Advances in Atomic Physics

The following sections are included:

• Foreword

• Contents

• Acknowledgments

The following sections are included:

• Purpose of this book
  • Organization of the book
  • Part 1 - Advances in spectroscopy
  • Part 2 - Perturbations of atomic levels by light
  • Part 3 - Multiphoton processes
  • Part 4 - Control of atomic motion. Cooling and trapping
  • Part 5 - Ultracold interactions and their control
  • Part 6 - Atomic interferometry. Entangled states
  • Part 7 - Quantum gases
  • Part 8 - A few frontiers of atomic physics

The following sections are included:

• Introduction

• The two interacting systems: atom and field
  • External and internal atomic variables
  • Classical versus quantum treatments of atomic variables
  • Classical description of field variables
  • Quantum description of field variables
  • Atom-field interaction Hamiltonian in the long wavelength approximation
  • Elementary interaction processes

• Basic conservation laws
  • Conservation of the total linear momentum
  • Conservation of the total angular momentum

• Two-level atom interacting with a coherent monochromatic field. The Rabi oscillation
  • A simple case: magnetic resonance of a spin ½
  • Extension to any two-level atomic system
  • Perturbative limit
  • Two physical pictures for Ramsey fringes

• Two-level atom interacting with a broadband field. Absorption and emission rates
  • Absorption rate deduced from a semiclassical treatment of the field
  • Physical discussion. Relaxation time and correlation time
  • Sketch of a quantum treatment of the absorption process
  • Extension to spontaneous emission

• Two-level atom interacting with a coherent monochromatic field in the presence of damping

The following sections are included:

• Introduction

The following sections are included:

• Introduction

• Double resonance
  • Principle of the method
  • Predicted shape for the double resonance curve
  • Experimental results
  • Interpretation of the Majorana reversal

• Optical pumping [Kastler (1950)]
  • Principle of the method for a J_g = ½→ J_e = ½ transition
  • Angular momentum balance
  • Double role of light

• First experiments on optical pumping

• How can optical pumping polarize atomic nuclei?
  • Using hyperfine coupling with polarized electronic spins
  • First example: optical pumping experiments with mercury-199 atoms
  • Second example: combining optical pumping with metastability exchange collisions for helium-3
  • A new application: magnetic resonance imaging of the lung cavities

• Brief survey of the main applications of optical methods

• Concluding remarks

The following sections are included:

• Introduction

• First experimental evidence of the importance of atomic coherences

• Zeeman coherences in excited states
  • How to prepare Zeeman coherences in excited states e?
  • Physical interpretation
  • How to detect Zeeman coherences in e?
  • Equation of motion of Zeeman coherences in e
  • Level crossing resonances in the excited state e
  • Pulsed excitation. Quantum beats
  • Excitation with modulated light
  • Modulation of the fluorescence light in a double resonance experiment. Light beats

• Zeeman coherences in atomic ground states
  • Hanle effect in atomic ground states
  • Detection of the magnetic resonance in the ground state by the modulation of the absorbed light

• Transfer of coherences

• Dark resonances. Coherent population trapping
  • Discovery of dark resonances
  • First theoretical treatment of dark resonances
  • Interpretation of the Raman resonance condition
  • A few applications of dark resonances

• Conclusion

The following sections are included:

• Introduction

• Low intensity limit. Perturbative approach
  • Lowest order process
  • Resonant scattering amplitude
  • Scattering of a light wave packet
  • First higher order processes

• Optical Bloch equations

• The dressed atom approach
  • The interacting systems
  • Uncoupled states of the atom-laser system
  • Effect of the coupling. Dressed states
  • Two different situations
  • Radiative cascade in the basis of uncoupled states
  • A new description of quantum dissipative processes

• Photon correlations. The quantum jump approach
  • The waiting time distribution
  • From the waiting time distribution to the second order correlation function
  • Photon antibunching

• Fluorescence triplet at high laser intensities
  • Limit of large Rabi frequency
  • Mollow fluorescence triplet
  • Widths and weights of the components of the Mollow triplet
  • Time correlations between the photons emitted in the two sidebands of the fluorescence triplet

• Conclusion

The following sections are included:

• Introduction

• Saturated absorption
  • Principle of the method
  • Crossover resonances
  • Recoil doublet

• Two-photon Doppler-free spectroscopy
  • Principle of the method
  • Examples of results
  • Comparison between saturated absorption and two-photon spectroscopy

• Recoil suppressed by confinement: the Lamb-Dicke effect
  • Intensities of the vibrational lines
  • Influence of the localization of the ion
  • Case of a harmonic potential
  • Historical perspective

• The shelving method
  • Single ion spectroscopy
  • Intermittent fluorescence
  • Properties of the detected signal
  • Observation of quantum jumps

• Quantum logic spectroscopy

• Frequency measurement with frequency combs

• Conclusion

The following sections are included:

• Introduction

The following sections are included:

• Introduction

• Light shift, light broadening and Rabi oscillation
  • Effective Hamiltonian
  • Weak coupling limit. Light shift and light broadening
  • High coupling limit. Rabi oscillation
  • Absorption rate versus Rabi oscillation
  • Semiclassical interpretation in the weak coupling limit
  • Generalization to a non-resonant excitation
  • Case of a degenerate ground state

• Perturbation of the field. Dispersion and absorption
  • Atom in a cavity
  • Frequency shift of the field due to the atom
  • Damping of the field

• Experimental observation of light shifts
  • Principle of the experiment
  • Examples of results

• Using light shifts for manipulating atoms
  • Laser traps
  • Atomic mirrors
  • Blue detuned traps: a few examples
  • Optical lattices
  • Internal state dependent optical lattices
  • Coherent transport

• Using light shifts for manipulating fields
  • Linear superposition of two field states with different phases
  • Non-destructive detection of photons

• Conclusion

The following sections are included:

• Introduction

• Spin ½ coupled to a high frequency RF field
  • Hamiltonian
  • Perturbative treatment of the coupling
  • Stimulated corrections
  • Radiative corrections

• Weakly bound electron coupled to a high frequency field
  • Effective Hamiltonian describing the modifications of the dynamical properties of the electron
  • Stimulated effects
  • Spontaneous effects. Vacuum fluctuations and radiation reaction

• New insights into radiative corrections
  • Examples of spontaneous corrections
  • Interpretation of the Lamb shift
  • Interpretation of the spin anomaly g − 2

• Conclusion

The following sections are included:

• Introduction

The following sections are included:

• Introduction

• Radiofrequency multiphoton processes
  • Multiphoton RF transitions between two Zeeman sublevels m_F and m_F + 2
  • Experimental observation on sodium atoms
  • Multiphoton resonances between two Zeeman sublevels m_F and m_F + 1

• Radiative shift and radiative broadening of multiphoton resonances
  • Energy levels of the atom+RF photons system. Transition amplitude
  • Pure single photon resonance. Simple anticrossing
  • Higher order anticrossing for a p-photon resonance (p < 1)
  • Application to the case of a spin ½ coupled to a σ-polarized RF field

• Optical multiphoton processes between discrete states
  • Introduction
  • Radiative shift of Doppler-free two-photon resonances
  • Stimulated Raman processes
    • Differences with spontaneous Raman processes
    • Doppler effect
    • Applications
  • Phase matching condition. Application to degenerate four-wave mixing

• Conclusion

The following sections are included:

• Introduction

• Multiphoton ionization
  • Parameters influencing the multiphoton ionization rate
  • Quantum interference effects in multiphoton ionization
  • Asymmetric line profiles in resonant multiphoton ionization

• Above threshold ionization (ATI)
  • Multiphoton transitions between states of the continuum
  • Consequences of the oscillatory motion of the electron in the laser field
  • Evidence for non-perturbative effects

• Harmonic generation
  • Physical interpretation
  • High order harmonic generation (HHG). Evidence for nonperturbative effects

• Tunnel ionization and recollision
  • The breakdown of perturbation theory
  • Keldysh parameter
  • Two-step quantum-classical model
  • Recollision
  • Full quantum treatments

• Conclusion

The following sections are included:

• Introduction

The following sections are included:

• Introduction
  • Order of magnitude of the force
  • Characteristic times
  • Validity of the concept of a mean force at a given point

• Calculation of the mean radiative force
  • Principle of the calculation
  • Hamiltonian and the rotating wave approximation
  • Heisenberg equations for the external variables. Force operator
  • Approximations. Mean radiative force
  • The two types of mean radiative forces: dissipative and reactive

• Dissipative force
  • Theoretical results
  • Physical interpretation
  • Application to the deflection and to the slowing down of an atomic beam
  • Fluctuations

• Reactive force
  • Theoretical results
  • Physical interpretation
  • Dressed atom interpretation

• Conclusion

The following sections are included:

• Introduction

• Doppler-induced friction force
  • Doppler effect in a red detuned laser plane wave
  • Low velocity behavior of the force
  • Idea of Doppler cooling for trapped ions
  • Idea of Doppler cooling for neutral atoms

• Two-level atom moving in a weak standing wave. Doppler cooling
  • Perturbative approach for calculating the force
  • Friction coefficient for a red-detuned weak standing wave
  • Momentum-energy balance. Entropy balance
  • Limits of Doppler cooling. Lowest temperature
  • Consistency of the various approximations
  • Spatial diffusion. Optical molasses

• Beyond the perturbative approach
  • Optical Bloch equations for a moving atom
  • Time lag of internal variables
  • Low velocity limit (k_Lv ≪Γ)
  • Higher velocities

• Dressed atom approach to atomic motion in an intense standing wave. Blue cooling
  • Energy and radiative widths of the dressed states
  • Friction mechanism
  • High intensity Sisyphus cooling
  • Experimental results

• Conclusion

The following sections are included:

• Introduction

• Sub-Doppler cooling
  • The basic ingredients of sub-Doppler cooling
  • Laser configuration and atomic transition
  • Light shifts and optical pumping for an atom at rest
  • Low intensity Sisyphus cooling for a moving atom
  • Characteristics of the friction force. Qualitative discussion
  • Quantum limits of sub-Doppler cooling

• Sub-recoil cooling
  • Physical mechanism
  • Velocity selective coherent population trapping (VSCPT)
  • Sub-recoil Raman cooling
  • Quantitative predictions for sub-recoil cooling

• Resolved sideband cooling of trapped ions

• Conclusion

The following sections are included:

• Introduction

• Trapping of charged particles
  • The Earnshaw theorem
  • The Penning trap
  • The Paul trap
  • Cooling of the trapped ions
  • High precision measurements performed with ultracold trapped ions

• Magnetic traps
  • Introduction
  • Quadrupole trap and Majorana losses
  • Ioffe-Pritchard trap
  • Time-averaged orbiting potential (TOP)
  • Loading neutral atoms in a magnetic trap

• Electric dipole traps
  • Induced dipole moment
    • Static induced dipole moment
    • Dynamic induced dipole moment
    • Near resonance induced dipole moment
  • Application of dipole forces to trapping
    • Historical perspective
    • The simplest optical trap
    • Transport of atoms with moving laser traps
    • Crossed dipole traps
  • Optical lattices

• Artificial orbital magnetism for neutral atoms
  • Introduction
  • Rotating a harmonically trapped quantum gas
  • Artificial gauge potential from adiabatic evolution

• Magneto-optical trap (MOT)

• Conclusion

The following sections are included:

• Introduction

The following sections are included:

• Introduction

• Quantum scattering: a brief reminder
  • Scattering amplitude
  • Scattering cross section
  • Partial wave expansion

• Scattering length
  • Low-energy limit
  • Scattering amplitude and scattering length
  • Square potential and resonances
  • Effective interactions and the sign of the scattering length

• Pseudo-potential
  • Motivation for introducing this pseudo-potential
  • Localized pseudo-potential giving the correct scattering length
  • Scattering amplitude. Validity of the Born approximation
  • Bound state of the pseudo-potential for a positive scattering length

• Delta potential truncated in momentum space
  • Expression of the potential
  • Determination of the new coupling constant
  • Comparison with the pseudo-potential

• Forward scattering
  • Gaussian incident wave and scattered wave
  • Interference of the incident and scattered waves in the far-field zone
  • Phase shift of the incident wave and mean field energy

• Conclusion

The following sections are included:

• Introduction

• Collision channels
  • Microscopic interactions
  • Quantum numbers of the initial collision state. Collision channels
  • Coupled channel equations
  • Two-channel model

• Qualitative discussion. Analogy between Feshbach resonances and resonant light scattering

• Scattering states of the two-channel Hamiltonian
  • Calculation of the dressed scattering states
  • Existence of a resonance in the scattering amplitude
    • Position and width of the resonance
    • Resonant value B₀ of the magnetic field
  • Asymptotic behavior of the dressed scattering states
    • Background scattering length
    • Effect of the inter-channel coupling on the asymptotic behavior
  • Scattering length. Feshbach resonance

• Bound states of the two-channel Hamiltonian
  • Calculation of the energy of the bound state
  • Wave function of the bound state
  • Halo states

• Producing ultracold molecules
  • Magnetic tuning of a Feshbach resonance
  • Photoassociation of ultracold atoms

• Conclusion

The following sections are included:

• Introduction

The following sections are included:

• Introduction

• De Broglie waves versus optical waves
  • Dispersion relations. Position and momentum distributions
  • Spatial coherences. Coherence length
  • Fragility of spatial coherences

• Young's two-slit interferences with atoms
  • Important parameters of Young's double-slit interferometer
  • Young's double-slit interferences with supersonic beams
  • Young's double-slit interferences with cold atoms
  • Can one determine which slit the atom passes through?

• Diffraction of atoms by material structures

• Diffraction by laser standing waves
  • New features compared to the diffraction by material gratings
  • Light-atom momentum exchange
  • Raman-Nath regime
  • Bragg regime

• Bloch oscillations
  • Review on the quantum treatment of a particle in a periodic potential
  • Implementation with cold atoms
  • Physical interpretations

• Diffraction of atomic de Broglie waves by time-dependent structures
  • Phase modulation of atomic de Broglie waves
  • Atomic wave diffraction and interference using temporal slits

• Conclusion

The following sections are included:

• Introduction

• Microwave atomic clocks with cold atoms
  • Principle of an atomic clock
  • Atomic fountains
  • Performances of atomic fountains
  • Cold atoms clocks in space
  • Tests of general relativity

• Extension of Ramsey fringes to the optical domain
  • Equivalence of the crossing of a laser beam with a coherent beam splitter
  • Spatial separation of the two final wave packets. Quenching of the interference
  • How to restore the interference signal?
  • Other possible schemes

• Calculation of the phase difference between the two arms of an atomic interferometer
  • Quantum propagator and Feynman path integral
  • Simple case of quadratic Lagrangians
  • Phase shift in the absence of external potentials and inertial fields
  • Phase shift due to external potentials and inertial fields in the perturbative limit

• Applications of atomic interferometry
  • Measurement of gravitational fields. Gravimeters
  • Measurement of rotational inertial fields
  • Measurement of h/M and α

• New perspectives opened by optical clocks

The following sections are included:

• Introduction

• Interference effects in double counting rates
  • Photodetection signals
  • Two-mode model for the light field
  • What are the “objects” which interfere in w_II?
  • Establishment of correlations between the two modes

• Entangled states
  • Definition
  • Schmidt decomposition of an entangled state
  • Information content of an entangled state

• Preparing entangled states
  • Entanglement between one atom and one field mode
  • Entanglement between two atoms
  • Entanglement between two separate cavity fields
  • Entanglement between two photons

• Entanglement and interference

• Entanglement and non-separability
  • The Einstein-Podolsky-Rosen (EPR) argument [Einstein et al. (1935)]
  • Bell's inequalities
  • Experimental results and conclusion

• Entanglement and which-path information

• Entanglement and the measurement process
  • Von Neumann model of an ideal measurement process
  • Difficulty associated with macroscopic coherences
  • A possible solution: coupling of with the environment
  • Simple example of pointer states
  • The infinite chain of Von Neumann

• Conclusion

The following sections are included:

• Introduction

The following sections are included:

• Introduction

• Quantum effects in collisions
  • S-matrix and T-matrix
  • Interfering scattering amplitudes for identical particles
  • Polarized Fermi gas at low temperature
  • Interference effects in forward and backward scattering
  • Identical spin rotation effect (ISRE)
  • A few examples of effects involving ISRE

• The first prediction of BEC in a gas
  • A new derivation of Planck's law for black body radiation
  • Extension of Bose statistics to atomic particles
  • The condensation phenomenon
  • Critical temperature
  • Variation of the number N₀ of condensed atoms with the temperature. Thermodynamic limit
  • Influence of dimensionality

• Conclusion

The following sections are included:

• Introduction

• First attempts on hydrogen
  • Spin polarized hydrogen as a quantum gas
  • Production of a spin polarized sample at low temperature
  • Difficulties associated with collisions
  • Need for other methods

• Second attempts on hydrogen
  • Wall free confinement. Magnetic trapping
  • Bose-Einstein condensation in a harmonic trap
  • New cooling method: evaporative cooling
  • Need for new detection method of polarized hydrogen

• The quest for BEC for alkali atoms
  • Difficulties associated with alkali atoms
  • Advantages of alkali atoms

• First observation of Bose-Einstein condensation
  • Time sequence
  • Signature of Bose-Einstein condensation
  • Subsequent observation on hydrogen

• Bose-Einstein condensation of other atomic species
  • Experimental improvements
  • Review of new condensates

• The first experiments on quantum degenerate Fermi gases
  • Ideal Fermi gas in a three-dimensional harmonic trap
  • Cooling fermions
  • Spatial distribution and Fermi pressure
  • Pairs of fermionic atoms

• Conclusion

The following sections are included:

• Introduction

• Mean field description of the condensate
  • Variational calculation of the condensate wave function
  • Stationary Gross-Pitaevskii equation
  • Expression of the various quantities in terms of the spatial density

• Condensate in a box and healing length
  • Condensate in a one-dimensional box
  • Healing length

• Condensate in a harmonic trap
  • Total energy and the different interaction regimes
  • Condensate with a positive scattering length and the Thomas-Fermi limit

• Condensate with a negative scattering length
  • Condition of stability in 3D
  • Solitonic solution in 1D
  • Collapse and explosion of a condensate in 3D with a negative scattering length

• Quantum vortex in an homogeneous condensate
  • Effective Gross-Pitaevskii equation
  • Properties of the velocity field

• Time-dependent problems
  • Time-dependent Gross-Pitaevskii equation
  • Analogy with hydrodynamic equations
    • Density field ρ and velocity field
    • Continuity equation. Evolution equations of and ρ
    • Quantum pressure
  • The two contributions to the kinetic energy: Thomas-Fermi approximation for time-dependent problems
  • Harmonic confinement
    • Scaling transformation
    • Ballistic expansion

• Conclusion

• Appendix: Normal modes of a harmonically trapped condensate
  • Isotropic trap
  • Cylindrically-symmetric trap
  • Scissors mode for anisotropic traps

The following sections are included:

• Introduction

• Atomic field operators and correlation functions
  • Brief reminder on second quantization
  • Atomic field operators
  • Examples of physical operators. Field correlation functions
  • Heisenberg equation of the field operator

• Calculation of correlation functions in a few simple cases
  • First-order correlation function for an ideal Bose gas in a box
  • Higher-order spatial correlation functions for an ideal gas of bosons above T_c
  • Correlation functions for a Bose-Einstein condensate
  • A few experimental results

• Relative phase of two independent condensates
  • Two condensates in Fock states
  • Phase states
  • Conjugate variable of the relative phase
  • Emergence of a relative phase in an interference experiment

• Long range order and order parameter
  • Long range order
  • Order parameter

• New effects in atom optics due to atom-atom interactions
  • Collapse and revival of first-order coherence due to interactions
  • An example of nonlinear effects in atom optics: Four-wave mixing with matter waves

• Conclusion

The following sections are included:

• Introduction

• Bogolubov approach for an homogeneous system
  • Second quantized Hamiltonian
  • Bogolubov quadratic Hamiltonian
  • Physical discussion
    • Elementary excitations. Quasi-particles
    • New ground state and quantum depletion
  • Energy of the ground state
  • Extension to inhomogeneous systems

• Landau criterion for superfluidity in an homogeneous system
  • Microscopic probe
  • Macroscopic approach

• Extension of Landau criterion for a condensate in a rotating bucket
  • The rotating bucket
  • Other possible states of the condensate: quantized vortices
  • Various threshold rotation frequencies

• Experimental study of vortices in gaseous condensates
  • Introduction
  • A few experimental results
  • Measuring the angular momentum per atom in a rotating condensate
  • Routes to vortex nucleation
    • Rotational motion of an interacting Bose-Einstein condensate
    • The experiment

• Conclusion

The following sections are included:

• Introduction

• Tests of fundamental theories

• Strongly interacting many-body systems

• Extreme light

The following sections are included:

• Introduction
  • Historical perspective
  • Atomic parity violation (APV)
  • Organization of this chapter

• The first cesium experiment
  • Principle of the experiment
  • Transition dipole moment
  • Existence of a chiral signal in the re-emitted light
  • Calibration of the parity violation amplitude

• Connection between the parity violation amplitude and the parameters of the electroweak theory
  • Non-relativistic limit of the weak interaction Hamiltonian
  • Calculation of the parity violation amplitude
  • Nuclear spin-dependent parity violating interactions. Anapole moment

• Survey of experimental results
  • Cesium experiments
  • Experiments using other atoms

• Conclusion about the importance of APV experiments

• Appendix: Testing time reversal symmetry by looking for electric dipole moments

The following sections are included:

• Introduction

• The double well problem for bosonic gases
  • Introduction
  • The Hubbard Hamiltonian
  • The superfluid regime
  • The insulator regime
  • Connection between the superfluid and insulator regimes
  • Production of Schrödinger cat states when interactions are attractive
  • Controlling the tunnelling rate with a modulation of the difference of the two potential depths

• Superfluid-Mott insulator transition for a quantum bosonic gas in an optical lattice
  • Bose Hubbard model
  • Qualitative interpretation of the superfluid-Mott insulator transition
  • Experimental observation

• Quantum fermionic gas in an optical lattice

• Feshbach resonances and Fermi quantum gases
  • Introduction
  • Brief survey of BCS theory
  • A simple model for the BEC-BCS crossover
  • Experimental investigation

• Conclusion

The following sections are included:

• Introduction

• Attosecond science
  • Mechanism of production of attosecond pulses
  • Multiple-cycle laser pulse. Train of attosecond pulses
  • Few-cycle laser pulse. Control of the carrier-envelope phase
  • Attosecond metrology
  • A few applications of attosecond pulses

• Ultra intense laser pulses
  • Q-switched lasers
  • Mode locking techniques
  • Chirped pulse amplification
  • A few applications of high intensity table-top lasers

• Conclusion

The following sections are included:

• An increasing control of atomic systems

• Higher frequency resolutions

• New schemes for overcoming the limits

• More stringent tests of fundamental laws

• From microscopic to macroscopic quantum systems

• Extracting information. A spectacular evolution

• An increasing and fruitful dialogue with other disciplines

The following sections are included:

• Bibliography

• Index