}({\bf R})\exp({\imath\omega t}).$
Bearing in mind that ${\bf E}_{0}=(A_{0x},0,A_{0z})\exp({-\imath\omega t})$,
we come to the result (6).

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