Molecules in Electromagnetic Fields E-Book

Roman V. Krems

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 579

Veröffentlichungsjahr: 2018

Beschreibung

A tutorial for calculating the response of molecules to electric and magnetic fields with examples from research in ultracold physics, controlled chemistry, and molecular collisions in fields

Molecules in Electromagnetic Fields is intended to serve as a tutorial for students beginning research, theoretical or experimental, in an area related to molecular physics. The author—a noted expert in the field—offers a systematic discussion of the effects of static and dynamic electric and magnetic fields on the rotational, fine, and hyperfine structure of molecules. The book illustrates how the concepts developed in ultracold physics research have led to what may be the beginning of controlled chemistry in the fully quantum regime.  Offering a glimpse of the current state of the art research, this book suggests future research avenues for ultracold chemistry. 

The text describes theories needed to understand recent exciting developments in the research on trapping molecules, guiding molecular beams, laser control of molecular rotations, and external field control of microscopic intermolecular interactions. In addition, the author presents the description of scattering theory for molecules in electromagnetic fields and offers practical advice for students working on various aspects of molecular interactions. 

This important text:

• Offers information on theeffects of electromagnetic fields on the structure of molecular energy levels
• Includes thorough descriptions of the most useful theories for ultracold molecule researchers
• Presents a wealth of illustrative examples from recent experimental and theoretical work
• Contains helpful exercises that help to reinforce concepts presented throughout text

Written for senior undergraduate and graduate students, professors, researchers, physicists, physical chemists, and chemical physicists, Molecules in Electromagnetic Fields is an interdisciplinary text describing theories and examples from the core of contemporary molecular physics. 

Leseprobe

Table of Contents

Cover

List of Figures

List of Tables

Preface

Goals of This Book

What This Book Is Not

How to Read This Book

Acknowledgments

Chapter 1: Introduction to Rotational, Fine, and Hyperfine Structure of Molecular Radicals

1.1 Why Molecules are Complex

1.2 Separation of Scales

1.3 Rotation of a Molecule

1.4 Hund's Cases

1.5 Parity of Molecular States

1.6 General Notation for Molecular States

1.7 Hyperfine Structure of Molecules

Chapter 2: DC Stark Effect

2.1 Electric Field Perturbations

2.2 Electric Dipole Moment

2.3 Linear and Quadratic Stark Shifts

2.4 Stark Shifts of Rotational Levels

Chapter 3: Zeeman Effect

3.1 The Electron Spin

3.2 Zeeman Energy of a Moving Electron

3.3 Magnetic Dipole Moment

3.4 Zeeman Operator in the Molecule‐Fixed Frame

3.5 Zeeman Shifts of Rotational Levels

3.6 Nuclear Zeeman Effect

Chapter 4: AC Stark Effect

4.1 Periodic Hamiltonians

4.2 The Floquet Theory

4.3 Two‐Mode Floquet Theory

4.4 Rotating Wave Approximation

4.5 Dynamic Dipole Polarizability

4.6 Molecules in an Off‐Resonant Laser Field

4.7 Molecules in a Microwave Field

4.8 Molecules in a Quantized Field

Chapter 5: Molecular Rotations Under Control

5.1 Orientation and Alignment

5.2 Molecular Centrifuge

5.3 Orienting Molecules Matters – Which Side Chemistry

5.4 Conclusion

Chapter 6: External Field Traps

6.1 Deflection and Focusing of Molecular Beams

6.2 Electric (and Magnetic) Slowing of Molecular Beams

6.3 Earnshaw's Theorem

6.4 Electric Traps

6.5 Magnetic Traps

6.6 Optical Dipole Trap

6.7 Microwave Trap

6.8 Optical Lattices

6.9 Some Applications of External Field Traps

Chapter 7: Molecules in Superimposed Fields

7.1 Effects of Combined DC Electric and Magnetic Fields

7.2 Effects of Combined DC and AC Electric Fields

Chapter 8: Molecular Collisions in External Fields

8.1 Coupled‐Channel Theory of Molecular Collisions

8.2 Interactions with External Fields

8.3 The Arthurs–Dalgarno Representation

8.4 Scattering Rates

Chapter 9: Matrix Elements of Collision Hamiltonians

9.1 Wigner–Eckart Theorem

9.2 Spherical Tensor Contraction

9.3 Collisions in a Magnetic Field

9.4 Collisions in an Electric Field

9.5 Atom–Molecule Collisions in a Microwave Field

9.6 Total Angular Momentum Representation for Collisions in Fields

Chapter 10: Field‐Induced Scattering Resonances

10.1 Feshbach vs Shape Resonances

10.2 The Green's Operator in Scattering Theory

10.3 Feshbach Projection Operators

10.4 Resonant Scattering

10.5 Calculation of Resonance Locations and Widths

10.6 Locating Field‐Induced Resonances

Chapter 11: Field Control of Molecular Collisions

11.1 Why to Control Molecular Collisions

11.2 Molecular Collisions are Difficult to Control

11.3 General Mechanisms for External Field Control

11.4 Resonant Scattering

11.5 Zeeman and Stark Relaxation at Zero Collision Energy

11.6 Effect of Parity Breaking in Combined Fields

11.7 Differential Scattering in Electromagnetic Fields

11.8 Collisions in Restricted Geometries

Chapter 12: Ultracold Controlled Chemistry

12.1 Can Chemistry Happen at Zero Kelvin?

12.2 Ultracold Stereodynamics

12.3 Molecular Beams Under Control

12.4 Reactions in Magnetic Traps

12.5 Ultracold Chemistry – The Why and What's Next?

Appendix A: Unit Conversion Factors

Appendix B: Addition of Angular Momenta

B.1 The Clebsch–Gordan Coefficients

B.2 The Wigner

‐Symbols

B.3 The Raising and Lowering Operators

Appendix C: Direction Cosine Matrix

Appendix D: Wigner D‐Functions

D.1 Matrix Elements Involving

‐Functions

Appendix E: Spherical Tensors

E.1 Scalar and Vector Products of Vectors in Spherical Basis

E.2 Scalar and Tensor Products of Spherical Tensors

References

Index

End User License Agreement

List of Tables

Chapter 6

Table 6.1 Summary of the external field traps developed for neutral molecules as of 2013. Only selected representative references are given. See text for a more comprehensive list of references.

Chapter 11

Table 11.1 Dependence of the scattering cross sections on the collision velocity

in the limit

for collisions in three dimensions (3D), two dimensions (2D), and in a quasi‐two‐dimensional geometry.

List of Illustrations

Chapter 2

Figure B.1 Schematic structure of the Hamiltonian matrix for two interacting particles in a spherically symmetric potential. Part (a) shows the matrix in the uncoupled angular momentum basis (B.5) and part (b) shows the matrix in the coupled angular momentum basis (B.6). Each square of the tables represents the

block of the radial matrix elements

. The empty squares show the blocks of the matrices, in which all matrix elements are zero. The shaded squares show the nonvanishing blocks of the matrices.

Chapter 1

Figure 1.1 Schematic diagram (not to scale) of the hierarchical structure of the electronic, vibrational, and rotational energy levels for a typical molecule.

Figure 1.2 Typical frequencies of electromagnetic field (in Hz) required to excite hyperfine, fine structure, rotational, vibrational, and electronic transitions in diatomic molecules.

Figure 1.3 The electronic potentials of the molecule OH arising from the interaction of the oxygen atom in two lowest‐energy electronic states labeled

and

with the hydrogen atom in the ground electronic state. The curves are labeled using standard spectroscopic notation [3]. In particular, the symbol X is the standard label for the lowest‐energy (ground) electronic state of the molecule. These potential energies were calculated in Ref. [6].

Figure 1.4 Couplings between different electronic states may affect the vibrational motion of molecules. These couplings become negligible when the electronic states are separated by a large amount of energy.

Figure 1.5 The vibrational energy levels of the OH molecule in the ground electronic state

. Only 10 lowest‐energy levels are shown. The inset shows the vibrational wave functions

for

,

, and

.

Chapter 2

Figure 2.1 The molecular potential and the dipole moment function of the molecule LiCs in the ground electronic state

.

Figure 2.2 Shifts of the energy levels

and

in the presence of a perturbation

that couples the states

and

.

Figure 2.3 The Stark shifts of the rotational energy levels of a diatomic molecule in a

electronic state with the permanent dipole moment

and the rotational constant

as functions of the electric field strength

.

Figure 2.4 The Stark shifts of the rotational energy levels of a diatomic molecule in a

electronic state. The energy levels presented are for the molecule CaH, which has

 cm

,

 cm

, and the dipole moment

Debye. The different panels correspond to different values of

in the limit of zero electric field. Bear in mind that

is not a good quantum number because the electric‐field‐induced interaction (2.36) couples states with different

.

Figure 2.5 The Stark shifts of the rotational energy levels of a diatomic molecule in a

electronic state. The energy levels presented are for the molecule OH, which has

 cm

,

 cm

,

 cm

,

 cm

and the dipole moment

Debye.

Chapter 3

Figure 3.1 A correlation diagram between the low‐ and high‐field limits for states from within the ground electronic state

and excited electronic state

of the molecule CaH. The dashed lines show the perturbing states of the

state coupled to the

states by the

operators. The states from within the

and

manifolds are labeled by the Hund's case (b) angular momentum quantum numbers in the low‐field limit and their projections on the direction of the field axis in the high‐field limit.

Figure 3.2 Zeeman levels of a CaF molecule in the rotational state characterized by

of the

electronic state: Full curves–accurate calculations; dashed lines–the magnitudes of the diagonal matrix elements given by Eq. (3.47).

Figure 3.3 Symbols ‐ measured frequency shift for the

(circles) and

(triangles) transitions; curves – direction cosine calculations.

Figure 3.4 Zeeman splitting of the hyperfine energy levels of

Rb

Cs(

) in the ground rotational state.

Chapter 4

Figure 4.1 A schematic diagram of the field‐free molecular energy levels

and

shown by solid lines and the states

and

shown by dashed lines. The time‐dependent operator

couples

with both

and

. The rotating wave approximation eliminates the coupling to

.

Figure 4.2 Stark shifts of the rotational energy levels of a

molecule in an off‐resonant microwave field.

Chapter 5

Figure 5.1 The rotational angular momentum of a linear molecule is perpendicular to the molecular axis. Here, we assume that

, as is the case for a molecule in a

electronic state. A molecule with the molecular axis oriented at an angle

with respect to the

‐axis can be in a superposition of angular momentum states with

and

projections, representing an aligned angular momentum state.

Figure 5.2 The expectation value of the orientation angle cosine of a rigid rotor in a DC electric field vs the dimensionless parameter

for three lowest energy states with

. The full lines show the results computed with Eq. (5.8) and the dotted lines with Eq. (5.9). Typical values of

for simple diatomic molecules are in the range of 0–12. The values of

are shown for higher values of

to illustrate that the limit of

is approached slowly, even for the lowest energy state.

Figure 5.3 The eigenstates of a quantum pendulum (a) and a rigid rotor in a DC field (b). The vertical lines show the energies and the curves – the square of the corresponding wave functions plotted as functions of the orientation angle

. The wave functions are not normalized and scaled for better visibility. The bound states of the rigid rotor are labeled by the quantum number

. The bound states and wave functions are calculated for

. The energy is in the units of

for the planar pendulum and in the units of

for the rigid rotor.

Figure 5.4 Energy levels of a rigid rotor in an off‐resonant laser field. The energy is in units of the rotational constant

and the molecule–field interaction strength is in units of

. The figure illustrates that the interaction with the laser field brings the molecular states of opposite parity together.

Figure 5.5 The eigenstates of a rigid rotor in an AC electric field. The horizontal lines show the energies and the curves – the square of the corresponding wave functions plotted as functions of the orientation angle

. The wave functions are not normalized and enhanced for better visibility. The bound states of the rigid rotor are labeled by the quantum number

. The bound states and wave functions are calculated for

. The energy is in the units of

.

Figure 5.6 An optical centrifuge for molecules. The spinning electric field is created by splitting a laser pulse at the center of its spectrum and applying an opposite frequency chirp to the two halves.

Chapter 6

Figure 6.1 The low‐field‐seeking and high‐field‐seeking states in the Stern–Gerlach experiment.

Figure 6.2 Paths of molecules. The two solid curves indicate the paths of two molecules having different moments and velocities and whose moments are not changed during passage through the apparatus. This is indicated by the small gyroscopes drawn on one of these paths, in which the projection of the magnetic moment along the field remains fixed. The two dotted curves in the region of the

magnet indicate the paths of two molecules the projection of whose nuclear magnetic moments along the field has been changed in the region of the

magnet. This is indicated by means of the two gyroscopes drawn on the dotted curves, for one of which the projection of the magnetic moment along the field has been increased, and for the other of which the projection has been decreased.

Figure 6.3 The electric field potential generated by a monopole (a), dipole (b), and quadrupole (c).

Figure 6.4 An illustration of a device called “centrifuge decelerator” for the production of slow molecules. The molecules are injected into the space between four electrodes bent into a spiral shape. The spiral is then rotated to decelerate molecules moving toward the center by the inertial force. The electrodes are bent upward at the end of the spiral, thus guiding only the slowest molecules (with translational temperature less than 1 K) toward the exit.

Figure 6.5 Schematic diagram of a Stark decelerator. The Stark energy of an ND

molecule in a low‐field‐seeking quantum state is shown as a function of position

along the molecular beam axis. The Stark energy has a period of

.

Figure 6.6 Scheme of the experimental setup used for the Stark deceleration of a beam of CO molecules in the original work. In this experiment, CO molecules are prepared in a single, metastable quantum state (

,

,

) by pulsed‐laser excitation of ground‐state CO molecules. The beam of metastable CO molecules is slowed down on passage through a series of 63 pulsed electric field stages. The time‐of‐flight distribution of the metastable CO molecules over the 54 cm distance from laser preparation to detection is measured via recording of the amount of electrons emitted from a gold surface when the metastable CO molecules impinge on it.

Figure 6.7 Schematic drawing of the electrodes for a cylindrically symmetric 3D rf trap. Typical dimensions are

m to

cm, with

100–500 V,

–50 V, and

kHz to 100 MHz.

Figure 6.8 Electrostatic quadrupole trap geometry in cross section. The figure has rotational symmetry about the

‐axis. Heavy shaded curves: electrode surfaces, held at constant potentials

. The ring radius is

and the end‐cap half‐spacing

. Dashed curves: surfaces of constant

with values

. Full curves: surfaces of constant

with values

. A particle whose electric polarizability is negative will have minimum potential energy at the origin, where

.

Figure 6.9 Magnetic trap for neutral atoms. (a) Spherical quadrupole trap with lines of the magnetic field. (b) Equipotentials of the trap (with field magnitudes indicated in millitesla) in a plane perpendicular to the coils.

Figure 6.10 Time evolution of the CaH spectrum in a magnetic trap. These spectra reveal that CaH molecules in the high‐field‐seeking state (negative frequency shifts) quickly leave the trap. The trapped molecules in the low‐field‐seeking state (positive frequency shifts) are confined and compressed toward the center of the trap.

Figure 6.11 Two‐dimensional optical lattice potential produced by interfering three oscillating electric fields propagating in the same plane along the directions given by Eq. (6.34). The potential can be controlled by changing the angle between the wave vectors of the interfering fields. The potential is shown for

(a) and

(b).

Chapter 7

Figure 7.1 Energy levels of the SrF(

) molecule as functions of a magnetic field in the presence of an electric field of 1 kV cm

. The rotational constant of SrF is 0.251 cm

, the spin–rotation interaction constant

is

cm

, and the dipole moment is 3.47 D. States

and

undergo an avoided crossing at the magnetic field value

. The value of

varies with the electric field.

Figure 7.2 Time‐dependent probability of the spin excitation to be localized on molecule four in a one‐dimensional array of seven SrF(

) molecules in an optical lattice with lattice spacing

nm for different magnetic fields near the avoided crossing shown in Figure 7.1. The electric field magnitude is 1 kV cm

. The value of

varies with the electric field.

Figure 7.3 (a, b) Energy levels of the SrF

molecule (

= 7.53 GHz,

= 74.7 MHz) in an electric field of

= 10 kV cm

as a function of magnetic field

; (c) frequency dependence of the ac field sensitivity for SrF in a linearly polarized microwave field for different electric fields; the lines of different color correspond to the

and

transitions. The dashed line represents the sensitivity to the magnetic field component of the ac field that can be achieved in experiments with atoms [294]; (d) same as in (c) but for the CaH

molecule (

= 128.3 GHz,

= 1.24 GHz).

Figure 7.4 Enhancement of the molecular axis orientation by an off‐resonant laser field. The last term in Eq. (7.2) brings the opposite parity states into closely spaced tunneling doublets, while the second last term orients the molecule with respect to the direction of the DC field by mixing the opposite parity states. The expectation value

is shown for the ground state of the molecule.

Figure 7.5 (a) The energy of the Floquet states for a rigid rotor in a superposition of a DC electric field directed along the

‐axis and a circularly polarized microwave field rotating in the

‐plane. The strength of the DC field is

. The intensity of the microwave field is represented by the parameter

. The frequency of the microwave field is

. (b) The modification of the

‐component of the molecule's dipole moment by the microwave field.

Chapter 8

Figure 8.1 Schematic structure of the Hamiltonian matrix in the total angular momentum representation (8.109) for two interacting particles in an external field. Each square of the table represents the block of the matrix elements corresponding to a set of two quantum numbers:

and

. The empty squares show the blocks of the matrices, in which all matrix elements are zero. The shaded squares show the nonvanishing blocks of the matrix. The bullets show the blocks of the matrix populated by the field‐induced couplings. In the absence of an external field, the Hamiltonian matrix is diagonal in

so only the blocks with

are nonzero.

Chapter 10

Figure 10.1 The effect of a resonance on a scattering cross section.

Figure 10.2 Schematic diagrams illustrating the mechanisms of Feshbach and shape scattering resonances. The resonances occur when a scattering state (with the energy shown by the dashed lines) interacts with a quasi‐bound state (shown by the full horizontal lines). The nature of the quasi‐bound states is different for the two types of resonances. See text for a detailed discussion.

Figure 10.3 Schematic diagrams illustrating the procedure for calculating the energy of the bound and quasi‐bound states.

Figure 10.4 (a) The

‐wave elastic scattering cross section for collisions of O

(

) molecules in the lowest high‐field‐seeking state

as a function of the magnetic field. (b) The minimal eigenvalue of the matching matrix

as a function of the magnetic field. The collision energy is 10

K. The projection of the total angular momentum of the system is

2. New resonances found using the analysis of the magnetic field dependence of

are marked by arrows.

Chapter 11

Figure 11.1 The logarithm of the cross section for the

transition in collisions of NH molecules with

He atoms as a function of the magnetic field and collision energy.

Figure 11.2 Rate constants versus electric field for OH–OH collisions with molecules initially in a particular Stark state. Shown are the collision energies 100 mK (Panel a) and 1 mK (Panel b). Solid lines denote elastic‐scattering rates, while dashed lines denote rates for inelastic collisions, in which one or both molecules change their internal state. These rate constants exhibit characteristic oscillations in field when the field exceeds a critical field of about 1000 V cm

.

Figure 11.3 Modification of a shape resonance by microwave fields. The elastic cross section is plotted as a function of the collision energy for zero microwave field (full line),

(dashed line), and

(dotted line). The Rabi frequency is

.

Figure 11.4 Magnetic field dependence of the Zeeman (full line) and hyperfine relaxation (dashed line) cross sections in collisions of YbF molecules with He atoms at zero electric field – (a),

kV cm

– (b), and

kV cm

– (c). The symbols in the (a) are the results of the calculations without the spin–rotation interaction. The collision energy is 0.1 K.

Figure 11.5 Effect of a scattering resonance on the chemical reaction of H atoms with LiF molecules at an ultralow temperature. The insets show the nascent rotational state distributions of HF molecules produced in the reaction as a function of the final rotational state

at electric field strengths of 0, 32, and 100 kV cm

(left) and 124, 125, and 125.75 kV cm

(right). Note the dramatic change in the shape of the distribution near the resonance electric field (right inset). All calculations were performed in the

‐wave scattering regime (at collision energy 0.01 cm

), where no resonances are present in the reaction cross sections as a function of collision energy.

Figure 11.6 The elastic scattering cross section for collisions of O

molecules in the lowest energy Zeeman state (in which each molecule has the spin angular momentum projection

) as a function of the magnetic field. The collision energy is

K

.

Figure 11.7 Potential energy of a molecule in the low‐field‐seeking and high‐field‐seeking Zeeman states in a magnetic trap. The strength of the trapping field

increases in all directions away from the middle of the trap. Since molecules in a high‐field‐seeking state are untrappable, collision‐induced relaxation from the low‐field‐seeking state to the high‐field‐seeking state leads to trap loss.

Figure 11.8 The rate constant for Zeeman relaxation in collisions of rotationally ground‐state NH(

) molecules in the maximally stretched spin state with

He atoms at zero temperature. Such field dependence is typical for Zeeman or Stark relaxation in ultracold collisions of atoms and molecules. The rate for the Zeeman relaxation vanishes in the limit of zero field. The variation of the relaxation rates with the field is stronger and extends to larger field values for systems with smaller reduced mass.

Figure 11.9 External field suppression of the role of centrifugal barriers in outgoing reaction channels. Incoming channels are shown by full curves; outgoing channels by broken curves. An applied field separates the energies of the initial and final channels and suppresses the role of the centrifugal barriers in the outgoing channels.

Figure 11.10 Decimal logarithm of the cross section for spin relaxation in collisions of CaD(

) molecules in the rotationally ground state with He atoms as a function of electric and magnetic fields. The fields are parallel. The collision energy is 0.5 K. The cross section increases exponentially near the avoided crossings.

Figure 11.11 Cross sections for spin‐up to spin‐down transitions in collisions of CaD molecules in the rotationally ground state with He atoms at two different angles

between the DC electric and DC magnetic fields. The magnetic field is 4.7 T. The positions of the maxima correspond to the locations of the avoided crossings depicted in Figure 7.1 that move as the relative orientation of the fields is changed.

Figure 11.12 Differential scattering cross sections for spin‐up to spin‐down inelastic transitions in collisions of CaD(

) radicals in the rotationally ground state with He atoms in a magnetic field

T at three different collision energies. The collision energy

cm

corresponds to a shape resonance arising from the

partial wave.

Figure 11.13 Differential scattering cross sections for spin‐up to spin‐down inelastic transitions in collisions of CaD(

) radicals in the rotationally ground state with He atoms in a magnetic field

T. The graphs show the cross sections for collisions in the absence of an electric field (a) and in a DC electric field with magnitude

kV cm

(b).

Chapter 12

Figure 12.1 Inelastic or reactive (a) and elastic (b) cross sections typical for atomic or molecular scattering near thresholds.

Figure 12.2 Collisions of ultracold molecules and atoms prepared in the lowest‐energy quantum state.

Figure 12.3 Collisions of ultracold molecules in a quasi‐2D geometry. The extracted loss‐rate constants for collisions of molecules in the same lattice vibrational level (squares) and from different lattice vibrational levels (circles) plotted for several dipole moments. Measured loss‐rate constants for molecules prepared in different internal states are shown as triangles.

Figure 12.4 Schematic illustration of minimum energy profiles for an A(

) + BC(

) chemical reaction in the singlet‐spin (lower curve) and triplet‐spin (upper curve) electronic states. Electric fields may induce nonadiabatic transitions between the different spin states and modify the reaction mechanism.

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Molecules in Electromagnetic Fields

From Ultracold Physics to Controlled Chemistry

Roman V. Krems

Copyright

This edition first published copyright year 2019

© 2019 copyright year John Wiley & Sons, Inc

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Library of Congress Cataloging-in-Publication Data

Names: Krems, Roman V., 1977- author.

Title: Molecules in electromagnetic fields : from ultracold physics to controlled chemistry / by Roman V. Krems.

Description: 1st edition. | Hoboken, NJ : John Wiley & Sons, 2019. | Includes index. |

Identifiers: LCCN 2017053373 (print) | LCCN 2017060297 (ebook) | ISBN 9781119387350 (pdf) | ISBN 9781119387398 (epub) | ISBN 9781118173619 (cloth)

Subjects: LCSH: Molecules. | Electromagnetic fields.

Classification: LCC QC173.3 (ebook) | LCC QC173.3 .K74 2018 (print) | DDC 539/.6-dc23

LC record available at https://lccn.loc.gov/2017053373

Cover design by Wiley

Cover image: © Roman Krems

List of Figures

Figure 1.1

Schematic diagram (not to scale) of the hierarchical structure of the electronic, vibrational, and rotational energy levels for a typical molecule.

Figure 1.2

Typical frequencies of electromagnetic field (in Hz) required to excite hyperfine, fine structure, rotational, vibrational, and electronic transitions in diatomic molecules.

Figure 1.3

The electronic potentials of the molecule OH arising from the interaction of the oxygen atom in two lowest‐energy electronic states labeled

and

with the hydrogen atom in the ground electronic state. The curves are labeled using standard spectroscopic notation [3]. In particular, the symbol X is the standard label for the lowest‐energy (ground) electronic state of the molecule. These potential energies were calculated in Ref. [6].

Figure 1.4

Couplings between different electronic states may affect the vibrational motion of molecules. These couplings become negligible when the electronic states are separated by a large amount of energy.

Figure 1.5

The vibrational energy levels of the OH molecule in the ground electronic state

. Only 10 lowest‐energy levels are shown. The inset shows the vibrational wave functions

for

,

, and

.

Figure 2.1

The molecular potential and the dipole moment function of the molecule LiCs in the ground electronic state

.

Figure 2.2

Shifts of the energy levels

and

in the presence of a perturbation

that couples the states

and

.

Figure 2.3

The Stark shifts of the rotational energy levels of a diatomic molecule in a

electronic state with the permanent dipole moment

and the rotational constant

as functions of the electric field strength

.

Figure 2.4

The Stark shifts of the rotational energy levels of a diatomic molecule in a

electronic state. The energy levels presented are for the molecule CaH, which has

 cm

,

 cm

, and the dipole moment

Debye. The different panels correspond to different values of

in the limit of zero electric field. Bear in mind that

is not a good quantum number because the electric‐field‐induced interaction (2.36) couples states with different

.

Figure 2.5

The Stark shifts of the rotational energy levels of a diatomic molecule in a

electronic state. The energy levels presented are for the molecule OH, which has

 cm

,

 cm

,

 cm

,

 cm

and the dipole moment

Debye.

Figure 3.1

A correlation diagram between the low‐ and high‐field limits for states from within the ground electronic state

and excited electronic state

of the molecule CaH. The dashed lines show the perturbing states of the

state coupled to the

states by the

operators. The states from within the

and

manifolds are labeled by the Hund's case (b) angular momentum quantum numbers in the low‐field limit and their projections on the direction of the field axis in the high‐field limit.

Figure 3.2

Zeeman levels of a CaF molecule in the rotational state characterized by

of the

electronic state: Full curves–accurate calculations; dashed lines–the magnitudes of the diagonal matrix elements given by Eq. (3.47).

Figure 3.3

Symbols ‐ measured frequency shift for the

(circles) and

(triangles) transitions; curves – direction cosine calculations.

Figure 3.4

Zeeman splitting of the hyperfine energy levels of

Rb

Cs(

) in the ground rotational state.

Figure 4.1

A schematic diagram of the field‐free molecular energy levels

and

shown by solid lines and the states

and

shown by dashed lines. The time‐dependent operator

couples

with both

and

. The rotating wave approximation eliminates the coupling to

.

Figure 4.2

Stark shifts of the rotational energy levels of a

molecule in an off‐resonant microwave field.

Figure 5.1

The rotational angular momentum of a linear molecule is perpendicular to the molecular axis. Here, we assume that

, as is the case for a molecule in a

electronic state. A molecule with the molecular axis oriented at an angle

with respect to the

‐axis can be in a superposition of angular momentum states with

and

projections, representing an aligned angular momentum state.

Figure 5.2

The expectation value of the orientation angle cosine of a rigid rotor in a DC electric field vs the dimensionless parameter

for three lowest energy states with

. The full lines show the results computed with Eq. (5.8) and the dotted lines with Eq. (5.9). Typical values of

for simple diatomic molecules are in the range of 0–12. The values of

are shown for higher values of

to illustrate that the limit of

is approached slowly, even for the lowest energy state.

Figure 5.3

The eigenstates of a quantum pendulum (a) and a rigid rotor in a DC field (b). The vertical lines show the energies and the curves – the square of the corresponding wave functions plotted as functions of the orientation angle

. The wave functions are not normalized and scaled for better visibility. The bound states of the rigid rotor are labeled by the quantum number

. The bound states and wave functions are calculated for

. The energy is in the units of

for the planar pendulum and in the units of

for the rigid rotor.

Figure 5.4

Energy levels of a rigid rotor in an off‐resonant laser field. The energy is in units of the rotational constant

and the molecule–field interaction strength is in units of

. The figure illustrates that the interaction with the laser field brings the molecular states of opposite parity together.

Figure 5.5

The eigenstates of a rigid rotor in an AC electric field. The horizontal lines show the energies and the curves – the square of the corresponding wave functions plotted as functions of the orientation angle

. The wave functions are not normalized and enhanced for better visibility. The bound states of the rigid rotor are labeled by the quantum number

. The bound states and wave functions are calculated for

. The energy is in the units of

.

Figure 5.6

An optical centrifuge for molecules. The spinning electric field is created by splitting a laser pulse at the center of its spectrum and applying an opposite frequency chirp to the two halves.

Figure 6.1

The low‐field‐seeking and high‐field‐seeking states in the Stern–Gerlach experiment.

Figure 6.2

Paths of molecules. The two solid curves indicate the paths of two molecules having different moments and velocities and whose moments are not changed during passage through the apparatus. This is indicated by the small gyroscopes drawn on one of these paths, in which the projection of the magnetic moment along the field remains fixed. The two dotted curves in the region of the

magnet indicate the paths of two molecules the projection of whose nuclear magnetic moments along the field has been changed in the region of the

magnet. This is indicated by means of the two gyroscopes drawn on the dotted curves, for one of which the projection of the magnetic moment along the field has been increased, and for the other of which the projection has been decreased.

Figure 6.3

The electric field potential generated by a monopole (a), dipole (b), and quadrupole (c).

Figure 6.4

An illustration of a device called “centrifuge decelerator” for the production of slow molecules. The molecules are injected into the space between four electrodes bent into a spiral shape. The spiral is then rotated to decelerate molecules moving toward the center by the inertial force. The electrodes are bent upward at the end of the spiral, thus guiding only the slowest molecules (with translational temperature less than 1 K) toward the exit.

Figure 6.5

Schematic diagram of a Stark decelerator. The Stark energy of an ND

molecule in a low‐field‐seeking quantum state is shown as a function of position

along the molecular beam axis. The Stark energy has a period of

.

Figure 6.6

Scheme of the experimental setup used for the Stark deceleration of a beam of CO molecules in the original work. In this experiment, CO molecules are prepared in a single, metastable quantum state (

,

,

) by pulsed‐laser excitation of ground‐state CO molecules. The beam of metastable CO molecules is slowed down on passage through a series of 63 pulsed electric field stages. The time‐of‐flight distribution of the metastable CO molecules over the 54 cm distance from laser preparation to detection is measured via recording of the amount of electrons emitted from a gold surface when the metastable CO molecules impinge on it.

Figure 6.7

Schematic drawing of the electrodes for a cylindrically symmetric 3D rf trap. Typical dimensions are

m to

cm, with

100–500 V,

–50 V, and

kHz to 100 MHz.

Figure 6.8

Electrostatic quadrupole trap geometry in cross section. The figure has rotational symmetry about the

‐axis. Heavy shaded curves: electrode surfaces, held at constant potentials

. The ring radius is

and the end‐cap half‐spacing

. Dashed curves: surfaces of constant

with values

. Full curves: surfaces of constant

with values

. A particle whose electric polarizability is negative will have minimum potential energy at the origin, where

.

Figure 6.9

Magnetic trap for neutral atoms. (a) Spherical quadrupole trap with lines of the magnetic field. (b) Equipotentials of the trap (with field magnitudes indicated in millitesla) in a plane perpendicular to the coils.

Figure 6.10

Time evolution of the CaH spectrum in a magnetic trap. These spectra reveal that CaH molecules in the high‐field‐seeking state (negative frequency shifts) quickly leave the trap. The trapped molecules in the low‐field‐seeking state (positive frequency shifts) are confined and compressed toward the center of the trap.

Figure 6.11

Two‐dimensional optical lattice potential produced by interfering three oscillating electric fields propagating in the same plane along the directions given by Eq. (6.34). The potential can be controlled by changing the angle between the wave vectors of the interfering fields. The potential is shown for

(a) and

(b).

Figure 7.1

Energy levels of the SrF(

) molecule as functions of a magnetic field in the presence of an electric field of 1 kV cm

. The rotational constant of SrF is 0.251 cm

, the spin–rotation interaction constant

is

cm

, and the dipole moment is 3.47 D. States

and

undergo an avoided crossing at the magnetic field value

. The value of

varies with the electric field.

Figure 7.2

Time‐dependent probability of the spin excitation to be localized on molecule four in a one‐dimensional array of seven SrF(

) molecules in an optical lattice with lattice spacing

nm for different magnetic fields near the avoided crossing shown in Figure 7.1. The electric field magnitude is 1 kV cm

. The value of

varies with the electric field.

Figure 7.3

(a, b) Energy levels of the SrF

molecule (

= 7.53 GHz,

= 74.7 MHz) in an electric field of

= 10 kV cm

as a function of magnetic field

; (c) frequency dependence of the ac field sensitivity for SrF in a linearly polarized microwave field for different electric fields; the lines of different color correspond to the

and

transitions. The dashed line represents the sensitivity to the magnetic field component of the ac field that can be achieved in experiments with atoms [294]; (d) same as in (c) but for the CaH

molecule (

= 128.3 GHz,

= 1.24 GHz).

Figure 7.4

Enhancement of the molecular axis orientation by an off‐resonant laser field. The last term in Eq. (7.2) brings the opposite parity states into closely spaced tunneling doublets, while the second last term orients the molecule with respect to the direction of the DC field by mixing the opposite parity states. The expectation value

is shown for the ground state of the molecule.

Figure 7.5

(a) The energy of the Floquet states for a rigid rotor in a superposition of a DC electric field directed along the

‐axis and a circularly polarized microwave field rotating in the

‐plane. The strength of the DC field is

. The intensity of the microwave field is represented by the parameter

. The frequency of the microwave field is

. (b) The modification of the

‐component of the molecule's dipole moment by the microwave field.

Figure 8.1

Schematic structure of the Hamiltonian matrix in the total angular momentum representation (8.109) for two interacting particles in an external field. Each square of the table represents the block of the matrix elements corresponding to a set of two quantum numbers:

and

. The empty squares show the blocks of the matrices, in which all matrix elements are zero. The shaded squares show the nonvanishing blocks of the matrix. The bullets show the blocks of the matrix populated by the field‐induced couplings. In the absence of an external field, the Hamiltonian matrix is diagonal in

so only the blocks with

are nonzero.

Figure 10.1

The effect of a resonance on a scattering cross section.

Figure 10.2

Schematic diagrams illustrating the mechanisms of Feshbach and shape scattering resonances. The resonances occur when a scattering state (with the energy shown by the dashed lines) interacts with a quasi‐bound state (shown by the full horizontal lines). The nature of the quasi‐bound states is different for the two types of resonances. See text for a detailed discussion.

Figure 10.3

Schematic diagrams illustrating the procedure for calculating the energy of the bound and quasi‐bound states.

Figure 10.4

(a) The

‐wave elastic scattering cross section for collisions of O

(

) molecules in the lowest high‐field‐seeking state

as a function of the magnetic field. (b) The minimal eigenvalue of the matching matrix

as a function of the magnetic field. The collision energy is 10

K. The projection of the total angular momentum of the system is

2. New resonances found using the analysis of the magnetic field dependence of

are marked by arrows.

Figure 11.1

The logarithm of the cross section for the

transition in collisions of NH molecules with

He atoms as a function of the magnetic field and collision energy.

Figure 11.2

Rate constants versus electric field for OH–OH collisions with molecules initially in a particular Stark state. Shown are the collision energies 100 mK (Panel a) and 1 mK (Panel b). Solid lines denote elastic‐scattering rates, while dashed lines denote rates for inelastic collisions, in which one or both molecules change their internal state. These rate constants exhibit characteristic oscillations in field when the field exceeds a critical field of about 1000 V cm

.

Figure 11.3

Modification of a shape resonance by microwave fields. The elastic cross section is plotted as a function of the collision energy for zero microwave field (full line),

(dashed line), and

(dotted line). The Rabi frequency is

.

Figure 11.4

Magnetic field dependence of the Zeeman (full line) and hyperfine relaxation (dashed line) cross sections in collisions of YbF molecules with He atoms at zero electric field – (a),

kV cm

– (b), and

kV cm

– (c). The symbols in the (a) are the results of the calculations without the spin–rotation interaction. The collision energy is 0.1 K.

Figure 11.5

Effect of a scattering resonance on the chemical reaction of H atoms with LiF molecules at an ultralow temperature. The insets show the nascent rotational state distributions of HF molecules produced in the reaction as a function of the final rotational state

at electric field strengths of 0, 32, and 100 kV cm

(left) and 124, 125, and 125.75 kV cm

(right). Note the dramatic change in the shape of the distribution near the resonance electric field (right inset). All calculations were performed in the

‐wave scattering regime (at collision energy 0.01 cm

), where no resonances are present in the reaction cross sections as a function of collision energy.

Figure 11.6

List of Tables

Table 6.1

Summary of the external field traps developed for neutral molecules as of 2013. Only selected representative references are given. See text for a more comprehensive list of references.

Table 11.1

Dependence of the scattering cross sections on the collision velocity

in the limit

Preface

Much of our knowledge about molecules comes from observing their response to electromagnetic fields. Molecule–field interactions provide a “lense” into the microscopic structure and dynamics of molecules. Molecule–field interactions also provide a knob for controlling molecules.

The focus of much recent research has been on controlling the translational and rotational motions of molecules by tunable fields. This effort has transformed molecular physics. New experimental techniques – unimaginable 10 to 20 years ago – have been introduced. We have learnt to interrogate molecules and molecular interactions with extremely high precision. Most importantly, this work has allowed – and stimulated! – us to ask new questions and has built bridges between molecular physics and other areas of physics.

For example, the interrogation of translationally controlled molecules is now considered to be the most viable route to determining the magnitude of the electric dipole moment of the electron. The outcome of such experiments may restrict, and maybe even resolve, the debate about the extensions of the Standard Model of particle physics. Interfacing with a completely different field, molecules trapped in optical lattices can be used as quantum simulators of a large variety of lattice spin models. Probing the structure and dynamics of such molecules is expected to identify the phases of many lattice spin models that are currently either unknown or under debate. Spectroscopy measurements of molecules in external field traps approach the fundamental accuracy limit of the nonrelativistic quantum mechanics, prompting quantum chemists to reconsider their toolbox for molecular structure calculations.

The effort aimed at controlling the three‐dimensional motion of molecules has resulted in many unique experiments. Molecules can be spun by cleverly crafted laser fields all the way until the centrifugal force pulls the nuclei apart, breaking chemical bonds. This provides potentially new sources of radiation and singular quantum objects – “superrotors” – for the study of collision physics and kinetics of chain reactions. Slow molecular beams, which can be grabbed and guided by external fields, provide new unique opportunities to study chemical encounters with extremely high control over the collision energy. The isolation of molecules from ambient environments into samples maintained at an ultracold temperature opens the pathway to studying chemistry near absolute zero, controlled chemistry and, as argued later in this book, a conceptually new platform for assembling complex molecules.

At the core of all this exciting research is the manipulation of molecules by external fields. This book is an attempt to describe basic quantum theory needed to understand and compute molecule–field interactions of relevance to the abovementioned work. The formal theory is accompanied with examples from the recent literature, predictions about what may be happening next, and the discussion of current and future problems that need to be overcome for new major applications of molecules in electromagnetic fields.

Goals of This Book

This book is intended to serve as a tutorial for students beginning research, experimental or theoretical, in an area related to molecular physics. The focus of the book is on theoretical approaches of relevance to the recent exciting developments in molecular physics briefly mentioned above.

There are two kinds of chapters in this monograph:

Chapters 1

–

4

and

8

–

10

are written to provide a detailed summary of rigorous theory of molecule–field interactions, quantum scattering theory, and the theory of scattering resonances. The goal of these chapters is practical – to provide enough details that will allow the reader to write computer codes for calculating the rotational energy levels of molecules, the AC and DC Stark shifts of molecules, the Zeeman shifts, molecular collision observables, and the features of field‐induced scattering resonances. These chapters also gradually introduce angular momentum and spherical tensor algebra with the application to problems involving molecules in electromagnetic fields. Each of the chapters is largely self‐contained. I attempt to present the complete derivations of all important equations so the material in these chapters requires little more than basic knowledge of quantum theory.

Chapters 5

–

7

and

11

–

12

illustrate the applications of field‐induced interactions for controlling the motion of molecules in three‐dimensional space, trapping molecules, controlling molecular collisions, and ultracold controlled chemistry. The purpose of these chapters is to illustrate the extent and power of field control of microscopic behavior of molecules. The discussion in these chapters is more descriptive, with references to relevant sources in the recent literature. However, this discussion relies in many places on equations derived in

Chapters 1

–

4

and

8

–

10

.

There are also five appendices at the end of the monograph. The main purpose of the appendices is to provide the theoretical background for understanding the details in Chapters 1–4 and 8–10. I hope that these appendices can serve as mini‐tutorials on angular momentum algebra, coordinate rotations, and spherical tensors.

What This Book Is Not

I have recently heard a famous scientist and author saying that “one never finishes a book; one abandons the writing.” I have now experienced this myself. There are many things I would have liked to do for this book, which I must abandon. To ensure that I do not mislead the reader, let me point out the following.

This book is not a comprehensive review. It would be impossible to cite all relevant papers. The references presented are isolated, sample articles, which should direct the reader either to the pioneering work or some of the most widely cited work. If your paper has not been referenced, and you think it should have, please forgive me.