# Refined Diameter Bounds under Curvature Dimension Conditions

Yi C. Huang  School of Mathematical Sciences, Nanjing Normal University,
Nanjing 210023, People’s Republic of China<EMAIL_ADDRESS>https://orcid.org/0000-0002-1297-7674 and  Ze Yang  School of Mathematical
Sciences, Nanjing Normal University, Nanjing 210023, People’s Republic of
China<EMAIL_ADDRESS>
###### Abstract.

In this article we derive an explicit diameter bound for graphs satisfying the
so-called curvature dimension conditions $CD(K,n)$. This refines a recent
result due to Liu, Münch and Peyerimhoff when the dimension $n$ is finite.

###### Key words and phrases:

Graph Laplacian, curvature dimension conditions, resistance distance,
combinatorial distance, diameter bounds

###### 2010 Mathematics Subject Classification:

Primary 53C21; Secondary 05C81, 35K05

Research of the authors is partially supported by the National NSF grant of
China (no. 11801274), the Visiting Scholar Program from the Department of
Mathematical Sciences of Tsinghua University, and the Open Project from Yunnan
Normal University (no. YNNUMA2403). YCH thanks in particular Xueping Huang and
Shiping Liu for helpful communications.

##  1\. Introduction

First, we prepare some notions about discrete geometric analysis (for further
materials in this rapidly growing area, one can refer to the monographs by
Bakry-Gentil-Ledoux [1] and Keller-Lenz-Wojciechowski [4]).

######  Definition 1.1 (Weighted graphs and nondegeneracy).

A triple $G=(V,w,m)$ is called a weighted (and measured) graph if $V$ is a
countable set, if $w:V^{2}\to[0,\infty)$ is symmetric, i.e.,

$w(x,y)=w(y,x),\quad\forall\,x,y\in V,$

and zero on the diagonal, i.e.,

$w(x,x)=0,\quad\forall\,x\in V,$

and if $m:V\to(0,\infty)$. We call $V$ the vertex set, $w$ the edge weight and
$m$ the vertex measure. If $w(x,y)>0$, we write $x\sim y.$ The measure $m$ on
$V$ is called non-degenerate, if $\inf_{x\in V}m(x)>0$.

######  Definition 1.2 (Locally finite graphs, vertex degree, and
connectedness).

A triple $G=(V,w,m)$ is called a locally finite graph if

$\deg(x):=\sum_{y\in V}w(x,y)<\infty,\quad\forall\,x\in V.$

Moreover, it is called a connected graph if for all $x,y\in V$ there exists a
sequence $\\{x_{i}\\}_{i=0}^{n}\subset V$ such that $x=x_{0}\sim
x_{1}\sim\cdots\sim x_{n}=y$.

######  Definition 1.3 (Laplacians on graphs).

We define the graph Laplacian $\Delta$ as

$\Delta f(x):=\frac{1}{m(x)}\sum_{y\in V}w(x,y)(f(y)-f(x)),\quad x\in V.$

Here, $G=(V,w,m)$ is a locally finite graph and $f:V\to\mathbb{R}.$

######  Definition 1.4 (Bakry-Émery curvature dimension conditions).

The Bakry-Émery operators $\Gamma$ and $\Gamma_{2}$ are defined via

$2\Gamma(f,g):=\Delta(fg)-f\Delta g-g\Delta f$

and

$2\Gamma_{2}(f,g):=\Delta\Gamma(f,g)-\Gamma(f,\Delta g)-\Gamma(g,\Delta f).$

We write for simplicity $\Gamma f=\Gamma(f,f)$ and
$\Gamma_{2}(f)=\Gamma_{2}(f,f)$.

A graph $G$ is said to satisfy the curvature dimension condition $CD(K,n)$ for
some $K\in\mathbb{R}$ and $n\in(0,\infty]$ if for all $f$,

$\Gamma_{2}(f)\geq\frac{1}{n}(\Delta f)^{2}+K\Gamma f.$

######  Definition 1.5 (Combinatorial distance).

Let $G=(V,w,m)$ be a connected graph. We define the combinatorial distance
$d:V^{2}\to[0,\infty)$ via

$\displaystyle d(x,y):=\min\\{n:\mbox{ there exist
}x=x_{0},x_{1},\cdots,x_{n}=y\mbox{ s.t. }x_{i}\sim
x_{i-1}\,\,\forall\,i=1\ldots n\\}$

and the combinatorial diameter as $\operatorname{diam}_{d}(G):=\sup_{x,y\in
V}d(x,y)$.

######  Definition 1.6 (Resistance distance).

Let $G=(V,w,m)$ be a locally finite graph. We define the resistance distance
$\rho:V^{2}\to[0,\infty)$ via

$\rho(x,y):=\sup\\{|f(y)-f(x)|:\left\|\Gamma f\right\|_{\infty}\leq 1\\}$

where $\left\|\Gamma f\right\|_{\infty}=\sup_{x\in V}(\Gamma f)(x)$, and the
resistance diameter as

$\operatorname{diam}_{\rho}(G):=\sup_{x,y\in V}\rho(x,y).$

By [5, Lemma 1.4], we have the following relationship between the
combinatorial and resistance distances. We also define

$\operatorname{Deg}(x):=\frac{\deg(x)}{m(x)}\text{ \,\,and
\,\,}\operatorname{Deg}_{\max}:=\sup_{x\in V}\operatorname{Deg}(x).$

######  Lemma 1.7 (Liu, Münch and Peyerimhoff).

Let $G=(V,w,m)$ be a locally finite connected graph with
$\operatorname{Deg}_{\max}<\infty$. Then for all $x_{0},y_{0}\in V$,

$d(x_{0},y_{0})\leq\sqrt{\frac{\operatorname{Deg}_{\max}}{2}}\rho(x_{0},y_{0}).$

The following explicit resistance distance bound is our main result.

######  Theorem 1.8.

Let $G=(V,w,m)$ be a locally finite, connected, complete graph with non-
degenerate vertex measure satisfying $CD(K,n)$ for $K>0$, $n<\infty$ and
$\operatorname{Deg}_{\max}<\infty$. Then for all $x_{0},y_{0}\in V$,

$\rho(x_{0},y_{0})\leq\sqrt{\frac{n}{K}}\left(\arcsin\frac{1}{\sqrt{\frac{Kn}{2\operatorname{Deg}\left(x_{0}\right)}+1}}+\arcsin\frac{1}{\sqrt{\frac{Kn}{2\operatorname{Deg}\left(y_{0}\right)}+1}}\right).$
(1.1)

######  Remark 1.9.

The completeness assumption is needed when using certain functional
consequences of the curvature dimension conditions, see Lemma 2.1 below. For
the definition of completeness of graphs, see [3, Sections 1 and 2.1] or [2,
Definition 2.9, Definition 2.13]. Our resistance distance bound refines a
recent result due to Liu, Münch and Peyerimhoff in the sense the two arcsin-s
in (1.1) sum to $\pi$ as
$\min\\{\operatorname{Deg}\left(x_{0}\right),\operatorname{Deg}\left(y_{0}\right)\\}\rightarrow\infty$.
(1.1) also holds when $n=\infty$, see [5, Theorem 2.1].

######  Remark 1.10.

For diameter bounds via a different curvature, see Steinerberger [6].

######  Corollary 1.11 (Diameter bounds under $CD(K,n)$).

Let $G=(V,w,m)$ be a locally finite, connected, complete graph with non-
degenerate vertex measure satisfying $CD(K,n)$ for $n<\infty$ and
$\operatorname{Deg}_{\max}<\infty$. Then

$\operatorname{diam}_{d}(G)\leq\sqrt{\frac{2n\operatorname{Deg}_{\max}}{K}}\arcsin\frac{1}{\sqrt{\frac{Kn}{2\operatorname{Deg}_{\max}}+1}}.$

##  2\. Proofs of Theorem 1.8 and Corollary 1.11

By [5, Lemma 2.3], we have the following consequence under $CD(K,n)$.

######  Lemma 2.1 (Liu, Münch and Peyerimhoff).

Let $G=(V,w,m)$ be a complete graph with non-degenerate vertex measure.
Suppose $G$ satisfies $CD(K,n)$. Then for all bounded $f:V\to{\mathbb{R}}$
with bounded $\Gamma f$,

$\Gamma P_{t}f\leq e^{-2Kt}P_{t}\Gamma f-\frac{1-e^{-2Kt}}{Kn}(\Delta
P_{t}f)^{2}.$

Here, $P_{t}$ denotes the heat semigroup associated to $\Delta$.

###### Proof of Theorem 1.8.

We follow the streamlined arguments for [5, Theorem 2.4]. By Lemma 2.1, we see
that $CD(K,n)$ is equivalent to

$\Gamma P_{t}f\leq e^{-2Kt}P_{t}\Gamma f-\frac{1-e^{-2Kt}}{Kn}(\Delta
P_{t}f)^{2}$

for all bounded function $f:V\to\mathbb{R}$. We fix $x_{0},y_{0}\in V$ and
$\varepsilon>0$. Then by the definition of $\rho$, there is a function
$f:V\to\mathbb{R}$ s.t. $f(y_{0})-f(x_{0})>\rho(x_{0},y_{0})-\varepsilon$ and
$\Gamma f\leq 1$. W.l.o.g., we assume that $f$ is bounded. When
$\operatorname{Deg}_{\max}<\infty$, we obtain

$\displaystyle\Gamma P_{t}f(x)$ $\displaystyle=\frac{\sum_{y\in
V}{w(x,y)(P_{t}f(y)-P_{t}f(x))^{2}}}{2m(x)}$
$\displaystyle\geq\frac{\left(\sum_{y\in
V}{\frac{w(x,y)(P_{t}f(y)-P_{t}f(x))}{m(x)}}\right)^{2}}{2\frac{\sum_{y\in
V}{w(x,y)}}{m(x)}}=\frac{\left(\Delta
P_{t}f(x)\right)^{2}}{2\operatorname{Deg}(x)},\quad x\in V.$

Combined with $CD(K,n)$ condition, we obtain

$\left(\frac{1}{2\operatorname{Deg}(x)}+\frac{1-e^{-2Kt}}{Kn}\right)(\Delta
P_{t}f(x))^{2}\leq e^{-2Kt}P_{t}(\Gamma f)(x),\quad x\in V.$

Taking square root and applying $\Gamma f\leqslant 1$, we have

$\displaystyle|\partial_{t}P_{t}f(x)|$ $\displaystyle=|\Delta P_{t}f(x)|$
$\displaystyle\leq\frac{e^{-Kt}}{\sqrt{\frac{1}{2\operatorname{Deg}(x)}+\frac{1-e^{-2Kt}}{Kn}}}=\sqrt{Kn}\frac{e^{-Kt}}{\sqrt{\left(\frac{Kn}{2\operatorname{Deg}(x)}+1\right)-e^{-2Kt}}},\quad
x\in V.$

When $T>0$, we have

$\displaystyle|P_{T}f(x)-f(x)|$
$\displaystyle\leq\int_{0}^{T}{\left|\partial_{t}P_{t}f(x)\right|dt}$
$\displaystyle\leq\int_{0}^{\infty}{\left|\partial_{t}P_{t}f(x)\right|dt}$
$\displaystyle\leq\int_{0}^{\infty}{\sqrt{Kn}\frac{e^{-Kt}}{\sqrt{\left(\frac{Kn}{2\operatorname{Deg}(x)}+1\right)-e^{-2Kt}}}dt}$
$\displaystyle=\sqrt{\frac{n}{K}}\arcsin\frac{1}{\sqrt{\frac{Kn}{2\operatorname{Deg}(x)}+1}},\quad
x\in V.$

Hence combined with the triangle inequality we obtain

$\displaystyle\rho(x_{0},y_{0})-\varepsilon$
$\displaystyle\leq|f(x_{0})-f(y_{0})|$
$\displaystyle\leq\left|P_{T}f(x_{0})-f(x_{0})\right|+\left|P_{T}f(x_{0})-P_{T}f(y_{0})\right|+\left|P_{T}f(y_{0})-f(y_{0})\right|$
$\displaystyle\leq\sqrt{\frac{n}{K}}\left(\arcsin\frac{1}{\sqrt{\frac{Kn}{2\operatorname{Deg}\left(x_{0}\right)}+1}}+\arcsin\frac{1}{\sqrt{\frac{Kn}{2\operatorname{Deg}\left(y_{0}\right)}+1}}\right)$
$\displaystyle\qquad\qquad\qquad+\left|P_{T}f(x_{0})-P_{T}f(y_{0})\right|$
$\displaystyle\xrightarrow{t\rightarrow\infty}\sqrt{\frac{n}{K}}\left(\arcsin\frac{1}{\sqrt{\frac{Kn}{2\operatorname{Deg}\left(x_{0}\right)}+1}}+\arcsin\frac{1}{\sqrt{\frac{Kn}{2\operatorname{Deg}\left(y_{0}\right)}+1}}\right).$

Due to $CD(K,n)$ which implies $\|\Gamma P_{T}f\|_{\infty}{\rightarrow}0$ as
$T\to\infty$, and since $G$ is connected,

$\left|P_{T}f(x_{0})-P_{T}f(y_{0})\right|{\rightarrow}0\text{\,\, as
\,\,}T\to\infty.$

Taking the limit $\varepsilon\to 0$ finishes the proof. ∎

###### Proof of Corollary 1.11.

By Lemma 1.7, we obtain

$\displaystyle d(x_{0},y_{0})$
$\displaystyle\leq\sqrt{\frac{\operatorname{Deg}_{\max}}{2}}\rho(x_{0},y_{0})$
$\displaystyle\leq\sqrt{\frac{n\operatorname{Deg}_{\max}}{2K}}\left(\arcsin\frac{1}{\sqrt{\frac{Kn}{2\operatorname{Deg}\left(x_{0}\right)}+1}}+\arcsin\frac{1}{\sqrt{\frac{Kn}{2\operatorname{Deg}\left(y_{0}\right)}+1}}\right)$
$\displaystyle\leq\sqrt{\frac{2\mathrm{\operatorname{Deg}_{\max}}n}{K}}\arcsin\frac{1}{\sqrt{\frac{Kn}{2\operatorname{Deg}_{\max}}+1}}.$

Taking the supremum over $x_{0},y_{0}\in V$ finishes the proof. ∎

## Compliance with ethical standards

Conflict of interest The authors have no known competing financial interests
or personal relationships that could have appeared to influence this reported
work.

Availability of data and material Not applicable.

## References

  * [1] Dominique Bakry, Ivan Gentil, and Michel Ledoux.  Analysis and Geometry of Markov Diffusion Operators, volume 348 of Grundlehren der mathematischen Wissenschaften.  2014\.
  * [2] Cao Gong and Yong Lin.  Equivalent properties for CD inequalities on graphs with unbounded Laplacians.  Chinese Annals of Mathematics, Series B, 38:1059–1070, 2017.
  * [3] Bobo Hua and Yong Lin.  Stochastic completeness for graphs with curvature dimension conditions.  Advances in Mathematics, 306:279–302, 2017.
  * [4] Matthias Keller, Daniel Lenz, and Radoslaw K. Wojciechowski.  Graphs and Discrete Dirichlet Spaces, volume 358 of Grundlehren der mathematischen Wissenschaften.  2021\.
  * [5] Shiping Liu, Florentin Münch, and Norbert Peyerimhoff.  Bakry–Émery curvature and diameter bounds on graphs.  Calculus of Variations and Partial Differential Equations, 57:1–9, 2018.
  * [6] Stefan Steinerberger.  Curvature on graphs via equilibrium measures.  Journal of Graph Theory, 103:415–436, 2023.


# The enduring relevance of the Jaynes-Cummings model: a personal perspective

Peter L. Knight1, Christopher C. Gerry2, Richard J. Birrittella3 and Paul M.
Alsing4
1Blackett Laboratory, Imperial College, London SW72AZ, UK
2Department of Physics and Astronomy, Lehman College, The City
University of New York, Bronx, New York, 10468-1589,USA
3Booz Allen Hamilton, 8283 Greensboro Drive, McLean, VA 22102, USA
4University at Albany-SUNY, Albany, NY 12222, USA

###### Abstract

In this short perspective article we present our personal highlights on how
the Jaynes-Cummings model has become a central model to describe spin-boson
couplings underpinning much of modern quantum optics. To the current authors,
the key contribution is a demonstration of a measurable effect that showed the
discreteness of the quantized radiation field.

##  I  Early History

We discuss how the Jaynes-Cummings model [1] has become the basic building
block of resonance physics, demonstrating the non-perturbative time-dependent
oscillation of transition probabilities when a single quantized field mode is
coupled to a two-level system. It underpins much of quantum optics and quantum
information processing [2] and 61 years after the original publication
continues to be a highly cited paper.

Atom-field resonant excitation has a long history [3] (and see [2] and
references therein). Rabi developed the semi-classical resonance model where
the field is described classically to describe atomic beam resonance
experiments. An essential assumption made was the rotating wave approximation
(RWA) [2] 111Going beyond the RWA has been an unwise theoretical obsession
hardly ever justified if the two-level approximation is employed: counter-
rotating non-resonant terms are always going to couple to other atomic levels,
unless dealing with a genuine spin with no other nearby levels., justified by
the closeness of a resonance and the lack of other nearby atomic levels. This
work built on earlier time-dependent work on non-adiabatic Majorana “flops”
[5]. Indeed, non-adiabatic transitions were studied even before Rabi’s
resonance model. The first suggestion came from C. Darwin (the Darwin’s
grandson!) in 1927 [6] and studied experimentally by Phipps and Stern and
theoretically by Guttinger and very thoroughly by Majorana [7] and confirmed
experimentally by Frisch and Segre (see [5] and references therein); hence
these were referred to by the atomic beams community as “Majorana flops” as
they created spin flips that “flopped” the focused atomic beam back to the
detector. Rabi, aware of these developments, then built on them to construct
his radio frequency resonance approach a few years later.

Jaynes, in a famous Stanford Microwave Lab report [8], extended the semi-
classical approach to describe maser action and shortly after brought in F.W.
Cummings, his then-graduate student, to analyze a fully quantized version of
the Rabi model assuming Fock (number) states of the field, and the resultant
paper is the now-famous Jaynes-Cummings Model (JCM) [1].

We denote the atomic resonance frequency and cavity frequency as $\omega_{0}$
and $\omega$, respectively, with Pauli raising and lowering operators to
describe absorption and emission, with an atom-field coupling strength denoted
by $\lambda$. In the RWA, the Jaynes-Cummings Hamiltonian is given by

$\hat{H}=\frac{1}{2}\hbar\omega_{0}\hat{\sigma}_{3}+\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\lambda\left(\hat{\sigma}_{+}\hat{\hat{a}}+\hat{\sigma}_{-}\hat{a}^{\dagger}\right).$
(1)

The JCM is ubiquitous in quantum optics as the fundamental spin-boson model
for a single photon interacting with a single two-level atom (in the RWA)
inside a cavity, spawning the field of cavity quantum electrodynamics (QED,
see e.g. Haroche and Raimond [9] and references therein). It’s investigations
and extensions in the field of quantum optics have been extensively explored.
A number of early papers investigated the interaction of a two level system
with a single quantized field mode including those by Paul [10] and Frahm[11].

Returning to the JCM Hamiltonian of Eq. (1), the probability that the system
remains in the initial state  $\ket{i}=\ket{e}_{a}\otimes\ket{n}_{f}$ (atom in
excited state $\ket{e}_{a}$ and field in Fock state $\ket{n}_{f}$ containing
$n$ photons)  is

$P_{i}(t)=|C_{i}(t)|^{2}=\cos^{2}\left(\lambda t\sqrt{n+1}\right),$ (2)

while the probability that it makes a transition to the state $\ket{f}$ is

$P_{f}(t)=1-P_{i}(t)=\sin^{2}\left(\lambda t\sqrt{n+1}\right).$ (3)

Cummings, in a later paper [12], addressed the role of photon statistics in
the JCM and showed how the distribution of photon numbers leads to a spread of
oscillatory Rabi frequencies and an inevitable dephasing of the otherwise
regular sinusoidal evolution. For the most classical of the pure quantized
field states, a coherent state of amplitude $\alpha$, given by

$\ket{\alpha}=e^{-\tfrac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\ket{n},$
(4)

with average photon number $\bar{n}=|\alpha|^{2}$, we have

$P_{n}=|\braket{n}{\alpha}|^{2}=e^{-|\alpha|^{2}}\frac{|\alpha|^{2n}}{n!}=e^{-\bar{n}}\frac{\bar{n}^{n}}{n!}.$
(5)

Thus the difference in occupation probabilities between the initial and final
states, known as the atomic inversion and denoted $W(t)$ is

$W(t)=\braket{{\hat{\sigma}}_{3}}=e^{-\bar{n}}\sum_{n=0}^{\infty}\frac{\bar{n}^{n}}{n!}\cos\left(2\lambda
t\sqrt{n+1}\right).$ (6)

This leads to a dephasing of the Rabi oscillations, now known as a Cummings
collapse, due to the spread of photon numbers. This is a unique quantum
feature that is important to note as even a “classical” state (coherent or
thermal) has such a spread: the evolution is photon-number sensitive in a
truly non-classical manner.

The collapse time found by Cummings is governed by the inverse of the one-
photon coupling strength $t_{c}=(2\lambda)^{-1}$. But this collapse is non-
dissipative$-$ what happens if one pursues the evolution for longer times? In
hindsight, it is clear that we should expect some kind of return of coherent
oscillations given the discreteness of photon occupation number. The
literature does contain precursors of what turned out to be revivals (Meystre
et al. [13], Stenholm [14], etc.) in numerical studies of the JCM albeit with
truncated numerics at this early stage in computational analysis in the early
1970s.

##  II  Collapse and Revivals

The systematic study of the effects of photon statistics on the long-term time
evolution of the JCM was pioneered by J. H. Eberly and his group, combining
sophisticated analytic and computational approaches [15, 16]. They were able
to describe in detail the emergence of revivals as well as collapses. The
numerics in their computational approach were this time converged. The new
feature they showed, that being the revivals of the atomic inversion, are
important as they showed how sensitive the atom-field coupling is to the
discreteness of photon numbers. They showed the revival time is given by

$t_{R}=(2\pi/\lambda)\bar{n}^{1/2}k,\;\;\;\;\bar{n}\gg 1,\;\;k=1,2,3,\dots\;.$
(7)

In Fig. (1) we plot the atomic inversion for an initial coherent state with
average photon number $\bar{n}=15$ as a function of scaled time $T=\lambda t$.

Figure 1: Atomic inversion evolution with the field prepared in an initial
coherent state with $\alpha=\sqrt{15}$ and the atom initially prepared in the
excited state, such that $\ket{\psi(0)}_{f,a}=\ket{\alpha,e}_{f,a}$. Here, we
plot the atomic inversion against scaled time $\lambda t$.

The relationship between the semi-classical Rabi formula for resonant
excitation and the Jaynes-Cummings model is worth mention. One might naively
expect that as the photon number increases, some kind of semi-classical limit
may be approached, yet this does not happen. Knight and Radmore [17] addressed
this; for a cavity field prepared in a coherent state, one can transform the
Hamiltonian by an inverse Glauber transformation [18] without approximation to
one where the cavity is occupied by a classical field of appropriate amplitude
but where the atom is also coupled to a vacuum field. As the classical field
drives the atom to an excited state, the atom, through quantum coupling to the
field creation operator, can generate a cascade of quantum generated photon
numbers: the first step in this cascade is the one-photon creation step and
for that reason the collapse is governed by the inverse of the one-photon Rabi
frequency.  Fleischhauer and Schleich [19] have shown how the Poisson
summation formula provides insight into the nature of the revivals.

##  III  Squeezed states

Shortly after the appearance of the paper by Naroznhy et al. [15] announcing
the collapse and revival phenomena in the Jaynes-Cummings model, Meystre and
Zubairy [20] studied the possible occurance of quadrature squeezing in the
evolution of the model with the field and the atom being at exact resonance
assuming the the initial state of the field to be a coherent state of average
photon number $\bar{n}(0)<10$ (this was in the days before squeezing was
observed in the laboratory.) These authors found that, indeed, quadrature
squeezing could occur during the evolution, but that the amount of squeezing
obtained was rather modest; being about 20% of the amount allowable below the
level of the vacuum noise. However, Kukliński and Madajczyk [21] found that
with a much greater average photon number, such as $\bar{n}(0)\sim 100$,
strong squeezing can be obtained even though they included field dampening in
their calculations. These results are demonstrated in Fig. (2), where the
usual quadrature operators are given by

$\hat{X}_{1}=\frac{1}{2}\left(\hat{a}+\hat{a}^{\dagger}\right),\;\;\;\;\;\;\hat{X}_{2}=\frac{i}{2}\left(\hat{a}-\hat{a}^{\dagger}\right),$
(8)

and we plot $s_{1}=\Delta^{2}\hat{X}_{1}-\tfrac{1}{4}$, where
$\Delta^{2}\hat{X}_{1}=\braket{\hat{X}_{1}^{2}}-\braket{\hat{X}_{1}}^{2}$ is
the quadrature variance and squeezing is observed for values $s_{1}<0$ with
the lower bound of $s^{(\text{min.})}_{1}=-1/4$. Interestingly, they found
that squeezing can be even stronger for the out-of-resonance case. Hillery
[22] subsequently showed for small initial photon number, the squeezing
obtained in this model will always be weak. He also pointed out that for some
quantum effects to be substantial, large photon numbers are required.

Figure 2: Quadrature squeezing with an initial coherent state with
$\bar{n}=10,40,100$. We plot only the variance for the the $\hat{X}_{1}$
quadrature noting that no squeezing occurs in $\hat{X}_{2}$ at any time. A
high degree of squeezing can be seen at later times for succesively higher
$\bar{n}$. We include the minimum squeezing values for each case; these are:
$-0.11,-0.185,\;\text{and}\;-0.213$, for increasing average photon numbers.
Figure 3: Mandel $Q$ parameter for different intensities of the initial field
$\bar{n}=10,40,100$. Note that for all initial coherent (Poissonian) field
intensities the state displays both super-, and sub-Poissonian statistics at
some point in the evolution. Note, the initial collapse time is insensitive to
the photon number $\bar{n}$, while the subsequent revival times are. The
revival of the varying statistics occurs at shorter times for smaller average
photon numbers.

For the same cases considered in Fig. (2), we display the corresponding Mandel
$Q$ parameter defined as

$Q=\frac{\Delta^{2}\hat{n}}{\braket{\hat{n}}}-1,$ (9)

where $\Delta^{2}\hat{n}$ represents the photon number fluctuations. For
states displaying Poissonian statistics, the photon number fluctuations are
equal to the average photon number and we have $Q=0$. For $Q>0$ and $Q<0$, the
state is said to have super$-$, and sub-Poissonian statistics, respectively
(the former having large photon number variances, an example of which being
thermal states, while the latter has comparably small variances like number
states, and is a strictly non-classical characteristic). We plot the Mandel
$Q$ parameter in Fig. (3) demonstrating variation in the field photon
statistics throughout the state evolution.

Alsing, Guo and Carmichael [23] calculated the quasienergies and steady states
of a coupled two-level atom and quantized electromagnetic cavity mode with the
cavity mode driven by a periodic classical field that are all on resonance.
The quasienergies resulted in shifted Jaynes-Cummings level splittings which
were reduced by the interaction with the driving field and vanished at a
threshold value of the driving field strength. Above this threshold, the
discrete quasienergies and normalizable steady states do not exist. For weak
driving fields below the threshold, the steady states were squeezed and
displaced Jaynes-Cummings eigenstates which gave rise to squeezing-induced
linewidth narrowing in the vacuum Rabi splitting for the ground state of the
driven Jaynes-Cummings system.

Figure 4: Evolution of the Wigner function for an initial coherent state with
$|\alpha|^{2}=9$. The distribution bifurcates and counter propagates in phase
space before recombining. When the two localized distributions are opposite
each other, the field is at its most cat-like. Note the negativity in the
Wigner function for scaled times $\lambda t>0$ throughout the evolution
suggesting non-classicality of the field.

In the cases mentioned in the previous paragraph, only continuous evolution
was involved, with atomic and field states generally entangled. However, as
shown by Gerry and Ghosh [24], it is possible to obtain a greater degree of
squeezing, even in the case of low photon number, provided selective
measurements are performed on the atom. The idea is that an atom passes
through the cavity and interacts with cavity field, then leaves the cavity
where it can be subjected to classical radiation fields which implement what
amounts to Ramsey pulses, after which the state of the atom is detected by
selective ionization. This process projects the cavity field into a pure
state, and it was found in [24] that squeezing up to 75% could be attained in
this matter. This procedure for projecting a cavity field into a pure state
with non-classical features, such as cat states of the form
$\ket{\psi_{\pm}}\propto\ket{\alpha}\pm\ket{-\alpha}$ has long been discussed
and perfected in the laboratory by Haroche and collaborators in the context of
the dispersive model of atom-field interactions (see [9]), but applied here to
the resonant case. The possible use of these techniques in the resonant model
was first proposed by Zagury and de Toledo Piza [25] who showed that large-
scale correlation effects could be obtained by the pre-selection and post-
selection of atomic states. Ghost and Gerry [26] later investigated the
occurence of sub-Poissonian statistics in the field by a sequence of properly
prepared and detected atoms. We note there has been recent activity in this
area [27, 28].

Figure 5: Expectation value of the field parity operator (purple, dashed)
plotted against scaled time $\lambda t$ for the initial conditions
$\alpha=\sqrt{15}$ and the atom initially prepared in the excited state, such
that $\ket{\psi(0)}_{f,a}=\ket{\alpha,e}_{f,a}$. We include a plot of the
atomic inversion (red,solid) for reference

The full range of the occurence of non-classical field effects in the Jaynes-
Cummings model can be found in the recent book by Larson and Mavrogordaros
[29].

##  IV  Atom-Field Schrödinger Cats

Further understanding can be obtained by studying the evolution of the field
state in phase space using quasi-probabilities. A single field mode in the JCM
interacting with a single two-level atom exhibits a two-component behaviour
easily described by the Schmidt basis [30, 31, 32]. A field initially in a
coherent state has a Gaussian Wigner function which evolves by bifurcating
into two counter-rotating components; the point in time at which the two
components are maximally separated coincides with the collapse time. Then,
when they come to a point 180 degrees from the starting point, we see a
revival. This is demonstrated in Fig. (4) for an initial coherent state with
$\alpha=3$. The JCM bifurcation was noted in early papers by Eiselt and Risken
[33, 34] who showed that the bifurcation could be seen in the evolution of
both the $Q$-function and Wigner function of the field.

At the point of maximal separation, which occurs at the midpoint between the
collapse and first revival of the Rabi oscillations, the atom-field system
becomes separable as was shown by Knight and Phoenix [30] and by Gea
Banacloche [31]. The field state at this time is approximately described by
(unnormalized) Schrödinger cat states of the form [35]

$\ket{\psi_{\pm}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sim}\ket{\beta}\pm\ket{-\beta},$
(10)

where $\ket{\pm\beta}$ are coherent states. Note that $\ket{\psi_{+}}$
contains only even photon numbers while $\ket{\psi_{-}}$ contains only odd. We
mention this because even though the atom and field states are approximately
separable at the aforementioned midpoint, the field is not static. That this
is so becomes apparent if one calculates the expectation value of the field
photon number parity operator, given by
$\hat{\Pi}=(-1)^{\hat{a}^{\dagger}\hat{a}}$, for the resonant JCM as was done
by Birrittella et al. [36]. The result is displayed in Fig. (5), where it is
evident that at the midpoint of the period where the Rabi oscillations of the
atomic inversion are quiescent, the photon number parity is rapidly
oscillating. This indicates that the field state at these times is rapidly
switching between states of even and odd parity, i.e. switching between even
and odd cat-like states. Ultimately, this can be attributed to quantum
interference efffects between the widely separated counter-rotating components
of the field. These oscillations in the photon number parity have been seen in
a recent experiment by the Haroche group [37].

But what happens when instead of an entirely closed loss-free system, we allow
dissipation? Barnett and Knight [38] were first to show revivals were very
sensitive to dissipation; collapses less so.

##  V  Experimental observations

The experimental investigations of the JCM came out of work on the radiative
properties of highly excited Rydberg atoms in cavity QED. Atoms prepared in a
very highly excited states have huge dipole moments, and combined with
cavities with very high quality factors allow the study of atom-field
couplings at low photon number (Rempe et al. [39], Haroche and Raimond [9]),
resulting in clear observations of both collapses and revivals of the Rabi
oscillations. But this is not the only way to realize a spin-boson JCM
Hamiltonian: another way uses laser-cooled trapped ions. A laser cooled ion in
an electromagnetic trap, such as an RF Paul trap [40] realizes the Jaynes-
Cummings interaction between internal states of the ion and the quantized
harmonic vibrational motion of the center-of-mass (CM) of the ion which is
constrained by the trap. The Jaynes-Cummings interaction is engineered by the
use of a tunable laser field directed along the axis of the trap. If $\nu$ is
the angular frequency of the vibrational motion of the CM of the ion, which is
determined by the trap parameters, and if $\omega_{0}$ is the angular
frequency associated with two relevant internal states of the ion, then with
the laser frequency tuned such that $\omega_{L}=\omega_{0}+\nu$, one ends up
with an interaction that contains, after discarding rapidly oscillation terms,
the Jaynes-Cummings interaction $-i\hat{a}^{\dagger}\hat{\sigma}_{-}+h.c.$
wherein the Bose operators $\hat{a},\hat{a}^{\dagger}$ are understood to
represent the quantized vibrational motion of the CM of the ion. This replaces
the quantized single-mode field: i.e. we have phonons instead of photons.
Indeed, with the choice $\omega_{L}=\omega_{0}-\nu$, the otherwise
unrealizeable “anti-Jaynes-Cummings” interaction
$-i\hat{a}^{\dagger}\hat{\sigma}_{-}+h.c.$ can be engineered.

As far as we are aware, the first laboratory realization of the JCM in the
context of trapped ions was reported by Meekhof et al. [41]. With the
vibrational motion of the ion prepared in coherent state by implementing a
displacement of the ion trap itself with the ion initially in its vibrational
ground state, Meekhoff et al. realized the coherent state JCM. Clear evidence
of the predicated collapse and revival of the Rabi oscillations is presented
in [41].

A significant physical implementation and translation of the JCM beyond the
field of quantum optics has been to circuit QED systems consisting of a single
superconducting qubit coupled to the electromagnetic field of a single mode
inside a microwave resonator (see the review article by Schmidt and Koch [42]
and references therein, with comparison/contrast to cavity QED systems).

##  VI  Closing Remarks

The Jaynes-Cummings model continues to be one of the key conceptual models of
current quantum optics: it continues to be very widely cited with no
diminution of its perceived importance.

In closing, it may seem ironic that E. T. Jaynes, passionate about looking for
ways to avoid field quantization  [43], was the creator of the most
fundamental model of atom-photon interaction; one widely adopted and confirmed
by experiment. Extensive reviews of the JCM exist, for example Shore and
Knight [44], and more recently and exhaustively by Larson and Mavrogordatos
[29], indicating its lasting importance.

In the 60+ years since the publication of the Jaynes-Cummings model, the paper
remains a seminal contribution to quantum optics, cited many times each year.
To the current authors, the key contribution is a demonstration of a
measurable effect that showed the discreteness of the quantized radiation
field. The Cummings collapse reflects the contribution of incommensurate
coupling strengths; of course any distribution of field strengths will lead to
dephasing analogous to a collapse, but the revival is truly dependent on the
photon number being discrete. This will happen even with fields showing little
sign otherwise of “non-classicality” including coherent and thermal fields. To
us, this is really the ‘take-home’ message of the Jaynes-Cummings model.

##  VII Acknowledgements

Any opinions, findings and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views
of their home institutions.

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Then, identifying $\mc E_{\alpha\beta}=\frac 1 2 (\partial_\alpha u_\beta +
\partial_\beta u_\alpha)$ the symmetric rank-2 elasticity tensor
(i.e. strain tensor), we identify $H'_{\rm ex}=H'_{ss\varepsilon}+\dots$, where

\begin{equation}
\label{eq:51}
H'_{ss\varepsilon} = \frac 1 2 \sum_{\mb r,\bs \eta} S_{\mb r+\bs \eta}^a S_{\mb  r}^b
\left [ \eta_\alpha
\partial_{\eta^\beta}
+ \eta_\beta
\partial_{\eta^\alpha} \right ] J^{ab}\Big |_{\bs \eta} \,
\mc E_{\alpha\beta}(\mb r) + \dots,
\end{equation}

where $a,b=x,y,z$ is a spin axis index, $\alpha,\beta=x,y,z$
is a spatial index, and “$+\dots$” encompasses terms featuring $\omega_{\alpha\beta}$
the anti-symmetric rank-2 elasticity tensor, as well as higher-order derivatives
of the displacement field.

Note that in this microscopic derivation, we identify $S^a_{\mb R} \mapsto S^a_{{\mb r}}$.
In fact, also expanding the magnetization fields (and not only the magnetic exchange)
with respect to displacement yields an interaction term which is formally of the same order as that derived here.
However, magnetization in an ordered magnet is a slow variable, while $J$ varies over distances of the order of the lattice parameter $\mf a$,
therefore such terms are quantitatively much smaller by a factor $O(k_B T \mf a/v_{\rm m})$,
both within and beyond the Born-Oppenheimer approximation.

Finally, we take the particular case of a square lattice with tetragonal symmetry,
and describe the spins in terms of $\rm m, n$ fields as in the main text,
namely $\mb S_{\mb r}  =  (-1)^{\mb r} \mu_0 \mb n(\mb r)
+ \mathfrak{a}^2 \mb m(\mb r) $.
We identify $H'_{ss\varepsilon} = \sum_{\mb r} \mc H'_{\rm tetra}(\mb r)+\dots$
where “$+\dots$” is made of rapidly oscillating (time-reversal breaking) terms,
and $\mc H'_{\rm tetra}$ is as displayed in Eq. (<ref>), with identification

\begin{eqnarray}
\label{eq:67}
\Lambda_{ab}^{({\rm m}),\alpha\beta}
&=& \frac 1 2 \sum_{\bs \eta} \left ( \eta_\alpha\partial_\beta +
\eta_\beta \partial_\alpha \right ) J^{ab} \Big |_{\bs
\eta} ,\\
\Lambda_{ab}^{({\rm n}),\alpha\beta}
&=& \frac 1 2 \sum_{\bs \eta} e^{i\bs \pi \bs \eta}
\left ( \eta_\alpha\partial_\beta + \eta_\beta
\partial_\alpha \right )
J^{ab} \Big |_{\bs \eta},
\end{eqnarray}

where the sum over $\bs \eta$ spans the whole direct (two-dimensional square)
lattice, and
$\bs \pi = \left ( \frac{\pi}{\mf a} , \frac{\pi}{\mf a}  \right )$ with $\mf a$
the square lattice parameter.

§.§ Contributions to intervalley couplings

In the main text, the $n_a,m_a$ fields live in the valleys identified by:

\begin{align}
\ell = 0 : &\qquad n_y, m_z\nonumber\\
\ell = 1 : &\qquad n_z, m_y.
\end{align}

Therefore, intervalley couplings are of the form
$\lambda_{ab;\xi\xi'}$ with
$\delta_{\xi\xi'}+\delta_{ab}=1$. More explicitly, using
Eq. (<ref>), they are:

\begin{eqnarray}
\lambda_{yz;00}^{\alpha\beta}&=&\Lambda_{yz}^{({\rm n}),\alpha\beta},\\
\lambda_{yz;11}^{\alpha\beta}&=&\Lambda_{yz}^{({\rm m}),\alpha\beta},\nonumber\\
\lambda_{yy;01}^{\alpha\beta} & = &
\frac{-1}{n_0}\left[ m^y_0 \Lambda^{({\rm m}), \alpha\beta}_{yx} +
\Lambda_{zx}^{({\rm m}), \alpha\beta} + m^y_0
\Lambda^{({\rm n}), \alpha\beta}_{yx}\right],\nonumber\\
\lambda_{zz;01}^{\alpha\beta} & = &
\frac{-1}{n_0}\left[ m^z_0 \Lambda^{({\rm m}), \alpha\beta}_{zx} +
\Lambda_{y x}^{({\rm m}), \alpha\beta} + m^z_0
\Lambda^{({\rm n}), \alpha\beta}_{zx}\right] .\nonumber
\end{eqnarray}

Also recall from Eq. (<ref>) that

\begin{align}
\Lambda_{yx}^{(\xi),xy}=\Lambda_{yx}^{(\xi),yx}= \Lambda_{xy}^{(\xi),xy}=\Lambda_{xy}^{(\xi),yx}  &=  \Lambda_6^{(\xi)},\nonumber\\
\Lambda_{zx}^{(\xi),xz}=\Lambda_{zx}^{(\xi),zx}
=\Lambda_{yz}^{(\xi),yz}=\Lambda_{yz}^{(\xi),zy} &= \Lambda_7^{(\xi)},
\end{align}

and all other values of $\alpha,\beta$ yield 0 for this set of lower
indices. From this, it is clear that the $\Lambda_7^{(\xi)}$ couplings
always mix valleys, regardless of $\bs m_0$, and contribute a
$\lambda_{yz;\xi\xi}$ term. This intervalley coupling is a
small contribution which does not contribute to $\mc T$
breaking. Meanwhile, the $\mc T$-odd $\lambda_{yy;01}$ and
$\lambda_{zz;01}$ intervalley couplings both contain contributions
from both $\Lambda_7^{(\xi)} m_0^z$ and $\Lambda_6^{(\xi)} m_0^y$.

§.§ Derivation of the gaps from a sigma model

Here we provide a heuristic microscopic argument for expressing the gaps in terms of spin-spin couplings.
We ignore spin-lattice coupling, and just consider corrections to the isotropic Heisenberg model.  We
assume the addition of a term of the XXZ anisotropy form:

\begin{equation}
\label{eq:gap2}
H_{\rm XXZ} = g J \sum_{\langle ij\rangle} \left( 2S_i^z S_j^z - S_i^x S_j^x
-S_i^y S_j^y\right).
\end{equation}

This is to be added to the isotropic Heisenberg model,
along with a Zeeman coupling to the transverse field.

Carrying out the long-wavelength expansion in terms of $\mb{m}$ and $\mb{n}$
fields, we obtain the corrected potential part (i.e. without gradient terms) of the
nonlinear sigma-model Eq. (<ref>):

\begin{align}
\label{eq:gap3}
\mathcal{H}_{\rm NLS}^{g,h} &= \frac{1}{2\chi} |\mb{m}|^2 + 2 g J
\mathfrak{a}^2  \left(2 \text m_z^2 - \text m_x^2 - \text m_y^2\right)  \\
& - 2 g J
\frac{\mu_0^2}{\mathfrak{a}^2} \left(2 \text n_z^2 -\text n_x^2 - \text n_y ^2\right)
-h_y \text m_y - h_z \text m_z .\nonumber
\end{align}

Note that the first term includes an $\text m_x^2$ term, which is
absent in the quadratic expansion describing linear spin waves in the main
text. Indeed this term is higher order in the small fluctuations around an
$x$-ordered state when carrying out a zero field spin
wave expansion, which was the case in the main text where the external field
had already been integrated out to yield the $\text n_a \text n_b$
mass term. We also included an external uniform field which
lies in the $y-z$ plane.

Expanding around the $x$-ordered state, using that $\text m_x = - \ms m_y \ms n_y - \ms m_z\ms n_z$ and
$\text n_x = 1 - \frac{1}{2} (\ms n_y^2+\ms n_z^2 + \frac{1}{n_0^2} (\ms m_y^2+\ms m_z^2))$, yields

\begin{align}
\label{eq:gap4}
\mathcal{H}_{\rm NLS}^{g,h}& = \frac{1}{2\chi} \left(\ms m_y^2+\ms m_z^2\right) + 2 g J
\mathfrak{a}^2  \left(2 \ms m_z^2  - \ms m_y^2\right)  - 2 g J
\frac{\mu_0^2}{\mathfrak{a}^2} \left(2 \ms n_z^2 - \ms n_y ^2\right)
-h_y \ms m_y - h_z \ms m_z  \nonumber\\
& +\left(\frac{1}{2\chi} - 2 g J
\mathfrak{a}^2 \right) \left( \ms m_y^2 \ms n_y^2 + \ms m_z^2 \ms n_z^2 + 2 \ms m_y \ms m_z \ms n_y
\ms n_z\right) + 2 g J
\frac{\mu_0^2}{\mathfrak{a}^2}\left[1 - \frac{1}{2} (\ms n_y^2+\ms n_z^2 +
\frac{1}{n_0^2} (\ms m_y^2+\ms m_z^2))\right]^2.
\end{align}

Note that the first term on the second line is of the form
$(\mb{m}_\perp\cdot\mb{n}_\perp)^2$, where the $\perp$ indicates the
components of the vectors normal to the ordering direction. Since we
in the next step shift the magnetization by its value induced by the
field, this is proportional to $(\mb{h}\cdot\mb{n})^2$, as is postulated
in the main text on symmetry grounds.

We now show this explicitly. We shift the definition
$\ms m_a = m_a + \chi_a h_a$ for $a=y,z$,
and expand the result to quadratic order in $m, n$.  Here
$\chi_z = (1/\chi + 4 g J \mathfrak{a}^2)^{-1}$ and
$\chi_y = (1/\chi - 2 g J \mathfrak{a}^2)^{-1}$.

This gives

\begin{align}
\label{eq:gap5}
\mathcal{H}_{\rm NLS}^{g,h} & = \frac{1}{2\chi} \left(m_y^2+m_z^2\right) + 2 g J
\mathfrak{a}^2  \left(2 m_z^2  - m_y^2\right)  - 2 g J
\frac{\mu_0^2}{\mathfrak{a}^2} \left(2 n_z^2 - n_y ^2\right)
\nonumber \\
& + \left(\frac{1}{2\chi} - 2 g J
\mathfrak{a}^2 \right) \left( \chi_y^2 h_y^2 n_y^2 + \chi_z^2 h_z^2
n_z^2 +2 \chi_y \chi_z h_y h_z n_y n_z\right) - 2 g J
\frac{\mu_0^2}{\mathfrak{a}^2} \left[ n_y^2+ n_z^2 + \cdots\right],
\end{align}

where the `$\cdots$' in the last brackets account for terms higher order in
field, magnetization fluctuations, etc.

The anisotropy coefficients, denoted by $\Gamma_{ab}$ in the text, can now be extracted.
The terms in $\mathcal{H}_{\rm NLS}^{g,h}$ which are quadratic in the $n_y,n_z$ fields read

\begin{align}
\label{eq:gap6}
\mathcal{H}_{nn} = \chi_y^2 h_y^2  \left(\frac{1}{2\chi} - 2 g J
\mathfrak{a}^2 \right) n_y^2 + \left[  \chi_z^2 h_z^2  \left(\frac{1}{2\chi} - 2 g J
\mathfrak{a}^2 \right) - 6 g J \frac{\mu_0^2}{\mathfrak{a}^2} \right]
n_z^2 + 2 \chi_y \chi_z h_y h_z \left(\frac{1}{2\chi} - 2 g J
\mathfrak{a}^2 \right) n_y n_z.
\end{align}

Note that the two terms proportional to $n_y^2$ from the right-most
contributions on each line of Eq. (<ref>) above canceled.  That
means the the coefficient of $n_y^2$ in Eq. (<ref>) vanishes if
$h_y=0$.  This occurs because of Goldstone's theorem and the assumed
XXZ form of the anisotropy: if the field is purely along the $z$
direction, XY symmetry of the Hamiltonian under rotations about the
$z$ axis is preserved, and this makes one of the spin wave modes
remain gapless.  Conversely, for a field along the $y$ direction, and
in the presence of anisotropy, both modes are generally gapped.

We can simplify the above expression if we assume $|g| \ll 1$, which
means $\chi_y^{-1} \approx \chi_z^{-1} \approx \chi^{-1} = 4\mf a^2J$
and therefore $1/\chi \gg g J\mathfrak{a}^2$; hence

\begin{align}
\label{eq:gap7}
\mathcal{H}_{nn} \approx \frac{\chi h_y^2}{2}  n_y^2 + \left[\frac{\chi h_z^2}{2} - 6 g J \frac{\mu_0^2}{\mathfrak{a}^2}\right]
n_z^2 +  \chi h_y h_z  n_y n_z.
\end{align}

The above shows that if $h_z$ is small or zero, stability
requires $g<0$.  This can be understood from the fact that, if the field
is along $y$, then $H_{\rm XXZ}$ is the only term, in the pure spin hamiltonian,
breaking explicitly the $O(2)$ symmetry in the $x-z$ plane.
It should therefore favor antiferromagnetic alignment along the $x$ axis,
which is the initial assumption of this derivation. It also proves the $\frac{\chi}{2}$
prefactor used in the main text.

The coefficients in Eq. (<ref>) give contributions to $\Gamma_{yy}$, $\Gamma_{zz}$
and $\Gamma_{yz}$, respectively. In this Appendix, as opposed to the more general expressions
given in the main text, we assume they are the only contribution.

Since taking the magnetic field purely along one of the two axes $y,z$ guarantees that $\Gamma_{yz}=0$,
so that (as explained in the main text) the two magnon valleys are independent,
let us assume that the field is along the $y$ axis.
Then one gap is $\Delta_1= |h_y|$, the Zeeman energy associated
with the field along $y$.  The other gap gets contributions both from
the anisotropy and the Zeeman energy associated
with the field along $z$.

Note that the anisotropy-induced gap involves the square
root of the anisotropy, i.e. $\Delta_0\big |_{h_z=0} = 4\sqrt{3|g|} J \mu_0$,
which is not necessarily very small for reasonably small values of $g$.

§ APPLICATION—SUPPLEMENTARY FIGURES

Here we present further calculations of scattering rates and (diagonal) thermal conductivity for the model of Sec. <ref>, as supplemental figures.

$|\mb k|={\mt {0.0625}}/\mf a$
$|\mb k|={\mt {0.125}}/\mf a$
$|\mb k|={\mt {0.25}}/\mf a$

$|\mb k|={\mt {0.5}}/\mf a$                             $|\mb k|={\mt {1.0}}/\mf a$
$|\mb k|={\mt {2.0}}/\mf a$

Diagonal scattering rate $D_{n\mb k}$ with respect to $\theta(\mb k)\in [0,\pi/2]$
(horizontal axis)
and $\phi(\mb k) \in [0,2\pi]$ (vertical axis) for fixed temperature $T=0.5\,T_0$, polarization
$n=0$, and momentum (1) $|\mb k|={\mt {0.0625}}/\mf a$, (2) $|\mb k|={\mt {0.125}}/\mf a$,
(3) $|\mb k|={\mt {0.25}}/\mf a$, (4) $|\mb k|={\mt {0.5}}/\mf a$, (5) $|\mb k|={\mt {1.0}}/\mf a$,
(6) $|\mb k|={\mt {2.0}}/\mf a$. Color scales vary from figure to figure. Note that the $C_4$ symmetry is
approximately preserved for small $|\mb k|$ but broken at large $|\mb k|$, as stated in the main text.
Also note how scattering processes at $\theta(\mb k)<\theta_-$
become allowed for $\omega_{n\mb k}\geq 2\Delta,2\Delta'$,
then dominant at large $|\mb k|$.

$n=0$                                 $n=1$                                 $n=2$

Diagonal scattering rate $D_{n\mb k}$ with respect to $\theta(\mb k)\in [0,\pi/2]$ (horizontal axis)
and $\phi(\mb k) \in [0,2\pi]$ (vertical axis) for fixed temperature $T=0.5T_0$, momentum
$|\mb k|={\mt {0.5}}/\mf a$, and polarizations (1) $n=0$, (2) $n=1$, (3) $n=2$.
Color scales are different in (1) and (2,3). Subfigure (1) is reproduced from the main text.
Note that with our choice of polarization vectors $\bs \varepsilon_{n,\mb k}$, results for $n=1$ and $n=2$
are simply related by the mirror symmetry $\phi \mapsto \pi -
\phi$.

$\phi(\mb k)=0$ ; $n=1$                     $\phi(\mb k)=\pi/2$ ; $n=0$                     $\phi(\mb k)=\pi/2$ ; $n=1$

Diagonal scattering rate $D_{n\mb k}$ with respect to $\theta(\mb k)\in [0,\pi/2]$ (horizontal axis)
and $|\mb k|\mf a$ (vertical axis) for fixed temperature $T= \mt{0.5} T_0$ and (1) $\phi(\mb k)=0$ and $n=1$,
(2) $\phi(\mb k)=\pi/2$ and $n=0$, (3) $\phi(\mb k)=\pi/2$ and $n=1$.
Color scales are different for the three subfigures. The $\phi(\mb k)=0$ and $n=0$ case
is displayed in the main text. Note that polarizations $n=1$ and $n=2$ yield the same results here.
Note also that the general features are the same for polarizations $n=1,2$ as for $n=0$: although the scattering rates
of $n=1,2$ polarizations
for energies $\omega_{n\mb k}\gtrsim 2\Delta$ are not as clearly visible as they are for $n=0$, they are finite (of order $10^{-4}$
in our units) and are only parametrically smaller than those for the $n=0$ polarization, due to purely geometrical factors
($\mc S_{n\mb k}^{q;\alpha\beta}$ in the main text).

Longitudinal thermal conductivity $\kappa_L$ with respect to temperature $T$, in log-log scale,
(left) for four different values of
$\gamma_{\rm ext}= 1\cdot 10^{-z}(v_{\rm ph}/\mathfrak{a}), z \in \llbracket 4, 7\rrbracket$,
from darker $(z=4)$ to lighter $(z=7)$ shade, (right) for four different values
$\gamma_{\rm ext}= 1\cdot 10^{-z}(v_{\rm ph}/\mathfrak{a}), z \in \llbracket 6, 9\rrbracket$,
from darker $(z=6)$ to lighter $(z=9)$ shade. Note that the two “bumps” come from the competition
between $\gamma_{\rm ext}$ and $D_{nn,\ell}$ for valley index $\ell = 0,1$, as explained in the main text.

$(q,q')=(-,-)$ ; $\phi(\mb k) = 0$              $(q,q')=(-,+)$ ; $\phi(\mb k) = 0$              $(q,q')=(-,+)$ ; $\phi(\mb k) = \pi/2$

Skew-scattering rates (1) $\mf W^{\ominus,--}_{n\mb k n'\mb k'}$ and (2,3) $\mf W^{\ominus,-+}_{n\mb k n'\mb k'}$,
with respect to $\theta(\mb k')\in [0,\pi/2]$ (horizontal axis) and
$\varphi(\mb k,\mb k')=\phi(\mb k')-\phi(\mb k)$ (vertical axis), for fixed magnetization $\bs m_0=\mt{0.05} \hat{\bs z}$, temperature $T=\mt{0.5} T_0$,
momentum $|\mb k'|=\mt{0.8}/\mf a$, $k_z = \mt{0.1}/\mf a$, and (1,2) $k_x=\mt{0.2}/\mf a$,
$k_y = 0$, (3) $k_x = 0$, $k_y=\mt{0.2}/\mf a$. The case $\mf
W^{\ominus,--}_{n\mb k n'\mb k'}, k_x=0, k_y = \mt{0.2}/\mf a$ is in the main text.
The colorbars are different for each figure and not linearly scaled. Note that thanks to anti-detailed-balance,
angular dependences of $\mf W^{\ominus,++}_{n\mb k n'\mb k'},\mf W^{\ominus,+-}_{n\mb k n'\mb k'}$
are identical to those of $\mf W^{\ominus,--}_{n\mb k n'\mb k'},\mf W^{\ominus,-+}_{n\mb k n'\mb k'}$,
respectively, for an isotropic phonon dispersion.


# MIMHD: Accurate and Efficient Hyperdimensional Inference Using Multi-Bit In-
Memory Computing

Arman Kazemi∗⋆, Mohammad Mehdi Sharifi∗, Zhuowen Zouψ, Michael Niemier∗, X.
Sharon Hu∗, Mohsen Imani†
∗University of Notre Dame, ψUniversity of California San Diego, †University of
California Irvine<EMAIL_ADDRESS>
###### Abstract

Hyperdimensional Computing (HDC) is an emerging computational framework that
mimics important brain functions by operating over high-dimensional vectors,
called hypervectors (HVs). In-memory computing implementations of HDC are
desirable since they can significantly reduce data transfer overheads. All
existing in-memory HDC platforms consider binary HVs where each dimension is
represented with a single bit. However, utilizing multi-bit HVs allows HDC to
achieve acceptable accuracies in lower dimensions which in turn leads to
higher energy efficiencies. Thus, we propose a highly accurate and efficient
multi-bit in-memory HDC inference platform called MIMHD. MIMHD supports multi-
bit operations using ferroelectric field-effect transistor (FeFET) crossbar
arrays for multiply-and-add and FeFET multi-bit content-addressable memories
for associative search. We also introduce a novel hardware-aware retraining
framework (HWART) that trains the HDC model to learn to work with MIMHD. For
six popular datasets and 4000 dimension HVs, MIMHD using 3-bit (2-bit)
precision HVs achieves (i) average accuracies of 92.6% (88.9%) which is 8.5%
(4.8%) higher than binary implementations; (ii) 84.1$\times$ (78.6$\times$)
energy improvement over a GPU, and (iii) 38.4$\times$ (34.3$\times$) speedup
over a GPU, respectively. The 3-bit MIMHD is 4.3$\times$ and 13$\times$ faster
and more energy-efficient than binary HDC accelerators while achieving similar
accuracies.

††publicationid: pubid: 978-1-6654-3922-0/21/$31.00 $\copyright$2021 IEEE

##  I Introduction

Hyperdimensional Computing (HDC) is an emerging computational framework based
on the observation that the brain operates in hyperdimensional spaces over
high-dimensional vectors to perform cognition and perception [1]. The
computational units of HDC are high-dimensional vectors called hypervectors
(HVs). HDC is applicable to a wide range of applications such as robotics [2],
machine learning, and cognitive computing. HDC achieves similar or higher
accuracy than convolutional neural networks (CNNs), support vector machines
(SVMs), and extreme gradient boosting for applications such as distributed
sensors [3], multimodal sensor fusion[4], and biomedical signal processing
[5].

Another important advantage of HDC over other machine learning models such as
CNNs and SVMs is the higher energy efficiency[6, 7]. HDC operates over high-
dimensional HVs and can greatly benefit from the high parallelism offered by
processing in-memory (PIM) architectures and offer even more energy and
latency improvements [8, 9]. In addition, the inherent robustness of HDC to
computational errors makes it amenable to PIM architectures based on emerging
devices with variability issues. However, previous PIM HDC works in the
literature has focused on binary implementations of HDC which suffer from
accuracy loss compared to floating point models and require dimensions higher
that 10,000 which thwarts achieving high energy efficiencies [10, 11].

We observe that using 2 or 3-bit HVs could improve HDC’s inference accuracy to
match that of a high precision model. This is inline with the fact that the
synapses in the brain have 4.7 bits of precision [12]. Furthermore, using
multi-bit HVs allows reducing the number of dimensions at iso-accuracy which
in turn offers significant energy savings. One of the most prominent
impediments to designing multi-bit HDC accelerators is associative search
based on cosine distance. Binary HDC implementations leverage Hamming distance
measurements using ternary content-addressable memories (TCAMs) to avoid
cosine distance. On the other hand, multiple binary storage elements are
required to store multi-bit HVs which reduces energy efficiency gains and adds
more complexity.

Recently, ferroelectric field-effect transistor(FeFET) multi-bit content-
addressable memories (MCAMs) were proposed that can perform single-step
associative search based on the MCAM distance metric [13]. Furthermore, multi-
level FeFETs for storage and computation have been experimentally demonstrated
[14]. These advances make FeFETs a suitable candidate for designing PIM based
multi-bit HDC accelerators.

In this paper, we introduce MIMHD, an in-memory implementation of HDC
inference. MIMHD achieves comparable accuracy as high-bit precision HDC while
offering significant energy and latency reduction. MIMHD directly represents
each dimension of a HV using multi-bit FeFETs and supports essential HDC
inference operations, i.e., encoding and associative search. MIMHD uses FeFET
crossbar arrays [14] to implement encoding, and FeFET MCAMs [13] for
associative search. We propose a novel hardware-aware retraining (HWART)
algorithm that trains the HDC model to compensate for possible accuracy loss
due to the MCAM distance metric. We evaluate the efficiency and accuracy of
MIMHD over a wide range of classification problems. For six popular datasets,
MIMHD using 2 and 3-bit precisions achieves (i) average accuracies of 88.9%
and 92.6% which is similar to high precision (8-bit) implementations, (ii)
84.1$\times$ and 78.6$\times$ energy improvement over a GPU, and (iii)
38.4$\times$ and 34.3$\times$ speedup over a GPU, respectively. 3-bit MIMHD is
4.3$\times$ and 13$\times$ faster and more energy-efficient than the binary
HDC accelerators [15].

##  II Background and Related Work

###  II-A Hyperdimensional Computing Basics

HDC is a neuro-inspired model of computation based on the observation that the
human brain operates on high-dimensional and distributed representations of
data [1] called hypervectors (HVs). Researchers have applied HDC to diverse
cognitive tasks, such as robotics [2], classification [16], bio-signal
processing [17], and regression [18]. Most of the previous work use binary
HVs. Below, we discuss an overview of HDC.

Figure 1: (a) An overview of HDC classification, (b) HDC encoding module.

####  II-A1 HDC Encoding

The first step in HDC is to map each data point into high-dimensional space.
The mapping procedure is often referred to as encoding (shown in Fig. 1b). The
encoding is such that the data points different from each other in the
original space are also different in the HDC space. To explain the encoding
process, assume an input feature vector (an image, voice, etc.) in the
original space $\vec{{F}}=\\{f_{1},~{}f_{2},\cdots,f_{n}\\}$ and
$\vec{F}\in\mathcal{R}^{n}$. HDC first linearly quantizes the feature vector
values into $m$ levels, $\\{\bar{q}_{1},\bar{q}_{2},\cdots,\bar{q}_{m}$}, and
assigns a level HV to each quantization level,
$\\{\vec{L}_{1},\vec{L}_{2},\cdots,\vec{L}_{m}\\}$. The level HVs are
generated such that $\vec{L}_{1}$ and $\vec{L}_{m}$ are orthogonal
$\delta(\vec{{L}}_{1},\vec{{L}}_{m})\simeq 0$, but the intermediate levels
follow a spectrum of similarity, depending on their physical closeness to
$\bar{q}_{1}$ and $\bar{q}_{m}$. The encoding module maps $\vec{{F}}$ into a
high-dimensional vector $\vec{\mathcal{H}}\in\\{0,1\\}^{D}$, where
$\mathcal{D}>>n$. The following equation and Fig. 1b show an encoding method
that maps an input vector into high-dimensional space [6]:

$\vec{\mathcal{H}}=\sum_{k=1}^{n}{L(f_{k})}*\vec{\mathcal{B}}_{k}$ (1)

where base HVs ($\vec{\mathcal{B}}_{k}$s) retain the spatial or temporal
location of the features. Base HVs are randomly chosen, hence orthogonal. Base
HVs are also of dimension $\mathcal{D}$. Function $L(f_{k})$ assigns the
corresponding level HV to the features.

####  II-A2 HDC Training

To find the universal property for each class in the training dataset, the
trainer module linearly combines HVs belonging to each class, i.e., adding the
HVs to create a single HV for each class. Once all HVs are combined, we treat
per-class accumulated HVs, called class HVs, as the learned model. Fig. 1a
shows HDC operation during single-pass training. Assuming a problem with $k$
classes,
$\mathcal{M}=\\{\vec{\mathcal{C}_{1}},\vec{\mathcal{C}_{2}},\cdots,\vec{\mathcal{C}_{k}}\\}$
represents the model. For example, after encoding all of training data
belonging to class/label $l$, the class HV $\vec{\mathcal{C}^{l}}$ can be
obtained by bundling (adding) all $\vec{\mathcal{H}}^{l}$s. Assuming there are
$\mathcal{J}$ inputs with label $l$:
$\vec{\mathcal{C}^{l}}=\sum_{j}^{\mathcal{J}}{\vec{\mathcal{H}^{l}_{j}}}$.

####  II-A3 HDC Inference

For inference, the model checks the similarity of each encoded test data with
the class HVs in two steps. The first step encodes the input (the same
encoding used for training) to produce a query HV $\vec{\mathcal{H}}$. Then,
as Fig. 1a shows, we compute the similarity ($\delta$) of $\vec{\mathcal{H}}$
and all class HVs. Query data is labeled with the label of the class HV with
the highest similarity.

###  II-B HDC Acceleration

Prior work showed how the binary HDC models could be accelerated using PIM
platforms [15, 19]. For example, work in [15, 8, 9] showed that PIM
architectures leveraging CAMs can accelerate HDC associative search by two
orders of magnitude. CAMs enable row-parallel associative search over binary
HVs with the Hamming distance metric. However, using non-binary HVs cannot
readily exploit current PIM architectures to accelerate HDC inference. A
multi-bit PIM structure that supports associative search could be highly
desirable for accelerating accurate HDC inference.

###  II-C Computing with Multi-Bit FeFETs

The FeFET device is a promising candidate for PIM architectures. FeFETs are
made by integrating a ferroelectric (FE) layer in the gate stack of a MOSFET
[20] (Fig. 2a). The threshold voltage ($V_{th}$) of the FeFET is determined by
the non-volatile polarization of the FE layer. Applying voltage pulses to the
gate of the FeFET changes the polarization of the FE layer and consequently
the $V_{th}$ of the FeFET device [14]. Fig. 2b shows the transfer
characteristics of a FeFET device programmed to 8 different $V_{th}$ states,
i.e., storing 3-bits of information [13].

FeFET crossbar arrays have been used to accelerate neural network inference
and training [21, 14]. Work in [13] proposed a FeFET multi-bit CAM (MCAM),
shown in Fig. 2c, that can perform parallel in-memory associative search based
on the MCAM distance metric. FeFET MCAM cells store different states by
programming $V_{th-Hi}$ and $\overline{V_{th-Lo}}$ and search using voltage
signals on the data lines ($DL$ and $\overline{DL}$). Fig. 2d shows the MCAM
distance metric for a single cell where the x-axis represents the distance
between the stored state and the search input, and the y-axis indicates the
conductance of the cell (which directly represents similarity). The
conductance grows exponentially with respect to the distance and saturates
towards the higher distance. This behavior results from the transfer
characteristics of the FeFETs. By sensing the match-lines (ML in Fig. 2c),
which are connected to all cells in a row (Fig. 2e), we can find the MCAM row
with the lowest conductance (data most similar to the search query). Although
the MCAM distance metric is not a previously known and commonly used metric
like cosine or Euclidean, it achieves high accuracies for different
applications [13]. Furthermore, technologies like resistive RAM are amenable
to multi-bit crossbar arrays, but they do not implement a CAM-based multi-bit
associative search with a useful distance metric.

Figure 2: (a) FeFET device schematic [20]; (b) Transfer characteristics of a
FeFET programmed to 8 distinct states; (c) Schematic of the FeFET MCAM cell
[13]; (d) MCAM distance metric from [13]; (e) MCAM array schematic.

##  III Proposed MIMHD

In this section, we present MIMHD, an accurate and efficient in-memory
platform for HDC inference. We exploit multi-bit FeFET devices to directly
support 1, 2, and 3-bit precision implementation of HDC. We leverage PIM to
implement encoding and associative search operations to achieve high energy
and latency efficiency. Furthermore, we introduce a novel hardware-aware
retraining framework (HWART) that trains the HDC model to adapt to the unique
features of FeFET MCAMs in MIMHD. The overall structure of MIMHD is shown in
Fig. 3 which consists of an encoding and an associative search module.

###  III-A Encoding

As discussed in Section II-A1, the goal of encoding is to transform the data
into a high-dimensional space. Using a multi-bit encoding, we increase the
information representation capacity of the HVs, compared to binary
implementations. The encoding module of MIMHD operates in multi-bit precision
and works with multi-bit level HVs and base HVs. To generate $m$ level HVs we
generate a random $\mathcal{D}$-dimensional $P$-bit ($P\in\\{1,2,3\\}$ denotes
the bit precision) HV to represent level 1. To generate the HV for level 2, we
randomly choose $\mathcal{D}/m$ dimensions and randomly assign them values
from $1$ to $2^{P}$. We continue this process until we have generated $m$
level HVs. Generating level HVs only happens once during training when $P$ and
$m$ are chosen. As such, $f_{k}$’s in equation 1 are represented with their
corresponding level HVs. Base HVs are chosen randomly (Section II-A1) and are
also fixed.

Fig. 3a shows the encoding module of MIMHD. We utilize FeFET crossbar arrays
to store the level HVs. We need to store a total of $n$ HVs of size
$m\times\mathcal{D}$ for a dataset with $n$ features. We consider $m=64$ to
support up to 64 level HVs and minimize accuracy loss due to HV quantization.
Given that $\mathcal{D}$ is in the order of thousands, we use $\mathcal{D}/64$
crossbar arrays of size $64\times 64$ for each feature since the computation
is independent for each dimension. Each $i-th$ group of crossbar arrays
($i\in[1,2,\dots,n]$) activates the rows containing the level HV corresponding
to $f_{i}$ and performs a multiplication between that level HV and the
$i_{th}$ base HV ($\vec{\mathcal{B}}_{i}$) as its input. Source lines ($SL$s)
add the resulting currents from these multiplications based on Kirchhoff’s law
and feed them to the analog-to-digital converters (ADCs). This current-based
addition ensures that we do not need to store intermediate values during the
encoding; avoiding expensive FeFET writes. The ADCs are used to generate a
$P$-bit $\mathcal{D}$-dimensional encoded HV.

Figure 3: MIMHD supports all required operations for HDC inference: (a) the
encoding module consists of $n$ crossbar array groups with $\mathcal{D}/64$
crossbar arrays of size $64\times 64$; (b) the associative search module
consists of MCAM blocks that store $k$ class HVs.

###  III-B Associative Search

Associative search is the final step in HDC inference where the class HVs are
compared with the query HV. Associative search is arguably the most
challenging operation of multi-bit HDC to implement with PIM. TCAMs can
realize associative search with Hamming distance, but they are only suitable
for binary implementations. Thus, we utilize FeFET MCAMs [13] for associative
search in MIMHD. Given the high dimensionality of the class HVs, we use
multiple MCAM arrays of size $64\times 64$. The MCAMs store the $k$ trained
class HVs in their rows (Fig. 3b). The encoded query is routed to the MCAMs as
input to perform associative search with the stored class HVs based on the
MCAM distance metric. We add the currents from the same rows of the different
MCAMs to get the distance for $\mathcal{D}$ dimensions following the
implementation in [9]. Then sense amplifies in [15] and used to detect the row
that is the most similar to the encoded query (lowest current) and report the
corresponding class as the result of the prediction.

The utilized MCAM cell (Fig. 2c) also supports Hamming distance search [22].
This allows MIMHD to support the binary implementation of HDC by programming
the FeFET crossbar arrays in a binary fashion. All peripherals that support
multi-bit operations in MIMHD support the binary operations as well. The only
difference between the binary and multi-bit operations is using Hamming
distance for associative search for the former and the MCAM distance metric
for the latter.

Figure 4: HWART framework adapts HDC models to work with the MCAM distance
metric supported by MIMHD.

###  III-C Hardware-Aware Retraining (HWART)

The HDC training method has been theoretically designed to work with the
cosine distance. MIMHD uses the MCAM distance metric for associative search,
which can result in a loss of classification accuracy. In order to compensate
for this possible quality loss, we propose HWART, a retraining method that
teaches the model to work with a new distance metric.

Fig. 4 shows an overview of the proposed HWART. We store two copies of the HDC
model: quantized and non-quantized. The non-quantized model allows for a fine-
grained accumulation of information and the quantized model mimics the forward
pass of inference. During the following two phases, we teach the model to work
with the quantization constraints: (i) feed-forward phase that checks the
similarity of an encoded training data point with the quantized model using
the MCAM distance metric, and (ii) model update phase that adjusts the non-
quantized HDC model for each mispredicted sample. For each encoded training
data point $\vec{\mathcal{H}}$, we check the similarity of $\vec{\mathcal{H}}$
with all class HVs stored in the quantized model to perform the prediction
($\mathbin{\vphantom{\circledast}\text{\ooalign{\smash{\raisebox{-2.58334pt}{\scalebox{2.3}{$\bullet$}}}\cr\smash{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bf\footnotesize$A$}}\cr}}}$).
This similarity search is based on the MCAM distance metric derived from the
FeFET
MCAM($\mathbin{\vphantom{\circledast}\text{\ooalign{\smash{\raisebox{-2.58334pt}{\scalebox{2.3}{$\bullet$}}}\cr\smash{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bf\footnotesize$B$}}\cr}}}$).
Then, we examine if the model returns the correct label $l$ for
$\vec{\mathcal{H}}$. If the model mispredicts it as label $l^{\prime}$, the
non-quantized model is updated as follows
($\mathbin{\vphantom{\circledast}\text{\ooalign{\smash{\raisebox{-2.58334pt}{\scalebox{2.3}{$\bullet$}}}\cr\smash{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bf\footnotesize$C$}}\cr}}}\mathbin{\vphantom{\circledast}\text{\ooalign{\smash{\raisebox{-2.58334pt}{\scalebox{2.3}{$\bullet$}}}\cr\smash{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bf\footnotesize$D$}}\cr}}}$).

$\begin{split}\vec{\mathcal{C}_{l}}\leftarrow\vec{\mathcal{C}_{l}}+\eta~{}(\delta_{l^{\prime}}-\delta_{l})\times\mathcal{\vec{H}}\\\
\vec{\mathcal{C}_{l^{\prime}}}\leftarrow\vec{\mathcal{C}_{l^{\prime}}}-\eta~{}(\delta_{l^{\prime}}-\delta_{l})\times\mathcal{\vec{H}}\end{split}$
(2)

where $\delta_{l}=\delta(\vec{\mathcal{H}},\vec{\mathcal{C}_{l}})$ and
$\delta_{l^{\prime}}=\delta(\vec{\mathcal{H}},\vec{\mathcal{C}_{l^{\prime}}})$
are the similarity of the query with correct and miss-predicted classes,
respectively. Thus, we ensure that we update the model based on how “far” a
training data point is miss-classified with the current model. In case of a
very far miss-prediction, $\delta_{l^{\prime}}>>\delta_{l}$, we make greater
changes to the non-quantized model; while in case of a marginal miss-
prediction, $\delta_{l^{\prime}}\simeq\delta_{l}$, the update makes lesser
changes to the non-quantized model.

After updating the non-quantized model over a batch of training data, we
update the quantized model by mapping each model element of non-quantized
model to the desired number of bits
($\mathbin{\vphantom{\circledast}\text{\ooalign{\smash{\raisebox{-2.58334pt}{\scalebox{2.3}{$\bullet$}}}\cr\smash{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bf\footnotesize$E$}}\cr}}}$).
We continue this adaptive iterative procedure until the classification
accuracy stabilizes (Fig. 5).

##  IV Evaluation

###  IV-A Experimental Setup

We implement MIMHD both in software and hardware simulation. In software, we
implemented and verified MIMHD functionality with a Python implementation. To
evaluate MIMHD accuracy, we design a cycle-accurate simulator based on our
framework in PyTorch [23]. Our framework translates all PyTorch code directly
to vector-based operations supported by our PIM architecture. For the hardware
design, we use NeuroSim [24] to obtain energy and latency results for the
crossbar arrays including all required peripherals. We use HSPICE and the
FeFETs Preisach model [20] to measure the energy consumption and latency of
the FeFET MCAMs and other peripherals. All the hardware components are based
on a 22nm technology node.

Figure 5: MIMHD accuracy during different HWART iterations.

We compare the energy and latency consumption of MIMHD with a NVIDIA GTX 1080
GPU running the HDC models with 8-bit precision. The performance and energy of
the GPU are measured by the `nvidia-smi` tool. We evaluate the accuracy and
efficiency of MIMHD on six popular datasets (listed in Table I), ranging from
small datasets collected in a small IoT network to large datasets which
includes hundreds of thousands of images of facial data.

TABLE I: Datasets ($n$: feature size, $k$: number of classes)

| $n$ | $k$ |  Train Size |  Test Size | Description
---|---|---|---|---|---
MNIST | 784 | 10 | 60,000 | 10,000 | Handwritten Recognition[25, 26]
UCIHAR | 561 | 12 | 6,213 | 1,554 | Activity Recognition(Mobile)[27]
ISOLET | 617 | 26 | 6,238 | 1,559 | Voice Recognition [28]
PAMAP | 75 | 5 | 611,142 | 101,582 | Activity Recognition(IMU) [29]
FACE | 608 | 2 | 522,441 | 2,494 | Face Recognition[30]
PECAN | 312 | 3 | 22,290 | 5,574 | Urban Electricity Prediction [31]

###  IV-B MIMHD Accuracy

MIMHD accuracy depends on three parameters: (i) the HV dimensionality that
determines the HV capacity and the level of redundancy, (ii) the precision of
each HV element, and (iii) the distance metric used for associative search.

####  IV-B1 Dimensionality

Fig. 6a compares HDC classification accuracy for different dimensionalities
for face detection applications. Other datasets exhibit similar trends. Our
evaluation shows that increasing dimensionality results in improving the
classification accuracy for all configurations. In the low dimensional space,
the binary model is much less accurate than the higher precision models. This
accuracy gap shrinks by scaling the dimensionality. For example, the 1-bit
precision model provides 17.9% and 4.2% lower accuracy than the 8-bit model in
$\mathcal{D}{=}1k$ and $\mathcal{D}{=}10k$, respectively. Fig. 6b reports the
maximum classification accuracy achieved by the HDC models using different bit
precisions and the lowest dimensionality that provides such accuracy. The
results are averaged over all tested datasets. Our evaluation shows that
MIMHD, using multi-bit elements, reduces the number of required dimensions to
achieve maximum accuracy compared to the binary model. MIMHD provides maximum
classification accuracy in $\mathcal{D}{=}4k$ ($\mathcal{D}{=}8k$) for 3-bit
(2-bit) precision. The results also indicate that, regardless of
dimensionality, the binary HDC model using Hamming distance cannot achieve the
maximum accuracy provided by the full precision (8-bit) model. In contrast,
MIMHD can provide the same accuracy as the full precision model while enabling
a fully in-memory implementation.

####  IV-B2 Hypervector Precision

Fig. 7a compares the classification accuracy of HDC models in different
precisions using different search metrics/hardware. All results are reported
using the same dimensionality ($\mathcal{D}{=}4k$). MIMHD is trained with
HWART. Results show that HDC classification accuracy highly depends on model
precision. MIMHD in 2-bit and 3-bit precisions provides, on average across all
datasets, 8.3% and 4.7% higher classification accuracy as compared to the
binary model. Moreover, 3-bit MIMHD can achieve the same accuracies as an
8-bit model using cosine distance.

####  IV-B3 Distance Metric

Fig. 7b shows the quality loss of models trained with cosine distance and
deployed on MIMHD, due to MIMHD’s using MCAM distance metric. The results
indicate that, in the same bit precision without HWART, 2 and 3-bit MIMHD
provides, on average, 1.77% and 2.7% lower accuracy than the 8-bit HDC model
with cosine distance. Note that the accuracy of 2-bit and 3-bit MIMHD models
are still 6.5% and 3.2% higher than the binary model using the Hamming
distance metric. As Fig. 7b shows, enhancing the model with HWART can
compensate for the quality loss caused by the MCAM distance metric. For
example, MIMHD 3-bit achieves a similar accuracy as HDC with 3 and 8-bit
precisions using cosine distance.

Figure 6: Impact of dimensionality on MIMHD accuracy.

###  IV-C MIMHD Efficiency

We compare the performance of MIMHD with a NVIDIA GTX 1080 GPU in terms of
energy and latency since GPUs are the most common computing fabrics for HDC.
We compare MIMHD, which is entirely FeFET-based, in 1, 2, and 3-bit precision
with the GPU to illustrate the efficiency trends of operating in different bit
precisions with the same technology. We further compare the performance of
MIMHD with state-of-the-art PIM-based HDC platforms: AHAM [9] that uses a
Loser-Take All circuit to support Hamming distance search, and SearcHD [15]
that exploits ganged-inverters and inverse CAM to support the same search
functionality. Our evaluations are performed in two setups: (i) Maximum
Accuracy, at maximum achievable accuracy of the designs, and (ii) Same
Accuracy, when all designs achieve the same accuracy, i.e., the maximum
accuracy of a binary design.

####  IV-C1 Energy

Fig. 8a shows the energy improvement results of MIMHD over the GPU in
different bit precisions and for different tasks (Table I). Due to its memory-
centric nature, MIMHD in all bit precisions, achieves high energy improvements
over the GPU by performing computations in-memory. The difference in energy
improvement of MIMHD in different bit precisions is mainly due to the overhead
of the ADCs as higher precision ADCs consume higher energy. On average, for
the considered tasks, MIMHD using 1, 2, and 3 bits achieves $90\times$,
$84\times$, and $78\times$ improvements over the GPU.

Figure 7: (a) Impact of HV precision on HDC classification accuracy, (b) MIMHD
quality loss vs. cosine metric with and without HWART retraining. Figure 8:
(a) and (b) show the energy improvement and speedup of MIMHD over the GPU for
$\mathcal{D}=4000$; (c) and (d) compare MIMHD with state-of-the-art binary HDC
accelerators in maximum accuracy and same accuracy setups in terms of energy
improvement and speedup over the GPU.

Fig. 8c compares the average energy improvement of AHAM [9], SearcHD [15], and
2 and 3-bit MIMHD over the GPU. The most important factor that affects these
results is the dimensionality. At Maximum Accuracy, 3-bit MIMHD operates in
$\mathcal{D}{=}4k$ while AHAM and SearcHD operate in $\mathcal{D}{=}16k$ (Fig.
6b), but still MIMHD achieves a higher accuracy. Dimensionality has a linear
relation with energy efficiency as it requires more resources, e.g., crossbar
arrays. At Same Accuracy, the 3-bit MIMHD outperforms other implementations by
a large margin as it only needs $\mathcal{D}{=}1k$ to achieve the same
accuracy as binary implementations. The 3-bit MIMHD outperforms the state of
the art, i.e., SearcHD, by $3.27\times$ and $13\times$ in Max Accuracy and
Same Accuracy setups, respectively.

####  IV-C2 Latency

Fig. 8b shows the speedup results of MIMHD in different bit precisions over
the GPU for the tasks in Table I. MIMHD leverages highly parallel computation
for encoding and associative search to achieve high speedups over the GPU. The
encoding is performed in $n$ crossbar arrays in parallel (Fig. 3a). The
single-step MCAM associative search increases speedup by alleviating memory
access overheads and searching across all classes in one pass. The difference
in speedup of MIMHD in different bit precisions is mainly due to the overhead
of the ADCs. For the considered tasks, MIMHD with 1, 2, and 3-bit precisions
achieves average improvements of $44\times$, $38\times$, and $34\times$ over
the GPU, respectively.

Fig. 8d compares the average speedup of AHAM [9], SearcHD [15], and MIMHD with
2 and 3-bit precisions over the GPU. Unlike energy improvement, speedup does
not scale linearly with respect to dimensionality, since a significant portion
of the computation happens in parallel. However, lower dimensinality reduces
some of the peripheral overhead which is reflected in the results. 3-bit MIMHD
outperforms the state of the art, i.e., SearcHD, by $2.6\times$ and
$4.3\times$ in Maximum Accuracy and Same Accuracy setups, respectively.

###  IV-D MIMHD Robustness

The technological and fabrication issues in highly scaled technology nodes add
a significant amount of noise to both memory and computing units [32, 33].
These issues add extra computational error sources, which degrades the quality
of inference. One of the main advantages of HDC is its high robustness to
noise and failure. In HDC, HVs are random and holographic with i.i.d.
components. Each HV stores the information across all its components so that
no component is more responsible for storing any piece of information than
another. This makes HDC robust against computational errors. However, HDC with
floating point precision HVs has a much lower robustness to hardware noise
compared to HDC with binary precision HVs. In high precision representation, a
bit-flip due to noise in the exponent, the most significant bits of mantissa,
or the sign bit can drastically change weight values, thus changing the
prediction results.

Models deployed on MIMHD inherit the intrinsic robustness of HDC. Table II
reports the quality loss of HDC using different bit precisions ($D{=}4k$). Our
evaluation indicates that HDC quality loss increases with HV precision. For
example, using a 10% random bit flip (noise) in the encoding and associative
search modules, 2 and 8-bit HDC experience 4.1% and 12.4% quality loss,
respectively. Thus, it is desired to operate in lower precisions. MIMHD
enables highly robust HDC inference by operating in low-precisions (1, 2, and
3 bits). HDC robustness also depends on HV dimensionality. Increasing
dimensionality from $D{=}4k$ to $D{=}10k$ leads to 3.4$\times$ lower noise
sensitivity. However, this boost in the robustness comes at the expense of
lower computation efficiency. Note that HDC is significantly less sensitive
than the existing learning solutions operating in floating point precision.
For example, considering 2% and 10% random noise in model parameters, neural
networks exhibit 9.4% and 40.0% quality loss, respectively [11].

TABLE II: HDC quality loss using different HV precisions.

Hardware Error | 1% | 2% | 5% | 10% | 15%
---|---|---|---|---|---
MIMHD | 1-bit | 0.0% | 0.0% | 0.9% | 3.1% | 5.2%
2-bit | 0.0% | 0.3% | 1.2% | 4.1% | 6.7%
3-bit | 0.1% | 0.3% | 2.0% | 5.9% | 9.3%
8-bit | 1.2% | 3.7% | 5.5% | 12.4% | 18.7%

##  V Conclusion

We proposed MIMHD, accurate and efficient in-memory implementation of HDC
inference. MIMHD uses multi-bit values to represent HV dimensions and exploits
FeFET crossbar arrays and FeFET MCAM arrays to perform in-memory encoding and
associative search. We also proposed a novel HWART algorithm which iteratively
retrains the HDC model to compensate for possible accuracy loss due to the
MCAM distance metric. Our evaluation shows that MIMHD provides comparable
accuracy to high precision models while enabling higher computation
efficiency.

## Acknowledgment

This work was supported in part by Semiconductor Research Corporation (SRC)
Task No. 2988.001, ASCENT, Department of the Navy, Office of Naval Research,
grant #N00014-21-1-2225, and a generous gift from Cisco.

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