# Spanish Built Factual Freectianary (Spanish-BFF): the first AI-generated
free dictionary

Miguel Ortega-Martín
dezzai, UCM
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Óscar García-Sierra
dezzai, UCM
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Alfonso Ardoiz
dezzai
<EMAIL_ADDRESS>
Juan Carlos Armenteros
dezzai
<EMAIL_ADDRESS>
Jorge Álvarez
dezzai
<EMAIL_ADDRESS>

Adrián Alonso
dezzai, URJC, DSL URJC
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

Dictionaries are one of the oldest and most used linguistic resources.
Building them is a complex task that, to the best of our knowledge, has yet to
be explored with generative Large Language Models (LLMs). We introduce the
"Spanish Built Factual Freectianary" (Spanish-BFF) as the first Spanish AI-
generated dictionary. This first-of-its-kind free dictionary uses GPT-3. We
also define future steps we aim to follow to improve this initial commitment
to the field, such as more additional languages.

##  1 Introduction

Dictionaries are among the earliest and most extensively utilized linguistic
resources. A dictionary is a list of alphabetically ordered words, although
sorting is not always necessary. There are many types of dictionaries, such as
monolingual, bilingual, general or domain-specific. Each kind has its
peculiarities (for example, structure or content). Here we build a general-
purpose monolingual Spanish dictionary, which provides the meaning of Spanish
lemmas. While it tries to cover the totality of the lemmas utilized in this
language, it may require more completeness. Nevertheless, as a free and open-
source tool, we promote contributions to this initiative.

This work contains a broad review of lexicography (section 2); an introduction
to Large Language Models (LLMs) in general and GPT-3 particularly (section 3);
our experimental set-up (section 4); an analysis of the generated dictionary
(section 5); the future lines of this project (section 6); the conclusions
(section 7); the limitations of our work (section 8); and the Ethics Statement
(section 9).

Our contributions to this paper are the following:

  * •

We build, to the best of our knowledge, the first free AI-generated
dictionary. In particular, it contains Spanish lemmas and definitions
generated with GPT-3.

  * •

We establish the future lines of work for the field and are also pleased to
provide an effective procedure to achieve further goals.

##  2 Lexicography

Lexicography is the study of dictionaries and how they are compiled. We can
distinguish between theoretical lexicography, which includes theories about
the structure and contents of dictionaries, and practical lexicography, which
deals with creating concrete dictionaries [1].

Building a dictionary is an arduous task that follows some guidelines and
lexicographic principles, ensuring an efficient consultation and understanding
[6]. A dictionary’s structure has three major components: outside matter (as
additional resources or use guidelines), macro-structure, and micro-structure
[4, 7]. Macro-structure refers to the list of lemmas and their organization.
The size and structure of this list depend on the type and field of dictionary
[4, 7]. Micro-structure refers to the linguistic data that each entry
contains.

Definitions are the essential elements of dictionaries. Traditionally a good
definition of a lemma contains the following information in order of
appearance: generic term, Part of Speech (POS) tags (the class the term
belongs to), a list of senses according to a predefined sorting rule and
eventually, some use cases. Furthermore, we can find other linguistic notes,
like spelling and pronunciation, base and inflected forms, morphological
information and other semantic knowledge (like synonyms, antonyms, hypernyms,
or hyponyms) [7].

As in many other linguistic fields, computers can contribute to lexicography,
where "the electronic storage of vast textual material in corpora and the
varied electronic presentation of lexicological and lexicographical work
represent a quantum leap" [7]. Computational lexicography has focused on
human-annotated dictionaries and lexicons generation from large amounts of raw
text. Traditionally, electronic dictionaries require considerable human,
economic and computational costs. As systems relied on computational lexicons
to match the tokens from an input text, dictionaries significantly contributed
to NLP. But modern neural LMs work without them and build tailored
vocabularies optimizing computational costs instead. Consequently, LMs
facilitate an essential open resource to the broad public where definitions
stop enduring to evolve according to trending use. Regardless, this
flexibility assumes the cost of generalizing.

Recently, some approaches employed dictionaries to create word embeddings and
benefit from lexicographic information that typically is not utilized when
training models on Internet corpora. Following this line, [5] and [11] use the
ensemble of words from the definition to compute its embedding. Besides,
Definition Modelling (DM) is an NLP task that proposes generating meanings
from word embeddings [10]. It uses Recurrent Neural Networks (RNNs) [15] and
performs qualitative and quantitative error analysis of the generated
definition, and [2] develops contextual glosses from words and phrases using a
BART model [8].

As far as we are concerned, the application of generative LMs to construct
comprehensive dictionaries has yet to be investigated.

##  3 GPT-3

Large Language Models (LLMs) are at the forefront of NLP. Particularly, GPT-3
[3] is one of the most famous encoder-decoder models. It generates text from a
given input based on vector representations of words or parts of words. Most
recent versions of this model aim to employ user intents to boost their
performance. For instance, IntructGPT [12] is a fine-tuned model of GPT-3 with
supervised learning, $1.3$ billion parameters and human feedback.
ChatGPT111https://openai.com/blog/chatgpt/ is another fine-tuned version of
GPT-3 having $175$ million parameters, yet trained to interact with users.
Prompting [12] is crucial in all of them. Indeed, a prompt is a piece of text
used to reduce the context of the input to improve the quality of the
generated text.

Although generative models have been widely used for many NLP tasks, their
capabilities for assembling an entire dictionary still need to be
investigated. For example, [9] explore GPT-3 to provide a meaning for new
words.

##  4 Experimental set-up

We use a curated list of $66353$ unique forged Spanish lemmas (including
neologisms) and generate a single definition for each. To benchmark the
results, we parse the output of queries of these lemmas to the "Diccionario de
la Lengua Española" 222https://dle.rae.es/ (DLE), which aims to contain all
Spanish words. We do not store, manipulate, or intend to make any commercial
use of these outputs whatsoever. This procedure is exclusively for research
purposes: to contrast the performance of the dictionary proposed against a
trusted source. Our first proposal neglects homonymy and polysemy. Although
POS tags of the lemmas are currently ignored, we restrict them to nouns,
verbs, adjectives and adverbs. However, we will aim to use this information
for further improvements.

Match size | Processed lemmas | Max tokens per prompt | Price () | ¢/lemma
---|---|---|---|---
$1$ | $400$ | $100$ | $0.60$ | $0.1500$
$3$ | $1179$ | $500$ | $0.78$ | $0.0662$
$5$ | $1290$ | $1000$ | $0.84$ | $0.0651$
$10$ | $1650$ | $2000$ | $0.90$ | $0.0545$
Table 1: Batch sizes experiments per half an hour.

The approach to generate the definitions consists of a prompt-based GPT-3
query submitted to the OpenAI API. The model selected for the generation is
"text-davinci-00". Our initial prompt was: Generate in Spanish a definition of
the word "[word]". However, we quickly realized that this could have been more
optimal in terms of time and money. Before submitting the entire amount of
lemmas, we tested different approaches for half an hour each, as shown in
Table 1. We improved the performance and produced the whole dictionary in
around $30$ hours for $40$ euros. The model parameter named temperature, also
known as creativity, is set by default at $0.5$ in all experiments. As
reported, a bigger batch also implies a higher maximum of generated tokens to
fit the entire content. Increasing the output length does not produce equally
more extended definitions, in any case.

The first version of the "Spanish Built Factual Freectianary" (Spanish-BFF) is
available at the Hugging Face333https://huggingface.co/datasets/MMG/spanishBFF
hub and at our Github444https://github.com/dezzai/Spanish-BFF.

##  5 Results

To explore the quality of the generated dictionary, we do a qualitative
analysis of definitions. We use the BLEU, Levenshtein and Jaccard scores and a
sentence-transformers model [14] to analyze those definitions quantitatively,
and eventually, we perform manual error analysis.

###  5.1 Qualitative analysis

When defining words, GPT-3 demonstrates favourable lexicographic qualities
across its various versions. In the case of nouns (Appendix A.1), it alludes
to its class. As seen in Appendix A.2, another verb is usually employed when
defining them. Regarding adjectives (Appendix A.3), structures like "que…"
("that…") and "se refiere a…" ("it refers to…") or synonyms of the defined
word are common. Adverbs are described using "de manera…" ("in a [adjective]
way") or with synonyms (Appendix A.4). Neologisms are remarkably defined when
they are borrowed from English terms and when they have been generated by
morphological processes, as seen in Appendix A.5.

###  5.2 Quantitative analysis

We divide the quantitative evaluation of the definitions into two approaches.
Firstly, we use the BLEU score [13], Levenshtein distance, Jaccard index, and
cosine similarity for lemmas with just one meaning to inspect the quality of
the description, as in [10]. Secondly, when further entries (polysemous words)
are available, we use cosine similarity to rank the purpose with the highest
resemblance to the generated one. From our lemmas, $44554$ have one
definition, and $21799$ are polysemous in DLE. As previously stated, we query
DLE to retrieve its reliable reference definitions for the sole purpose of
this contrast. We use alternative handpicked definitions from other resources
for neologisms that still need to be added to DLE.

####  5.2.1 Monosemy

Metric | Score
---|---
Cumulative BLEU | $0.0083$
1-gram BLEU | $0.1069$
Levenshtein | $53.22$
Jaccard | $0.0812$
Table 2: Metrics for lemmas with just one sense in DLE. | Measure | Mean | Std Dev
---|---|---|---
DLE | words | $10.2$ | $8.7$
characters | $59.9$ | $50.2$
Spanish-BFF | words | $8.3$ | $5.1$
characters | $49.1$ | $28.4$
Table 3: Statistical distribution (mean and standard deviation) of the
definitions’ lengths.

There exist $44554$ lemmas with a single definition, with metrics reported in
Table 2. As expected, the cumulative BLEU score (1-grams, 2-grams, 3-grams,
and 4-grams equally weighted) is deficient. As a reference, best model
achieves $35.78$ for English [10]. Levenshtein’s and Jaccard’s scores also
exhibit flawed values. Undoubtedly, the gold definitions we use to compare the
generated output have a very high standard. Besides, on average, GPT-3
definitions are shorter than the corresponding DLE entries, as shown in Table
3. And DLE also contains additional information, such as POS tags, etymology,
or domain, that is out of this procedure’s scope and should be withdrawn. In
addition, GPT-3 was presumably trained on diverse texts, some of which likely
have less lexicographic quality. Consequently, some lexicographical rules and
errors are prominently reprised, see section 5.3.

POS tag | Cosine Similarity | % of total
---|---|---
All | $0.3598$ | $100$
Nouns | $0.2886$ | $57.41$
Adjectives | $0.4746$ | $26.26$
Verbs | $0.3866$ | $14.01$
Adverbs | $0.6623$ | $2.32$
Table 4: Cosine similarity for lemmas with just one sense in DLE.

We use the model named "distiluse-base-multilingual-cased-v2"
555https://huggingface.co/sentence-transformers/distiluse-base-multilingual-
cased to provide the cosine similarity using sentence-transformers, see Table
4. Since this metric relies on contextual embeddings, the results positively
improve. Specifically, GPT-3 performs better defining adverbs or adjectives
than verbs or nouns.

####  5.2.2 Polysemy

There are $21799$ lemmas with two or more meanings. As seen in Figure 1, the
likelihood of matching deeper definitions decreases rapidly. This decline
makes sense since both dictionaries are based on the frequency of use. Yet,
some DLE lemmas rely on the chronological order, disrupting the expected
ranking. While shallow definition matches indicate good health (as GPT-3 is
quicker at capturing the statistical trend of meanings), deeper matches show
the tergiversation of the mainstream connotations. On average, the most
similar definition, from all the senses provided for a lemma at DLE, has a
mean cosine similarity of $0.443$ with the generated one, which is higher than
for monosemous words, as shown in Table 4.

Figure 1: Highest cosine similarity among predictions and DLE definitions by
ID

###  5.3 Error analysis

By manual error analysis, we obtained the following type of errors:

  * •

Around $11$ % of the definitions start with "A [lemma] is…". When defining a
word, whatever the defined lemma, it must not appear in the description (see
Appendix B.1). But, this straightforward formula does not involve circular
logic. Thus, it is easy to post-process these cases to deliver better results.

  * •

Sometimes, a word is defined as an almost similar spelt word: for instance,
"re", which could be described as a musical note, could also be defined as
"monarca" instead, a synonym of "rey", "king" in Spanish. In Appendix B.2, we
can inspect some examples of the drawbacks of subword tokenizing, which are
consolidated using quotation marks around the lemmas to avoid splitting.

  * •

When nouns are defined as conjugated verbs (Appendix B.3), for instance,
"pare" ("stop") is a noun derived from the verb "parar" ("to stop"), but GPT-3
defines it as the verb.

  * •

There are cases of language interference (Appendix B.4) in which the Spanish
word is poorly defined in English, primarily due to highly uncommon lemmas.

  * •

Occasionally, part of the meaning is accurate (Appendix B.5). For instance,
"maquis", a Spanish guerrilla during Civil War, is defined by GPT-3 as where
this guerrilla used to fight.

  * •

Despite all, some terms are just completely wrong. For example, the word
"yangüés" is a demonym wrongly defined as a bird (Appendix B.6).

  * •

The specific meaning of some neologisms is not adequately captured, and the
definition alludes to a related term instead (Appendix B.7).

Indeed, most of these errors can be partially solved with better prompting.
Queries such as Genera en español la definición para la palabra literal
"[palabra]" (Generate in Spanish the literal definition for the word "[word]")
averts the subword tokenizing and skips some nouns-as-verbs errors.

##  6 Future

We understand this is a naive approach to creating a dictionary and must fix
numerous lexicographic rules. However, this was just the first attempt to
commit to the task, which we consider extremely useful for NLP and
lexicography. In the future, we aim to solve these issues by, for instance,
taking homonymy and polysemy into account, using POS tags, semantics roles,
example sentences, domain, and more. We also plan to do the same for other
languages. Furthermore, we intend to use ChatGPT and other instruct models to
find the best approach.

##  7 Conclusions

Building a dictionary is a complex task that, to our knowledge, has yet to be
fully explored with LLMs. Here we introduce the first Spanish dictionary
generated with GPT-3. The "freectianary" mourns some flaws (some lexicographic
rules are broken and other mistakes related to language models) but also
contains some promising aspects (reliance on some different lexicographic
rules). In this paper, we define the future steps we aim to follow to improve
our initial commitment to the task, which goes from improving the Spanish
dictionary to generating dictionaries for other languages.

##  8 Limitations

This approach is based on a list of lemmas, so we understand it is limited to
languages with substantial resources, such as Spanish. Another option could be
using a corpus and a lemmatizer, but we should note that not all languages
have these resources.

##  9 Ethics statement

We understand the possibilities that models like GPT-3 can imply for industry
and future academic research. We intend to contribute to a better
understanding and development of NLP and promote responsible use.

## References

  * Bergenholtz and Gouws [2012] Henning Bergenholtz and Rufus H. Gouws. 2012.  What is lexicography?  _Lexikos_ , 22:31–42.  ISBN: 2224-0039.
  * Bevilacqua et al. [2020] Michele Bevilacqua, Marco Maru, and Roberto Navigli. 2020.  Generationary or “how we went beyond word sense inventories and learned to gloss”.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 7207–7221.
  * Brown et al. [2020] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D. Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, and Amanda Askell. 2020.  Language models are few-shot learners.  _Advances in neural information processing systems_ , 33:1877–1901.
  * Hausmann and Wiegand [1989] Franz Josef Hausmann and Herbert Ernst Wiegand. 1989.  Component parts and structures of general monolingual dictionaries: A survey.  _1989-1991_.
  * Hill et al. [2016] Felix Hill, Kyunghyun Cho, Anna Korhonen, and Yoshua Bengio. 2016.  Learning to understand phrases by embedding the dictionary.  _Transactions of the Association for Computational Linguistics_ , 4:17–30.  Publisher: MIT Press.
  * Jackson [2013] Howard Jackson. 2013.  _Lexicography: an introduction_.  Routledge.
  * Kirkness [2004] Alan Kirkness. 2004.  Lexicography.  _The handbook of applied linguistics_ , pages 54–81.
  * Lewis et al. [2019] Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Ves Stoyanov, and Luke Zettlemoyer. 2019.  Bart: Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension.  _arXiv preprint arXiv:1910.13461_.
  * Malkin et al. [2021] Nikolay Malkin, Sameera Lanka, Pranav Goel, Sudha Rao, and Nebojsa Jojic. 2021.  GPT Perdetry Test: Generating new meanings for new words.  In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 5542–5553.
  * Noraset et al. [2017] Thanapon Noraset, Chen Liang, Larry Birnbaum, and Doug Downey. 2017.  Definition modeling: Learning to define word embeddings in natural language.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 31.  Issue: 1.
  * Ortega-Martín [2021] Miguel Ortega-Martín. 2021.  _Grafos de vinculación semántica a partir del definiens del DUE_.  Ph.D. thesis, Universidad Complutense de Madrid.
  * Ouyang et al. [2022] Long Ouyang, Jeff Wu, Xu Jiang, Diogo Almeida, Carroll L. Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, and Alex Ray. 2022.  Training language models to follow instructions with human feedback.  _arXiv preprint arXiv:2203.02155_.
  * Papineni et al. [2002] Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002.  Bleu: a method for automatic evaluation of machine translation.  In _Proceedings of the 40th annual meeting of the Association for Computational Linguistics_ , pages 311–318.
  * Reimers and Gurevych [2019] Nils Reimers and Iryna Gurevych. 2019.  Sentence-bert: Sentence embeddings using siamese bert-networks.  _arXiv preprint arXiv:1908.10084_.
  * Schmidt [2019] Robin M Schmidt. 2019.  Recurrent neural networks (rnns): A gentle introduction and overview.  _arXiv preprint arXiv:1912.05911_.

##  Appendix A Adequate examples

###  A.1 Nouns

  * •

"exotismo": "La cualidad de lo que es exótico o extraño, especialmente en
relación con la cultura y la naturaleza." ("exoticism": "The quality of what
is exotic or strange, especially in relation to culture and nature.")

  * •

"camisón": "Una prenda de vestir ligera y suelta, normalmente de algodón, que
se usa para dormir." ("nightgown": "A light, loose-fitting article of
clothing, usually cotton, worn for sleeping.")

  * •

"pillería": "Acto de robar o tomar algo sin permiso." ("mischief": "Act of
stealing or taking something without permission.")

###  A.2 Verbs

  * •

"guiar": "dirigir una persona o un grupo." ("to lead": "direct a person or a
group.")

  * •

"parir": "dar a luz a un bebé." ("to give birth": "to give birth to a baby."

  * •

"coagular": "formar coágulos." ("to coagulate": "to form clots.")

###  A.3 Adjectives

  * •

"arrendable": "Que se puede alquilar." ("rentable": "That can be rented.")

  * •

"candente": "Ardiente o intenso." ("glowing": "Fiery or intense.")

###  A.4 Adverbs

  * •

"peculiarmente": "de manera particular." ("peculiarly": "in a particular
way.")

  * •

"otrora": "antiguamente." ("once": "formerly.")

###  A.5 Neologisms

  * •

"antibotellón": "Antibotellón es una iniciativa para prevenir la reunión de
grandes grupos de personas en espacios públicos para beber alcohol y evitar
los disturbios asociados." (Antibotellón is an initiative to prevent the
gathering of large groups of people in public spaces to drink alcohol and
avoid the associated riots)

  * •

"mainstream": "Lo que es ampliamente aceptado o popular en una cultura, una
comunidad o una sociedad." ("mainstream": "What is widely accepted or popular
in a culture, a community or a society." )

  * •

"phising": "El phising es una forma de fraude cibernético que consiste en el
uso de correos electrónicos o mensajes de texto fraudulentos para obtener
información confidencial, como nombres de usuario y contraseñas." ("phising":
"Phishing is a form of cyber fraud that involves the use of fraudulent email
or text messages to obtain sensitive information such as usernames and
passwords.")

  * •

"fomo": "FOMO es un acrónimo de "miedo a perderse", y se refiere a la ansiedad
que uno puede sentir cuando ve que otros están disfrutando de algo que él o
ella no está haciendo." ("fomo": "FOMO is an acronym for "fear of missing
out", and refers to the anxiety one may feel when seeing others enjoying
something that he or she is not doing.")

  * •

"nomofobia": "Es el miedo excesivo a estar desconectado de la tecnología,
especialmente de los teléfonos inteligentes." ("nomophobia": "It is the
excessive fear of being disconnected from technology, especially
smartphones.")

  * •

"googlear": "Buscar información en Internet usando el motor de búsqueda de
Google." ("to google": "Search for information on the Internet using the
Google search engine.")

  * •

"polimedicar": "Polimedicar es un término médico que se refiere a la práctica
de tratar a un paciente con varias medicinas o terapias al mismo tiempo."
("polymedicate": "Polymedicate is a medical term that refers to the practice
of treating a patient with several medicines or therapies at the same time.")

##  Appendix B Errors

###  B.1 Lack of lexicographic quality

  * •

"campeonato": "Un campeonato es una competición para determinar quién es el
mejor en algo." ("championship": "A championship is a competition to determine
who is the best at something.")

  * •

"negacionismo": "El negacionismo es una actitud consistente en negar o
discutir la existencia de un hecho o una realidad." ("denialism": "Denialism
is an attitude consisting of denying or discussing the existence of a fact or
reality.")

###  B.2 Tokenizer issues

  * •

"re": "monarca" ("re": "monarch")

  * •

"tona": "Una tona es una medida de volumen equivalente a 1000 kilogramos."
("cream": "A ton is a measure of volume equal to 1000 kilograms.")

  * •

"adoquiera": "adquirir algo." ("anywhere": "to acquire something.")

###  B.3 Noun as a conjugated verb

  * •

"paragranizo": "Verbo que significa proteger con una granizada." (*hail net:
"Verb that means to protect with a hailstorm.")

  * •

"pare": "Pare es un verbo que significa detener algo, detener a alguien o
interrumpir una acción." ("stop -noun-": "Stop is a verb that means to stop
something, stop someone, or interrupt an action.")

  * •

"lleve": "El verbo llevar significa transportar algo de un lugar a otro."
(*money from public transactions: "The verb to carry means to transport
something from one place to another.")

###  B.4 English used in the definition

  * •

"relva": "Grass." (*Raise the fallow.: "Grass")

  * •

"sueldacostilla": "Paycheck." (*A plant: "Paycheck")

###  B.5 Somehow related

  * •

"maquis": "Una zona de bosque o maleza en la que los combatientes se ocultan
para luchar contra una ocupación militar." (*Spanish guerrillas: "An area of
forest or brush where combatants hide to fight against a military
occupation.")

  * •

"baleario": "Una región insular del mar Mediterráneo, compuesta por las islas
Baleares." ("balearic": "An insular region of the Mediterranean Sea, made up
of the Balearic Islands.")

  * •

"zampabollos": "Persona que se dedica a vender bollos por la calle." (*someone
who eats too much: "Person who is dedicated to selling buns on the street.")

###  B.6 Just bad

  * •

"ll": "Abreviatura de la expresión L̈lámame.̈" ("ll": "Abbreviation for the
expression C̈all me.̈")

  * •

"yangüés": "Pájaro de la familia de los estorninos." (*a demonym: "Bird of the
starling family.")

  * •

"napoleón": "Napoleon."

  * •

"menos": "menos." ("less": "less.")

###  B.7 Neologisms

  * •

"pagafantas": "Persona que invita a los demás a salir o a tomar algo sin tener
intención de pagar la cuenta." ("simp": "Person who invites others to go out
or have a drink without intending to pay the bill.")

  * •

"telonear": "Ver una película en una pantalla grande, como una sala de cine."
("playing before the main band in a concert": "Watch a movie on a big screen,
like a movie theater.")

  * •

"chandalismo": "El chandalismo se refiere al comportamiento de personas que,
sin ningún motivo, destruyen o dañan bienes públicos o privados." ("use and
abuse of wearing tracksuit": "Chandalism refers to the behavior of people who,
without any reason, destroy or damage public or private property." )


# On the Fiber Characters of $\mathbb{F}^{*}_{p^{m}}$ and
related Polynomial Algebras

Michele Elia Politecnico di Torino Corso Duca degli Abruzzi 24, I - 10129
Torino – Italy; e-mail<EMAIL_ADDRESS>
###### Abstract

Let $p$ be a prime, $m$ be a positive integer ( $m\geq 1$, and $m\geq 2$ if
$p=2$), and $\chi_{n}$ be a multiplicative complex character on
$\mathbb{F}^{*}_{p^{m}}$ with order $n|(p^{m}-1)$. We show that a partition
$\mathcal{A}_{1}\cup\mathcal{A}_{2}\cup\cdots\cup\mathcal{A}_{n}$ of
$\mathbb{F}^{*}_{p^{m}}$ is the partition by fibers of $\chi_{n}$ if and only
if these fibers satisfy certain additive properties. This is equivalent to
showing that the set of multivariate characteristic polynomials of these
fibers, completed with the constant polynomial $1$, is the basis of an
$(n+1)$-dimensional commutative algebra with identity in the ring
$\mathbb{Q}[x_{1},\ldots,x_{n}]/\langle
x_{1}^{p}-1,\ldots,x_{n}^{p}-1\rangle$.

Mathematics Subject Classification (2000): 11A15, 11N69, 11R32

Key words:  nth power residue, cyclotomic coset, character, polynomial ring.

##  1 Introduction

In 1952, Perron gave some additive properties of the fibers of the quadratic
character on $\mathbb{F}_{p}$. Specifically in [12], he showed that if
$\mathfrak{A},~{}\mathfrak{B}\subset\mathbb{F}_{p}$ are the subsets of
quadratic residues and non-residues, respectively, and letting
$~{}d_{p}=\frac{p-1}{4}~{}~{}\mbox{if}~{}~{}p=1\bmod
4,~{}~{}\mbox{and}~{}~{}d_{p}=\frac{p+1}{4}~{}~{}\mbox{if}~{}~{}p=3\bmod
4~{}$, then

  1. 1.

Every element of $\mathfrak{A}$ [respectively $\mathfrak{B}$] can be written
as the sum of two elements of $\mathfrak{A}$ [respectively $\mathfrak{B}$] in
exactly $d_{p}-1$ ways.

  2. 2.

Every element of $\mathfrak{A}$ [respectively $\mathfrak{B}$] can be written
as the sum of two elements of $\mathfrak{B}$ [respectively $\mathfrak{A}$] in
exactly $d_{p}$ ways.

It was natural to inquire just how strong this result is, and to what extent
it may hold for any character $\chi_{n}$, other than $\chi_{2}$. In [10] it is
shown that these additive properties uniquely characterize the even partition
of $\mathbb{F}_{p}$ into quadratic residues and non-residues. In [11], the
even restriction is removed, and the result is generalized to fibers of
arbitrary multiplicative character $\chi_{n}$ on $\mathbb{F}_{p}$ ($n$ being a
divisor of $(p-1)$), with suitable cyclotomic numbers in place of the
constants $d_{p}$ above. Lastly, in [5], the generalization of the even
partition (i.e. by the quadratic character $\chi_{2}$) to every finite field
of odd characteristic, that is, the partition of $\mathbb{F}_{p^{m}}$ into
squares and non-squares, is discussed and settled. Perron’s view is
attractive, but the formulation of the problem purely in terms of
characteristic polynomials and their algebras permits a full description and
proof of facts that occur in every finite field. The purpose of this paper is
to prove this definitive result.

##  2 Preliminary results

Let $\mathbb{F}_{p^{m}}$ be a finite field with $p^{m}$ elements generated by
a root $\gamma$ of a primitive irreducible polynomial
$p(x)=x^{m}+a_{m-1}x^{m-1}+\ldots+a_{0}$ over $\mathbb{F}_{p}$. Let
$\mathcal{B}=\\{1,\gamma,\ldots,\gamma^{m-1}\\}$ be a basis of
$\mathbb{F}_{p^{m}}$, any non-zero element $\beta\in\mathbb{F}_{p^{m}}$ is
represented either as a power $\gamma^{h}$ or in the basis $\mathcal{B}$ as
$\sum_{i=1}^{m}b_{i}\gamma^{i-1}$ with $\beta_{i}\in\mathbb{F}_{p}$. In the
following, $\beta$ will be interchangeably indicated with the $m$-dimensional
vector $\mathbf{b}=[b_{1},b_{2},\ldots,b_{m}]\in\mathbb{F}^{m}_{p}$, whenever
necessary.
A multiplicative complex character is an isomorphism
$\chi:\mathbb{F}^{*}_{p^{m}}\rightarrow\mathcal{C}_{p^{m}-1}$ between the
multiplicative cyclic group $\mathbb{F}^{*}_{p^{m}}$ and the complex
multiplicative group $\mathcal{C}_{p^{m}-1}$ of the units of order $p^{m}-1$
in the complex field $\mathbb{C}$. Let $n>1$ be a non-trivial positive divisor
of $p^{m}-1$, that is $n\cdot s=p^{m}-1$ (if $p=2$ then $m$ must be greater
than $1$), then the subset consisting of the powers of $\rho=\gamma^{n}$ is a
cyclic subgroup of order $s$ of $\mathbb{F}^{*}_{p^{m}}$.
Let $\zeta_{n}$ be a primitive $n$th complex root of unity, i.e. $\zeta_{n}$
satisfies the $n$th cyclotomic polynomial. A character of order $n$ is
explicitly defined as the mapping $\chi_{n}:\gamma\rightarrow e^{\frac{2\pi
i}{n}}=\zeta_{n}$, that is

$\chi_{n}(\gamma^{n~{}t+h})=e^{\frac{2\pi i(n~{}t+h)}{n}}=e^{\frac{2\pi
i}{n}h}=\zeta^{h}_{n}~{}~{}\forall
t\in\mathbb{Z}~{}~{},~{}~{}\mbox{and}~{}~{}h\in\\{0,1,\ldots,n-1\\}~{}~{}.$

For each integer $0\leq k\leq(n-1)$ let $\mathcal{A}_{k}$ be the fiber
$\chi^{-1}(\zeta_{n}^{k-1})$, then the fiber $\mathcal{A}_{1}=\chi^{-1}(1)$ is
the subgroup of $\mathbb{F}^{*}_{p^{m}}$ consisting of the $n$-th powers of
$\gamma$, and the fiber $\mathcal{A}_{k}=\chi^{-1}(\zeta_{n}^{k-1})$, with
$k>1$, is clearly the coset $\gamma^{k-1}\mathcal{A}_{1}$. We have
$|\mathcal{A}_{k}|=\frac{p^{m}-1}{n}=s$, and for each $1\leq k\leq n$, the
corresponding multivariate characteristic polynomial is

$q_{k}(\mathbf{x})=q_{k}(x_{1},\ldots,x_{n})=\sum_{\beta\in\mathcal{A}_{k}}\prod_{i=1}^{m}x^{b_{i}}_{i}\in\mathbb{Z}[x_{1},\ldots,x_{n}]~{}~{}.$

The set of fibers $\mathcal{A}_{1},\ldots,\mathcal{A}_{n}$ form a partition of
$\mathbb{F}^{*}_{p^{m}}=\\{1,\gamma,\gamma^{2},\ldots,\gamma^{p^{m}-2}\\}$,
thus, defining the polynomial $q_{0}(\mathbf{x})=1$ which is the
characteristic polynomial of the set $\\{0\\}$, we have

$\sum_{k=0}^{n}q_{k}(\mathbf{x})=\prod_{i=1}^{m}\frac{x_{i}^{p}-1}{x_{i}-1}~{}~{}.$

The following lemmas and theorem show that the set of these $n+1$ multivariate
polynomials is the basis of an algebra of dimension $n+1$ in the polynomial
ring $\mathfrak{R}_{n}[\mathbf{x}]=\mathbb{Q}[\mathbf{x}]/\langle
x_{1}^{p}-1,\ldots,x_{n}^{p}-1\rangle$, where $\langle
x_{1}^{p}-1,\ldots,x_{n}^{p}-1\rangle$ denotes the ideal generated by the
polynomials included in brackets.

Since the fiber $\mathcal{A}_{1}$ is a sub-group of order
$s=\frac{p^{m}-1}{n}$ of $\mathbb{F}^{*}_{p^{m}}$, and the remaining fibers
are its cosets, which form a partition of $\mathbb{F}^{*}_{p^{m}}$, the
following proposition easily follows

######  Proposition 1.

The set $\\{q_{0}(\mathbf{x}),q_{1}(\mathbf{x}),\ldots,q_{n}(\mathbf{x})\\}$
of $n+1$ multivariate polynomials is a basis of a $\mathbb{Q}$-subspace
$\mathbf{V}_{n+1}$ of dimension $n+1$ in the $p^{m}$-dimensional vector space
$\mathbb{Q}[\mathbf{x}]/\langle x_{1}^{p}-1,\ldots,x_{n}^{p}-1\rangle$ of
multivariate polynomials of degree at most $p-1$ in each variable $x_{i}$.

The elements of $\mathcal{A}_{1}$ have the following properties:

######  Lemma 1.

Let $p$ be an odd prime, and assuming the above hypotheses, we have

  1. 1.

If $s$ is even, for any $\beta\in\mathcal{A}_{1}$ there exists a
$\alpha\in\mathcal{A}_{1}$ such that $\beta+\alpha=0$.

  2. 2.

If $s$ is odd, there exists a coset $\mathcal{E}=\eta\mathcal{A}_{1}$ such
that for any $\beta\in\mathcal{A}_{1}$ there is a $\alpha\in\mathcal{E}$ such
that $\beta+\alpha=0$.

Let $p=2$, then

  1. 3.

In $\mathbb{F}_{2^{m}}$, any element $\beta$ is the opposite of itself, i.e.
$\beta+\beta=0$.

###### Proof.

Consider the primitive element $\gamma$ of $\mathbb{F}_{p^{m}}$, then

  1. 1.

If $s$ is even, the elements of $\mathcal{A}_{1}$ are all the roots of
$X^{s}-1$, which splits as $(X^{s/2}-1)(X^{s/2}+1)$. Let $\eta=\gamma^{n}$
denote a root of $X^{s/2}+1$, and $\beta=\gamma^{2nt}$ be any root of
$X^{s/2}-1$. Since $\eta^{s/2}=-1$, we have

$\beta\eta^{s/2}=-\beta=\gamma^{2nt}\gamma^{ns/2}=\gamma^{(2t+s/2)n}\in\mathcal{A}_{1}~{}~{},$

therefore $\beta+\gamma^{(2t+s/2)n}=0$, i.e. $\alpha=\gamma^{(2t+s/2)n}$.

  2. 2.

If $s$ is odd, no power of any element in $\mathcal{A}_{1}$ is equal to $-1$.
However, let $\theta=\gamma^{n}$ be a generator of the cyclic group
$\mathcal{A}_{1}$, then an $\eta=\gamma^{t}\in\mathbb{F}_{p^{m}}$ certainly
exists such that $\theta+\eta=0$. Consider the coset $\eta\mathcal{A}_{1}$,
therefore for any $\beta=\theta^{u}\in\mathcal{A}_{1}$, the element
$\zeta=\eta\theta^{u-1}$ is such that $\beta+\zeta=0$ because we have

$\beta+\zeta=\theta^{u}+\eta\theta^{u-1}=\theta^{u-1}(\theta+\eta)=0~{},$

i.e.
$\alpha=\gamma^{t}\theta^{u-1}=\gamma^{t}\gamma^{n(u-1)}=\gamma^{t+n(u-1)}$.

  3. 3.

If $p=2$, then we trivially have $\beta+\beta=0$, thus in any fiber
$\mathcal{A}_{k}$ in $\mathbb{F}_{2^{m}}$, the sum of every element with
itself is $0$, and the sum of two elements that are not in the same fiber is
always different from zero.

∎

The immediate goal is to show that $\mathbf{V}_{n+1}$ is actually a
$\mathbb{Q}$-sub-algebra of $\mathbb{Q}[\mathbf{x}]/\langle
x_{1}^{p}-1,x_{2}^{p}-1,\ldots,x_{m}^{p}-1\rangle$.

######  Lemma 2.

The following properties hold for the sums of elements of cosets in
$\mathbb{F}^{*}_{p^{m}}$ with odd $p$:

  1. 1.

If a fixed $u\in\mathbb{F}_{p^{m}}$ can be expressed as the sum
$\alpha_{1}+\alpha_{2}=u$, with $\alpha_{1}\in\mathcal{A}_{i}$ and
$\alpha_{2}\in\mathcal{A}_{j}$, then every element of the coset
$\mathcal{A}_{k(u)}=u\mathcal{A}_{1}$ can be expressed as the sum of two
elements, one from $\mathcal{A}_{i}$, and one from $\mathcal{A}_{j}$.

  2. 2.

As a direct consequence of the previous point, the product
$q_{i}(\mathbf{x})q_{j}(\mathbf{x})$ is a linear combination of the basis
elements of $\mathbf{V}_{n+1}$.

###### Proof.

The proof of claim 1 is immediate, assuming $\alpha_{1}+\alpha_{2}=u$, we have

$\alpha(\alpha_{1}+\alpha_{2})=\alpha\alpha_{1}+\alpha\alpha_{2}=\alpha
u~{}~{}\forall~{}\alpha\in\mathcal{A}_{1}~{}~{},$

and the conclusion follows from the definition of the coset
$\mathcal{A}_{k(u)}=u\mathcal{A}_{1}$, and group closure.

The proof of claim 2 is a little more elaborate. Due to the definition of the
monomials $m(\mathbf{x})$ that form part of the definition of the polynomials
$q_{k}(\mathbf{x})$, and the correspondence
$m(\mathbf{x})\leftrightarrow\eta\in\mathbb{F}_{p^{m}}$, the product of two
monomials in the ring $\mathbb{Q}[\mathbf{x}]/\langle
x_{1}^{p}-1,x_{2}^{p}-1,\ldots,x_{m}^{p}-1\rangle$ corresponds to the sum of
the corresponding elements in $\mathbb{F}_{p^{m}}$. Now the product
$q_{i}(\mathbf{x})q_{j}(\mathbf{x})$ consists of $s^{2}$ distinct monomials,
which can be partitioned into groups of $s$ monomials, each group
corresponding to some polynomial $q_{k(i,j)}(\mathbf{x})$ by the previous
claim 1; the conclusion follows, by linearity. ∎

######  Theorem 1.

Let $2\leq n|(p^{m}-1)$, $p$ prime, $m$ positive integer ($m\geq 2$ if $p=2$),
and $s=\frac{p^{m}-1}{n}$. The $\mathbb{Q}$-vector space $\mathbf{V}_{n+1}$ of
Proposition 1 is a $\mathbb{Q}$-sub-algebra of the residue ring
$\mathbb{Q}[\mathbf{x}]/\langle
x_{1}^{p}-1,x_{2}^{p}-1,\ldots,x_{m}^{p}-1\rangle$. In particular, for every
$1\leq i,j\leq n$ there exist integers $c_{ijk}$ such that

$q_{i}(\mathbf{x})q_{j}(\mathbf{x})\bmod\langle
x_{1}^{p}-1,x_{2}^{p}-1,\ldots,x_{m}^{p}-1\rangle=c_{ij0}+\sum_{k=1}^{n}c_{ijk}q_{k}(\mathbf{x})~{}~{}.$
(1)

The coefficients $c_{ij0}$ can be explicitly expressed considering $p$ odd and
$p=2$ separately:

  * a)

$p$ odd

    1. 1.

$c_{ii0}=s$ and $c_{ij0}=0$ for every $j\neq i$ if $s$ is even;

    2. 2.

$c_{ii0}=0$ and $c_{ij0}=s$ for a suitable pair $j\neq i$, if $s$ is odd.

  * b)

$p=2$, in this case $s$ is always odd, and we have

    1. 1.

$c_{ii0}=s$, and $c_{ij0}=0$ for every $j\neq i$.

###### Proof.

The $\mathbb{Q}$-vector space $\mathbf{V}_{n+1}$ is a sub-algebra of
$\mathbb{Q}[\mathbf{x}]/\langle
x_{1}^{p}-1,x_{2}^{p}-1,\ldots,x_{m}^{p}-1\rangle$ by Lemma 2.

In general it does not seem possible to obtain a closed form for all constants
$c_{ijk}$ holding for every $p$ and every $m$, except for the following
exceptions.
Let $\mathbf{1}$ be the all-ones $m$-dimensional vector, then we have

$q_{i}(\mathbf{1})q_{j}(\mathbf{1})=c_{ij0}+\sum_{k=1}^{n}c_{ijk}q_{k}(\mathbf{1})~{}~{},$

since $q_{i}(\mathbf{1})=s$, we have $s^{2}=c_{ij0}+s\sum_{k=1}^{n}c_{ijk}$;
this equation implies that $s|c_{ij0}$; since the integer $c_{ij0}\leq s$, it
follows that $c_{ij0}$ is either $0$ or $s$. If $p$ is odd, by Lemma 1 it
follows that

  1. 1.

$c_{ii0}=s$ and $c_{ij0}=0$ for every $j\neq i$ if $s$ is even;

  2. 2.

$c_{ii0}=0$ and $c_{ij0}=s$ for a suitable pair $j\neq i$, if $s$ is odd.

If $p=2$, then $s$ is necessarily odd, however in $\mathbb{F}_{p^{m}}$ every
element is the opposite of itself, then letting $m_{\beta}(\mathbf{x})$ be the
monomial associated to $\beta$, it follows that
$m_{\beta}(\mathbf{x})^{2}\bmod\langle
x_{1}^{2}-1,x_{2}^{2}-1,\ldots,x_{m}^{2}-1\rangle$ is the monomial associated
to $\beta+\beta=0$, that is the monomial $1$; it follows that

  1. 1.

$c_{ii0}=s$ and $c_{ij0}=0$ for every $j\neq i$.

∎

Theorem 1 shows that the vector space $\mathbf{V}_{n+1}$ is a commutative sub-
algebra with identity of the ring of residue polynomials
$\mathbb{Q}[\mathbf{x}]/\langle
x_{1}^{p}-1,x_{2}^{p}-1,\ldots,x_{m}^{p}-1\rangle$. As observed in the proof
of Theorem 1, in general it seems that the structure constants cannot be given
in closed form for every prime $p$, extension degree $m$, and power residue
exponent $n$, thus the computational aspects for obtaining numerical values of
every $c_{ijk}$ may be of interest.

##  3 Computation of the structure constants

The structure constants $c_{ijk}$ are easily found in closed form for $n=2$,
$m=1$, and any odd $p$; however, for every $m\geq 2$ and $n>2$, in general
these constants must be numerically computed by means of convenient
algorithms. We briefly, describe two different computational methods.

###  3.1 Direct method

For fixed $i,j$, equation (1) can be directly used to compute the structure
constants. A consistent linear system of $n$ equations in the $n$ unknowns
$c_{ijk}$, $k=1,\ldots,n$, can be obtained by comparing the coefficients of
equal multivariate monomials on the two sides of (1); actually, we would
obtain a consistent linear system of $n^{2}$ linear equations in $n$ unknowns.
The search for the solution could present some difficulty because the product

$q_{1}(x_{1},\ldots,x_{2})q_{1}(x_{1},\ldots,x_{2})\bmod\langle
x_{1}^{p}-1,\ldots,x_{n}^{p}-1\rangle$

consists of $n^{2}$ monomials in some order that, a priori, we do not know.
They must all be computed, but only $n$ are used. When $n$ is small, as in the
following examples, the method is very efficient, but when $n$ is large,
$n^{2}$ multivariate monomials must be sorted according to some ordering
criterion: this computational issue is left as an open problem.

######  Example 1.

Let $p=3$ and $m=2$, thus $p^{m}-1=8$ and $n$ may be $2$ or $4$, which are the
only proper divisors of $8$. Let $m(z)=z^{2}+z+2$ be a primitive quadratic
polynomial over $\mathbb{F}_{3}$. Let $\alpha$ be a root of $m(z)$, the $9$
elements of $\mathbb{F}_{9}$ are

$\begin{array}[]{|c|cr|}\hline\cr 0&=&0\\\ 1&=&1\\\ \alpha&=&0+\alpha\\\
\alpha^{2}&=&1+2\alpha\\\ \alpha^{3}&=&2+2\alpha\\\ \alpha^{4}&=&2\\\
\alpha^{5}&=&0+2\alpha\\\ \alpha^{6}&=&2+\alpha\\\ \alpha^{7}&=&1+\alpha\\\
\hline\cr\end{array}$

#### Case 1: $\mathbf{n}=2,~{}s=4$

; we have two fibers (cosets)

$\mathcal{A}_{1}=\\{1,1+2\alpha,2,2+\alpha\\}~{}~{},~{}~{}\mathcal{A}_{2}=\\{\alpha,2+2\alpha,2\alpha,1+\alpha\\}~{},$

and the corresponding characteristic multivariate polynomials are

$q_{1}(x_{1},x_{2})=x_{1}+x_{1}x_{2}^{2}+x_{1}^{2}+x_{1}^{2}x_{2}~{}~{}~{}~{},~{}~{}~{}~{}q_{2}(x_{1},x_{2})=x_{2}+x_{1}^{2}x_{2}^{2}+x_{2}^{2}+x_{1}x_{2}~{}~{}.$

The structure constants of the polynomial algebra of $\mathbf{V}_{3}$, with
basis $\\{1,q_{1}(x_{1},x_{2}),q_{2}(x_{1},x_{2})\\}$, are identified by the
system

$\left\\{\begin{array}[]{lcl}q_{1}(x_{1},x_{2})q_{1}(x_{1},x_{2})\bmod\langle
x_{1}^{3}-1,x_{2}^{3}-1\rangle&=&c_{110}+c_{111}q_{1}(x_{1},x_{2})+c_{112}q_{2}(x_{1},x_{2})\\\
q_{1}(x_{1},x_{2})q_{2}(x_{1},x_{2})\bmod\langle
x_{1}^{3}-1,x_{2}^{3}-1\rangle&=&c_{120}+c_{121}q_{1}(x_{1},x_{2})+c_{122}q_{2}(x_{1},x_{2})\\\
q_{2}(x_{1},x_{2})q_{2}(x_{1},x_{2})\bmod\langle
x_{1}^{3}-1,x_{2}^{3}-1\rangle&=&c_{220}+c_{221}q_{1}(x_{1},x_{2})+c_{222}q_{2}(x_{1},x_{2})\\\
\end{array}\right.$

where the constants with the third index equal to $0$ are known by Theorem 1
$c_{110}=4$, $c_{120}=0$, and $c_{220}=4$.
To find the remaining $6$ constants with the direct method we compute
$q_{i}(x_{1},x_{2})q_{j}(x_{1},x_{2})\bmod\langle
x_{1}^{3}-1,x_{2}^{3}-1\rangle$ and subtract
$(c_{ij1}q_{1}(x_{1},x_{2})+c_{ij2}q_{2}(x_{1},x_{2}))+c_{ij0}$, obtaining
three multivariate polynomials which must be identically zero

$\left\\{\begin{array}[]{lcl}4+(2-c_{112})x_{1}^{2}x_{2}^{2}+(1-c_{111})x_{1}^{2}x_{2}+(1-c_{111})x_{1}^{2}+(1-c_{111})x_{1}x_{2}^{2}+(2-c_{112})x_{1}x_{2}&&\\\
~{}~{}~{}+(1-c_{111})x_{1}+(2-c_{112})x_{2}^{2}+(2-c_{112})x_{2}-4&=&0\\\
(2-c_{122})x_{1}^{2}x_{2}^{2}+(2-c_{121})x_{1}^{2}x_{2}+(2-c_{121})x_{1}^{2}+(2-c_{121})x_{1}x_{2}^{2}+(2-c_{122})x_{1}x_{2}&&\\\
~{}~{}~{}~{}~{}+(2-c_{121})x_{1}+(2-c_{122})x_{2}^{2}+(2-c_{122})x_{2}&=&0\\\
4+(1-c_{222})x_{1}^{2}x_{2}^{2}+(2-c_{221})x_{1}^{2}x_{2}+(2-c_{221})x_{1}^{2}+(2-c_{221})x_{1}x_{2}^{2}+(1-c_{222})x_{1}x_{2}&&\\\
~{}~{}~{}~{}~{}+(2-c_{221})x_{1}+(1-c_{222})x_{2}^{2}+(1-c_{222})x_{2}-4&=&0\\\
\end{array}\right.$

From the first equation we obtain $c_{111}=1,~{}c_{112}=2$, from the second
equation we obtain $c_{121}=2,~{}c_{122}=2$, and from the third equation
$c_{221}=2,~{}c_{222}=1$, which allows us to write the multiplication table
with the coefficients of the linear combinations (the trivial multiplications
by $q_{0}(x_{1},x_{2})=1$ are not reported)

$\begin{array}[]{l|ccc}&q_{0}(x_{1},x_{2})&q_{1}(x_{1},x_{2})&q_{2}(x_{1},x_{2})\\\
\hline\cr q_{1}(x_{1},x_{2})q_{1}(x_{1},x_{2})&4&1&2\\\
q_{1}(x_{1},x_{2})q_{2}(x_{1},x_{2})&0&2&2\\\
q_{2}(x_{1},x_{2})q_{2}(x_{1},x_{2})&4&2&1\\\ \end{array}$

#### Case 2: $\mathbf{n}=4,~{}s=2$

; we have four cosets

$\mathcal{A}_{1}=\\{1,2\\}~{}~{},~{}~{}\mathcal{A}_{2}=\\{\alpha,2\alpha\\}~{}~{},~{}~{}\mathcal{A}_{3}=\\{2+\alpha,1+2\alpha\\}~{}~{},~{}~{}\mathcal{A}_{4}=\\{2+2\alpha,1+\alpha\\}~{},$

and, correspondingly, the characteristic multivariate polynomials are

$q_{1}(x_{1},x_{2})=x_{1}+x_{1}^{2}~{}~{},~{}~{}q_{2}(x_{1},x_{2})=x_{2}+x_{2}^{2}~{}~{},~{}~{}q_{3}(x_{1},x_{2})=x_{1}x_{2}^{2}+x_{1}^{2}x_{2}~{}~{},~{}~{}q_{4}(x_{1},x_{2})=x_{1}^{2}x_{2}^{2}+x_{1}x_{2}~{}~{}.$

The multiplication table can be conveniently written as a $4\times 4$ table,
where rows and columns are orderly indexed by the polynomials
$q_{i}(\mathbf{x})$, and the entries are five-tuples of integers which are the
five coefficients of the linear combinations

$\begin{array}[]{|c||c|c|c|c|}\hline\cr&q_{1}&q_{2}&q_{3}&q_{4}\\\
\hline\cr\hline\cr q_{1}&[2,1,0,0,0]&[0,0,0,1,1]&[0,0,1,0,1]&[0,0,1,1,0]\\\
\hline\cr q_{2}&[0,0,0,1,1]&[2,0,1,0,0]&[0,0,1,0,1]&[0,0,1,1,0]\\\ \hline\cr
q_{3}&[0,0,1,0,1]&[0,0,1,1,0]&[2,0,0,1,0]&[0,0,1,1,0]\\\ \hline\cr
q_{4}&[0,0,1,1,0]&[0,0,1,1,0]&[0,0,1,0,1]&[2,0,0,0,1]\\\ \hline\cr\end{array}$

For instance we have

$\left\\{\begin{array}[]{lcl}q_{1}(x_{1},x_{2})q_{1}(x_{1},x_{2})\bmod\langle
x_{1}^{3}-1,x_{2}^{3}-1\rangle&=&2+q_{1}(x_{1},x_{2})\Rightarrow[2,1,0,0,0]\\\
q_{1}(x_{1},x_{2})q_{2}(x_{1},x_{2})\bmod\langle
x_{1}^{3}-1,x_{2}^{3}-1\rangle&=&q_{3}(x_{1},x_{2})+q_{4}(x_{1},x_{2})\Rightarrow[0,0,0,1,1]\end{array}\right.$

######  Example 2.

Let $m(z)=z^{4}+z+1$ be a $4$-degree primitive polynomial over
$\mathbb{F}_{2}$. Let $\alpha$ be a root of $m(z)$, the $16$ elements of
$\mathbb{F}_{16}$ are

$\begin{array}[]{|c|cr|}\hline\cr 0&=&0\\\ 1&=&1\\\ \alpha&=&\alpha\\\
\alpha^{2}&=&\alpha^{2}\\\ \alpha^{3}&=&\alpha^{3}\\\ \alpha^{4}&=&1+\alpha\\\
\alpha^{5}&=&\alpha+\alpha^{2}\\\ \alpha^{6}&=&\alpha^{2}+\alpha^{3}\\\
\alpha^{7}&=&1+\alpha+\alpha^{3}\\\ \alpha^{8}&=&1+\alpha^{2}\\\
\alpha^{9}&=&\alpha+\alpha^{3}\\\ \alpha^{10}&=&1+\alpha+\alpha^{2}\\\
\alpha^{11}&=&\alpha+\alpha^{2}+\alpha^{3}\\\
\alpha^{12}&=&1+\alpha+\alpha^{2}+\alpha^{3}\\\
\alpha^{13}&=&1+\alpha^{2}+\alpha^{3}\\\ \alpha^{14}&=&1+\alpha^{3}\\\
\hline\cr\end{array}$

In this case $n$ may be $3$ or $5$; only $n=3$ is considered, being fully
illustrative.

#### Case: $n=3$, $s=5$

; we have three cosets

$\left\\{\begin{array}[]{lcl}\mathcal{A}_{1}&=&\\{1,\alpha^{3},\alpha^{2}+\alpha^{3},\alpha+\alpha^{3},1+\alpha+\alpha^{2}+\alpha^{3}\\}\\\
\mathcal{A}_{2}&=&\\{\alpha,1+\alpha,1+\alpha+\alpha^{3},1+\alpha+\alpha^{2},1+\alpha^{2}+\alpha^{3}\\}\\\
\mathcal{A}_{3}&=&\\{\alpha^{2},\alpha+\alpha^{2},1+\alpha^{2},\alpha+\alpha^{2}+\alpha^{3},1+\alpha^{3}\\}\end{array}\right.$

and correspondingly three characteristic multivariate polynomials

$\left\\{\begin{array}[]{lcl}q_{1}(x_{1},x_{2},x_{3},x_{4})&=&x_{1}+x_{4}+x_{3}x_{4}+x_{2}x_{4}+x_{1}x_{2}x_{3}x_{4}\\\
q_{2}(x_{1},x_{2},x_{3},x_{4})&=&x_{2}+x_{1}x_{2}+x_{1}x_{2}x_{4}+x_{1}x_{2}x_{3}+x_{1}x_{3}x_{4}\\\
q_{3}(x_{1},x_{2},x_{3},x_{4})&=&x_{3}+x_{2}x_{3}+x_{1}x_{3}+x_{2}x_{3}x_{4}+x_{1}x_{4}\end{array}\right.~{}~{}.$

Let $\mathbf{x}=[x_{1},x_{2},x_{3},x_{4}]$, a basis of $\mathbf{V}_{4}$ is
$\\{1,q_{1}(\mathbf{x}),q_{2}(\mathbf{x}),q_{3}(\mathbf{x})\\}$, and the
structure constants of the polynomial algebra can be computed from the
following system of six equations

$\left\\{\begin{array}[]{lcl}q_{1}(\mathbf{x})q_{1}(\mathbf{x})&=&c_{110}+c_{111}q_{1}(\mathbf{x})+c_{112}q_{2}(\mathbf{x})+c_{113}q_{3}(\mathbf{x})\\\
q_{1}(\mathbf{x})q_{2}(\mathbf{x})&=&c_{120}+c_{121}q_{1}(\mathbf{x})+c_{122}q_{2}(\mathbf{x})+c_{123}q_{3}(\mathbf{x})\\\
q_{1}(\mathbf{x})q_{3}(\mathbf{x})&=&c_{130}+c_{131}q_{1}(\mathbf{x})+c_{132}q_{2}(\mathbf{x})+c_{133}q_{3}(\mathbf{x})\\\
q_{2}(\mathbf{x})q_{2}(\mathbf{x})&=&c_{220}+c_{221}q_{1}(\mathbf{x})+c_{222}q_{2}(\mathbf{x})+c_{223}q_{3}(\mathbf{x})\\\
q_{2}(\mathbf{x})q_{3}(\mathbf{x})&=&c_{230}+c_{231}q_{1}(\mathbf{x})+c_{232}q_{2}(\mathbf{x})+c_{233}q_{3}(\mathbf{x})\\\
q_{3}(\mathbf{x})q_{3}(\mathbf{x})&=&c_{330}+c_{331}q_{1}(\mathbf{x})+c_{332}q_{2}(\mathbf{x})+c_{333}q_{3}(\mathbf{x})\\\
\end{array}\right.$

Now $c_{110}=c_{220}=c_{330}=5$, and $c_{120}=c_{130}=c_{230}=0$, then we have
to compute only $18$ constants instead of $24$. Proceeding as in the previous
example we obtain all structure constants $c_{ijk}$ and write the
multiplication table where the coefficients of the linear combinations for
$q_{i}(\mathbf{x})q_{j}(\mathbf{x})$ are reported in the corresponding row
(the trivial multiplications by $q_{0}(x_{1},x_{2})=0$ are not reported)

$\begin{array}[]{l|cccc}&q_{0}(\mathbf{x})&q_{1}(\mathbf{x})&q_{2}(\mathbf{x})&q_{3}(\mathbf{x})\\\
\hline\cr q_{1}(\mathbf{x})q_{1}(\mathbf{x})&5&0&2&2\\\
q_{1}(\mathbf{x})q_{2}(\mathbf{x})&0&2&2&1\\\
q_{1}(\mathbf{x})q_{3}(\mathbf{x})&0&2&1&2\\\
q_{2}(\mathbf{x})q_{2}(\mathbf{x})&5&2&0&2\\\
q_{2}(\mathbf{x})q_{3}(\mathbf{x})&0&1&2&2\\\
q_{3}(\mathbf{x})q_{3}(\mathbf{x})&5&2&2&0\\\ \end{array}$

###  3.2 A numerical method based on cyclotomic fields

Let $\mathbb{Q}(\zeta_{p})$ be the cyclotomic field of $p$-th roots of unity,
with $\zeta_{p}$ denoting a primitive root of unity, that is a root of the
cyclotomic polynomial of degree $p-1$. Thus $\mathbb{Q}(\zeta_{p})$ is an
extension of degree $p-1$ of $\mathbb{Q}$. Let $\mathfrak{G}_{p}$ denote the
multiplicative cyclic group generated by $\zeta_{p}$. Let
$\mathbf{u}=(\zeta^{i_{1}}_{p},\zeta^{i_{2}}_{p},\ldots,\zeta^{i_{m}}_{p})$
denote an $n$-tuple of elements of $\mathfrak{G}_{p}$, thus from the
evaluation of equation (1) for $\mathbf{x}=\mathbf{u}$, we get a polynomial in
$\zeta_{p}$ that is equal to $0$

$c_{ij0}+\sum_{k=1}^{n}c_{ijk}q_{k}(\mathbf{u})-q_{i}(\mathbf{u})q_{j}(\mathbf{u})=0~{}~{}.$
(2)

We thus obtain a system of $p$ linear equations with integral coefficients in
$n$ unknowns. If $n\leq p$ a solution is easily obtained, since it certainly
exists by Theorem 1. If $n>p$ we need more linear equations, then we consider
the equations obtained using $\ell$ different vectors $\mathbf{u}$, with the
aim of getting $n$ linearly independent equations.

######  Example 3.

Reconsider the problem of example 1. Its solutions by this second method are
obtained working in $\mathbb{Q}(\zeta_{3})$ with $\zeta_{3}$ a primitive
complex cubic root of unity.
Take $\mathbf{u}_{0}=(1,1)$, and $\mathbf{u}_{1}=(1,\zeta_{3})$; in this case
we obtain two equations using (2), considering that $c_{110}=4$, $c_{120}=0$,
and $c_{220}=4$, $q_{1}(1,1)=q_{2}(1,1)=4$, $q_{1}(1,\zeta_{3})=1$, and
$q_{2}(1,\zeta_{3})=-2$. Thus we can write the system

$\left\\{\begin{array}[]{lcl}4+4c_{111}+4c_{112}&=&16\\\
4+c_{111}-2c_{112}&=&1\end{array}\right.$

Solving for $c_{111},c_{112}$ we obtain $c_{111}=1,c_{112}=2$.
Similarly, we obtain all structure constants summarized in the following table

$\begin{array}[]{l|ccc}&1&q_{1}(x_{1},x_{2})&q_{2}(x_{1},x_{2})\\\ \hline\cr
q_{1}(x_{1},x_{2})q_{1}(x_{1},x_{2})&4&1&2\\\
q_{1}(x_{1},x_{2})q_{2}(x_{1},x_{2})&0&2&2\\\
q_{2}(x_{1},x_{2})q_{2}(x_{1},x_{2})&4&2&1\\\ \end{array}$

###  3.3 A new proof of Perron’s original observations

The history of the $\mathbb{F}^{*}_{p^{m}}$ partition by the fibers of a given
character began with Perron’s characterization of the sets or quadratic
residues and non-residues in prime fields, and several independent proofs have
since been given. A ”new” proof is obtained by specializing the general
results given above, and holds for every finite field of odd characteristic .
Consider the prime field $\mathbb{F}_{p^{m}}$, $p$ odd, and the character
$\chi_{2}$ of order $2$ defined over $\mathbb{F}^{*}_{p^{m}}$. Let
$\mathcal{R}$ and $\mathcal{N}$ be the subsets of $\mathbb{F}^{*}_{p^{m}}$ of
squares and non-squares, respectively, that is $\mathcal{R}=\chi^{-1}(1)$ and
$\mathcal{N}=\chi^{-1}(-1)$. The corresponding characteristic polynomials are

$q_{\mathcal{R}}(\mathbf{x})=\sum_{\beta\in\mathbb{F}^{*}_{p^{m}}}\frac{1+\chi_{2}(\beta)}{2}\prod_{i=1}^{m}x_{i}^{b_{i}}~{}~{},~{}~{}q_{\mathcal{N}}(x)=\sum_{\beta\in\mathbb{F}^{*}_{p^{m}}}\frac{1-\chi_{2}(\beta)}{2}\prod_{i=1}^{m}x_{i}^{b_{i}}~{}~{},$

depending on whether $\frac{p^{m}-1}{2}$ is odd or even, we have

$\begin{array}[]{l}\frac{p^{m}-1}{2}~{}~{}\mbox{odd}~{}~{}~{}~{}\left\\{\begin{array}[]{lcl}q_{\mathcal{R}}(\mathbf{x})q_{\mathcal{R}}(\mathbf{x})\bmod\langle
x_{1}^{p}-1,\ldots,x_{m}^{p}-1\rangle&=&~{}~{}~{}~{}~{}0+a_{11}q_{\mathcal{R}}(\mathbf{x})+b_{11}q_{\mathcal{N}}(\mathbf{x})\\\
q_{\mathcal{R}}(\mathbf{x})q_{\mathcal{N}}(\mathbf{x})\bmod\langle
x_{1}^{p}-1,\ldots,x_{m}^{p}-1\rangle&=&\frac{p-1}{2}+a_{12}q_{\mathcal{R}}(\mathbf{x})+b_{12}q_{\mathcal{N}}(\mathbf{x})\\\
q_{\mathcal{N}}(\mathbf{x})q_{\mathcal{N}}(\mathbf{x})\bmod\langle
x_{1}^{p}-1,\ldots,x_{m}^{p}-1\rangle&=&~{}~{}~{}~{}~{}0+a_{22}q_{\mathcal{R}}(\mathbf{x})+b_{22}q_{\mathcal{N}}(\mathbf{x})\\\
\end{array}\right.\\\ \\\
\frac{p^{m}-1}{2}~{}~{}\mbox{even}~{}~{}~{}~{}\left\\{\begin{array}[]{lcl}q_{\mathcal{R}}(\mathbf{x})q_{\mathcal{R}}(\mathbf{x})\bmod\langle
x_{1}^{p}-1,\ldots,x_{m}^{p}-1\rangle&=&\frac{p-1}{2}+a_{11}q_{\mathcal{R}}(\mathbf{x})+b_{11}q_{\mathcal{N}}(\mathbf{x})\\\
q_{\mathcal{R}}(\mathbf{x})q_{\mathcal{N}}(\mathbf{x})\bmod\langle
x_{1}^{p}-1,\ldots,x_{m}^{p}-1\rangle&=&~{}~{}~{}~{}~{}0+a_{12}q_{\mathcal{R}}(\mathbf{x})+b_{12}q_{\mathcal{N}}(\mathbf{x})\\\
q_{\mathcal{N}}(\mathbf{x})q_{\mathcal{N}}(\mathbf{x})\bmod\langle
x_{1}^{p}-1,\ldots,x_{m}^{p}-1\rangle&=&\frac{p-1}{2}+a_{22}q_{\mathcal{R}}(\mathbf{x})+b_{22}q_{\mathcal{N}}(\mathbf{x})\\\
\end{array}\right.\end{array}$ (3)

Let $\mathbf{u}_{o}$ be the vector of all ones, then we have

$q_{\mathcal{R}}(\mathbf{u}_{o})=\sum_{\beta\in\mathbb{F}^{*}_{p^{m}}}\frac{1+\chi_{2}(\beta)}{2}=\frac{p^{m}-1}{2}~{}~{},~{}~{}q_{\mathcal{N}}(\mathbf{u}_{o})=\sum_{\beta\in\mathbb{F}^{*}_{p^{m}}}\frac{1-\chi_{2}(\beta)}{2}=\frac{p^{m}-1}{2}$
$q_{\mathcal{R}}(-\mathbf{u}_{o})=\sum_{\beta\in\mathbb{F}^{*}_{p^{m}}}\frac{1+\chi_{2}(\beta)}{2}(-1)^{\sum_{i=1}^{m}b_{i}}=t~{},~{}q_{\mathcal{N}}(-\mathbf{u}_{o})=\sum_{\beta\in\mathbb{F}^{*}_{p^{m}}}\frac{1-\chi_{2}(\beta)}{2}(-1)^{\sum_{i=1}^{m}b_{i}}=-t$

If $q_{\mathcal{R}}(-\mathbf{u}_{o})=0$, it is necessary to use a vector
$\mathbf{u}$ different from $-\mathbf{u}_{o}$: there are $2^{m}-2$ possible
choices for $\mathbf{u}\neq-\mathbf{u}_{o}$, and one of them certainly works
because of Theorem 1.

## References

  * [1] E. Bach, J. Shallit, Algorithmic Number Theory, vol.1, Cambridge: MIT Press, 1996.
  * [2] B.C. Berndt, R.J. Evans, K.S. Williams, Gauss and Jacobi Sums, Wiley, New York, 1998.
  * [3] D.A. Cox, Galois Theory, Wiley, Hoboken, 2004.
  * [4] L. E. Dickson, Algebras and their Arithmetics, Dover, New York, NY, 1960.
  * [5] M. Elia, On a Problem of Perron, JPANTA, Volume 42, Number 2, 2019, pp. 255-266, ISSN: 0972-5555
http://dx.doi.org/10.17654/NT042020255

  * [6] A Frölich, M. Taylor, Algebraic Number Theory, , Cambridge University Press, 1994.
  * [7] C.F. Gauss, Disquisitiones Arithmeticae, New York: Springer-Verlag, 1986.
  * [8] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, Oxford at the Clarendon Press, 1971.
  * [9] Hua Loo Keng, Introduction to Number Theory, New York: Springer, 1981.
  * [10] C. Monico, M. Elia, Note on an additive characterization of quadratic residues modulo $p$, J. Comb. Inf. Syst. Sci., 31 (2006), pp. 209-215.
  * [11] C. Monico, M. Elia, An Additive Characterization of Fibers of Characters on $\mathbb{F}_{p}^{*}$, Int. J. Algebra 4 (2010), pp. 109-117.
  * [12] O. Perron, Bemerkungen uüber die Verteilung der quadratischen Reste, Math. Z. 56 (1952), pp. 122-130.
  * [13] B.L. van der Waerden, Modern Algebra, 2 vol., New York: Ungar, 1966.
  * [14] H. Weyl, Algebraic Theory of Numbers, Princeton University Press, Princeton, 1998.
  * [15] A. Winterhof, On the distribution of powers in finite fields, Finite Fields Appl. 4 (1998), pp. 43-54.


# Dynamical control of two-level system’s decay and long time freezing

Wenxian Zhang  The State Key Laboratory for Advanced Materials and Devices,
Department of Optical Science and Engineering, Fudan University, Shanghai
200433, China  Ames Laboratory and Iowa State University, Ames, Iowa 50011,
USA  Jun Zhuang  The State Key Laboratory for Advanced Materials and Devices,
Department of Optical Science and Engineering, Fudan University, Shanghai
200433, China

###### Abstract

We investigate with exact numerical calculation coherent control of a two-
level quantum system’s decay by subjecting the two-level system to many
periodic ideal $2\pi$ phase modulation pulses. For three spectrum intensities
(Gaussian, Lorentzian, and exponential), we find both suppression and
acceleration of the decay of the two-level system, depending on difference
between the spectrum peak position and the eigen frequency of the two-level
system. Most interestingly, the decay of the two-level system freezes after
many control pulses if the pulse delay is short. The decay freezing value is
half of the decay in the first pulse delay.

###### pacs:

03.67.Pp, 03.65.Yz, 02.60.Cb

##  I Introduction

It is of fundamental interests to study the quantum control of spontaneous
emission which inspired the development of the quantum electrodynamics during
the last century Milonni (1994); Scully and Zubairy (1997); Allen and Eberly
(1975). The unwanted spontaneous emission often sets the ultimate limit of
precise quantum measurement and many proposals have been made to suppress it
Frishman and Shapiro (2003); Evers and Keitel (2002); Frishman and Shapiro
(2001); Agarwal et al. (2001a). One way widely applied to control the
spontaneous emission of a small quantum system, such as a neutral atom, is to
place it either into or nearby a microcavity such that only a single mode or
several modes of the cavity are resonant to the eigenfrequency of the quantum
system Vogel and Welsch (2006). In this way the structure of the vacuum is
modified and the spontaneous emission could be controlled by the properties of
the cavity. Another way to control the spontaneous emission is dynamical
control of the quantum system such that the coupling between the quantum
system and the vacuum is effectively modified. For example, dynamical
decoupling of the quantum system from the vacuum via either phase modulation
or amplitude modulation could in principle extend the coherent time of the
system Itano et al. (1990, 1991). Analogous to the quantum Zeno effect (ZE)
which says frequent measurements of a quantum system would prevent the decay
of an unstable quantum system Misra and Sudarshan (1977); Itano et al. (1990,
1991), the extension of the coherent time through coherent modulations of the
quantum system are often called ZE as well. Under some unfavorable conditions,
it is also possible that frequent modulations lead to acceleration of the
spontaneous emission, the so-called quantum anti-Zeno effect (AZE) Kofman and
Kurizki (2000); Agarwal et al. (2001b); Fischer et al. (2001); Kofman and
Kurizki (2001).

In practice people employ both ways separately or combination of them to
realize the optimal control of a quantum system. G. S. Agarwal et al.’s work
Agarwal et al. (2001a, b) is among many of such works. In their paper, a two-
level model system with both structured vacuum and free space vacuum are
investigated. They found significant suppression of spontaneous emission rate
for structured vacua but either suppression or acceleration may appear in a
free space vacuum, depending on the frequency of the control pulses which is
$2\pi$ phase modulation pulses. They demonstrated for a free space vacuum that
ZE shows up for $\omega_{0}\tau=1$ while AZE appears for $\omega_{0}\tau=\pi$
where $\omega_{0}$ is the eigenfrequency of the two-level system and $\tau$ is
the delay time between control pulses. At the end they argued that AZE is
possible for $\tau>\omega_{0}^{-1}$ Agarwal et al. (2001a).

A puzzle arises if one accepts the AZE condition because neither ZE or AZE
appear if one leaves the quantum system alone in which case obviously
$\tau\gg\omega_{0}^{-1}$. In this paper we revisit this problem by adopting
the exact solution of a two-level quantum system which does not require the
weak coupling and short time approximations compared to Ref. Agarwal et al.
(2001a). By investigating several typical structured vacuum, we show the
conditions for quantum ZE and AZE and the boundary between them.

Another puzzle is the ZE effect of a large number of pulses. According to the
ZE, at a fixed evolution time $t$, the survival probability of the initially
excited state approaches 1 with infinite number of pulses $N\rightarrow\infty$
(the pulse delay $\tau=t/N$ approaches 0). It is unclear what the road map
looks like as $N$ increases. Two ways might be possible. One way is that the
rate of decay depends solely on the pulse delay $\tau$ (independent on $N$)
and approaches zero if $\tau\rightarrow 0$ Scully et al. (2003). Another way
is that the decay rate depends only on the number of pulses $N$ (independent
on $\tau$) and the decay rate becomes zero after initial several pulses. For
the latter one, we expect to see decay freezing at large number of pulses. We
will find which way is the correct one in this paper.

The paper is organized as follows. Section II reviews briefly Rabi oscillation
of a two-level system under a perturbation and Sec. III develops an exact
formula of Rabi oscillation under periodic pulses. In Sec. IV after
establishing a quasi-level picture of the pulsed two-level system with
constant spectrum intensity, we investigate the quantum Zeno and anti-Zeno
effect for three spectrum intensities, including the Gaussian, Lorentzian, and
exponential one. The boundary between quantum Zeno and anti-Zeno effect is
given. We discuss the decay freezing as pulse delay $\tau$ getting small in
Sec. V. Finally, conclusion and discussion are given in Sec. VI.

##  II Rabi oscillations of a two-level system

Let us consider a simple model of a two-level quantum system with constant
coupling to its environment modes. Such a model serves as the base of further
discussions for more complicated systems with mode-dependent coupling. The
Hamiltonian of the system with detuning $\delta=\omega-\omega_{0}$ (we set
$\hbar=1$ for convenience) and coupling strength $\nu$ is described by Scully
and Zubairy (1997); dip

$\displaystyle H$ $\displaystyle=$ $\displaystyle{\delta\over
2}(|e\rangle\langle e|-|g\rangle\langle g|)+\nu(|e\rangle\langle
g|+|g\rangle\langle e|),$ (1)

where $\omega_{0}=E_{e}-E_{g}$ is the difference between eigenfrequencies of
the excited state $|e\rangle$ and the ground state $|g\rangle$, with
respectively eigenenergy $E_{e}$ and $E_{g}$, and $\omega$ is the frequency of
the external mode coupled to the two-level system picturenote . Utilizing
Pauli matrices, the Hamiltonian can be rewritten as

$\displaystyle H$ $\displaystyle=$
$\displaystyle{\delta^{\prime}}\sigma_{z}+\nu\sigma_{x}$ (2)

with $\sigma_{z}=|e\rangle\langle e|-|g\rangle\langle g|$,
$\sigma_{x}=|e\rangle\langle g|+|g\rangle\langle e|$, and
$\delta^{\prime}=\delta/2$.

The evolution operator of the coupled two-level system is

$\displaystyle U$ $\displaystyle=$ $\displaystyle d_{1}|e\rangle\langle
e|+d_{1}^{*}|g\rangle\langle g|+d_{2}(|e\rangle\langle g|+|g\rangle\langle
e|)$ (3)

where $d_{1}=\cos\Omega t-i(\delta^{\prime}/\Omega)\sin\Omega t$ and
$d_{2}=-i(\nu/\Omega)\sin\Omega t$ with $\Omega^{2}=\delta^{\prime
2}+\nu^{2}$. The transition probability at time $t$ from $|e\rangle$ to
$|g\rangle$ is

$\displaystyle p_{eg}(t)$ $\displaystyle=$
$\displaystyle{\nu^{2}\over\Omega^{2}}\sin^{2}(\Omega t)$ (4)

providing the initial state is $\psi(0)=|e\rangle$. The system revives at
times such that $\Omega t=k\pi$ with $k$ being an integer.

##  III Controlled Rabi oscillations of a two-level system

Figure 1: Diagram of a coupled two-level system subjected to ideal $2\pi$
pulses.

By applying an ideal $2\pi$ pulse (or parity kick, see Fig. 1), which is very
strong in amplitude and short in time but gives a $\pi$ phase shift solely to
the excited state $|e\rangle$ utilizing an auxiliary state $|a\rangle$ Agarwal
et al. (2001a), the state of the system changes according to

$\displaystyle x|e\rangle+y|g\rangle$
$\displaystyle\underrightarrow{\;\;2\pi\;{\rm pulse}\;\;}$
$\displaystyle-x|e\rangle+y|g\rangle$ (5)

where $|x|^{2}+|y|^{2}=1$. We denote such a pulse as $Z$ pulse hereafter

$\displaystyle Z(x|e\rangle+y|g\rangle)=-x|e\rangle+y|g\rangle.$ (6)

In fact, $Z=-\sigma_{z}=|g\rangle\langle g|-|e\rangle\langle e|$.

The evolution operator for the $2\pi$ pulse and the free evolution $\tau$ is

$\displaystyle U^{(1)}$ $\displaystyle=$ $\displaystyle
ZU=\left(\begin{array}[]{cc}d_{1}&d_{2}\\\
-d_{2}&-d_{1}^{*}\end{array}\right).$ (9)

For $N$ such operations, the evolution operator at time $t=N\tau$ becomes

$\displaystyle U^{(N)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}d_{1}&d_{2}\\\
-d_{2}&-d_{1}^{*}\end{array}\right)^{N}.$ (12)

Let $d_{1}=d_{1r}+id_{1i}$ and $d_{2}=id_{2i}$. After some straightforward
simplifications with the use of Pauli matrices, one easily obtains

$\displaystyle U^{(N)}$ $\displaystyle=$ $\displaystyle i^{N}(\cos
N\lambda-i\sigma_{\alpha}\sin N\lambda)$ (13)

where $\sin\lambda=\sqrt{1-d_{1i}^{2}}$ and
$\sigma_{\alpha}=(1/\sin\lambda)(d_{1r}\sigma_{z}-d_{2i}\sigma_{y})$. Note
that $\lambda$ depends on $\tau$ instead of $t=N\tau$.

The transition probability from the initial state $|e\rangle$ to the ground
state $|g\rangle$ is

$\displaystyle p_{eg}^{\prime}(t=N\tau)$ $\displaystyle=$
$\displaystyle|\langle
g|U^{(N)}|e\rangle|^{2}=p_{eg}(\tau){\sin^{2}N\lambda\over\sin^{2}\lambda}.$
(14)

The above result is exact for any coupling strength and pulse delay $\tau$.
For weak coupling $\nu\ll\delta$, $\Omega\approx\delta^{\prime}$ and
$\lambda\approx\delta^{\prime}\tau+\pi/2+2k\pi$ with $k$ an integer, then

$\displaystyle p_{eg}^{\prime}$ $\displaystyle\approx$
$\displaystyle{\nu^{2}\over\delta^{\prime
2}}\tan^{2}\delta^{\prime}\tau\sin^{2}N\delta^{\prime}\tau$ (15)

for even $N$, which is exactly Eq. (9) in Agarwal et al.’s paper Agarwal et
al. (2001a).

##  IV Quantum Zeno and anti-Zeno effects

The above results are applicable only to single mode bath/environment which
couples to the central two-level system. In general the bath has multimodes
and the coupling may depend on the mode, e.g., the dipolar coupling between an
atom and an electric and magnetic field. For a many-mode bath, the transition
probability of the two-level system subjected to control pulses is in general
given by

$\displaystyle p_{eg}^{\prime}$ $\displaystyle=$
$\displaystyle\sum_{n}{\nu^{2}\over\Omega_{n}^{2}}\sin^{2}\Omega_{n}\tau{\sin^{2}N\lambda_{n}\over\sin^{2}\lambda_{n}}$
(16)

with $n$ the bath mode index mmnote . Note that $\Omega_{n}$ and $\lambda_{n}$
become mode dependent in the many-mode bath case and $\nu$ is constant here
but will be considered as mode dependent in Sec. IV.4. Assuming the bath
spectrum is dense, we turn the summation over mode index $n$ into the
integration over mode frequency $\omega$,

$\displaystyle p_{eg}^{\prime}$ $\displaystyle=$ $\displaystyle\int
d\omega\rho(\omega){\nu^{2}\over\Omega^{2}}\sin^{2}\Omega\tau{\sin^{2}N\lambda\over\sin^{2}\lambda}$
(17)

where $\rho(\omega)$ is the density of states of the bath. For $N=1$ we get
the free decay results (no control pulse). Before considering the real
physical systems which usually have mode dependent coupling, we study several
toy models with constant coupling strength $\nu=\nu_{0}$ for all bath modes to
gain some ideas about the effect of control pulses.

###  IV.1 Uniform spectrum intensity

Figure 2: Control pulse effect with fixed pulse delays. The parameters are
$\omega_{0}=1,\nu_{0}=0.001,\omega_{c}=100$, and $\tau=10,4,3,2,1$ from top to
bottom for solid lines. The dashed line denotes the free decay. Time is in
units of $1/\omega_{0}$ hereafter.

Taking $\rho(\omega)=\rho_{0}\equiv 1/\omega_{c}$ if $\omega\in[0,\omega_{c}]$
and $\rho(\omega)=0$ otherwise with $\omega_{c}$ denoting the cutoff frequency
of the bath, Fig. 2 shows typical free decay and controlled decay of the
excited state. Except at very short times, Fig. 2 shows that the transition
probability linearly depends on the total evolution time. Utilizing the linear
dependence, one defines decay rate (Einstein constant) at long time $t$ as

$\displaystyle A$ $\displaystyle\equiv$ $\displaystyle{\partial
p_{eg}(t)\over\partial t}=2\pi\rho_{0}\nu_{0}^{2}$ (18)

for free decay Agarwal et al. (2001a) and

$\displaystyle A^{\prime}$ $\displaystyle\equiv$
$\displaystyle{1\over\tau}{\partial p_{eg}^{\prime}\over\partial
N}={2\pi\rho_{0}\nu_{0}^{2}\over\tau^{2}}\sum_{k}{1\over\nu_{0}^{2}+(\omega_{k}-\omega_{0})^{2}/4}$
(19)

for controlled decay, where $\omega_{k}$ is the $k$th resonant mode frequency
Kofman and Kurizki (2001). For weak coupling $\nu_{0}\ll\pi/\tau$, the $k$th
resonant mode lies approximately at
$\omega_{k}\approx\omega_{0}+(2k+1)\pi/\tau$ where $\sin\lambda\ll 1$ and
$\sin
2N\lambda/\sin\lambda\approx\pi\delta(\lambda)\approx(2\pi/\tau)\delta(\omega-\omega_{k})$
for large $N$. Note that the two nearest neighbor peaks around $\omega_{0}$
with $k=0$ and $k=-1$ contribute equally about $40\%$ among all the peaks
Agarwal et al. (2001a); Kofman and Kurizki (2001).

If we assume the bath has only positive frequency and concentrate on the two
dominant peaks, we find

$\displaystyle{A^{\prime}\over A}$ $\displaystyle\simeq$
$\displaystyle\left\\{\begin{array}[]{cc}0,&\omega_{c}\tau<\pi\\\
{1/2},&\omega_{0}\tau<\pi<\omega_{c}\tau\\\ 1,&{\rm
otherwise}\end{array}\right.$ (23)

and $p_{eg}^{\prime}/p_{eg}$ also shows step-like behavior as depicted in Fig.
3. Moreover, Fig. 3 exhibits only quantum Zeno effect, i.e., suppression of
the decay by the control pulses. The decay are completely suppressed,
$A^{\prime}=0$, once the control pulse frequency is larger than the cutoff
frequency.

Figure 3: (Color online) Control pulse effect vs number of pulses $N$ (main
panel) and relative positions of resonant modes to $\omega_{0}$ and
$\omega_{c}$ (upper three panels A, B, and C corresponds to three regions in
the main panel, respectively). The parameters are
$N\tau=100,\omega_{0}=1,\nu_{0}=0.001$, and $\omega_{c}=100$. Two big steps
shows two major peaks out of the lower and upper cutoff frequency,
respectively. These small steps shows minor peaks going out of the cutoff
frequencies. Circles are for the frequency-dependent coupling case.

###  IV.2 Gaussian spectrum intensity

A Gaussian spectrum intensity has a form as

$\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2\pi}\;\Gamma}\;e^{-(\omega-\omega_{m})^{2}/2\Gamma^{2}},\;\;\omega\geq
0$ (24)

where $\Gamma$ and $\omega_{m}$ denotes the width and the position of the
maximal intensity, respectively.

####  IV.2.1 $\omega_{0}=\omega_{m}$

In this case we expect only quantum Zeno effect to appear, $A^{\prime}\leq A$,
because it is easy to check that

$\displaystyle{A^{\prime}\over A}$ $\displaystyle=$
$\displaystyle{4\over\pi^{2}}\sum_{k}{1\over(2k+1)^{2}}\;{\rho(\omega_{k})\over\rho(\omega_{0})}\leq
1$ (25)

where we have used the large $N$ and weak coupling assumptions. By considering
the two dominant peaks, we further obtain that

$\displaystyle{A^{\prime}\over A}$ $\displaystyle\approx$
$\displaystyle{8\over\pi^{2}}\;e^{-\pi^{2}/2\Gamma^{2}\tau^{2}}$ (26)

for small $\tau$ such that $\Gamma\tau\lesssim\pi$. Equation (26) shows that
the decay rate decreases rapidly with the control pulse frequency in a
Gaussian form. For exceedingly small $\tau$ which satisfies
$\Gamma\tau\ll\pi$, $A^{\prime}$ is essentially zero, which means the
transition is inhibited and the survival probability of the initial state
saturates. The red dashed line in Fig. 4 demonstrates the quantum Zeno effect
and the prohibition of the transition.

Figure 4: (Color online) Reduced transition probability at fixed time
($t=N\tau=100$) vs number of pulses $N$ for Gaussian spectrum intensity for
$\omega_{0}=\omega_{m}$ (red dashed line) and $\omega_{0}=\omega_{m}-2$ (blue
solid line). Circles and crosses are for the corresponding frequency-dependent
coupling case. The vertical dot-dashed line shows the boundary between quantum
Zeno and anti-Zeno effects for $\omega_{0}=\omega_{m}-2$. Other parameters are
$\nu_{0}=0.001,\omega_{c}=100,\omega_{m}=\omega_{c}/2$, $\Gamma=1$.

####  IV.2.2 $|\omega_{0}-\omega_{m}|\gtrsim\Gamma$

For $|\omega_{0}-\omega_{m}|<\Gamma$, the results are similar to
$\omega_{0}=\omega_{m}$ and exhibit only quantum Zeno effect. To observe
substantially enhancement of the transition, i.e., the quantum anti-Zeno
effect Kofman and Kurizki (2000), $|\omega_{0}-\omega_{m}|\gtrsim\Gamma$ is
required. More specifically, by taking the biggest peak around $\omega_{0}$,
one obtains the necessary condition for quantum anti-Zeno effect as

$\displaystyle|\omega_{0}-\omega_{m}|\geq\sqrt{4\ln{\pi\over
2}}\;\Gamma\approx 1.3\Gamma.$ (27)

The blue solid line in Fig. 4 shows the quantum Zeno and anti-Zeno effects for
$|\omega_{0}-\omega_{m}|=2$. Both quantum Zeno (large $N$ region) and anti-
Zeno (small $N$ region) effects are observed. At fixed time $t=N\tau$, large
$\tau$ (small N) gives anti-Zeno effect while small $\tau$ (large N) gives
Zeno effect. The boundary between quantum Zeno and anti-Zeno effect is about
$2|\omega_{m}-\omega_{0}|\tau\approx\pi$ if
$|\omega_{m}-\omega_{0}|\gg\Gamma$. Strongest anti-Zeno effect is obtained at
$|\omega_{0}-\omega_{m}|\tau=\pi$ where one of the two main resonant modes
lies near $\omega_{m}$.

###  IV.3 Lorentzian and exponential spectrum intensity

The spectrum intensity for Lorentzian and exponential are, respectively,

$\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle{\Gamma/\pi\over(\omega-\omega_{m})^{2}+\Gamma^{2}},\;\omega\geq
0$ (28) $\displaystyle\rho(\omega)$ $\displaystyle=$ $\displaystyle{1\over
2\Gamma}e^{-|\omega-\omega_{m}|/\Gamma},\;\omega\geq 0.$ (29)

Similar to Gaussian spectral intensity, only the quantum Zeno effect is
observed if $\omega_{0}=\omega_{m}$ for the Lorentzian and exponential shape
(Fig. 5). The transition probability is essentially inhibited once
$\tau\ll\tau_{c}$ where $\tau_{c}\sim 1/\omega_{c}$. Once
$|\omega_{0}-\omega_{m}|\gtrsim\Gamma$, we find both quantum Zeno and anti-
Zeno effects and the boundary between Zeno and anti-Zeno regime is determined
approximately by $2|\omega_{m}-\omega_{0}|\tau\approx\pi$. The peak position
of the quantum anti-Zeno effect lies about at
$|\omega_{0}-\omega_{m}|\tau\approx\pi$. As shown in Fig. 5, the reduced
transition probability in the exponential case has a narrower peak width than
that in the Lorentzian case. Two small secondary peaks are noticeable in the
anti-Zeno region of the exponential spectrum case. These peaks are due to the
second and third quasi-level resonant to $\omega_{m}$. They satisfy the
condition, respectively, $|\omega_{0}-\omega_{m}|\tau\approx 3\pi,5\pi$.

Figure 5: (Color online) Same as Fig. 4 except that the spectrum intensity is
Lorentzian (top) and exponential (bottom).

###  IV.4 Frequency dependent coupling

The widely adopted dipolar coupling in spontaneous emission of a two level
atom or molecule has a frequency dependence as
$\nu(\omega)=\nu_{0}\sqrt{\omega}$ where $\nu_{0}$ is taken as a constant,
which depends on the dipole matrix elements Scully and Zubairy (1997). We will
consider the same coupling spectrum intensity as before, i.e.,

$\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle{1\over\omega}{1\over\sqrt{2\pi}\;\Gamma}\;e^{-(\omega-\omega_{m})^{2}/2\Gamma^{2}},\;\omega\geq
0$ (30) $\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle{1\over\omega}{\Gamma/\pi\over(\omega-\omega_{m})^{2}+\Gamma^{2}},\;\omega\geq
0$ (31) $\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle{1\over\omega}{1\over
2\Gamma}e^{-|\omega-\omega_{m}|/\Gamma},\;\omega\geq 0$ (32)

for Gaussian, Lorentzian, and exponential density of state, respectively. As
shown in Figs. 4 and 5, the frequency dependent coupling has little effect on
the performance of the control pulses.

##  V Decay freezing

Figure 6: Decay freezing (left column) and dependence of the differential
freezing value on pulse delay (right column) for the Gaussian (top row),
Lorentzian (middle row), and exponential (bottom row) spectrum. Only results
of odd number pulses are plotted for a better view. The dashed lines show
decay without control pulses. Crosses of dashed lines and solid lines in the
left column panels denote the specific pulse delay $\tau$.

By inspecting Eq. (17), the controlled transition probability freezes if
$N\lambda\gg 1$ and the freezing value is

$\displaystyle\lim_{N\rightarrow\infty}p_{eg}^{\prime}$ $\displaystyle=$
$\displaystyle{1\over 2}\int
d\omega\rho(\omega){\nu^{2}\over\Omega^{2}}\sin^{2}\Omega\tau{1\over\sin^{2}\lambda},$
(33)

where we have replaced the rapidly oscillating integrand $\sin^{2}(N\lambda)$
with its average $1/2$. The left column of Fig. 6 from numerical calculation
indeed shows that the transitions freeze at long times for three different
cases at small $\tau$. The smaller the $\tau$ is, the smaller the freezing
value is. Freezing of the transition probability is equivalent to freezing of
the survival probability of the excited state, $f=1-p^{\prime}_{eg}$.

Moreover, for small enough pulse delay $\tau$, we have $\sin^{2}\lambda\approx
1$ thus

$\displaystyle\lim_{N\rightarrow\infty}p_{eg}^{\prime}$ $\displaystyle\approx$
$\displaystyle{1\over 2}\int
d\omega\rho(\omega){\nu^{2}\over\Omega^{2}}\sin^{2}\Omega\tau={1\over
2}p_{eg}(\tau).$ (34)

The above relation indicates that the freezing value of the transition
probability at long times is one half of the free transition value at $t=\tau$
which is the pulse delay. Clearly, the road map to the quantum ZE is that the
decay rate becomes zero after initial pulses (decay freezing) and the freezing
value of the survival probability $f$ approaches 1 as $\tau$ decreases.

Define differential freezing value
$\varepsilon(\tau)=p_{eg}^{\prime}(N\tau=1000)-{1\over 2}p_{eg}(\tau)$, which
describes the difference of the transition probability after many pulses and
one half of the decay in the first pulse delay. The right column of Fig. 6
shows that $\varepsilon$ decreases with increasing of $1/\tau$ (decreasing of
$\tau$), confirming the analytical results of Eq. (34).

In fact, decay freezing exists not only in the model system we consider in the
paper, but also exists in many other pulse-controlled systems, such as gated
semiconductor quantum dot Zhang et al. (2007a, b, 2008); Lee et al. (2008);
Liu et al. (2007), spin-boson model Viola and Lloyd ; Faoro and Viola (2004),
and nuclear spins Haeberlen (1976). The basic idea behind the decay freezing
is that the control pulses creates an effective preferred direction along
which the decay is frozen. In terms of Pauli matrix (c.f. Eq. 2), the
preferred direction created by control pulses in the model we consider is $z$.

##  VI Conclusion and discussion

By investigating three models of coupling spectrum intensity (Gaussian,
Lorentzian and exponential), we demonstrate that a two-level system subjected
to many ideal $2\pi$ pulses exhibits both quantum Zeno and anti-Zeno effect,
depending on the relative position of $\omega_{0}$ to the peak position
$\omega_{m}$ of the spectrum and the pulse delay $\tau$. Instead of decreasing
the decay rate, the pulsed two-level system shows decay freezing after many
pulses at small $\tau$ and the freezing value of the survival probability of
the initial excited state approaches 1 (no decay) with decreasing $\tau$.

In this paper, all the spectrums have single peak and we observe only single
quantum Zeno and/or anti-Zeno region. Under some special circumstances where a
multiple peaks spectrum exists, one would expect multiple quantum Zeno and
anti-Zeno regions. We have also assumed that the spectrum of the structured
vacuum is time independent where the back action exerted on the vacuum by the
two-level system has been neglected. A full quantum version of the coupling
between the two-level system and the vacuum could possibly change the picture
of the controlled decay at long times but the short time behavior would be
intact in the weak coupling regime, because the back action is weak and needs
a long time to manifest its effect on the two-level system dynamics.

We consider only periodic pulse sequence ($\tau$ is fixed) in this paper. In
principle, other pulse sequences with varying $\tau$ may also give similar
quantum Zeno and anti-Zeno effect. They may even have additional advantages
Uhrig (2007); Lee et al. (2008). In addition, the strength and the duration of
the $2\pi$ pulses are finite in practice. One could minimize the finite pulse
effect by employing the phase alternation techniques Slichter (1992) or the
Eulerian protocols Viola (2004).

##  VII Acknowledgments

W. Z. is grateful for many helpful discussions with V. V. Dobrovitski, S. Y.
Zhu, and T. Yu. This work was partially carried out at the Ames Laboratory,
which is operated for the U. S. Department of Energy by Iowa State University
under Contract No. W-7405-82 and was supported by the Director of the Office
of Science, Office of Basic Energy Research of the U. S. Department of Energy.
Part of the calculations was performed at the National High Performance
Computing Center of Fudan University.

## References

  * Milonni (1994) P. W. Milonni, _The Quantum Vacuum: An Introduction to Quantum Electrodynamics_ (Academic Press, New York, 1994).
  * Scully and Zubairy (1997) M. O. Scully and M. S. Zubairy, _Quantum Optics_ (Cambridge University Press, Cambridge, 1997).
  * Allen and Eberly (1975) L. Allen and J. H. Eberly, _Optical Resonance and Two-level Atoms_ (Dover Publication, New York, 1975).
  * Frishman and Shapiro (2003) E. Frishman and M. Shapiro, Phys. Rev. A 68, 032717 (2003).
  * Evers and Keitel (2002) J. Evers and C. H. Keitel, Phys. Rev. Lett. 89, 163601 (2002).
  * Frishman and Shapiro (2001) E. Frishman and M. Shapiro, Phys. Rev. Lett. 87, 253001 (2001).
  * Agarwal et al. (2001a) G. S. Agarwal, M. O. Scully, and H. Walther, Phys. Rev. Lett. 86, 4271 (2001a).
  * Vogel and Welsch (2006) W. Vogel and D.-G. Welsch, _Quantum Optics_ (Wiley-VCH, Weinheim, 2006), 3rd ed.
  * Itano et al. (1990) W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 41, 2295 (1990).
  * Itano et al. (1991) W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 43, 5168 (1991).
  * Misra and Sudarshan (1977) B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
  * Kofman and Kurizki (2000) A. G. Kofman and G. Kurizki, Nature(London) 405, 546 (2000).
  * Agarwal et al. (2001b) G. S. Agarwal, M. O. Scully, and H. Walther, Phys. Rev. A 63, 044101 (2001b).
  * Fischer et al. (2001) M. C. Fischer, B. Gutiérrez-Medina, and M. G. Raizen, Phys. Rev. Lett. 87, 040402 (2001).
  * Kofman and Kurizki (2001) A. G. Kofman and G. Kurizki, Phys. Rev. Lett. 87, 270405 (2001).
  * Scully et al. (2003) M. O. Scully, S.-Y. Zhu, and M. S. Zubairy, Chaos, Solitons and Fractals 16, 403 (2003).
  * (17) We have assumed the dipolar interaction has a symmetry such that $\langle e|\vec{d}\cdot\vec{E}|g\rangle=\langle g|\vec{d}\cdot\vec{E}|e\rangle$, where $\vec{d}$ and $\vec{E}$ are the dipole operator of the atom and the electric field of the light, respectively.
  * (18) We adopt the Schrödinger picture instead of the usual interaction picture (e.g., Ref. Agarwal et al. (2001a)) for our convenience. Both pictures are equivalent in principle.
  * (19) For weak couplings, which are the case in the decay problem we consider, high order nonlinear interaction between the two-level atom and multiple electric field modes is neglected.
  * Zhang et al. (2007a) W. Zhang, V. V. Dobrovitski, L. F. Santos, L. Viola, and B. N. Harmon, Phys. Rev. B 75, 201302(R) (2007a).
  * Zhang et al. (2007b) W. Zhang, N. P. Konstantinidis, K. A. Al-Hassanieh, and V. V. Dobrovitski, J. Phys.: Condens. Matter 19, 083202 (2007b).
  * Zhang et al. (2008) W. Zhang, N. P. Konstantinidis, V. V. Dobrovitski, B. N. Harmon, L. F. Santos, and L. Viola, Phys. Rev. B 77, 125336 (2008).
  * Lee et al. (2008) B. Lee, W. M. Witzel, and S. Das Sarma, Phys. Rev. Lett. 100, 160505 (2008).
  * Liu et al. (2007) R.-B. Liu, W. Yao, and L. J. Sham, New J. Phys. 9, 226 (2007).
  * (25) L. Viola and S. Lloyd, eprint quant-ph/9809058.
  * Faoro and Viola (2004) L. Faoro and L. Viola, Phys. Rev. Lett. 92, 117905 (2004).
  * Haeberlen (1976) U. Haeberlen, _High resolution NMR in solids: Selective averaging_ (Academic Press, New York, 1976).
  * Uhrig (2007) G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007).
  * Slichter (1992) C. P. Slichter, _Principles of Magnetic Resonance_ (Springer-Verlag, New York, 1992).
  * Viola (2004) L. Viola, J. Mod. Opt. 51, 2357 (2004).