Modern Atomic and Optical Physics II Lecture Notes, Lecture notes of Quantum Mechanics

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Modern Atomic and Optical Physics II

Lectures by M.D. Lukin

Notes taken and typed by L. Childress

Notes last updated December 2016

Contents

Chapter 1

Preface: Quantum mechanics

and atomic physics review

1.1 Introduction

In this course, we will approach the field of atomic physics by presenting a unified picture of coherent evolution and environmental decoherence and dissipation. The course will develop around three themes: firstly, we will elucidate theoretical techniques for simultaneously treating both coherent and dissipative processes. We will also consider experimental methods for studying atomic systems, including, for example discussion of high resolution spectroscopy, laser trapping and cooling and preparation of quantum states of ions. Finally, we will incorporate examples which illustrate how these experimental and theoretical techniques find application in current research. The basic philosophy of the course is to develop a theoretical under- standing of ”simple” systems first, that is systems where only a few states, a few particles, or a few quanta are relevant. This will lead us to explore some non-trivial quantum dynamics and study the possibility of controlling realistic quantum systems, including imperfections in ”real world” experi- ments. Once these basic building blocks are mastered, we will consider more complex systems by combining elements we already understand at a basic physical level. To understand the dynamics of the systems we will study in this course, we will need to treat both dissipation and coherent quantum evolution on an equal footing.

1.2 Review of quantum mechanics

1.2.1 States

We briefly review basic quantum mechanics and some elementary results of atomic physics. The most complete knowledge of the system is contained in the state vector |ψ〉, which is an element of a Hilbert space H. We can expand the state vector |ψ〉 in any orthonormal basis {|n〉} of the Hilbert space,

|ψ〉 =

n

|n〉〈n|ψ〉. (1.1)

For convenience, we write cn = 〈n|ψ〉 for the projection of the state |ψ〉 on the basis state |n〉, which gives us the expansion coefficients of the state |ψ〉 in the basis {|n〉}. The probability of finding the system in the state |n〉 is then Pn = |cn|^2.

1.2.2 Superpositions - Interference

Quantum mechanics allows for superpositions of states, in particular, for two states

|ψ 1 〉 =

n

c(1) n |n〉 and (1.2)

|ψ 2 〉 =

n

c(2) n |n〉, (1.3)

their superposition √^12 (|ψ 1 〉 + |ψ 2 〉) has the unique property that the proba- bility of being in state |n〉 is not in general given by the sum of probabilities. Instead it is given by the absolute value of the sum of amplitudes

Pn =

∣c(1) n +^ c(2) n

2 6 =

(P (^) n(1) + P (^) n(2) ) (1.4)

This inequality occurs as a result of the interference term between the two components of state vector, given in this case by c(1) n ∗c(2) n.

1.2.3 Observables

Observables in quantum mechanics are represented by Hermitian operators acting on states of the Hilbert space. An operator Oˆ is Hermitian when for any two states |ψ 1 , 2 〉 in the Hilbert space, we have

〈ψ 1 | Oˆ|ψ 2 〉 = (〈ψ 2 | Oˆ|ψ 1 〉)∗. (1.5)

state can be expanded in the the basis of eigenstates of Hˆ, and in that basis the time evolution of the state vector takes a particularly simple form, namely

|ψ(t)〉 =

n

e−iEnt/ℏcn|En〉 (1.12)

corresponding to the initial state |ψ(0)〉 with expansion coefficients cn = 〈En|ψ(0)〉. In the basis of its eigenstates, the Hamiltonian is diagonal Hˆ = ∑ n |En〉En〈En|. The time evolution equation can also be formally integrated to yield the time evolved state |ψ(t)〉 = exp[−i Ht/ˆ ℏ]|ψ(0)〉 = Uˆ (t)|ψ(0)〉, where Uˆ (t) is the time evolution operator.

1.2.6 Time evolution: Heisenberg picture

In Heisenberg’s picture, the state vector describing the initial state of the system does not evolve in time, rather the observables describing possible measurements on the system at time t, depend explicitely on time and evolve according to Heisenberg’s equation of motion

iℏ

d dt Oˆ(t) = [ O,ˆ Hˆ] + iℏ ∂^

Oˆ

∂t

1.3 Basic Atomic Physics

The physical systems associated with atomic physics include neutral atoms, ions, and molecules as well as more modern“artificial atoms” such as quan- tum dots fabricated in semiconductor heterostructures. The structure of these systems determines their interaction with applied fields, whether they be electric, magnetic, oscillating, static, classical, or quantum. These inter- actions allow us to manipulate the system via applied fields, or, conversely, use matter to control the electromagnetic fields. Such techniques are at the heart of atomic physics, but to apply them we must understand the physical structure of the system under consideration.

1.3.1 Example: the hydrogen atom

One of the simplest systems to consider is hydrogen, which consists of one electron bound by the Coulomb force to a proton. The energy of the system is given by

Hˆ = pˆ

2 2 m

e^2 ˆr

where ˆp represents the (operator-valued) electronic momentum and ˆr repre- sents the distance between the electron and the proton. The stationary solutions of Schrodinger’s equation for this system are of two types. There is an infinite set of discrete bound states with energies En = − R n∞ 2 and a continuum set of unbound states with energies E > 0. The constant R∞ is called Rydberg’s constant; while its numerical value can be found in any textbook on quantum mechanics, it is a good exercize to know how to derive an expression for it from basic principles. To estimate the binding energy R∞, we note that in the ground state Heisenberg’s inequality is approximately obeyed as an equality. In partic- ular, the uncertainty in momentum ∆p and the uncertainty in position ∆r obey ∆p · ∆r ∼ ℏ. Since the mean value of the momentum must be zero for a bound state, we can replace 〈p^2 〉 by ∆p^2 ∼ (ℏ/∆r)^2 in the Hamilto- nian. Similarly, the average distance 〈r〉 should also be approximately ∆r, yielding an average energy

E =

ℏ^2

2 m∆r^2

e^2 ∆r

Minimizing the energy with respect to ∆r, we find ∆r ∼ ℏ 2 me^2 , so that the ground state energy is approximately given by

E 0 = −R∞ ∼ −

me^4 2 ℏ^2

In fact, this turns out to be equal to the exact ground state energy for an infinitely heavy nucleus.

1.3.2 Level degeneracies

Although we have enumerated the energies of the bound states in hydro- gen, we have not addressed whether or not these states are degenerate. In particular, suppose that there is an operator Aˆ which commutes with the Hamiltonian Hˆ, i.e. [ A,ˆ Hˆ] = 0. We can then find a common set of eigen- vectors |En, Al〉 such that

Aˆ|En, Al〉 = Al|En, Al〉 (1.17) Hˆ|En, Al〉 = En|En, Al〉. (1.18)

The energy En is the same for all states |En, Al〉, and it does not depend on Al. This level degeneracy is associated with the underlying symmetry which renders A invariant under Hamiltonian evolution. Consider the effect

ated total electronic angular momentum J = L + S. Therefore, relativistic effects break the degeneracy of each principal level n such that different J values have different energies. The first contribution to the fine structure is a relativistic correction to the kinetic energy. The second contribution is referred to as the spin- orbit term and physically represents the interaction between the intrinsic magnetic dipole of the electron (i.e., the electron’s spin) and the internal magnetic field of the atom which is related to the orbital angular momentum of the electron. The final correction is referred to as the Darwin term, and is a relativistic correction to the potential energy. The Darwin term acts only at the origin and thus is only nonzero for L = 0 states. To illustrate how such relativistic effects can result in the splitting of levels with different J, we examine the spin-orbit coupling below. For details on the kinetic energy and Darwin term, one can refer to any introductory quantum mechanics text. The interaction energy of the spin-orbit effect is:

VSL = A Sˆ · L.ˆ (1.19)

VSL commutes with S^2 , L^2 , J^2 and Jz , where Jˆ = Sˆ + Lˆ, so we now label atomic states by J^2 and Jz rather than by Lz and Sz (which do not commute with VSL and are thus not good quantum numbers). As an example, consider sodium (Na). The ground state of Na has zero orbital angular momentum (| Lˆ| ≡ L = 0), and therefore no fine structure since J = S can only take on one value. For the first excited state (L = 1), J = 1 + 1/2 = 3/2 or J = 1 − 1 /2 = 1/2. What is the energy splitting between the J = 1/2 and J = 3/2 states? Using the fact that

Jˆ · Jˆ = J^2 = L^2 + S^2 + 2 Sˆ · Lˆ (1.20) ⇒ Sˆ · L = J^2 − L^2 − S^2 , (1.21)

we have VSL = A Sˆ · Lˆ = (A/2)(J^2 − L^2 − S^2 ) (1.22)

This implies that the energy splitting between the J = 3/2 and J = 1/ 2 state is given by

∆E 3 / 2 − ∆E 1 / 2 =

A

[(9/ 4 − 1 / 4 − 1) − (1/ 4 − 1 / 4 − 1)] =

A

· 2 = A. (1.23)

Traditionally, the transition from the 3 S 1 / 2 ground state of Na to the 3 P 1 / 2 excited state is called the D1 line whereas the transition from the 3 S 1 / 2 ground state of Na to the 3 P 3 / 2 excited state is called the D2 line.

Remark concerning the wavefunctions of hydrogen-like atoms

The wavefunction of a hydrogen-like atom can be written as ψnlm(ρ, θ, φ) = Rnl(ρ)Ylm(θ, φ) where Rnl(ρ) contains the radial dependence, and the angu- lar dependence is contained in the spherical harmonics

Ylm(θ, φ) = P (^) lm (cos θ)eimφ, (1.24)

where P (^) lm (cos θ) is the associated Legendre polynomial of degree l and order m. It is important to remember: (1) as ρ → 0, Rnl ∼ ρl, which vanishes unless l = 0, and (2) under the parity transformation (ˆr → −ˆr, i.e., φ → φ + π, θ → π − θ), Ylm(π − θ, φ + π) = (−1)lYlm(θ, φ). From this and the fact that the radial function Rnl(ρ) is insensitive to the parity transformation, we see that the parity of the ψnlm wavefunctions is (−1)l.

1.3.5 Hyperfine interaction

The interaction of the magnetic dipole of the valence electron with the nu- clear spin Iˆ is referred to as the hyperfine interaction. The magnetic field produced by a nucleus with magnetic moment mN is given by

B(R) =

μ 0 4 π

3 n(n · mN ) − mN R^3

8 π 3 mN δ(R)

where R is the vector distance from the nucleus and n is the unit vector in the direction of R. The second term is a strong contact term which is non-zero only for l = 0 states, whereas the first term is a weaker long-range (∼ 1 /R^3 ) term which is non-zero for all other l. The result is that the hyperfine interaction is generally strongest for S states with small r. The interaction energy may be approximated as

VHF S = AHF S Iˆ · Jˆ (1.26)

Taking into account both the fine and hyperfine interactions, the appro- priate conserved quantum number is the total angular momentum Fˆ given by Fˆ = Iˆ + Sˆ + L.ˆ (1.27)

Structure of the alkali ground state

As an example, consider the ground state of hydrogen for which L = 0, I = 1/2, and S = 1/2. Since L = 0, F 2 = I^2 + S^2 + 2 Sˆ · Iˆ, and HHF S = (AHF S /2)(F 2 − I^2 − S^2 ). Allowed values of F = 0, 1 lead to hyperfine shifts

letting J 1 = S = 1/2, J 2 = I = 1/2, and J = J 1 + J 2 = F (we’re assuming L = 0), we have

We will primarily be concerned with low values of the magnetic field, where the Zeeman effect can be treated as a perturbation to the fine and hy- perfine interactions. In this case, we use the |F, mF 〉 basis as our zero-order wavefunctions, and the Zeeman shift ∆EZ can be shown to be proportional to the applied magnetic field according to the relation

∆EZ = gF μB mF Bz , (1.37)

where μB is the Bohr magneton, gF is the Lande g-factor in the presence of nuclear spin given by

gF = gJ

F (F + 1) + J(J + 1) − I(I + 1)

2 F (F + 1)

Simliarly, gJ is the Lande g-factor without considering nuclear spin given by

gJ = 1 +

J(J + 1) + S(S + 1) − L(L + 1)

2 J(J + 1)

Note: A good reference on these topics is Vanier and Audoin, ”Quantum Physics of Atomic Frequency Standards.”

1.3.7 Interaction with optical fields

In this section we consider the interaction of isolated neutral atoms with optical fields. Such atoms alone have no net charge and no permanent electric dipole moment. In the presence of a static electric field E, however, the atoms do develop an electric dipole moment d which can then interact with the electric field with an interaction energy U given by

U = −d · E. (1.40)

In our semi-classical treatment of this interaction, we treat the atom quan- tum mechanically and therefore consider dˆ as an operator, but treat the electromagnetic field classically and so consider E as a vector of complex numbers. This dipole interaction has two effects: (1) when the field is off-resonant with the atomic transitions, the dominant effect is that the atomic energy levels undergo an energy shift (∼ E^2 ), (2) when the field is near-resonant with an atomic transition, the dominant effect is transitions between atomic levels.

Selection rules

The Hamiltonian for the dipole interaction interaction, can be written in terms of a complete basis of states {|a〉} for the atom:

Hˆ = −dˆ · E (1.41)

aa′

|a′〉〈a|〈a′|dˆ|a〉 · E. (1.42)

Note that the quantum state |a〉 is shorthand for a complete description in terms of quantum numbers na, la, sa, etc. Transitions are possible between states |a〉 and |b〉 only when the matrix element

μa′a ≡ 〈a′|dˆ|a〉 6 = 0. (1.43)

There are many restrictions on the states |a〉, |a′〉 for which μa′a 6 = 0; these are referred to as selection rules. For example, parity arguments re- strict the possible L values between which transitions are allowed. Since the scalar quantity

μa′a ∝

ψL′^ r ψLdV (1.44)

is invariant under the parity operation r → −r, ψL → (−1)LψL, μ is nonzero only if (−1)L+L′+1^ = 1, leading to the selection rule

∆L = ± 1. (1.45)

Of course, this argument really only implies that ∆L is odd. To prove that it must be specifically ±1, one can take the matrix element of both sides of equation [L^2 , [L^2 , ˆr]] = 2ℏ^2 (ˆrL^2 + L^2 ˆr) (1.46)

between 〈α′, L′, m′ L| and |α, L, mL〉. One can then derive the requirement ∆L = ±1.

longer a good quantum number but is replaced by J. However, the optical field still couples only to the orbital angular momentum L of the states. To calculate the angular matrix element then one must use the Clebsch-Gordan coefficients to express the |J, mJ 〉 states in terms of |L, mL〉|S, mS 〉 states. One must use a similar procedure (twice) for atoms with hyperfine structure for which F is the good quantum number (see Metcalf and van der Straten, ”Laser Cooling and Trapping”, for example). Now we move on to the selection rules for the magnetic quantum numbers which are related to the polarization of the incident electric field. We can write the electric field as a sum of linearly polarized components

E = ˆxEx + ˆyEy + ˆzEz, (1.55)

or as a sum of right and left circularly polarized and ˆz linearly polarized components E = ˆ+E+ + ˆ−E− + ˆzEz, (1.56)

where ˆ± = ∓(ˆx ± iˆy)/

From Eq. 1.54 and knowledge of spherical harmonics, one immediately sees that the mL selection rules are given by

σ+^ light (⇒ q = +1) ⇒ ∆mL = +1 (1.58) σ−^ light (⇒ q = −1) ⇒ ∆mL = − 1 (1.59) zˆ light (⇒ q = 0) ⇒ ∆mL = 0, (1.60)

The same selection rules apply as well for mJ and mF with an additional restriction. For J = J′, ∆mJ = 0 with mJ = 0 is forbidden, because

〈J, mJ |z|J, mJ 〉 ∼ mJ (1.61)

(where J here can stand for J or F ). Note that Eq. 1.61 is true because all the functions in this matrix element are symmetric about z.

Chapter 2

Basic processes in AMO

physics and quantum optics

2.1 Atomic Transitions

In the introductory lecture we have reviewed the somewhat complex level structure of hydrogenic atoms. Selection rules and energy conservation allow us to pick a subset of transitions, so that only a few energy levels are involved in the relevant dynamics. In this lecture we will focus in detail on the simplest atom-field system, namely that in which only two atomic energy levels interact with an applied laser field.

2.1.1 The Two-Level Atom

It is valid to approximate an atom as a two-level system when the external field is weak and nearly on resonance with a single atomic transition. In this case only the ground state and the near-resonant excited level will have appreciable probability of occupation. We will label these states | 1 〉 and | 2 〉 and designate their energy differency by ℏω. We consider the evolution of this system when illuminated by a nearly- monochromatic classical electromagnetic field

E(r, t) = E(r, t)e−i(νt−k·r)^ + E(r, t)∗ei(νt−k·r), (2.1)

where the slowly-varying envelope E(r, t) varies on a timescale much slower than the optical frequency ν ≈ ω and a length scale much larger than the optical wavelength 2π/|k|. This electric field interacts with the atomic dipole, given by an operator dˆ = eˆr, which may be conveniently represented