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Home , Coherent state

Coherent states and the

Paul Eastham

February 23, 2012 ... and its consequences Uncertainty in the EM field Interference experiments and statistics Classical and non-classical states of light ← this lecture Light-matter interactions Dipole coupling for an atom Jaynes-Cummings model and Rabi oscillations Motivation: electric fields in number states

Number states become classical waves?

So far : number/Fock states |ni of a single-mode field. – “n ”. Do we recover classical wave as n gets large? No : Eˆ = E(aˆ + aˆ†) ⇒ hEi is always zero. Motivation: electric fields in number states

Number states: fields and uncertainties

Suppose we have a state |ni at t=0 – fields at later time? Use Heisenberg equation, e.g. daˆ i = [aˆ, Hˆ ] and Hˆ = ω(aˆ†aˆ + 1/2) ~ dt ~ ⇒ aˆ(t) = e−iωt aˆ(0) ≡ e−iωt aˆ ⇒ Eˆ (t) = E(aeˆ −iωt + aˆ†eiωt )

So

hn|Eˆ (t)|ni = Ehn|(aeˆ −iωt + aˆ†eiωt )|ni = 0. Motivation: electric fields in number states

Number states: fields and uncertainties

hn|Eˆ (t)|ni = Ehn|(aeˆ −iωt + aˆ†eiωt )|ni = 0.

Also:

hn|Eˆ 2(t)|ni = E2hn|(aeˆ −iωt + aˆ†eiωt )2|ni = E2hn|(aˆaˆ† + aˆ†a|ni = E2(2n + 1). Motivation: electric fields in number states

Number states: fields and uncertainties

Electric Field

Time Coherent states as eigenstates of the annihilation

Coherent states: motivation and definition

Approx. to classical wave should have hEˆ i= 6 0 ∼ cos(ωt), Since Eˆ ∝ aˆ + aˆ†, try finding eigenstate of aˆ. Eigenvalue equation aˆ|λi = λ|λi ⇒ eigenstates

∗ † |λi = e−λ λ/2eλa |n = 0i ∞ n ∗ X λ = e−λ λ/2 √ |ni. ! n=0 n Properties of coherent states Useful maths Coherent states: formal properties

Right eigenstates of non-Hermitian operator aˆ. Continuous complex eigenvalue λ. State hλ| is the left eigenstate of aˆ† (with eigenvalue λ∗) Normalised hλ|λi = 1.

−|λ |2/2−|λ |2/2+λ∗λ Not orthogonal hλ1|λ2i = e 1 2 1 2 . 1 R ˆ Complete π |λihλ|d(<(λ))d(=(λ)) = 1. Properties of coherent states Photon number distribution Coherent states: physical properties

Average photon number hλ|a†a|λi = |λ|2. Probability of n photons P(n) = |hn|λi|2 2n 2 |λ| = e−|λ| . n! – (as before) – with average n=|λ|2.

0.6 0.12

0.5 0.10

0.4 0.08

0.3 0.06

0.2 0.04

0.1 0.02

0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9101112131415161718192021222324252627282930 Properties of coherent states Fields and uncertainties Coherent states: fields and uncertainties

Suppose we have a mode in a coherent state |λi at time zero, and field operator

Eˆ (t) = E(aeˆ −iωt + aˆ†eiωt ) Then

hλ|Eˆ (t)|λi = E(2<(λ) cos(ωt) + 2=(λ) sin(ωt)).

hλ|Eˆ (t)2|λi − hλ|Eˆ |λi2 = E2. Properties of coherent states Fields and uncertainties Coherent states: electric fields

Electric Field

Time Coherent states are minimum uncertainty states Visualising states in quadratures Quadrature operators

A related visualisation of single-mode states by considering the “quadrature operators”

ˆ 1 † ˆ X1 = √ (aˆ + aˆ ) ∝ Ex , 2 ˆ −i † ˆ X2 = √ (aˆ − aˆ ) ∝ By . 2

– For a single mode field, dimensionless Ex and By . – For a , dimensionless position and momentum operators. – ∼ real and imaginary parts of aˆ Coherent states are minimum uncertainty states Visualising states in quadratures Quadrature operators: time dependence

Time evolution from Heisenberg equation

ˆ 1 −iωt † iωt X1(t) = √ (aeˆ + aˆ e ) 2 ˆ −i −iωt † iωt X2(t) = √ (aeˆ − aˆ e ), 2 ⇒ ˆ ˆ ˆ X1(t) = X1(0) cos(ωt) + X2(0) sin(ωt) ˆ ˆ ˆ X2(t) = −X1(0) sin(ωt) + X2(0) cos(ωt). Coherent states are minimum uncertainty states Visualising states in quadratures Quadrature visualisations - classical oscillators Coherent states are minimum uncertainty states Quadrature picture of coherent states Coherent states: quadrature representation

For the coherent state |λi,

√ 1 ∗ hX1i = √ (λ + λ ) = 2<(λ) 2 √ −i ∗ hX2i = √ (λ − λ ) = 2=(λ) 2 And the uncertainties 1 ∆X 2 = hX 2 i − hX i2 = 1,2 1,2 1,2 2 Coherent states are minimum uncertainty states Quadrature picture of coherent states Coherent states: quadrature visualisation Coherent states are minimum uncertainty states Quadrature picture of coherent states Coherent states: uncertainty minimisation

Remember uncertainty ∆A∆B ≥ |c|/2 if [A, B] = c. For quadrature operators [X1, X2] = i, so ∆X1∆X2 ≥ 1/2. For coherent states – This bound is saturated – “minimum uncertainty state”

The uncertainty is symmetrically distributed ∆X1 = ∆X2. – this is the sense in which coherent states are the closest state to a classical wave Generation of coherent states (non-exam)

Physical origin of coherent states

Laser light is a close approximation to a coherent state When we have matter (atoms) etc., add Z 3 Hint = − d r E.P

– to the Hamiltonian of the EM field (see later in course).

Suppose classical electric dipoles P → (−P0/2) cos(iωt) but quantum E → Ee(aˆ + aˆ†). Generation of coherent states (non-exam)

Physical origin of coherent states

For one mode of the field we now have something like

† iωt −iωt H0 + (Ee.P0)(aˆ + aˆ )(e + e )

Suppose at time t = 0 field is in the |0i and switch on the oscillator. What happens to the field? Generation of coherent states (non-exam)

Physical origin of coherent states

Simplify: field mode of frequency ω Switch to “” operators aˆ → aeˆ −iωt . State vector obeys d|ψi i = Hˆ |ψi, ~ dt int where ˆ † Hint ≈ (Ee.P0)(aˆ + aˆ ). Integrating

iEe.P − 0 t(aˆ+aˆ†) |ψi = e ~ |0i

∗ † itEe.P ≡ e−α aˆ+αaˆ |0i α = − 0 ~ Generation of coherent states (non-exam)

Physical origin of coherent states

iEe.P − 0 t(aˆ+aˆ†) |ψi = e ~ |0i

∗ † itEe.P ≡ e−α aˆ+αaˆ |0i α = − 0 ~ Gives :

2 − |α| αa† ⇒ |ψi = e 2 e |0i.

= a coherent state (of amplitude α ∝ t - why?) Summary

Summary

Number states do not approach the expected classical limit at large n. Important class of states which do are the coherent states – defined as eigenstates of the annihilation operator aˆ. In these states - Poisson distributon of photon number Expected hEi and hBi follow a classical wave with The smallest fluctuations consistent with the uncertainty [X1, X2] = i Coherent states are generated by classical oscillating currents or dipoles ←