# An ab initio framework for understanding and controlling quantum
fluctuations
in highly multimoded light-matter systems

Shiekh Zia Uddin1,2,†, Nicholas Rivera1,3,†, Devin Seyler2, Yannick
Salamin1,2, Jamison Sloan2, Charles Roques-Carmes2,4, Shutao Xu5, Michelle Y.
Sander5,6,7, and Marin Soljačić1,2 $\dagger$ These authors contributed equally
to this work. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA
02139, USA.
2Research Laboratory of Electronics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA.
3Department of Physics, Harvard University, Cambridge, MA 02138, USA.
4 E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305,
USA.
5Department of Electrical and Computer Engineering and BU Photonics Center,
Boston University, Boston, MA 02215, USA.
6Division of Materials Science and Engineering, Boston University, Brookline,
MA 02446, USA.
7Department of Biomedical Engineering, Boston University, Boston, MA 02215,
USA.

###### Abstract

Quantum mechanics imposes fluctuations onto physical quantities, leading to
sources of noise absent in the classical world. For light, quantum
fluctuations limit many applications requiring high sensitivities,
resolutions, or bandwidths. In many cases, taming quantum fluctuations
requires dealing with highly multimode systems with both light and matter
degrees of freedom $-$ a regime which has traditionally eluded mechanistic
insights, and for which general rules are largely lacking. In this work, we
introduce and experimentally test a new theoretical framework for describing
quantum noise in multimode systems of light and matter, called quantum
sensitivity analysis. The framework leads to new general rules and mechanisms
for quantum noise propagation $-$ and accurately models all known quantum
noise phenomena in nonlinear optics. We develop experiments to test unexplored
aspects of our theory in the quantum noise dynamics of ultrafast multimode
systems. For example, in physical effects related to supercontinuum
generation, we observe and account for a proliferation of ultra low-noise
_pairs_ of wavelengths, despite that individual wavelengths are very noisy due
to strong nonlinear amplification of vacuum fluctuations. We then show that by
taking advantage of the spectral dynamics of quantum noise, it is possible to
generate quantum light states, such as squeezed states, even with very noisy
and complex light states $-$ by exploiting the spectral dynamics of vacuum
fluctuations undergoing nonlinearity and Raman scattering. Effects like these
can widely extend the range of sources that can be used for quantum metrology,
bringing quantum optics to higher powers and more complex sources. Broadly,
the framework we have developed is likely to enable many new approaches for
realizing diverse light sources across the entire electromagnetic spectrum,
whose performance approaches ultimate limits set by quantum mechanics.

Among the most important results in quantum mechanics is the Heisenberg
uncertainty principle, which enforces a minimum level of variation of physical
quantities. The electromagnetic (EM) field is subject to such an uncertainty
principle, having a minimum product of the fluctuations of the electric field
and vector potential. A major consequence is that even in the absence of any
photons (the “vacuum” state), the EM field cannot be exactly zero. These so-
called “vacuum fluctuations” account for fundamental effects in quantum
electrodynamics, such as spontaneous emission dirac1927quantum , Lamb shifts
lamb1947fine , as well as Casimir and van der Waals forces
casimir1948attraction .

These vacuum fluctuations have important consequences even for macroscopic
states of light such as laser light, which contains many photons in it. In the
macroscopic case, the light can be described as having quantum fluctuations
“on top” of an otherwise well-defined, oscillating EM field
haus2000electromagnetic . These fluctuations manifest as familiar phenomena
like intensity noise, phase noise, and temporal jitter (for pulses). Such
noises famously lead to limits in a wide variety of applications including
interferometry caves1981quantum ; barsotti2018squeezed , sensing
lawrie2019quantum ; degen2017quantum , biological imaging taylor2013biological
; taylor2016quantum ; casacio2021quantum , optical communication systems
yamamoto1986preparation ; rappaport2011state , optical coherence tomography
corwin2003fundamental ; rao2021shot , analog-to-digital conversion
kim2016ultralow , and many more. As a result, there has been an enduring
interest in understanding and controlling the fluctuations of light, and
understanding the fundamental noise limits imposed by quantum mechanics.

Understanding and controlling the noise of light is the subject of quantum
optics, a field which historically, was initiated by the study of the quantum
fluctuations of starlight and laser beams brown1956correlation ;
glauber1963quantum ; arecchi1965measurement . Much of the understanding and
intuition in quantum optics thus far has come from systems involving one or
few modes, such as continuous-wave beams and one or few modes of an
electromagnetic resonator. Recently, it has proved important to control the
noise of light in settings that deviate from these relatively clean few-mode
descriptions of light fabre2020modes . Recent experiments have brought into
focus the quantum noise and statistics of ultrafast pulses of light, such as
pulses in integrated nonlinear waveguides and parametric oscillators ra2020non
; nehra2022few , soliton microcombs in integrated photonics kues2019quantum ;
guidry2022quantum , supercontinuum light sources corwin2003fundamental , light
in random media peruzzo2010quantum ; lib2022quantum , and even more complex
settings. In all of the examples mentioned above, optical nonlinearities play
a major role. Going beyond optical frequencies, quantum optics questions are
now being explored with sources at extreme ultraviolet and X-ray wavelengths,
such as free-electron lasers saldin2010statistical ; gorobtsov2018seeded  and
coherent X-ray sources such as in high-harmonic generation gorlach2020quantum
; lewenstein2021generation , where many modes (e.g.,
temporal/spatial/polarization) play a role.

Currently, there is a lack of general rules for noise dynamics in systems with
many nonlinearly interacting degrees of freedom. In some special cases, such
as those involving solitons, or long pulses at the zero-dispersion wavelength,
or systems with a small number of degrees of freedom, it is possible to make
analytical statements (see as examples haus2000electromagnetic ;
bergman1991squeezing ; drummond1993quantum ; friberg1996observation ;
sizmann1999v ; levandovsky1999perturbation ; paschotta2004noise ;
herr2012universal ), but beyond these limited contexts, general laws of noise
are lacking, and so are constructive principles for engineering noise in
complex systems. There has been a resurgence of interest $-$ both
theoretically and experimentally $-$ in both the classical and quantum noise
properties of nonlinear multimode systems, in both the macroscopic
guidry2022quantum ; ng2023quantum ; guidry2023multimode  and mesoscopic
regimes yanagimoto2021efficient ; yanagimoto2022onset ; rivera2023creating ;
sloan2023driven . This new wave of interest in ultrafast quantum nonlinear
optics may lead to answers to a variety of unresolved questions. One major
class of questions concerns the role of more complex phenomena known in
nonlinear media, such as higher-order dispersion and Raman scattering, which
are important in the ultrafast regime, and whose effects have elicited
conflicting answers spaelter1998propagation ; werner1999raman ; sizmann1999v ;
corney2008simulations . Generally speaking, the role of material degrees of
freedom, be they phonons or even charges zhang2023fundamental , on realizing
quantum effects in light propagation, remain unclear. Other important
questions include the structure of multimode quantum correlations, and the
potential for squeezing and quantum optical engineering in settings where
complex effects become more important (at higher powers which enhance
nonlinear effects).

The lack of general-purpose rules parallels a similar situation in classical
nonlinear optics of strongly spatially multimode systems cristiani2022roadmap
; wright2022physics . There, a flurry of activity has gone into new frameworks
for making general statements on nonlinear multimode classical systems, such
as frameworks for identifying reduced models wright2020mechanisms , as well as
optical thermodynamics wu2019thermodynamic ; pourbeyram2022direct ;
marques2023observation . In general, much more is known about classical
dynamics in nonlinear multimode systems than the quantum noise dynamics of
those same systems.

Here, we address this knowledge gap with a new theoretical framework for
quantum noise dynamics in multimode nonlinear systems with strong interactions
between modes. Our framework allows predicting quantum noise effects in _any_
system where the classical dynamics are known, and allows the development of
previously unidentified, closed-form general rules for noise propagation in
nonlinear systems. To further illustrate the theory, we designed experiments
to test previously unexplored aspects of quantum noise. For example, in the
physics of strong frequency-conversion (closely related to supercontinuum
generation), we show a remarkable pair-interference that leads to a
proliferation of low-noise pairs of wavelengths, despite the fact that
individual frequencies of light are very noisy. The pairs can be 100 times
less noisy than the individual wavelengths, and are essentially quantum-
limited. In another class of experiments, we show how by taking advantage of
the finite spectral range of vacuum fluctuations, it is possible to generate
squeezed states of light $-$ states of light with lower fluctuations than
classical light states $-$ even with very noisy inputs. In particular, we show
that it is possible to generate 4 dB of directly measured squeezing with
amplified sources with over 10 dB excess noise, which can in principle widely
extend the range and application of quantum sources. In the Discussion
section, we outline some further areas where the framework can be immediately
and fruitfully applied.

##  I Connecting quantum noise to classical nonlinear light-matter dynamics

Figure 1: Quantum sensitivity analysis of light-matter interactions. (a) A
generic optical system can be specified as a set of linear and nonlinear
optical components with input and output ports. The inputs can be populated
with light (e.g., electromagnetic field modes of any type) or matter
excitations (e.g., two-level gain, phonons, electrons, etc.), or can be
“dark”, indicating vacuum field inputs. The inputs undergo interactions in a
nonlinear medium, as well as scattering due to linear optical components. (b)
The crux of the framework is to analyze the sensitivity of observables
$X(\\{\alpha,\alpha^{*}\\})$ to vacuum level shifts (quantum fluctuations) in
the input fields $\alpha(0),\alpha^{*}(0)$. Larger (smaller) gradients
translate to larger (smaller) noise in $X$. (c-f) Application of the framework
to ultrafast nonlinear pulse propagation. (c) Simulation of the classical
spectrum of a 185 fs (full width at half maximum) sech-pulse at 1550 nm
propagating through 10 meters of a single-mode fiber with anomalous group
velocity dispersion. (d) Dynamics of wavelength-dependent photon-number
fluctuations with increasing peak power, showing a high-noise and strongly
modulated signature near the pump wavelength, with a more diffuse peak at long
wavelengths due to Raman phonons in silica. (e) Wavelength-dependent noise
normalized to the shot-noise level: in general, pump wavelengths are highly
noisy, displaying strong nonlinear amplification of vacuum fluctuations at
higher powers (bottom). Raman-shifted wavelengths are far less noisy. (f)
Quantum sensitivity analysis in the two cases, showing that at higher powers,
each output wavelength gets increasingly influenced by vacuum fluctuations at
a wide range of input wavelengths, leading to higher noise. Simulation
parameters are in the SI.

We summarize the essential results of our theory of quantum noise in multimode
nonlinear systems. Detailed derivations, as well as additional examples, are
shown in the Supplementary Information (SI). Consider a general system with
both electromagnetic field and matter degrees of freedom (see Fig. 1a).
Electromagnetic degrees can refer to any type of solution to Maxwell’s
equation: plane waves modes, cavity modes, temporal modes, etc. Matter degrees
of freedom are also broadly defined: these can include a two-level transition
in a gain medium, phonons in a fiber or waveguide, or even electrons (see SI
for examples of gain and phonons). Defining the collection of independent
degrees of freedom (and their conjugates) as $\bm{\alpha},\bm{\alpha}^{*}$, we
define the classical theory of the system as an input-output relation,
relating the degrees of freedom at the final time
($\bm{\alpha}(t),\bm{\alpha}^{*}(t)$) to those at the initial time
($\bm{\alpha}(0),\bm{\alpha}^{*}(0)$). In particular, we write:

$\bm{\alpha}(t)=F(\bm{\alpha}(0),\bm{\alpha}^{*}(0)),$ (1)

where $F$ is some arbitrary function of the light and matter variables. In the
SI we present examples of this function related to models of linear
dissipation, laser amplifiers, as well as second- and third-order
nonlinearities.

In what follows, we consider the quantum noise dynamics of this system. We
consider the most common case in which the quantum fluctuations are small
compared to the mean values of the quantities, permitting an expansion of the
quantum dynamics in powers of the fluctuations $\delta a_{i},\delta
a_{i}^{\dagger}$. This is true in nearly all cases where nonlinearities are
triggered by a number of photons (or other excitations) which is much larger
than one. In this limit, we claim the following rule for determining the final
quantum state of the system: associate the classical degrees of freedom
$\alpha_{i},\alpha_{i}^{*}$ with bosonic annihilation and creation operators
$a_{i},a_{i}^{\dagger}$ 111This can also be done for most matter degrees of
freedom, in the limit where the number of matter particles is large in the
characteristic spatial scale of variation of the modes of interest. This is
the essence of what is today called macroscopic quantum electrodynamics. See
reviews such as Ref. rivera2020light  for more conceptual details and Ref.
scheel2008macroscopic  for more technical details.. Then, the output operators
$a_{i}(t),a_{i}^{\dagger}(t)$ can be written in terms of the input operators
$a_{i}(0),a_{i}^{\dagger}(0)$ as a Bogoliubov transformation. In particular:

$\displaystyle a_{i}(t)$
$\displaystyle=\alpha_{i}(t)+\sum\limits_{j}\mu_{ij}\delta
a_{j}(0)+\nu_{ij}\delta a_{j}^{\dagger}(0),\text{where}$
$\displaystyle\mu_{ij}$
$\displaystyle=\frac{\partial\alpha_{i}(t)}{\partial\alpha_{j}(0)},\text{and
}\nu_{ij}=\frac{\partial\alpha_{i}(t)}{\partial\alpha^{*}_{j}(0)}.$ (2)

The term $\alpha_{i}(t)$ is simply the classical solution. The Bogoliubov
coefficients $\mu_{ij},\nu_{ij}$ are the Jacobian matrix (or gradients) of the
classical dynamics: they are the sensitivities of the classical final state of
Eq. (1) to changes in the classical initial conditions. The final
_wavefunction_ can also be written down in this case, following the standard
rule quesada2022beyond .

###  I.1 The quantum optical law of total variance

We exploit the connection between quantum noise and classical dynamics to
present a general expression for the noise in _any_ system observable (e.g.,
intensity, phase, quadrature, correlation functions, etc.) in terms of
derivatives of classical equations of motion with respect to classical initial
conditions. Define an observable $X(t)=X(\bm{\alpha}(t),\bm{\alpha}^{*}(t))$.
The mean value is governed by the classical dynamics for some fixed initial
conditions, while the variance $(\Delta X(t))^{2}$ is governed by the
_derivative_ of the classical prediction with respect to the initial
conditions: $\partial X(t)/\partial\bm{\alpha}(0),\partial
X(t)/\partial\bm{\alpha}^{*}(0)$. In particular:

$\displaystyle(\Delta X(t))^{2}$ $\displaystyle=v^{T}Cv,\text{where}$
$\displaystyle v$ $\displaystyle=\left(\frac{\partial
X(t)}{\partial\bm{\alpha}(0)}~{}\frac{\partial
X(t)}{\partial\bm{\alpha}^{*}(0)}\right)^{T}$ $\displaystyle C$
$\displaystyle=\begin{pmatrix}\langle\delta\bm{\alpha}\delta\bm{\alpha}\rangle&\langle\delta\bm{\alpha}\delta\bm{\alpha}^{*}\rangle\\\
\langle\delta\bm{\alpha}^{*}\delta\bm{\alpha}\rangle&\langle\delta\bm{\alpha}^{*}\delta\bm{\alpha}^{*}\rangle\end{pmatrix}$
(3)

The correlation matrix $C$ contains the statistics of the initial state, which
allows straightforward inclusion of the effects of excess noise (thermal or
technical), multimode correlations (e.g., entanglement), and phase-sensitive
correlations (e.g., from squeezed states of light).

This rule, we call the _quantum optical law of total variance_ , because it
essentially prescribes a “quadrature-addition” rule for quantum noise $-$ in
parallel to the law of total variance in classical statistics. In classical
statistics, the variance of a sum of independent random variables adds in
quadrature. Here, we find the same rule for quantum noise, where the relevant
noises are the coefficients of $C$ and they are weighted by the sensitivities
$\partial X(t)/\partial\alpha(0),\partial X(t)/\partial\alpha^{*}(0)$, since
we are considering in general a nonlinear function of the quantum random
variables. The quadrature-addition is especially clear in the simple but
important case in which the initial fluctuations of all modes only have
uncorrelated vacuum noise (meaning each mode is in a coherent state). In that
case:

$(\Delta X(t))^{2}=(\partial
X(t)/\partial\bm{\alpha}(0))^{\dagger}\left(\frac{\partial
X(t)}{\partial\bm{\alpha}(0)}\right)=\sum_{k}\Big{|}\frac{\partial
X(t)}{\partial\alpha_{k}(0)}\Big{|}^{2}.$ (4)

It must be emphasized that the power of our framework is in specifying states
and observables in terms of _purely_ classical dynamics, which in basically
all cases are more well understood than the corresponding quantum noise
dynamics, and are solvable with existing numerical or analytical techniques.
In previous works on quantum noise in nonlinear photonics, the noise dynamics
in this regime are determined often by identifying the system Hamiltonian,
writing Heisenberg equations, expanding the equations to lowest order in
operator fluctuations, and solving the resulting equations. We want to
emphasize that two methods (the usual, and the one we present here) come to
the same results. In the SI, we show how to reproduce the results of many
textbook noise models of linear and nonlinear devices with our framework.
Moreover, our framework allows _efficient_ numerical computation of quantum
noise: namely, the sensitivities in Eq. (3) can be calculated by numerical
adjoint methods (e.g., molesky2018inverse ; hughes2018adjoint ;
maclaurin2015autograd ; chen2018neural ), taking advantage of automatic
differentiation methods to calculate the noise in an observable much faster
than black-box quantum noise numerical methods based on stochastic
differential equations 222Namely, when $X$ follows from a differential
equation (such as the nonlinear Schrodinger equation, $(\Delta X)^{2}$ can be
calculated with two differential equation solves, rather than solving over a
large ensemble of random initial conditions, solving enough times for
acceptable statistical convergence of observables).. This allows us to
calculate and model noise over a very wide range of parameters and physical
effects, allowing us to develop many new insights, which will be the subject
of the next sections.

##  II Quantum noise under ultrafast nonlinear pulse dynamics

As we will now show, the quantum optical law of total variance immediately
allows new insights into quantum noise dynamics in complex multimode systems,
and allows one to readily employ classical insights and design rules to
control quantum noise. Before getting into specific systems, let us mention
one general insight coming from Eq. (4): the role of connectivity in
determining noise. The noise of any quantity $X$, according to Eq. (4) is a
quadrature-sum of derivatives of $X$ with respect to initial conditions of all
possible modes. For a resonator, this may involve looking at initial
conditions of one or few modes. For a system like a nonlinear waveguide, one
needs to sum over all possible wavevectors, spatial, and polarization modes.
From Eq. (4), we can say that if $X$ is sensitive to many different initial
modes (which happens in more complex interactions, like supercontinuum
generation, or dynamics in multimode fibers), then one can expect the
fluctuations to be very high, and the quantity to be noisy. The higher the
connectivity, the noisier the outputs get.

We now give an example of this general expectation. In Fig. 1c-f, we use our
framework to predict, and provide new insights, into quantum noise dynamics in
perhaps one of the most widely-studied cases of third-order nonlinearity in
nonlinear optics: an ultrafast pulse propagating in a nonlinear fiber. The
classical spectrum is shown in Fig. 1c for the case of an initially launched
pulse with an energy above the fundamental soliton. The spectrum shows
characteristic dynamics such as soliton fission and Raman scattering as the
peak power of the initial pulse is increased, which are characteristic
features of the early stages of complex phenomena such as supercontinuum
generation dudley2010supercontinuum . The noise in the spectral intensity is
also shown in Fig. 1d as a function of wavelength and power, illustrating a
strong modulation near the wavelengths of pumping (1550 $\pm$ 5 nm), and a
smooth, broad feature corresponding to the “Raman soliton” (adopting similar
terminology as in dudley2010supercontinuum  and other works). In Fig. 1e, we
normalize these spectral noises to the shot-noise level (equal to the number
of photons in the corresponding spectral bin). Remarkably, despite considering
the initial pulse to be in a multi-mode coherent state, so that each
wavelength is at the shot noise level, the output wavelengths near the pump
are far-above the shot-noise level. Over a small span of powers, as a soliton
begins to break off from the incident pulse (due to Raman scattering), the
excess noise near the pump wavelengths reaches nearly two-orders of magnitude
above the shot-noise level (at higher powers, this excess noise factor stays
relatively steady). On the other hand, the longer wavelength Raman soliton is
not subject to this noise amplification, and maintains a fairly low noise
closer to the shot-noise level.

The quantum sensitivity analysis, shown in Fig. 1f, directly elucidates the
physical mechanism. The gradient of the spectral intensity with respect to
classical initial conditions, $|\partial
n(\lambda_{\text{out}})/\partial\alpha(\lambda_{\text{in}})|^{2}$, plotted in
Fig. 1f, governs the intensity noise. According to the quantum optical law of
total covariance in Eq. (4), the noise in $n(\lambda_{\text{out}})$ is given
by the quadrature sum along a row of the maps in Fig. 1f. For the higher-power
case (bottom), the pump wavelengths display a much larger sensitivity to
vacuum noise at a much wider range of wavelengths: the two sensitivity maps
have the same color scale. That said, the spectral intensity has not changed
much, leading to a pronounced increase in excess noise. That the spectrum does
not change much is an indication that increasing power to the fission point is
mostly changing the sensitivity of the fission to the initial conditions of
the pump. Moreover, we see at the Raman soliton wavelengths a far weaker (and
smoother) sensitivity to initial conditions of the pump (as well as
sensitivity to nearby wavelengths to the Raman soliton). In this framework,
the strong influence of input pump wavelength fluctuations on the output
spectrum near the pump wavelength (in other words, the strong connectivity)
can be used to account for the noise amplification.

Figure 2: Spectral decorrelation of vacuum fluctuations in nonlinear optics.
(a) Intensity noise as a function of wavelength (similar to Fig. 1d) for an
injected peak power of 2.5 kW. Blue represents noise for injected coherent
states (same as bottom curve in Fig. 1d). Red represents a case where 20 dB
phase-insensitive noise (e.g., from amplified spontaneous emission) is added
to the input at a narrow band of wavelengths around the pump wavelength
$\lambda_{1}$. Green represents a case where 20 dB is added to all wavelengths
(“global added noise”). For the global case, as expected, the output noise at
all wavelengths goes up by 20 dB. For the narrow-band case, a noise “kick”,
the effect is more complex in spectrum, and for some wavelengths
$\lambda^{*}$, there is no effect of added noise at the pump wavelength.
Meanwhile (simulation not shown here), if narrow-band noise is added far away
from the pump wavelength, at $\lambda_{2}$, then the output noise is only
enhanced in a small wavelength window around $\lambda_{2}$. Simulation
parameters are same as in Fig. 1. (b) Visual illustration of the simulation
results shown in (a).

From the discussion above, we expect the consequences of this intuition to go
beyond the parameters simulated in Figs. 1c-f. In particular, nonlinear
amplification of vacuum fluctuations is known to be important in the physics
of supercontinuum generation, where it is known that even with coherent-state
inputs, the output wavelengths are extremely noisy corwin2003fundamental ;
rao2021shot . This leads to fundamental noise limitations in supercontinuum
generation, which impacts various applications of supercontinuum in materials
characterization and the life sciences (e.g., optical coherence tomography
huang1991optical ; rao2021shot ). Understanding and controlling this noise
could lead to a variety of new applications. We argue that in many ways, the
phenomena observed in supercontinuum and those simulated here are of a common
origin, and that _any_ process of strongly cascaded frequency-spreading is
expected to display a large sensitivity to initial conditions and large noise
$-$ unless the system has fixed points or has weak memory of the initial
conditions. In the SI, we show analytically that this strong excess noise also
applies to pure self-phase modulation (Kerr nonlinearity with no dispersion
and no Raman), and grows rapidly with input power. Supercontinuum, being a
case of strongly cascaded four-wave-mixing and Raman processes, falls under
this umbrella $-$ and we argue that much of what we see here, and in the
experiments that follow, is common to supercontinuum, despite the lower
frequency-broadening.

Figure 3: High-resolution quantum noise and correlation mapping of light
undergoing strong nonlinear interaction. (a) Experiment to probe wavelength-
dependent quantum noise and correlations. An ultrafast laser passes through a
nonlinear medium (in this case, a single-mode optical fiber), undergoing
strong spectral broadening. The spectrum in (b) shows nearly 100 nm frequency
broadening, as well as evidence of Raman scattering and soliton fission,
signatures of the early stage of supercontinuum generation. The light after
this process is spectrally separated and, using an arbitrarily reconfigurable
filter, single wavelengths or pairs of wavelengths can be isolated, and their
noise can be measured. (b, right) Noise of individual wavelengths for
different incident powers. Notably, a part of the noise stays localized around
the pump wavelength, while another part continuously red-shifts, due to the
Raman effect (black dashed lines). (c) Noises of pairs of wavelengths, for
different input powers. Due to non-trivial anticorrelations between
wavelengths (right-most panel of (c)), pairs can have remarkably lower noises
than the individual wavelengths. In fact, despite the individual wavelengths
being measured to have over 20 dB excess noise, the pairs can have noises
within a few decibels the quantum shot-noise level (QNL). These pairs can have
over 20 dB less noise than expected for uncorrelated wavelengths (right). (d)
Mechanism of pair-interference, illustrated via quantum sensitivity analysis.
Pairs with low noise have opposite influence from vacuum fluctuations over a
remarkably broad band (star), while high-noise pairs have parallel influence
over a broad band (hexagon), as indicated by the anti-aligned/aligned phases
of the red and blue curves, which show the sensitivities of the two
wavelengths $\lambda_{1},\lambda_{2}$ to changes in inputs. (Right) Simulated
pair noise for similar conditions to Fig. 1 (at 3.5 kW input power), roughly
corresponding to the experimental conditions, and showing very good agreement
with experiment (colored boxes show regions of pairs in correspondence between
theory and experiment in (c) and (d)). In (a): ND, neutral density filter;
$\lambda/2$, half-wave plate; PBS, polarizing beam splitter; G, grating; M,
mirror; DMD, digital micromirror device; L, lens; PD, photodiode; ESA,
electronic spectrum analyzer.

The quantum sensitivity analysis also suggests new questions and readily
reveals new effects. For example, in Fig. 2, we consider the question: how
does the wavelength-dependent output noise respond to added noise at the
input? Fig. 2a shows the result of a local versus global addition of noise on
the spectral noise dynamics (a noise “kick”). For example, adding 20 dB noise
to all wavelengths (as a broadband phase-insensitive optical amplifier might
do) leads to 20 dB extra noise relative to the case of initial coherent
states, as expected. On the other-hand, for a localized noise kick (here of 10
nm bandwidth, e.g., from a narrow-band amplifier similar to an erbium-doped
fiber amplifier), output wavelengths near $\lambda_{1}$ get 20 dB added noise.
But the Raman-shifted wavelengths receive much less noise (and we find in
simulations that an even narrower-band noise of the same strength would lead
to even less noise at the Raman wavelengths). Even more strikingly, some
wavelengths $\lambda^{*}$ just away from the added noise experience almost no
noise-enhancement, despite being the at the peak of the noise distribution for
the coherent state case (blue curve). In the framework of quantum sensitivity
analysis, this is because $|\partial
n(\lambda^{*})/\partial\alpha(\lambda_{1})|^{2}=0$. This shows a case where
the spectral dynamics of noise can cause large amounts of added noise to have
a weak influence. The results of these simulations are also schematically
summarized in Fig. 2b.

##  III Proliferation of shot-noise limited pairs of wavelengths

In what follows, we present experiments to test the theoretical framework
developed here, showing that the amplification of vacuum fluctuations typical
of individual wavelengths can break down strongly for _pairs_ of wavelengths.
Not only this, but the resulting noise of these pairs can be at the quantum
shot-noise level. This is a result of strong and complex spectral correlations
between pairs of wavelengths. Such correlations have been of interest in both
the first spalter1998observation  and second-wave of interest in ultrafast
quantum noise dynamics wetzel2012real ; guidry2022quantum , because they
provide information on the potential for entanglement and squeezing of the
output light (this will be discussed further in Fig. 4). They also act as a
spectroscopic signature of quantum noise dynamics. At the same time, such
correlations have proven elusive to fully characterize (as well as
understand), due to a need for quantum noise measurements which not only are
quantum-limited, but also are high-resolution in spectrum $-$ because of a
strong fine structure characteristic of these correlations. We have built
high-resolution, quantum-noise-limited experiments to probe these
correlations. Quantum sensitivity analysis directly equips us to understand
them.

In Fig. 3a we show an experiment to probe spectral noise and correlation
dynamics after nonlinear broadening in a single-mode fiber (further
experimental details provided in SI). Light from a pulsed laser passes through
10 meters of a single-mode polarization-maintaining fiber (details of fiber in
SI), providing nonlinearity. Afterwards, it is passed through a grating-mirror
pair, demultiplexing the wavelengths, and is then sent to a digital
micromirror device, which acts as an arbitrarily programmable spectral filter,
allowing us to pass one, two, or any subset of wavelengths of the outgoing
pulse into the detector. The classical and noise spectrum dynamics in this
experiment as a function of incident power are shown in Fig. 3b 333The power-
dependence acts as a proxy for the dependence on propagation distance, hence
we refer to the power-dependent evolution as “dynamics”.. The spectra show
soliton fission and Raman signatures (as well as other complex spectral
signatures attributed to the initial pulse spectrum, which has a background in
addition to the soliton). The noise dynamics, similar to simulations, show a
lingering feature at the pump wavelength, followed by a red-shifting feature
due to Raman. In Fig. 3c we present intensity noises of _pairs_ of
wavelengths. Over very small changes in power, we see large and non-monotonic
changes in the measured pair-noises, revealing a strong “fine-structure” in
the noise dynamics. For the highest power we consider (100 mW average power),
we see a “cascade” of wavelength scales in the correlations: the Raman-soliton
region has smooth variations with respect to the pairs of wavelengths, while
near the pump-wavelength, the features become extremely fine, and start to
fall outside of the resolution of our correlation measurements. Intriguingly,
the Raman-shifted soliton still displays non-trivial correlation with
frequencies near the pump, indicating persistent correlations between the pump
and Raman wavelengths, despite the complex photon scattering and generation
processes at play in the classical spectrum of Fig. 2b. This is also what
would be expected from the sensitivities in Fig. 1f (bottom).

Strikingly, we see a variety of pairs of wavelengths (marked with white
circles), where the noise is extremely low, despite that the noises of the
individual wavelengths are well-above the shot noise level. Although it is not
clear on a linear scale, the magnitudes are in some cases below -155 dBm/Hz.
The shot noise level is determined from attenuation measurements to be -160
dBm/Hz at 1 mW optical power. In fact, these points in white-circles are
within a few decibels of the shot-noise limit, indicating that they lie about
20 dB below the noise of the individual wavelengths, which carry lower
intensity. The mechanism of this “pair-noise-interference” is explained in (d)
using quantum sensitivity analysis. The quantum optical law of total variance,
applied to the pair noises, reads:

$\Delta(n_{\lambda}+n_{\lambda^{\prime}})^{2}=\sum_{\lambda^{\prime\prime}}\left[\Big{|}\frac{\partial
n_{\lambda}}{\partial\alpha_{\lambda^{\prime\prime}}}\Big{|}^{2}+\Big{|}\frac{\partial
n_{\lambda^{\prime}}}{\partial\alpha_{\lambda^{\prime\prime}}}\Big{|}^{2}+2\text{Re}\frac{\partial
n_{\lambda}}{\partial\alpha_{\lambda^{\prime\prime}}}\frac{\partial
n_{\lambda^{\prime}}}{\partial\alpha^{*}_{\lambda^{\prime\prime}}}\right].$
(5)

We write the integral over continuous frequencies schematically as a sum over
wavelengths to more intuitively illustrate the point (integral expressions are
given in the SI). This formula indicates that these points of strong pair-
interference require $n_{\lambda}$ and $n_{\lambda^{\prime}}$ to be
_oppositely_ impacted by vacuum fluctuations at wavelength
$\lambda^{\prime\prime}$, for _many_ $\lambda^{\prime\prime}$. It is not
enough for $n_{\lambda}$ and $n_{\lambda^{\prime}}$ to be oppositely impacted
by vacuum fluctuations from one special wavelength: this needs to hold for
many wavelengths, in order to counteract the strong single-wavelength noises
(the modulus-squared terms in the equation). In Fig. 3d, we show insets of the
quantum sensitivities for two pairs of wavelengths: one with very low noise
(star) and one with enhanced noise (hexagon). The quantum-sensitivity analysis
clearly affirms this criterion for pair interference. To understand this
condition, let us consider the fundamental dynamics leading to frequency-
conversion in third-order nonlinear systems. In the absence of Raman
scattering, the Kerr effect leads to two major types of four-wave mixing
processes: a “scattering-type” process, where one photon is created and one is
destroyed, and a “pair generation” process, where two photons are created or
destroyed. For the scattering-type process, a vacuum fluctuation seed leads to
one photon number going up and one going down. Meanwhile, the pair-generation
process leads to the two-photon numbers being impacted the same way by a seed
vacuum fluctuation.

Beyond the qualitative understanding coming from the quantum sensitivity
analysis, we also compare the numerical predictions of the pair noises to
experiment for the 100 mW case in (c). The predictions are in excellent
correspondence with the experiments, indicating that the noise rule we
introduce: “differentiation” of classical dynamics, accounts very well for the
complex correlations we observe in experiments.

Before moving to the next section, we point out that many of the conclusions
drawn here are largely expected to hold for other third-order nonlinear pulse-
propagation systems, including nonlinear waveguides and resonators in
integrated photonics, where Kerr nonlinearity, dispersion, and Raman
scattering co-exist. In fact, some features even seem to mirror recent work on
soliton microcombs in integrated resonators guidry2022quantum , indicating the
generality of our findings. Compared to prior work measuring multimode
correlations in fiber supercontinuum (e.g., wetzel2012real ; godin2013real ),
we are working in a regime near the shot-noise limit, and we do not attenuate
the pulses after the fiber, which leads to fundamental changes in the pair
noises, due to the addition of uncorrelated vacuum fluctuations. For example,
in the limit of large attenuation, the state approaches a multi-mode coherent
state, and the ratios plotted on the right-most panel of Fig. 3c would be 1.
Compared to seminal work on “quantum” multi-mode correlations in fibers
spalter1998observation , we push into the regime of strong spectral
broadenings, strong soliton fission, and strong Raman $-$ and we arrive at an
excellent agreement between theory and experiment. As a result, our
correlation measurements reveal a variety of features which were not
previously observed.

###  III.1 Squeezing of high-power noisy light states

Figure 4: Squeezing of noisy high-power light states by correlation
engineering. (a) Schematic. Light after nonlinear propagation has nontrivial
correlations imparted (colormap shows measured covariance matrix of output
light of 100 mW average power). By passing this light through a spectral
filter, low-power but positively-correlated parts of the spectrum can be
filtered out, leading to squeezing. (b) Histogram of photon-number noise for
40000 different filters realized by an arbitrarily reconfigurable spectral
filter. Intriguingly, the average closely follows what is expected from linear
loss, while some points in the histogram can be nearly 10 dB below. (c)
Measured noise relative to shot noise (QNL) for these filters (left) compared
to the theoretical framework introduced in this work (right), showing strong
agreement on major features. Surprisingly, up to 4 dB squeezing can be
realized despite the initial states having 10 dB excess noise. Colors of dots
visually illustrate level above (red) or (below) shot noise on the y-axis.

In this section, we show another consequence of the general framework:
squeezing of light with strong excess noise. Squeezing is a quantum-mechanical
effect in which the uncertainty in a certain observable is reduced relative to
some nominal classical value. For example, in quadrature squeezing (e.g., what
is used in gravitational-wave detection caves1981quantum ;
ligo2011gravitational ), the variance in the amplitude or phase quadrature
goes below the vacuum level of $1/2$. In photon-number squeezing, which has
applications in high-precision absorption measurements and optical
communications haus2000electromagnetic , the photon-number variance goes below
the shot-noise level. In either case, generating squeezed light proceeds by
sending some pump light through a second- or third-order nonlinear medium.
Typically, low-noise light is needed: otherwise, while the noise may be
reduced by the nonlinear medium, the noise does not go below the shot-noise
level. This places a substantial restriction on the types of lasers that can
be used: for example, lower-cost commercial amplified sources typically cannot
be used, due to quantum noise coming from amplified spontaneous emission of
the gain medium. If one were able to squeeze noisy sources, it would enable a
strong extension of both the light sources, and possible applications of
quantum effects such as squeezing.

In a second set of experiments, we exploit quantum sensitivity analysis to
demonstrates photon-number squeezing of light emerging from soliton fission
and Raman interactions, even though the initial light has a large excess noise
of 10 dB $-$ a regime that is largely difficult to achieve squeezing in. We
attribute the occurrence of squeezing to the Raman effect, which spectrally
decorrelates red-shifted photons from the noisy pump wavelengths, similar to
Fig. 2. In this sense, the mechanism for the squeezing we identify is a type
of optomechanically-assisted squeezing (distinct from the usual optomechanical
squeezing considered in optical cavities subject to radiation-pressure forces
aspelmeyer2014cavity ). In simulations, we find that when the Raman shift is
too small compared to the bandwidth of amplifier noise, squeezing disappears.
Meanwhile, we also find that when squeezing does happen, adding large noise at
certain wavelengths, which increases the level of excess noise, has not much
impact on the squeezing magnitude.

In Fig. 4a, we illustrate an experiment to prepare photon-number squeezed
light: by passing light through an arbitrarily reconfigurable spectral filter
(similar to Fig. 3a), light emerging on the other side can in some cases be
squeezed. This happens when the filter rejects frequencies which carry low
power, but are strongly positively-correlated with other frequencies. This
causes the variance to drop faster than the mean. Defining $T$ as the filter
(taken to be a vector of zeros and ones), the output photon noise is
$T^{T}CT$, where $C$ is the covariance matrix of the light, shown for our case
in Fig. 4a. Meanwhile, the output photon number is $T^{T}N$, where $N$ is the
initial spectrum.

Photon-number squeezing happens for filters such that $T^{T}CT<T^{T}N$. Such
spectral filtering was first observed in seminal papers such as
friberg1996observation , and then probed by several authors experimentally
spaelter1998propagation ; spalter1998observation ; sizmann1999v ;
hirosawa2005photon  and theoretically levandovsky1999perturbation ;
werner1999raman  for fundamental solitons. In general, such questions were
largely studied for clean cases of shot-noise limited inputs, involving
fundamental solitons, and weak frequency-broadening. Since the maximal
spectral filtering is a strong function of the multimode quantum correlations
that build up, and because theories of these quantum correlations have been
largely absent, there has been a lack of general statements about when
squeezing is possible, and where the limits may be. Moreover, in most cases,
theory could not be compared against experiment. For clean fundamental
solitons undergoing limited broadening, it was shown that squeezing could be
achieved by high-pass or low-pass filters, realized by a diffraction grating
and knife-edge. For the types of complex correlations we observe in our
experiment, it is largely unclear what the best filter is, also given the
noise level. Thus, in our experiments, we use the programmable filter of Fig.
3 to sample the space of possible filters. A histogram of the noise after
40000 filters is shown in Fig. 4b. Intriguingly, the average noise follows
exactly what is expected by linear loss with the same average transmission.
Since the noise transformation by linear loss depends on the initial noise
bachor2019guide , we can fit this to show that the incident light is 10 dB
above the shot-noise level. The noises relative to shot noise are shown in (c)
and have a minimum at 4 dB below the shot noise level, optimized for an
average filter transmission (integrated over all wavelengths) of about 10%. In
Fig. 4c (right), we compare this against the theoretical predictions, using as
the theoretical correlation matrix one similar to what we predicted in Fig.
3d. The general trends and squeezing magnitude are well-captured. Some
differences are present, which is also expected given that the correlations we
predict have some differences from experiment, and the squeezing dynamics are
a sensitive function of these correlations.

##  IV Discussion

To summarize, we have developed and experimentally validated a new framework,
quantum sensitivity analysis, for predicting the flow of quantum noise in
complex multimode systems. One of the major results of the framework, the
quantum-optical law of total variance, allows one to characterize the effect
of quantum noise in different degrees of freedom on the output, and allows one
to leverage classical calculations and understanding to predict the dynamics
of quantum noise, even in complex settings. This simple prescription not only
leads to new analytical insights, but also realizes strong agreement with
experiments $-$ in several cases shown here, enabling some of the first direct
comparisons of theory and experiment (e.g., in Figs. 3, 4). One of the major
effects: spectral decorrelation of vacuum fluctuations, leads to the
possibility of employing techniques such as squeezing on noisy intense pulses,
which could extend the range of sources (and therefore applications) for which
quantum optical engineering can be fruitfully used.

Let us briefly address some future directions which are natural with the
framework introduced here. One of the main areas where our framework yields
immediate insights is in determining the effect of changing the initial pump
state and statistics. For example, by biasing the system with a coherent seed
(similar to injection locking haus1984waves  or techniques recently applied to
parametric oscillators roques2023biasing ), one moves along the hypersurface
shown in Fig. 1b. By adding squeezed light, one introduces the possibility of
suppressing the effect of vacuum fluctuations. This is similar to how in
gravitational-wave detection, phase sensitivity can be improved by injecting
squeezed light into the empty arm of a Michelson interferometer, suppressing
the effect of vacuum fluctuations caves1981quantum . Now, one may start to ask
questions for the first time about optimal biasing and squeezed-light
injection into complex nonlinear systems (e.g., supercontinuum systems) to
minimize noise for various applications. Further, the gradient and adjoint-
based framework is a natural fit with optimization methods, enabling the
discovery of optimal pumps, and optimal _geometries_ for maximizing squeezing,
even under fabrication constraints. We expect this to be important in the
current endeavors in realizing integrated squeezed light sources.

Finally, the framework we developed here also allows the consideration of
material degrees of freedom, allowing application of this framework to
describe the noise dynamics of more complex sources, such as conventional
lasers, free-electron lasers, and systems which exploit strongly interacting
photonic quasiparticles (e.g., excitons and exciton-polaritons). In all such
cases, the question of the noise contribution from these matter degrees of
freedom, in quantum optical experiments, is still largely open for
exploration.

##  V Acknowledgements

We acknowledge useful discussions with Erich Ippen and Ido Kaminer. N.R.
acknowledges the support of a Junior Fellowship from the Harvard Society of
Fellows. Y. S. acknowledges support from the Swiss National Science Foundation
(SNSF) through the Early Postdoc Mobility Fellowship No. P2EZP2-188091. J.S.
acknowledges previous support of a Mathworks Fellowship, as well as previous
support from a National Defense Science and Engineering Graduate (NDSEG)
Fellowship (F-1730184536). This material is based upon work also supported in
part by the U. S. Army Research Office through the Institute for Soldier
Nanotechnologies at MIT, under Collaborative Agreement Number
W911NF-23-2-0121. We also acknowledge support of Parviz Tayebati.

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# Electron gases in toroidal shells: mode coupling and state functions

M. Encinosa<EMAIL_ADDRESS>J. Williamson
<EMAIL_ADDRESS>
###### Abstract

Eigenvalues and wave functions describing free electron gases in toroidal
shells are determined using a basis set expansion natural to the system
geometry. Couplings between azimuthal and poloidal modes are found to be
appreciable at lower values of ${m_{\phi}}$, modifying single particle density
distributions within the shells. Thermodynamic state functions are calculated
and contrasted with those of a solid toroidal volume.

###### keywords:

torus, toroidal shell, confined thermodynamics

##  1 Introduction

Nanoscale shape engineering has emerged as a vibrant subject of investigation
within several areas of physics [1, 2, 3]. As fabrication methods become
increasingly sophisticated, the geometry of nano-materials is no longer
synonymous with only their overall shapes, but is also now inclusive of
periodic corrugations [4, 5], ripples [6, 7] and twist distortions [8].

An intriguing question that arises for nano-systems with novel configurations
concerns how their properties are affected by curvature and confinement
mechanisms. Toroidal structures have been, and likely will continue to be,
subjects of study due to their potential applications as traps [9, 10] for
cold molecules, biosensors [11, 12], and qubits [13]. Additionally, torodial
structures provide laboratories for exploring local curvature[14, 15, 16] and
flux quantization effects. In principle a magnetic field through a conducting
ring, torus or any genus one object can induce persistent currents (Gauss-
Laguerre beams may serve as means of generating poloidal modes giving rise to
toroidal moments in extended objects [17]).

In [18], the Schrödinger equation (SE) for spinless charge carriers in a
toroidal volume (TV) was derived and solved using finite element methods on a
disk within the volume; azimuthal symmetry then allowed for the solution to be
known everywhere. In [19] solutions for the same system were found by reducing
the SE to single variable form with effective potentials accounting for the
coupling between radial, poloidal and azimuthual degrees of freedom. In this
work, the emphasis is shifted towards understanding the properties of an
electron gas confined within a toroidal shell (TS), anticipating circumstances
in which analytic expressions for wave functions may be of importance.

This paper is organized as follows: Section 2 presents a brief review of the
derivation of the TV SE and describes the basis set used in this work. Section
3 presents spectra and wave functions, and illustrates the rapidity with which
toroidal degrees of freedom begin to display substantial coupling. Eigenvalues
are employed to calculate the heat capacity, free energy and degeneracy
pressure within a shell and are compared to those calculated for a TV. Section
4 presents conclusions and some remarks regarding extensions of this work.

##  2 The Hamiltonian and basis set

The SE for a shell is the same as for a volume, however the solutions are
modified due to the different boundary conditions. To keep this work self-
contained, the derivation of the SE for a solid torus is given below.

A solid torus with major radius $R$ minor radius $a$, poloidal angle $\theta$,
and azimuthal angle $\phi$ is defined by the set of points
${\textbf{r}}(\rho,\theta,\phi)$ which satisfy

${\textbf{r}}(\rho,\theta,\phi)=W(\rho,\theta)\,\hat{\boldsymbol{\rho}}+\rho\,\sin\theta\,\hat{\textrm{{k}}},$
(1)

where

$W(\rho,\theta)=R+\rho\,\cos\theta$ (2)

The metric and Laplacian can be derived from Eq. (1) by standard methods [20].
However, results will be shown in section 3 that include magnetic field
effects, so it is preferable to first write the gradient. In terms of unit
vectors $\hat{{n}}$ (normal to a toroidal surface at fixed $\rho$),
$\hat{\theta}$ (tangent to the surface defined by constant $\rho$) and
$\hat{\phi}$ in the azimuthal direction, the gradient is

$\boldsymbol{\nabla}=\hat{\boldsymbol{\rm
n}}\frac{\partial}{\partial\rho}+\hat{\boldsymbol{\theta}}\frac{1}{\rho}\frac{\partial}{\partial\theta}+\hat{\boldsymbol{\phi}}\frac{1}{W(\rho,\theta)}\frac{\partial}{\partial\phi}$
(3)

which will be used in the time-independent Schrödinger equation (in the
Coulomb gauge)

$\bigg{(}\nabla^{2}+2i\frac{e}{\hbar}~{}\textbf{A}\cdot\boldsymbol{\nabla}-\frac{e^{2}}{\hbar^{2}}\textbf{A}^{2}+\frac{2m_{e}E}{\hbar^{2}}\bigg{)}\Psi=0,$
(4)

where $e$ and $m_{e}$ are the electron charge and mass. It proves convenient
to define the parameters

$\begin{split}u=&\,\frac{\rho}{R}\\\ \alpha=&\,\frac{a}{R}\\\
\gamma_{0}=&\,B_{0}\pi R^{2}\\\ \gamma_{\mbox{\tiny
N}}=&\,\frac{\pi\hbar}{e}\\\ \epsilon=&\,\frac{2m_{e}ER^{2}}{\hbar^{2}}\\\
\tau_{0}=&\,\frac{\gamma_{0}}{\gamma_{\mbox{\tiny N}}}\\\
W(u,\theta)=&\,1+u\cos\theta,\end{split}$

after which, expansion of Eq. (4) takes the dimensionless form

$\begin{split}\bigg{[}\frac{\partial^{2}}{\partial
u^{2}}+&\frac{1}{u}\frac{\partial}{\partial
u}+\frac{1}{u^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+\frac{\cos\theta}{W(u,\theta)}\frac{\partial}{\partial
u}+\frac{1}{W^{2}(u,\theta)}\frac{\partial^{2}}{\partial\phi^{2}}\\\
&-\frac{\sin\theta}{uW(u,\theta)}\frac{\partial}{\partial\theta}-\frac{\tau_{0}}{i}\frac{\partial}{\partial\phi}-\frac{1}{4}\tau^{2}_{0}W^{2}(u,\theta)+\epsilon\bigg{]}\psi=0\end{split}$
(5)

The three leading terms in Eq. (5) comprise part of Bessel’s equation. The
$\theta\rightarrow-\theta$ invariance of Eq. (5) allow for its solutions to be
separated into even $\rm cos[n\theta]$ and odd $\rm sin[n\theta]$ parities,
providing a natural choice for a basis expansion in terms of Bessel functions
of the first kind,

$\chi{{}^{\pm}_{\bar{n}\bar{\nu}m_{\phi}}}(u,\theta)=\sum_{n\nu}\frac{1}{b_{n\nu}t_{n}}C{{}^{n\nu}_{\bar{n}\bar{\nu}m_{\phi}}}J_{n}\biggl{(}\frac{x_{n\nu}u}{\alpha}\biggr{)}\begin{pmatrix}\cos
n\theta\\\ \sin n\theta\end{pmatrix}$ (6)

with $b_{n\nu}$ and $t_{n}$ Bessel function and trigonometric normalizations.

The first order derivative in $\theta$ term can be eliminated with an
integrating factor. The basis set elements then have the form

$\psi^{\pm}_{\bar{n}\bar{\nu}m_{\phi}}(u,\theta,\phi)=\frac{1}{\sqrt{W(u,\theta)}}\chi^{\pm}_{\bar{n}\bar{\nu}m_{\phi}}(u,\theta)\exp[im_{\phi}\phi]$
(7)

Only one term in Eq. (5) is not separable into a product form in $u$ and
$\theta$. In [19] the $\theta$-dependence of this term was integrated out by
contour methods to yield effective potentials in the radial variable defined
here as $\Lambda_{\bar{n}n}^{\pm}(u)$. For a given ${m_{\phi}}$ the matrix
elements reduce to

$\begin{split}H_{\bar{n}\bar{\nu},n\nu}=&-\bigg{[}\biggl{(}\frac{x_{n\nu}}{\alpha}\biggr{)}^{2}+\tau_{0}m_{\phi}+\frac{\tau_{0}^{2}}{4}\bigg{]}\delta_{\bar{n}n}\delta_{\bar{\nu}\nu}\\\
&-\bra{\bar{n}\bar{\nu}}(\frac{1}{4}-m_{\phi}^{2})\Lambda_{\bar{n}n}^{\pm}(u)+\frac{1}{2}\tau^{2}_{0}uf^{\pm}_{1}(n,\bar{n})+\frac{1}{8}\tau^{2}_{0}u^{2}f^{\pm}_{2}(n,\bar{n})\ket{{n\nu}}\end{split}$
(8)

where

$f^{\pm}_{k}(\bar{n},n)\equiv\int_{0}^{2\pi}d\theta\begin{pmatrix}\cos\bar{n}\theta\cos
k\theta\cos n\theta\\\ \sin\bar{n}\theta\sin k\theta\sin n\theta\end{pmatrix}$
(9)

reduces to a sum over Kronecker delta functions in $\bar{n},k$, and $n$.
Explicit forms for $\Lambda_{\bar{n}n}^{\pm}(u)$ are presented in the
appendix.

The outer wall of the torus occurs at $u=\alpha$. To create a shell, define
$0<s\leq 1$ as the parameter that sets the position of its interior wall.
Excluding the origin $u=0$ requires

$J_{n}\bigg{(}\frac{x_{n\nu}u}{\alpha}\bigg{)}\rightarrow
A_{n\nu}J_{n}\bigg{(}\frac{\eta_{n\nu}u}{\alpha}\bigg{)}+B_{n\nu}Y_{n}\bigg{(}\frac{\eta_{n\nu}u}{\alpha}\bigg{)}\equiv
T_{n}\bigg{(}\frac{\eta_{n\nu}u}{\alpha}\bigg{)}$ (10)

with $Y_{n}$ a Bessel function of the second kind. It is then a standard
exercise to obtain the $\eta_{n\nu}$ appearing in Eq. (10) from (suppressing
indices for clarity)

$AJ_{n}(s\alpha)+BY_{n}(s\alpha)=0$ (11) $AJ_{n}(\alpha)+BY_{n}(\alpha)=0$
(12)

The ratio $B/A$ of irregular to regular solutions can be obtained for a given
$\alpha$ satisfying Eqs. (11) and (12).

In this work, 100 basis functions were employed to find eigenvalues and wave
functions for both positive and negative parities. There are no terms Eq. (5)
that mix parity states. The positive state basis set comprises functions with
$n=(0,1,2,3,4)$ coupled to the first twenty associated zeros
$\nu=(1,2,...,20)$ found by solving Eqs. (11) and (12). The negative parity
states have $n=(1,2,3,4,5)$. In accordance with and to facilitate comparison
with previous work [18, 19], a GaAs effective mass $m=.067m_{e}$ is assumed.

##  3 Results

In Table 1, representative zeros of $T_{n}$ for aspect ratio $\alpha=0.5$ and
$s=0.25$, $s=0.5$, $s=0.75$ are shown next to multiples of $\pi$. Inclusion of
the irregular Bessel function leads to the $\eta_{n\nu}$ being reasonably
approximated with a hard wall form $\sim n\pi/(1-s)$ even for an thicker shell
($s=0.25$). This observation may prove useful for developing approximations
that circumvent finding roots of the $T_{n}$. Table 2 illustrates $B_{n\nu}$
coefficients for the Table 1 basis states normalized to $A_{n\nu}=1$. Patterns
that emerge among the $B_{n\nu}$ cluster into the same $n$-values of the
$\eta_{n\nu}$ and are only useful as a rough consistency check.

Tables 3 and 4 contain examples of eigenstates generated from solving Eq. (5)
in terms of the basis at a given $m_{\phi}$ for the full ($s=0$) torus. Two
representative states were selected to illustrate the general behavior
observed within the substantially larger set of results. It was expected the
lowest energy eigenstates would be substantially altered as $m_{\phi}$
increased, but as evidenced in Table 4, this was generally not the case.
Tables 5 and 6 are results for an $s=0.75$ shell. The interplay of confinement
and mode coupling within a region corresponding to a thin film of thickness
12.5 nm is substantial, and appears to be a consequence of the
$\Lambda_{\bar{n}n}(u)$ potential assuming its maximum values (of order unity)
near $u\sim\alpha$. In contrast, for the full torus, the $J_{n}$ basis states
overlap near $\rho\sim 0$ where $\Lambda_{\bar{n}n}(u)$ is smaller and are
oscillatory over $0\leq s\leq\alpha$.

Figures 1-4 show volume and shell density plots for the $n\nu m_{\phi}=010$
and $n\nu m_{\phi}=01(10)$ states indicated in Tables 3 and 5 (in the interest
of conciseness, only positive parity results are presented). Figures 5-8
present the same for $n\nu=33$ states. A cross-sectional plot in general will
not exhibit symmetry about $\theta=\pi/2$; the larger radius of curvature at
the inner rim leads to a left-right asymmetry for $|\psi{{}^{+}_{n\nu
m_{\phi}}}|^{2}$ . The results indicate increasing $m_{\phi}$ shifts densities
rightward towards $\theta=0$ mimicking a centrifugal force whose effect is
more pronounced in shells than for volumes.

The results above indicate that calculation of thermodynamic quantities
requires a more comprehensive determination of the nano-scale torus spectrum
than the method performed in [19]. Additionally, the extent of mode coupling
on the spectrum in that work was underestimated. Here, a sufficient number of
eigenvalues, numbering in the several hundreds, are computed to insure
convergence of the partition function

$Z(\tau)=\sum_{N}\exp[-{E_{N}}/{\tau}]$ (13)

up to 100 K. Results for only one shell with $s=0.75$ are given here, leaving
a more detailed study of the interplay between aspect ratio and shell
thickness for more extensive future analysis. Further, in anticipation of
applying the methods developed here to the case of the toroidal trap described
in [9], the dimensions of that trap $R=875\ {\rm\AA},a=250\ {\rm\AA}$ are
adopted in what follows.

The effect of confinement to a shell is apparent in the results for
thermodynamic variables. In Figure 9 the specific heat determined from

$C_{V}(\tau)=-\frac{\partial U}{\partial\tau}$ (14)

indicates $C_{V}^{sh}>C_{V}^{vol}$ for $\tau<{\rm 5\ K}$; this is a reflection
of the energy scale of the shell being much larger than of the volume, as
activation of the ground state for small but increasing $\tau$ yields a
pronounced effect. The plot shows that both geometries converge to their
assumed limits at approximately $30{\rm\ K}$ with crossover at $5\ {\rm K}$.
In Figures 10 and 11, free energies $F(\tau)=-\tau{\rm Ln}[Z(\tau)]$ are shown
to contrast the scaling that arises from confinement. The degeneracy pressure
will also depend on the volume but is not immediately accessible from

$p(\tau)=-\tau\frac{\partial F(\tau)}{\partial V}$ (15)

To arrive at a value for $p(\tau)$, first express the volume derivative as

$\frac{\partial}{\partial
V}=\frac{1}{2\pi^{2}R^{3}\alpha}\frac{\partial}{\partial\alpha}$ (16)

for the full torus and

$\frac{\partial}{\partial
V}=\frac{1}{2\pi^{2}R^{3}(1-s)^{2}\alpha}\frac{\partial}{\partial\alpha}$ (17)

for the shell. With $R$ held fixed, five $a$ values corresponding to a
$\Delta\alpha=0.05$ were taken (in Angstroms,
$a=(162.49,206.24,250.00,293.75,337.49$) for $\tau=1-5{\rm\ K}$. A five point
derivative formula and interpolating polynomial generated by Mathematica as a
consistency check were used to evaluate the derivative at $250{\rm\ \AA}$
resulting in pressure in the interval $0.01{\rm\ K}\leq\tau\leq 5{\rm\ K}$,

$p_{v}(\tau)=5.57(10^{-8}){\rm\ K}/{\rm\AA^{3}},$ (18)
$p_{s}(\tau)=1.27(10^{-7}){\rm\ K}{\rm/\AA^{3}}$ (19)

Finally, the effect of a magnetic field on $C_{V}$ was not intended to be a
primary focus of this work, but it bears noting that results are easily
obtainable. Figures 12 and 13 demonstrate the sensitivity of the specific heat
to moderate fields of the form $\bf B(r)$ = $B_{0}\hat{{k}}$ at low
temperatures.

##  4 Conclusions

The goal of this work in its preliminary stages was focused on the
thermodynamics of electron gases in toroidal shells. It became apparent that
more basis states and eigenvalues would be necessary than the number employed
in [19] for a volume. The first half of this paper provided the methods by
which those eigenvalues and wave functions were computed and indicates the
spectrum strongly depends on $m_{\phi}$ in a way not captured by a simple
approximation, thus requiring a reanalysis of earlier results. Once the wave
functions were acquired, density plots for representative states provided
visual insight into an effective centrifugal force causing shifts from the
inner rim of the torus towards its outer rim for moderate values of
$m_{\phi}$.

With the eigenvalues acquired in sufficient number and accuracy, the latter
half of this paper focused on thermodynamic state functions with convergence
questions within the temperature ranges considered here moot. $F(\tau)$ and
$p(\tau)$ evidenced unsurprising behavior, however, the $C_{V}(\tau)$ plots
are indicative of how confinement affects the magnitude and spacing of the
spectrum and consequently, the specific heat.

There are three length scales associated with a toroidal shell after re-
scaling by $R$; $1,\alpha$, and $s\alpha$. For both volume and shell, the
azimuthal energy scales as 1; however, the volume radial/poloidal energies
scale as $1/\alpha^{2}$ while for thin shells the radial energy scales as
$1/\alpha^{2}(1-s)^{2}$, and the nearly decoupled poloidal energy as
$1/\alpha^{2}$. Upon examining the eigenvalues it becomes clear for the volume
case fewer $m_{\phi}$ are needed to match and surpass contributions made by
the Bessel (radial/poloidal) $x_{n\nu}$ at a given temperature. For shells,
however, fewer radial and poloidal contributing terms are required, but many
more $m_{\phi}$ before there are roughly equivalent contributions to the
particular energy of a state; this appears to be the origin of the $C_{V}$
behavior shown in Figure 9.

There are several directions for further investigation suggested by this work.
As stated previously, conciseness motivated inclusion of only one shell
thickness, chosen to approximate values of recently fabricated thin films
[21], but a systematic study of aspect ratio and its interplay with shell
thickness would give a measure of which variable is dominant in fixing the
properties of a structure for given $R,a$ and $s$. Additionally, while proof
of principle results were presented here for the influence of a magnetic
field, interesting results may emerge again depending on the values of $R,a$,
$s$ with a more thorough treatment, particularly if vacuum in the region
$0\leq u\leq s\alpha$ is replaced with a magnetic or conducting material.
Finally, a description of gold toroidal traps by the methods presented here
was thought to be prohibitive given the magnitude of eigenvalues required to
reach the Fermi surface. However, results related to this work suggest that
while a simple approximation method will not accurately represent the spectra,
there may be sufficient predictability within it to render another type of
approximation feasible, making calculating persistent current and other low-
temperature phenomena for metals with Fermi levels $\sim 5\ e\rm V$ possible.

$s$ | $.25$ | $.5$ | $.75$
---|---|---|---
$\eta_{n\nu}$ | Comp. | Approx. | Comp. | Approx. | Comp. | Approx.
$\eta_{01}$ | 4.098 | 4.189 | 6.246 | 6.283 | 12.553 | 12.566
$\eta_{12}$ | 8.536 | 8.378 | 12.625 | 12.666 | 25.153 | 25.132
$\eta_{23}$ | 13.121 | 12.566 | 19.045 | 18.850 | 37.765 | 37.699
$\eta_{34}$ | 17.747 | 16.755 | 25.477 | 25.133 | 50.381 | 50.265
$\eta_{45}$ | 22.389 | 20.944 | 31.912 | 31.416 | 62.999 | 62.831

Table 1: Extended Bessel roots computed from Eqs. (11) and (12) compared to an
approximate value for a hard wall of thickness 1-s.

$s$ | $.25$ | $.5$ | $.75$
---|---|---|---
$\psi_{n\nu}$ | $B_{n\nu}$ | $B_{n\nu}$ | $B_{n\nu}$
$\psi_{01}$ | -7.045 | 0.892 | 0.955
$\psi_{12}$ | 16.991 | -0.838 | -0.933
$\psi_{23}$ | 11.395 | 1.871 | 1.263
$\psi_{34}$ | 14.130 | -0.276 | -0.659
$\psi_{45}$ | 28.219 | 23.6908 | 1.859

Table 2: Irregular Bessel function coefficient $B_{n\nu}$ indicated in Eq.
(10) for $s=0.25,0.5$ and $0.75$ with $A_{n\nu}=1$

.

$m_{\phi}$ |  $|\hskip 30.00005pt\psi{{}^{+}_{01m_{\phi}}}(u,\theta)$, $\alpha=0.5,\hskip 20.00003pts=0$
---|---
$0$ | $|-1.00\chi{{}^{+}_{010}}(u,\theta)$
$5$ | $|\ \ \ \ 0.974\chi{{}^{+}_{015}}(u,\theta)+0.222\chi{{}^{+}_{225}}(u,\theta)$
$10$ | $|-0.841\chi{{}^{+}_{015}}(u,\theta)-0.527\chi{{}^{+}_{225}}(u,\theta)$

Table 3: Full torus ground state from Eq. (5) for increasing $m_{\phi}$
neglecting terms of order $\sim 10^{-2}$.

$m_{\phi}$ |  $|\hskip 30.00005pt\psi{{}^{+}_{33m_{\phi}}}(u,\theta)$, $\alpha=0.5,\hskip 20.00003pts=0$
---|---
$0$ | $|\ \ \ \ 1.00\chi{{}^{+}_{330}}(u,\theta)$
$5$ | $|\ \ \ \ 0.994\chi{{}^{+}_{335}}(u,\theta)-0.256\chi{{}^{+}_{145}}(u,\theta)+0.055\chi{{}^{+}_{435}}(u,\theta)$
$10$ | $|-0.917\chi{{}^{+}_{335}}(u,\theta)-0.527\chi{{}^{+}_{145}}(u,\theta)-0.209\chi{{}^{+}_{435}}(u,\theta)$

Table 4: The (33) state evolution as $m_{\phi}$ is increased.

$m_{\phi}$ |  $|\hskip 30.00005pt\psi{{}^{+}_{01m_{\phi}}}(u,\theta)$, $\alpha=0.5,\hskip 20.00003pts=0.75$
---|---
$0$ | $|-1.00\chi{{}^{+}_{010}}(u,\theta)$
$5$ | $|\ \ \ \ 0.743\chi{{}^{+}_{015}}(u,\theta)+0.654\chi{{}^{+}_{115}}(u,\theta)-0.141\chi{{}^{+}_{215}}(u,\theta)$
$10$ | $|\ \ \ \ 0.639\chi{{}^{+}_{015}}(u,\theta)+0.697\chi{{}^{+}_{115}}(u,\theta)-0.315\chi{{}^{+}_{215}}(u,\theta)$

Table 5: Shell s = 0.75 ground state from Eq. (5) for increasing $m_{\phi}$
neglecting terms of order $\sim 10^{-2}$.

$m_{\phi}$ |  $|\hskip 30.00005pt\psi{{}^{+}_{33m_{\phi}}}(u,\theta)$, $\alpha=0.5,\hskip 20.00003pts=0.75$
---|---
$0$ | $|-1.00\chi{{}^{+}_{330}}(u,\theta)$
$5$ | $|-0.232\chi{{}^{+}_{035}}(u,\theta)+0.366\chi{{}^{+}_{135}}(u,\theta)+0.481\chi{{}^{+}_{235}}(u,\theta)-0.641\chi{{}^{+}_{435}}(u,\theta)+0.411\chi{{}^{+}_{435}}(u,\theta)$
$10$ | $|\ \ \ 0.319\chi{{}^{+}_{03(10)}}(u,\theta)-0.463\chi{{}^{+}_{13(10)}}(u,\theta)-0.489\chi{{}^{+}_{23(10)}}(u,\theta)+0.502\chi{{}^{+}_{43(10)}}(u,\theta)+0.438\chi{{}^{+}_{43(10)}}(u,\theta)$

Table 6: Shell s = 0.75 ground state from Eq. (5) for increasing $m_{\phi}$
neglecting terms of order $\sim 10^{-2}$. Figure 1:
$|\psi{{}^{+}_{010}(u,\theta)}|^{2}$ for the full torus with $\alpha=0.5$.
Figure 2: $|\psi{{}^{+}_{01(10)}(u,\theta)}|^{2}$ for the full torus with
$\alpha=0.5$. Figure 3: $|\psi{{}^{+}_{010}(u,\theta)}|^{2}$ for shell
$s=0.75$, $\alpha=0.5$. Figure 4: $|\psi{{}^{+}_{01(10)}(u,\theta)}|^{2}$ for
shell $s=0.75$, $\alpha=0.5$. Figure 5: $|\psi{{}^{+}_{330}(u,\theta)}|^{2}$
for the full torus, $\alpha=0.5$. Figure 6:
$|\psi{{}^{+}_{33(10)}(u,\theta)}|^{2}$ for the full torus, $\alpha=0.5$.
Figure 7: $|\psi{{}^{+}_{330}(u,\theta)}|^{2}$ for shell $s=0.75$,
$\alpha=0.5$. Figure 8: $|\psi{{}^{+}_{33(10)}(u,\theta)}|^{2}$ for shell
$s=0.75$, $\alpha=0.5$. Figure 9: $C_{V}(\tau)$ as a function of temperature
in Kelvin for $\alpha=0.5$ with $s$ as indicated. Figure 10: $F(\tau)$ as a
function of temperature in Kelvin for $\alpha=0.5$, $s=0$. Figure 11:
$F(\tau)$ as a function of temperature in Kelvin for $\alpha=0.5$, $s=0.75$.
Figure 12: $C_{V}(\tau)$ as a function of temperature in Kelvin for
$\alpha=0.2875$, $s=0$ for $B=0,0.5$ and $1.0\ \rm T$. Figure 13: Heat
capacity as a function of temperature in Kelvin for $\alpha=0.2875$, $s=0.75$
for for $B=0,0.5$ and $1.0\ \rm T$.

##  5 Appendix

Unit vectors cited in section 2 are

$\displaystyle\hat{\textbf{n}}$
$\displaystyle=\cos\theta\,\hat{\boldsymbol{\rho}}+\sin\theta\,\hat{\textbf{k}}$
(A.1) $\displaystyle\hat{\boldsymbol{\theta}}$
$\displaystyle=-\sin\theta\,\hat{\boldsymbol{\rho}}+\cos\theta\,\hat{\textbf{k}}$
(A.2) $\displaystyle\hat{\boldsymbol{\phi}}$
$\displaystyle=-\sin\phi\,\hat{\textbf{i}}+\cos\phi\,\hat{\textbf{j}}\,$ (A.3)

The $\Lambda_{\bar{n}n}^{\pm}(u)$ are generated from

$\displaystyle\Lambda_{\bar{n}n}^{+}(u)$
$\displaystyle=\int_{0}^{2\pi}d\theta\frac{\cos\bar{n}\theta\cos\theta\cos
n\theta}{(1+u\cos\theta)^{2}}$ (A.4) $\displaystyle\Lambda_{\bar{n}n}^{-}(u)$
$\displaystyle=\int_{0}^{2\pi}d\theta\frac{\sin\bar{n}\theta\sin\theta\sin
n\theta}{(1+u\cos\theta)^{2}}$ (A.5)

Let $w(u)=\sqrt{1-u^{2}}$ and define

$\displaystyle I_{1}$
$\displaystyle=\frac{1}{[w(u)]^{3}}\big{[}-\frac{1}{u}(1-w(u))\big{]}^{\bar{n}+n}\big{[}({\bar{n}+n})w(u)+1\big{]}$
(A.6) $\displaystyle I_{2}$
$\displaystyle=\frac{1}{[w(u)]^{3}}\big{[}-\frac{1}{u}(1-w(u))\big{]}^{|{\bar{n}-n}|}\big{[}(|{\bar{n}-n})|w(u)+1\big{]}$
(A.7)

Eqs. (A.6) and (A.7) are corrected forms of the corresponding expressions in
[19]. Then

$\Lambda_{\bar{n}n}^{\pm}(u)=\pi(I_{2}\pm I_{1})$ (A.8)

##  6 Acknowledgements

M.E. would like to thank Lewis Johnson for providing software support for this
work.

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