Physics 566: Quantum Optics I Problem Set #6

Due: Wednesday Oct. 13, 2021 Problem 1: Light forces on atoms (20 Points) Electromagnetic fields can exert forces on atoms. This is force can be dissipative (the basis of laser cooling) or conservative (the basis for optical trapping, such as optical lattices). Suppose we are given a monochromatic, uniformly polarized laser field of the form . The interaction of this field with a two-level atom

is described by the Hamiltonian in the rotating frame,

,

where R is the center of mass position of the atom, and .

Assuming the internal state of the atom relaxes to its steady state much faster than the atom moves, we can neglect the quantum mechanics of the atom's center of mass and treat its motion as a classical point particle (this is known as the "semiclassical model"). The force on the atom is the defined by the expectation value

,

where is the “internal state” of the atoms according to the optical Bloch equations.

(a) Under these condition show that the mean force on the atom is, , where

is the "dissipative force" and

is the "reactive force",

with u and v the components of the Bloch vector in the rotating frame relative to the incident phase .

(b) Show that in steady state, the rate at which that laser does work on the atom, averaged over an optical period is:

E(x, t) = ε LE0 (x)cos(ωL t + φ(x))

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HAL(R) = �~�|eihe|+ ~⌦(R)

2

⇣e�i�(R)|eihg|+ e

i�(R)|gihe|⌘

!Ω(R) = − e d ⋅ "εL g E0 (R)

F = − ∇HAL (R) = −Tr ∇HAL (R) ρ(t)( )

ρ(t)

F = Fdiss + Freact

Fdiss = −

12!v(t) Ω(R)∇φ(R)

Freact = − 1

2u(t)∇Ω(R)

ρgee− iφ (R ) = (u + iv) / 2

, where is the photon scattering rate.

Interpret this result. (c) For the case of a plane wave , show that in steady-state:

. This is known as "radiation pressure" or the “scattering force” - interpret.

(d) For the "reactive force" consider the case of weak saturation, s<<1. Show that

,

where the optical “dipole potential” is

.

Interpret the physical meaning of . Problem 2: Dark states (25 points) Let us consider again a three level “lambda system”

The two ground states are resonantly coupled to the excited state, each with a different Rabi frequency. Taking the two ground states as the zero of energy, then in the RWA (and in the rotating frame) the Hamiltonian is

(a) Find the “dressed states” of this system (i.e. the eigenstates and eigenvalues of the total atom laser system). You should find that one of these states has a zero eigenvalue,

dWdt s.s

= Ω0ω L

2vs.s = γ sω L γ s = Γρee

s.s.

E(R,t) = ε LE0 cos(ω Lt − k ⋅R)

Fdiss = γ skL

Freact = −∇U (R)

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U(R) = ~�s(R)

2= �1

4Re(↵) |E0(R)|2

U(R)

HRF =

2

Ω1 g1 e + e g1( ) +Ω2 g2 e + e g2( )⎡⎣ ⎤⎦

This particular superposition is called a “dark state” or uncoupled state because the laser field does not couple it to the excited state. (b) Adiabatic transfer through the “nonintuitive” pulse sequence. Suppose we want to transfer population from to . A robust method is to use adiabatic passage, always staying in the local dark state. This can then be on resonance. If we apply a slowly varying pulse overlapped, but followed by shown below, we accomplish this transfer

Sketch the dressed state eigenvalues a function of time. Explain the conditions necessary to achieve the adiabatic transfer. (c) As discussed in lecture, when including spontaneous emission of the excited state, the atom will “relax” to the dark state. This is known as coherent population trapping (CPT). In class we solved this under the conditions of adiabatic elimination. Let’s return to this here, under the condition of strong coupling. Consider, for simplicity, the case that

, and the atom decays with equal rates to the two ground sublevels:

Write the master equation in the basis , where

is the dark-state and is the bright-state. Show that

Comment on this representation.

ψ Dark =Ω2 g1 −Ω1 g2

Ω12 +Ω2

2

�

g1

�

g2

�

Ω2(t)

�

Ω1(t)

Ω1 =Ω2 =Ω

D =g1 − g2

2, B =

g1 + g22

, e⎧⎨⎩

⎫⎬⎭

D B

dρdt

= −i

Heff ρ − ρH †eff( ) + Lfeed[ρ]

Heff = −i Γ

2e e + 2Ω

2e B + B e( ), Lfeed[ρ]=

Γ2

e ρ e B B + D D( )

Ω ΩΓ2

Γ2

(d) Show that the equations of motion for the density matrix in this basis are

, ,

, ,

,

and the steady state solution is , i.e., the system relaxes to the dark-state. Epilogue: The relaxation to the dark state is somewhat mysterious from the equations of motion since a spontaneous decay of along one of the two paths sketched above CANNOT land us in the dark state – we land in or . Actually, we relax to the dark state when we DO NOT see a spontaneous decay. Not seeing spontaneous emission is information too. We’ll return to this later when we study “quantum trajectories.” Problem 3: Autler-Townes (25 Points) Consider a 3-level atom, with one ground state, coupled through two different laser fields to two excited states in a “V-configuration”. For the transition, , the laser weakly excites the atom, and can be detuned from resonance. The transition is tuned on resonance and can be highly excited. The goal of this problem is to study the effect of the strong coupling on the optical response on the transition. The density matrix for the atom evolves according to the Master Equation,

,

where the “effective” non-Hermitian Hamiltonian in the RWA is,

!ρee = −Γρee + i

2Ω2

ρeB − ρBe( ) !ρBB = + Γ

2ρee − i

2Ω2

ρeB − ρBe( )

ρeB = − Γ

2ρeB − i

2Ω2

ρBB − ρee( ) ρDD = + Γ

2ρee

!ρeD = − Γ

2ρeD − i

2Ω2

ρBD !ρBD = −i 2Ω2

ρeD

ρ s.s. = D D

g1 g2

�

3 ↔ 1

�

3 ↔ 2

�

3 ↔ 2

�

3 ↔ 1

�

ddt

ˆ ρ = − i

ˆ H effˆ ρ − ˆ ρ ˆ H eff

†( ) + Γ1ρ11 + Γ2ρ22( ) 3 3

�

3

�

2

�

1

�

Ω2

�

Ω1

�

Γ2

�

Γ1

�

Δ1

.

(a) Show that the equations of motion of the density matrix elements are,

(b) Under the assumption of weak excitation of , in order to find the response on the

transition, we need only retain terms to first order in . Coherence is dominated by the strong field, . Further we will assume . Show that in steady state in these approximations,

, , .

(c) Put all this together to show that the population excited into is

Plots as a function of , normalized in units of , for different coupling

strengths on the auxiliary transition, , are shown below.

Heff = − Δ1 +

i2Γ1

⎛⎝⎜

⎞⎠⎟ 1 1 − i

2Γ2 2 2 + Ω1

23 1 + 1 3( ) + Ω2

23 2 + 2 3( )

ρ11 = −Γ1ρ11 −i2Ω1 ρ31 − ρ13( ), ρ22 = −Γ2ρ22 −

i2Ω2 ρ32 − ρ23( ),

ρ11 + ρ22 + ρ33 = 1

ρ13 = iΔ1 −Γ12

⎛⎝⎜

⎞⎠⎟ ρ13 −

i2Ω1 ρ33 − ρ11( ) + i

2Ω2ρ12,

ρ23 = − Γ2

2ρ23 −

i2Ω2 ρ33 − ρ22( ) + i

2Ω1ρ21,

ρ12 = iΔ1 −Γ1 + Γ2

2⎛⎝⎜

⎞⎠⎟ ρ12 +

i2Ω2ρ13 −

i2Ω1ρ32,

�

1

�

3 ↔ 1

�

Ω1

�

ρ12

�

Ω2

�

Γ2 << Γ1

ρ11 = −Ω1

Γ1Im(ρ13)

�

ρ13 ≈ −Ω1

2Δ1 + iΓ1( ) ρ33 + Ω2

2Δ1 + iΓ1( ) ρ12

�

ρ12 ≈Ω2

2Δ1 + iΓ1( ) ρ13

�

1

�

ρ11 ≈ −Ω12

Γ1ρ33 Im

2Δ1 + iΓ12Δ1 + iΓ1( )2 −Ω2

2

⎡

⎣ ⎢

⎤

⎦ ⎥

�

ρ11

�

Δ1

�

Ω12

Γ12 ρ33

�

Ω2

�6 �4 �2 2 4 6

0.2

0.4

0.6

0.8

1.0

�6 �4 �2 0 2 4 6

0.2

0.4

0.6

0.8

1.0

�6 �4 �2 0 2 4 6

0.2

0.4

0.6

0.8

1.0

�

Ω2 /Γ1 = 0

�

Ω2 /Γ1 = 2

�

Ω2 /Γ1 = 5

�

ρ11

�

ρ11

�

ρ11

�

Δ1 /Γ1

�

Δ1 /Γ1

�

Δ1 /Γ1

For no coupling, we see the familiar Lorentzian lineshape. As the coupling increases so that , we see the line split into a doublet known as the Autler-Townes splitting. A more intuitive understanding of the origin of the Autler-Townes doublet is to think about the “dressed states” of atom+laser. The strong laser field on the dresses the atom. The weak laser then probes the absorption from the dressed states. (d) Find the dressed states of the two-level system, coupled on resonance (diagonalize of the Hermitian part of the Hamiltonian matrix in the subspace). Based on the eigenvalues and eigenvectors, explain the doublet. (e) Explain the physical difference between Autler-Townes splitting and EIT. Under what conditions are they equivalent?

�

Ω2 > Γ1

�

3 ↔ 2

�

3 ↔ 2

�

2× 2

�

3 , 2