Light Matter Interaction - Semi-classical

Light Matter Interaction-Semi-classical The picture where atoms are quantized, but light is still kept classical – semi-classical approximation The time-dependent Schrodinger equation The weak-field limit: Einstein’s B coefficient The strong-field limit: Rabi oscillations The Bloch sphere

Introduction 1913 - Bohr postulated that a quantum of light of angular frequency ω is absorbed or emitted whenever an atom jumps between two quantized energy levels E1 and E2 E2− E1 = ħω 1916–7 Einstein introduced the Einstein coefficients to quantify the rate at which the absorption and emission of quanta occur Also, discovered the process of stimulated emission - the basis for laser operation What happens to the irradiated atom before the absorption transition is complete?

The two-level atom approximation When light with angular frequency ω coincides with one of the optical transitions of the atom, a resonant interaction between the transition and the light field exists The other levels of the atom, which only weakly interact with the light because they are off-resonance – are neglected In the classical picture the light beam induces dipole oscillations in the atom, which then re-radiate at the same frequency If the light frequency corresponds to the natural frequency of the atom, the magnitude of the dipole oscillations will be large and the interaction between the atom and the light will be strong

Coherent superposition states • resonant interaction between an atom and a light field involves the concept of coherent superposition states • Consider a quantum system with two levels: a two-level atom or a spin 1/2 nucleus in a magnetic field • The wave function of the system is of the form c1and c2describe the wave function amplitude coefficients for the two states of the atom or nucleus • a measurement would give with probability |c1|2 for level 1 and |c2|2 for level 2

consider a gas of N0 identical two-level particles (e.g. two-level atoms or spin 1/2 nuclei) with N1 particles in the lower level and N2 in the upper level - a statistical mixture • by setting |c1|2 = N1/N0 and |c2|2 = N2/N0 we would obtain the same results as for repeated measurements on a gas of N0 particles in the superposition state • what, then, is the difference? • each of the particles in the superposition state is in some sense simultaneously in the |1>and |2>states, and this leads to the possibility of wave function interference • in the statistical mixture, by contrast, a given particle is either in level 1 or in level 2, and no wave function interference can occur

analogy can be made with interference effects in light beams • consider two overlapping light beams of the same frequency with phases φ1 and φ2, respectively. The resultant field is: the beams interfere if they are coherent, when Δφ = φ2− φ1 = cons. at a point in space at some positions where Δφ = (even integerxπ) - a bright fringe and where Δφ = (odd integer x π) - a dark fringe if the beams are incoherent - phases vary randomly with time - no interference the powers of the beams just add together giving |ε|2 = | ε1|2 + | ε2|2 analogy with the two-level superposition states can be made by setting c1→ ε1 exp(−iφ1) and c2→ ε2 exp(−iφ2) wave function interference can occur for a definite phase relationship for c1 and c2 this occurs in the superposition state, but not in a statistical mixture, where the different particle wavefunctions have random phases with respect to each other in the treatment, the light pulse links the phases of the upper and lower levels of an atom – wavefunction coherence

The density matrix • the elements of the density matrix are ρij= <cic∗j> ciis the wave function amplitude for the ith quantum level, and i and j run over all the quantum states of the atom for a two level system the difference between statistical mixtures and coherent superpositions are the off-diagonal terms (ρ12 and ρ21)in the density matrix - non-zero off-diagonal elements in a statistical mixture, each atom will either have |c1| = 1 and |c2| = 0 or vice versa

The time-dependent Schr¨odinger equation We want to solve the time-dependent Schr¨odinger equation for a two-level (E1 and E2) atom in the presence of a lightwave of angular frequency ω assume that the light is very close to resonance with the transition

Let’s split the Hamiltonian into a time-indpendent Ĥ0- the atom in dark and a perturbation for light-atom interaction For the two-level atom two solutions for the unperturbed system

The general solution for the time dependent eq. is and for a two level system Substituting this wavefunction into the time-dependent Schr¨odinger equation and subsituting for Ĥ

implying that On multiplying by ψ∗1, integrating over space, and making use of the orthonormality of the eigenfunctions, requiring that dij is the Kronecker delta function

Similarly, on multiplying by ψ∗2 and integrating let’s consider explicitly the form of the perturbation in the semi-classical approach, the light–atom interaction is given by the energy shift of the atomic dipole in the electric field of the light e - magnitude of the electron charge e0- amplitude of the light wave arbitrarily choose the x-axis as the polarization direction

the perturbation matrix elements are the dipole matrix element mijis and x is an odd parity operator and atomic states have either even or odd parities, therefore, μ11 = μ22 = 0 the dipole matrix element represents a measurable quantity and must be real, implying μ21 = μ12, because μ21 = μ∗12

let’s introduce the Rabi frequency These equations have to be solved to understand the behavior of the atom in the light field two distinct types of solution: the weakfield limit and the strong-field limit

The weak-field limit: Einstein’s B coefficient • The weak-field limit applies to low-intensity light sources such as blackbody lamps • The atom is initially in the lower level and the lamp is turned on at t = 0, implying that c1(0) = 1 and c2(0) = 0 • With a low-intensity source, the electric field amplitude will be small and the perturbation weak • The number of transitions expected is therefore small, and it will always be the case that c1(t) >> c2(t) • Therefore, c1(t) = 1 for all t

according to the rotating wave approximation, the second term is neglected since δω<<(ω + ω0), the second term is much smaller than the first For a beam tuned to exact resonance with the transition, δω is equal to zero the probability that the atom is in the upper level increases as t2 in Einstein’s approach the transition probability is time independent

Let’s re-examine the assumptions of our analysis • It was assumed that the atomic transition line is perfectly sharp • we know that all spectral lines have a finite width Δω • we consider the interaction between the atom and a broad-band • source such as a black-body lamp • Such a broad-band source can be specified by the spectral energy density u(ω), which must satisfy Let’s integrate the eq. for the probability over the spectral line

making the approximation that the spectral line is sharp compared to the broad-band spectrum of the lamp • u(ω) does not vary significantly within the integral, allowing to replace u(ω) by a constant value u(ω0) and to evaluate the integral • the limiting value for tΔω→∞ is u(ω0)2πt • Hence we obtain the probability that the atom is in the upper level increases linearly with time this eq. can be related to that defining the Einstein B coefficient which implies that the transition probability per unit time per atom is Bω12u(ω0)

in the analysis it was assumed that the atomic dipole moment was aligned parallel to the polarization vector of the light • however, in a gas of atoms, the direction of the atomic dipoles is random • if the angle between the polarization and a particular dipole is θ, then we need to take the average of (μ12 cos θ)2 for all the atoms in the gas • using <cos2 θ> = 1/3, we then replace μ212 by μ212/3 to obtain the transition probability rate W12 the weak-field limit is equivalent to the Einstein analysis enables calculation of the B coefficient from the atomic wave functions

The strong-field limit: Rabi oscillations • previously, assumed that the light field was weak so that the population of the excited state was always small and the approximation c1(t) ≈ 1 was valid for all t • the population of the upper level is significant • this condition applies when the light–atom interaction is strong, meaning strong electric fields - powerful laser beams • to find a solution in the strong-field limit we make two simplifications: • the rotating wave approximation to neglect the terms that oscillate at ±(ω +ω0) • we only consider the case of exact resonance with δω = 0

with the simplifications differentiate the first line and substitute from the second to find describes oscillatory motion at angular frequency ΩR/2

if the particle is in the lower level at t = 0 so that c1(0) = 1 and c2(0) = 0 the solution is and the probability for finding the electron in the upper and lower levels is The time dependence of the probabilities is shown in the Fig. At t = π/ΩR the electron is in the upper level, whereas at t = 2π/ΩR it is back in the lower level The process then repeats itself with a period equal to 2π/ΩR

The electron thus oscillates back and forth between the lower and upper levels at a frequency equal to ΩR/2πThe oscillatory behavior in response to the strong-field is called Rabi oscillation or Rabi flopping • I.I. Rabi, Phys. Rev. 51, 652 (1937). • Rabi’s original derivation applied to oscillating electromagnetic fields tuned to resonance with the Zeeman-split levels of a spin-1/2 nucleus • The RF field tips the spin vector from down to up and then back to down again, a process equivalent to the Rabi flopping • Rabi’s work was the precursor NMR techniques • Nobel Prize for Physics in 1944.

For light that is not exactly resonant with the transition Δω is the detuning the frequency of the Rabi oscillations increases but their amplitude decreases as the light is tuned away from resonance for transitions in the visible-frequency range – observation of Rabi flopping requires powerful laser beams - pulsed, so that the electric field amplitude varies with time therefore, the Rabi frequency ΩR/2π also varies with time, and so it is useful to define the pulse area Θ A pulse with an area equal to π is aπ-pulse An atom with c1 = 1 at t = 0 is promoted to the excited state with c2 = 1 by a π-pulse will end in the ground state if it interacts with a 2π-pulse

what are the time-scales involved in Rabi flopping processes? • ExampleA powerful beam of light is incident on monatomic hydrogen and is tuned to resonance with the 1s → 2p transition at 137 nm. • (a) Calculate the Rabi oscillation period when the optical intensity is 10 kWm−2 and the light is polarized in the z-direction • (b) Calculate the optical intensity required to make the Rabi oscillation period equal to the radiative lifetime of the 2p level, namely 1.6 ns. • Solution • (a) The atomic dipole moment for this transition is −0.74ea0 = −6.32x10-30 Cm The intensity of a light beam is related to its electric field amplitude • In a gas we may take n ≈ 1, and so we find E0 = 2.7x 103Vm−1 • On substituting into the Rabi frequency: the Rabi flopping period is 2π/ΩR = 38 ns • (b) A Rabi oscillation period of 1.6 ns corresponds to ΩR =3.9 x109 rad s−1 E0 = 6.5 x104 Vm−1, and I = 5.7 MWm−2

Damping • the example shows why it is difficult to observe Rabi oscillations at low powers - the oscillation period is longer than the radiative lifetime, and we would expect random spontaneous emission to destroy the coherence of the superposition states • thus higher powers are needed to shorten the Rabi flopping period • Spontaneous emission is one of the damping mechanisms for the Rabi oscillations • consideration of damping is very important for determining the experimental conditions under which Rabi oscillations can be observed

The damping processes for Rabi flopping are characterized by two time constants, T1 and T2called - longitudinal relaxation and transverse relaxation, respectively • T1 damping is determined by population decay • T2 damping is related to dephasing processes • the T1 (longitudinal) damping processes is related to the spontaneous tendency of the atom (in the excited state) to decay to lower levels and randomly break the coherence of the electronic wavefunction - interrupt the Rabi flopping • The rate of these types of damping process is governed by the lifetime tof the upper level, which itself is determined by both the radiative and non-radiative decay rates The upper limit on T1 is set by the radiative lifetime tR of the excited state includes transitions both to the resonant lower level and to other non-resonant levels

The T2 (transverse) damping processes is related to elastic or near-elastic collisions which breaks the phase of the wavefunction without altering the population of the excited state mechanismIn a gas, collisions can occur between the atoms or with the walls whereas in a solid there can be interactions with impurities or phonons By randomizing the phase of the wavefunction, the collisions destroy any Rabi flopping which rely on phase coherencedephasing can occur by two distinct mechanisms:population decay and population-conserving scattering processesthe total dephasing rate is: in solids at room temperature, the pure dephasing rate is much faster than the population decay rate (T2’ <<T1) decoherence is by scattering When T2’ >> T1, the decoherence is then governed primarily by T1, which itself is determined by the lifetime of the upper level

If the damping rate is g, the probability that the electron is in the upper level |c2(t)|2 is given by when g the formula reduces to the undamped case For light damping - the electron performs a few damped oscillations and approaches the asymptotic limit |c1|2 = |c2|2 =1/2 expected from the Einstein analysis of a pure two-level system in the strong-field limit At high optical power levels the spontaneous emission rate is negligible and the rates of stimulated emission and absorption equal - identical upper and lower level populations for strong damping - equivalent to the weak-field limit, we can make g/ΩR large by turning down the electric field of the light beam

Experimental observations of Rabi oscillations • Rabi oscillations are strongly damped except when In gases typical values of gfor optical-frequency transitions 107–109 s−1 In solids the dephasing times are often shorter and gcan be 1012 s−1 high damping rates make demonstration of Rabi oscillations difficult The observation of the oscillations requires a time resolution shorter than 1/ΩR, while the short Rabi oscillation periods require large electric field amplitudes

The first experimental evidence of Rabi oscillations came from the observation of self-induced transparency - S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1969). • if the pulse area is equal to 2π, then the atoms are left in the ground state at the end of the pulse • This implies that there is no net absorption, and so a medium that absorbs strongly at low powers would become transparent to a 2π-pulse: hence ‘self-induced transparency’ • The condition to observe the phenomenon is that the pulse duration should be shorter than the damping time, and that the pulse area should be equal to an integer multiple of 2π McCall and Hahn performed their experiments on the absorption of a ruby crystal excited resonantly with nanosecond pulses from a ruby laser. The ruby crystal was held to suppress damping by phonon scattering • They confirmed that the crystal did indeed become more transparent as the pulse area (determined by the energy of the pulse) approached 2π

The first direct evidence of Rabi oscillations came from experiments of Gibbs H. M. Gibbs, Phys. Rev. Lett. 29,459 (1972) and Phys. Rev. A 8, 446 (1973) • fluorescence emitted by Rb atoms excited resonantly by short pulses from a mercury laser • By placing the Rb atoms in a superconducting magnet, one of the hyperfine components of the MJ= −1/2 → +1/2 line of the 5 2S1/2→ 5 2P1/2 transition with a dipole moment of 1.45 x 10-29Cm could be tuned to resonance with the laser by the Zeeman effect • The upper level could decay either to the +1/2 or −1/2 levels of the 5s state, with radiative lifetimes of 42 and 84 ns, respectively, - a total radiative lifetime of (1/42 + 1/84)−1 = 28 ns • With a low density of atoms to prevent dephasing by collisions, the right conditions to observe Rabi flopping were present for pulses significantly shorter than 28 ns. The oscillations were then detected by measuring the fluorescence from the upper level as a function of the pulse area Θ • When the pulse area was equal to odd integer multiples of π the atoms ended up in the excited state at thecompletion of the pulse, and then decayed to the ground state by spontaneous emission, producing a strong fluorescence signal • when the pulse area was equal to even integer multiples of π the atoms were in the ground state at the completion of the pulse, and no fluorescence occurred

Mollow triplets - the frequency space equivalent of the Rabi oscillations in the time domainB. R. Mollow, Phys. Rev. 188 (1969). • Mollow demonstrated that the coherent oscillatory motion of the electrons in the strong-field limit beat with the fundamental transition angular frequency ω0 and produce side bands in the emission spectrum at ω0±ΩR The AC Stark interaction between a two-level atom and an intense resonant light field splits the bare atom states into doublets separated by the Rabi frequency ΩR R. E. Grove, F. Y. Wu, and S. Ezekiel, Phys. Rev. A 15, 227 (1977).

Light Matter Interaction - Semi-classical