Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

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Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
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Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
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Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
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I am looking forward to hearing from you.

Kind regards,

David Rain
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Science and Education Publishing (http://www.sciepub.com)

Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

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I am looking forward to hearing from you.

Kind regards,

David Rain
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Science and Education Publishing (http://www.sciepub.com)

Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

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Science and Education Publishing (http://www.sciepub.com)

Generalized Optical Bloch Equations for a General Three-Level Atom Interacting with a Linearly Polarized Light in the Rotating Wave and Electric Dipole Approximations

Mesﬁn Woldeyohannes

Department of Chemistry & Physics, Western Carolina University, 112 Apodaca Science Building, Cullowhee, NC, USA

Abstract: The nine (9) elements of the density matrix of a general three-level atom interacting with a classically described linearly polarized light are parameterized by eight (8) variables related to the populations of the atomic levels, and the coherences (interference effects) between the levels. The dynamical equations of these eight parameters (that is, the optical Bloch equations for a general three-level atom) are derived under the electric dipole as well as the rotating wave approximations, and under the assumption that the matrix elements of the atomic electric dipole are all real. The general optical Bloch equations so derived are shown to contain the well-known optical Bloch equations for three-level atoms in the , and cascade configurations as well as those for two-level atoms as special cases. Expressing the interaction of light with a three-level atom in all possible configurations as well as with a two-level atom by a single set of generalized optical Bloch equations is the main result of this work.

Keywords: Optical Block equations, two-level atom, three-level atom, Ʌ-configuration, V-configuration, cascade configuration

Received June 16, 2024; Revised July 17, 2024; Accepted July 24, 2024

Citation: Mesﬁn Woldeyohannes, Generalized Optical Bloch Equations for a General Three-Level Atom Interacting with a Linearly Polarized Light in the Rotating Wave and Electric Dipole Approximations. International Journal of Physics, 12 (4). 164-174. January 2024. doi:10.12691/ijp-12-4-4

1. Introduction

The optical Bloch Equations for an atom-ﬁeld interaction system are the equations of motion for the elements of the density matrix of the system [1,2]. In the case of a two-level atom interacting with light, these equations are formally identical to the dynamical equations of a spin one-half particle in an oscillatory magnetic ﬁeld. As a result, the spin vector formalism of Bloch developed for magnetic resonance [3] can be directly applied to optical resonance problems involving two-level atoms [1], hence the term optical Bloch equations.

In this work we derive the optical Bloch equations governing the interaction of a general three-level atom with a linearly polarized light, under the electric dipole as well as the rotating wave approximations, and under the assumption that the matrix elements of the atomic electric dipole are all real. The electric dipole approximation is invoked by assuming that the external light is in the visible region so that its wavelength is much larger than the size of the interacting atom. The rotating wave approximation is applied by transforming the 8 parameters of the density matrix of the general three-level atom into another set of 8 parameters through anorthogonal transformation matrix representing “pure rotation” in an 8-dimensional state space, and then ignoring the rapidly varying terms. The matrix elements of the electric dipoles associated with the atomic transitions can be made real by adjusting the arbitrary phases of the wave functions of the relevant atomic levels, provided the transitions between the levels obey the ∆m = 0 selection rule [1]. The optical Bloch equations for the special cases of three-level atoms in the V, Λ, and cascade conﬁgurations as well as those for two-level atoms are easily derived from the general Bloch equations by setting to zero all quantities non-relevant for each of those special cases.

The remainder of this paper is organized in three sections as follows. Section (2) gives a general description of the operators needed to describe the interaction of a general n-level atom with a classically described electromagnetic ﬁeld in the electric dipole approximation. This description is then applied in section (3) to derive the optical Bloch equations for a general three-level atom with arbitrary arrangement of atomic levels. Section (4) deals with the derivation of the optical Bloch equations for the special cases of three-level atoms in the V, Λ, and cascade conﬁgurations as well as those for two-level atoms from the general Bloch equations of section (3). Finally, section (5) provides a brief conclusion of the paper.

2. General Description of Operators

2.1. Atom-ﬁeld Interaction Hamiltonian in the Electric Dipole Approximation

The Hamiltonian of a bare atom (with no radiation present) can be written as [2,4]

[picture] | (2.1.)

where and are, respectively, the eigenvectors and eigenvalues of [2.4] satisfying the orthonormality () and closure () relations, whereas are the atomic operators satisfying the commutation relation in which stands for the Kronecker delta.

Like the bare atom Hamiltonian , any linear operator can be expressed in terms of the atomic operators by applying the closure relation twice as

[picture] | (2.2)

Here, the expansion coefficients are the matrix elements of in the basis . If is a Hermitian operators (as any observable should be), it is represented by a Hermitian matrix [5], one in which all the diagonal elements are real, and any two elements symmetric with respect to the diagonal are complex conjugates of each other ().

The interaction of an atom with a radiation field is represented by an interaction Hamiltonian [2] given by Eq.2.3. In this equation, is the atomic dipole moment vector which is expanded in terms of the atomic operators in accordance with Eq.2.2, are the matrix elements of , is the magnitude of , and is the direction (unit vector) of . The matrix elements are in general complex, and (since is Hermitian) . Moreover, since is a vector operator, its diagonal matrix elements are all zero [5].

[picture] | (2.3)

In the full quantum mechanical treatment of atom-field interaction, the atom is treated quantum mechanically, and the radiation field is taken to be a quantized field [7,8,9]. In this work, we employ the semiclassical approach in which the atom is described quantum mechanically whereas the radiation field is taken to be a purely classical field (by assuming the field to be sufficiently strong). Moreover, we assume the radiation field to be a monochromatic (or quasi-monochromatic) laser light with central frequency which is assumed to be in the visible region. We also assume the light to be linearly polarized in the direction of a unit vector , and propagating in the direction of the wave vector . Taking the nucleus of the atom as the origin of coordinates, the electric field of the applied laser light can be expressed as

[picture] | (2.4)

Here, is a real amplitude of the electric field, is the central frequency of the field, and is the corresponding wave number. If is truly monochromatic, will be constant in time, whereas, more generally, varies with time.

When the applied light is in the optical (visible) region so that the wavelength of the light is much greater than the size of the atom, the spatial variation of the electric field of the light across the atom can be ignored to a good approximation, and this is what is meant by electric dipole approximation [5]. Invoking this approximation in Eq.2.4, we ignore in the exponent, and in to write the applied electric field as

[picture] | (2.5)

Using Eq.2.5 in Eq.2.3 and expanding the result according to Eq.2.2, we obtain

[picture] | (2.6)

The quantity has the dimensions of frequency and is called the Rabi frequency of the external laser radiation. It quantifies the interaction of the external laser radiation with the relevant atomic transition and is zero when (because , in accordance Eq. 2.3)

[picture] | (2.7)

In general, the laser field envelope defined by Eq. 2.4 depends on time, and so do the quantities and which are dependent on as defined by Eq.2.6. However, if the external laser field is purely monochromatic, is a constant independent of time so that and are also constants independent of time. For most applications, if varies sufficiently slowly within an optical period, it (and, therefore, both and ) can be taken to be constants to a good approximation.

Combining Eq.2.1 and Eq. 2.6, we finally obtain

[picture] | (2.8)

for the total Hamiltonian describing atom-field interaction in the electric dipole approximation. In connection with the coefficients of the interaction Hamiltonian , we introduce the atomic transition frequency , and the detuning frequency by

[picture] | (2.9)

The atomic transition frequency is the frequency difference between any two atomic levels and as shown in Fig.1, whereas the detuning frequency is the frequency difference between and the radiation frequency .

2.2. State Vectors and Density Operators in General

In this paper, we choose to work in the Schrödinger picture where the basis vectors are stationary but the state vectors vary in time according to the time dependent Schrödinger equation. Each time dependent state vector can be expanded on the time independent basis uniquely as in Eq. 2.10.

Here, the expansion coefficient gives the probability amplitude for finding the atom in the eigenstate and is, in general, complex. We assume that is normalized () so that the coefficients satisfy the probability conservation condition .

[picture] | (2.10)

The density operator of a system is such that, characterizing the system by a state vector is completely equivalent to characterizing it by the corresponding acting in that state space, in the sense that all physical information that can be obtained from can also be obtained from [5]. For each state vector , the corresponding density operator and its time evolution are given by,

[picture] | (2.11)

where is the total Hamiltonian of the atom-radiation system given by Eq. 2.8. The density operator can be expanded in terms of the atomic operators in accordance with Eq. 2.2 to obtain,

[picture] | (2.12a)

[picture] | (2.12b)

[picture] | (2.12c)

[picture] | (2.12d)

Here are elements of the density matrix, and are the expansion coefficients of the state vector given by Eq. 2.10. The diagonal elements of the density matrix are clearly real, and the trace of the density matrix is equal to unity which follows directly from the probability condition of Eq. 2.10. The off-diagonal elements of the density matrix are in general complex, and satisfy , since is a Hermitian operator, as can be easily deduced from its definition by Eq. 2.11. From the definition of the density operator (Eq. 2.11), the orthonormality condition (), and Eq. 2.12c for the trace of the density matrix, it readily follows that,

[picture] | (2.13)

The diagonal elements the density matrix represent the probability of finding the atom in its eigenstate state and, therefore, are known as the populations. The non-diagonal elementsexpress the interference effects between the states and when the atom is in a coherent linear superposition of these states and, therefore, are known as the coherences. According to Eq. 2.12b, is non-zero only when both and are non-zero. Therefore, the density matrix can have coherences (non-diagonal elements) only between populated states.

Figure 1. (a) Two-level atom, (b) three-level atom in the V-conﬁguration, (c) three-level atom in the Ʌ-conﬁguration, and (d) three-level atom in the cascade configuration. For a three-level atom in the V-conﬁguration, the upper levels |3> and |2> are close to each other but far from the ground level |1>, and the transition |3> ßà |2> is not dipole allowed. On the other hand, for a three-level atom in the Λ-conﬁguration the lower levels |2> and |1> are close to each other but far from the third (highest) level |3>, and the transition |3> ßà |2> is not dipole allowed. For a three-level atom in the cascade configuration, the transition |3> ßà |1> is not dipole allowed

3. Optical Bloch Equations for a General Three-Level Atom

3.1. Parametric Equations of the Density Matrix Elements of a General Three-Level Atom

For a three-level atom, the state space is a 3- dimensional space spanned by the eigenvectors , and , Figure. 1. The most general (normalized) state vector of this 3-dimensional state space is given by the linear superposition of these eigenvectors as

[picture] | (3.1b)

In the three state basis arranged in this order, any operator is represented by a matrix in accordance with Eq. 2.2. In particular, the density operator of Eq. 2.11 corresponding to the state vector

of Eq. 3.1 is represented by a density matrix whose elements are, according to Eq. 2.12b, .

[picture] | (3.2)

Noting that the diagonal elements of the density matrix are real, and add up to unity (Eq. 2.12), whereas the non-diagonal elements are related by complex conjugation, the nine elements of density matrix of Eq. 3.2 can be parameterized by a new set of eight variables as

[picture] | (3.3)

Comparing Eqs. 3.2 and 3.3, we obtain the following expressions for in terms of

[picture] | (3.4a)

[picture] | (3.4b)

[picture] | (3.4c)

[picture] | (3.4d)

[picture] | (3.4e)

[picture] | (3.4f)

[picture] | (3.4g)

[picture] | (3.4h)

where and denote real and imaginary parts, respectively.

From Eq. 3.4, we see that all eight parameters of the density matrix are real. The parameter and are related to the populations of the upper levels and . When the atom is certainly in level so that , we obtain and . On the other hand, when the atom is certainly not in level so that , we obtain and . Therefore, the population parameters and vary between 2 and -1.

[picture] | (3.5a)

[picture] | (3.5b)

Apart from the population parameters and , all the other six parameters of the density matrix are coherence parameters related to interference effects between the atomic levels , and . The quantity

[picture] | (3.6)

is related to the population inversion between the upper levels , and is relevant for three-level atoms in the configuration, as will be seen in section (4). Taking the square of the density matrix in parameter form (Eq. 3.3), and then applying Eq. 2.13 to equate the trace of this square matrix to 1, we obtain,

[picture] | (3.7)

which can also be obtained directly from Eq. 3.4.

For a general three-level atom, the density matrix given by Eq. 3.3 is expanded in terms of the atomic operators of Eq. 2.1, in accordance with Eq. 2.12a as

[picture] | (3.8)

Moreover, the total Hamiltonian of a general three-level atom is obtained from the general Hamiltonian of an n-level atom (Eq. 2.8) to be,

[picture] | (3.9)

where we have used Eq. 2.7 to set for . Using Eq. 3.8 and Eq. 3.9 in Eq. 2.11, we obtain, after considerable careful accounting of terms, the following eight coupled equations for time evolutions of the eight parameters of the density matrix of a general three-level atom defined by Eq. 3.3

[picture] | (3.10a)

[picture] | (3.10b)

[picture] | (3.10c)

[picture] | (3.10d)

[picture] | (3.10e)

[picture] | (3.10f)

[picture] | (3.10g)

[picture] | (3.10h)

Here, a dot over a symbol represents time derivative, and is the frequency difference between the atomic levels and , as defined by Eq. 2.9.

Eqs. 3.10 are general equations applicable for any three-level atom interacting with a radiation field in the optical region under the electric dipole approximation. In these equations, all coefficients containing and are of the form,

[picture] | (3.11)

and, therefore, are real, indicating that Eqs.3.10 are equations for real variables with real coefficients. These equations are simplified a great deal if we assume the atomic dipole matrix elements , , and associated with the atomic transitions , , and , and (as defined by Eq. 2.3) are all real so that

[picture] | (3.12)

where , and (defined by Eq. 2.6) are now real. Using Eq. 3.12 in Eq. 3.10, we obtain

[picture] | (3.13a)

[picture] | (3.13b)

[picture] | (3.13c)

[picture] | (3.13d)

[picture] | (3.13e)

[picture] | (3.13f)

[picture] | (3.13g)

[picture] | (3.13h)

These equations govern the time evolutions of the matrix elements of the density matrix of a general three-level atom (Eq. 3.3) when the matrix elements of the dipole moment of the atom are assumed to be real. They can be cast in matrix form as

[picture] | (3.14)

where is a vector in an 8-dimensional space given in column form as

[picture] | (3.15)

and is an coefficient matrix given by

[picture] | (3.16)

According to Eq. 3.7, the norm of vector is given by,

[picture] | (3.17)

Therefore,is not constant but depends on the population parameters and . However, if we introduce the normalized population parameters and by

[picture] | (3.18a)

so that

[picture] | (3.18b)

the norm of the new vector defined by

[picture] | (3.19)

will be a constant and is given by,

Therefore, there is a conservation law associated with Eq. 3.13 when the population parameters and are replaced by their normalized versions and . This constraint on is nothing but another way of expressing the probability conservation of Eq.2.10 in terms of the parameters of the density matrix.

[picture] | (3.20)

3.2. Rotating Wave Approximation for a General Three-Level Atom

Eq. 3.13 contain terms which vary rapidly at the optical frequencies , , and . Switching to not so rapidly varying variables will render the dynamics of the system more transparent. This can be achieved by an orthogonal transformation - a linear transformation in which the sum of the squares of the new variables is equal to the sum of the squares of the old variables [10]. The necessary and sufficient condition for a linear transformation to be orthogonal is that the transformation matrix be orthogonal (that is, the transpose of be equal to its inverse ). If is orthogonal, then its determinant is necessarily . If , the transformation is called proper rotation because it involves only rotation, and no reflection. On the other hand, if the transformation involves reflection on all or some of the axes in addition to rotation [10].

An orthogonal matrix representing pure (proper) rotation in an 8-dimensional state space is given by,

[picture] | (3.21)

For the matrix M given by Eq. 3.21, direct multiplication shows that (where 1 is the identity matrix) so that . Moreover, , proving our assertion that is orthogonal, and, therefore, represents proper rotation. Under , the 8-dimensional vector of Eq. 3.15 orthogonally transforms to another 8-dimensional vector which, by the very definition of orthogonal transformation, has the same norm as

[picture] | (3.22a)

[picture] | (3.22b)

[picture] | (3.22c)

[picture] | (3.22d)

Differentiating both sides of the transformation equation with respect to time, rearranging results, and then using from Eq. 3.14, we obtain

[picture] | (3.23)

Multiplying both sides of Eq. 3.23 from the right by , we finally obtain

[picture] | (3.24)

This is the equation of motion for the components of the transformed vector obtained from the original vector by the orthogonal transformation . It is a general equation valid for any orthogonal transformation of the form , independent of the dimension of vector , and the corresponding sizes of the coefficient matrix and the transformation matrix.

Using Eq. 3.16 for the coefficient matrix , and Eq. 3.21 for the transformation matrix in Eq. 3.24, we obtain (after considerable algebra) the following eight equations for the time evolution of the eight components of vector .

[picture] | (3.25a)

[picture] | (3.25b)

[picture] | (3.25c)

[picture] | (3.25d)

[picture] | (3.25e)

[picture] | (3.25f)

[picture] | (3.25g)

[picture] | (3.25h)

In these equations, is the detuning of the atomic transition frequency from the applied laser field frequency , as defined by Eq. 2.9

In Eqs. 3.25, the terms containing , , and represent high frequency evolutions, and, therefore, average out to zero over the evolution of the slowly varying terms containing the detuning frequencies . For this reason, these rapidly varying terms can be ignored to a good approximation, and this is what is meant by the rotating wave approximation (RWA) [1]. Therefore, when the RWA is invoked in Eqs. 3.25, we obtain,

[picture] | (3.26a)

[picture] | (3.26b)

[picture] | (3.26c)

[picture] | (3.26d)

[picture] | (3.26e)

[picture] | (3.26f)

[picture] | (3.26g)

[picture] | (3.26h)

Eqs. 3.26h-h are the principal equations of this paper. They are the optical Bloch equations for a general three-level atom interacting with a linearly polarized light in the visible region under the electric dipole and rotating wave approximations, and under the assumption that the matrix elements of the atomic dipole are all real. They reduce to the Bloch equations for all possible special cases of a general three-level atom as well as to the Bloch equations for a two-level atom as shown in the next section.

4. Optical Bloch Equations for Special Cases of a General Three-Level Atoms

4.1. Optical Bloch Equations for Two-Level Atom

A general three-level atom with three atomic transitions , , and reduces to a two-level atom [Fig. 1(a)] when only one of these three transitions is dipole allowed. The density matrix of such a two-level atom can be written in terms of the parameters of the density matrix of a three-level atom (Eq. 3.3) as follows.

[picture] | (4.1a)

[picture] | (4.1b)

[picture] | (4.1c)

From Eq. 4.1c we see that, when a three-level atom is considered as a two-level atom consisting of levels and , the parameters and must be equal and opposite so that the trace of the density matrix is equal to unity, in accordance with Eq. 2.12d.

[picture] | (4.2)

For a two-level atom consisting of levels and , the only relevant quantities in Eqs. 3.26a-h are those related to the transition , namely , , , and . Similarly, for a two-level atom consisting of levels and , the only relevant quantities in Eqs. 3.26a-h are those related to the transition , namely , , , and . Likewise, for a two-level atom consisting of levels and , the only relevant quantities in Eqs. 3.26a-h are those related to the transition , namely , , , , and where, according to Eq. 4.2, . The general Bloch equations for a general three-level atom (Eqs. 3.26a-h) reduce to those of a two-level atom when all quantities non-relevant to the atom are set to zero. As a result, we obtain,

[picture] | (4.3a)

[picture] | (4.3b)

[picture] | (4.3c)

These optical Bloch equations are identical to the optical Bloch equations derived in Allen and Eberly [1] specifically for two-level atoms, showing that our general Bloch equations for three-level atoms contain those for two-level atoms as special cases.

4.2. Optical Bloch Equations for Three-Level Atoms in the , , and Cascade Configurations

A three-level atom is said to be in the so called configuration, if the upper levels and are close to each other but far from the ground level , and if these upper levels are of the same symmetry so that the atomic transition is not dipole allowed, Figure 1(b). Therefore, for a three-level atom in the configuration, the only relevant quantities in Eqs. 3.26a-h are , , , and (associated with the atomic transition ) as well as , , , and (associated with the atomic transition ). As a result, Eqs. 3.26a-h for a general three-level atom reduce (as they should) to those of a three-level atom in the configuration when all quantities non-relevant to the configuration are set to zero.

[picture] | (4.4)

In the case of a three-level atom in the configuration, the lower levels and are close to each other but far from the third (highest) level , and these lower levels are of the same symmetry so that the atomic transition is not dipole allowed, Figure 1(c). Therefore, in such a case, the only relevant quantities in Eqs. 3.26a-h are , , , and (associated with the atomic transition ) as well as , , , and (associated with the atomic transition ). As a result, Eqs. 3.26a-h for a general three-level atom reduce to those of a three-level atom in the configuration when all quantities non-relevant to the configuration are set to zero.

[picture] | (4.5)

The main difference between the Bloch equations for and configurations is that, in addition to the individual populations and of the upper levels and , the configuration (because of its allowed transition) involves the population difference between these upper levels, as indicated by the second term on the RHS of Eq. 4.5e.

For a three-level atom in the cascade configuration, there is no direct transition from to , Figure 1(d). Therefore, in such a case, the only relevant quantities in Eqs. 3.26a-h are , , , and (associated with the atomic transition ) as well as , , , and (associated with the atomic transition ). As a result, Eqs. 3.26a-h for a general three-level atom reduce to those of a three-level atom in the cascade configuration when all quantities non-relevant to the configuration are set to zero.

Eqs. 4.4, 4.5, and 4.6 are identical to the optical Bloch equations derived separately for -configuration,-configuration, and cascade configuration [7,8], showing once again that our general Bloch equations for three-level atoms contain those for , , and cascade configurations as special cases.

[picture] | (4.6)

5. Conclusions

We derived a single set of optical Bloch equations for a general three-level atom from which the optical Bloch equations for three-level atoms in the , , and cascade configurations as well as those for two-level atoms can be easily obtained as special cases. These general optical Bloch equations govern the interaction of a general three-level atom (and all its special cases) with linearly polarized light under the electric dipole and rotating wave approximations and under the assumption that all relevant electric dipoles are real.

References

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[3]   | F. Bloch, Nuclear Induction, Phys. Rev. 70, 460-474 (1946).
[4]   | Woldeyohannes, M and John, S, Coherent control of spontaneous emission near a photonic band edge, PhD Tutorial, J. Opt. B, 5, R43-R82 (2003).
[5]   | Cohen-Tannoudji C., Diu B., and Lalö F., Quantum Mechanics, Volume 1, (Wiely, Toronto, 1977).
[6]   | Loudon R, The Quantum Theory of Light, (Oxford, Calendron, 1983).
[7]   | Meystre P. and Sargen M., Elements of Quantum Optics, (Springer, New York, 1991).
[8]   | Scully M. O. and Zubary S., Quantum Optics, (Cambridge University Press, 1997).
[9]   | Walls D. F. and Milburn G. J., Quantum Optics, (Springer, New York, 1995).
[10]   | Boas M. L, Mathematical Methods in the Physical Sciences, (Wiely, New York, 1991).
[11]   | Woldeyohannes, M and John, S, Coherent control of spontaneous emission near a photonic band edge, Phys. Rev. A, 60, 5046-66 (1999).
[12]   | Woldeyohannes M., Idehenre I, and Hardin T., Coherent control of cooperative spontaneous emission from two identical three-level atoms in a photonic crystal, J. Opt. 17 085105(19pp) (2015).