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Coherent States and the Classical Limit Coherent states and the classical limit Paul Eastham February 23, 2012 ::: and its consequences Uncertainty in the EM field Casimir effect Interference experiments and photon 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 jni of a single-mode field. – “n photons”. Do we recover classical wave as n gets large? No : E^ = E(a^ + a^y) ) hEi is always zero. Motivation: electric fields in number states Number states: fields and uncertainties Suppose we have a state jni at t=0 – fields at later time? Use Heisenberg equation, e.g. da^ i = [a^; H^ ] and H^ = !(a^ya^ + 1=2) ~ dt ~ ) a^(t) = e−i!t a^(0) ≡ e−i!t a^ ) E^ (t) = E(ae^ −i!t + a^yei!t ) So hnjE^ (t)jni = Ehnj(ae^ −i!t + a^yei!t )jni = 0: Motivation: electric fields in number states Number states: fields and uncertainties hnjE^ (t)jni = Ehnj(ae^ −i!t + a^yei!t )jni = 0: Also: hnjE^ 2(t)jni = E2hnj(ae^ −i!t + a^yei!t )2jni = E2hnj(a^a^y + a^yajni = E2(2n + 1): Motivation: electric fields in number states Number states: fields and uncertainties Electric Field Time Coherent states as eigenstates of the annihilation operator Coherent states: motivation and definition Approx. to classical wave should have hE^ i 6= 0 ∼ cos(!t), Since E^ / a^ + a^y, try finding eigenstate of a^. Eigenvalue equation a^jλi = λjλi ) eigenstates ∗ y jλi = e−λ λ/2eλa jn = 0i 1 n ∗ X λ = e−λ λ/2 p jni: ! n=0 n Properties of coherent states Useful maths Coherent states: formal properties Right eigenstates of non-Hermitian operator a^. Continuous complex eigenvalue λ. State hλj is the left eigenstate of a^y (with eigenvalue λ∗) Normalised hλjλi = 1. −|λ j2=2−|λ j2=2+λ∗λ Not orthogonal hλ1jλ2i = e 1 2 1 2 . 1 R ^ Complete π jλihλjd(<(λ))d(=(λ)) = 1. Properties of coherent states Photon number distribution Coherent states: physical properties Average photon number hλjayajλi = jλj2. Probability of n photons P(n) = jhnjλij2 2n 2 jλj = e−|λj : n! – Poisson distribution (as before) – with average n=jλj2. 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 jλi at time zero, and field operator E^ (t) = E(ae^ −i!t + a^yei!t ) Then hλjE^ (t)jλi = E(2<(λ) cos(!t) + 2=(λ) sin(!t)): hλjE^ (t)2jλi − hλjE^ jλi2 = 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” ^ 1 y ^ X1 = p (a^ + a^ ) / Ex ; 2 ^ −i y ^ X2 = p (a^ − a^ ) / By : 2 – For a single mode field, dimensionless Ex and By . – For a harmonic oscillator, dimensionless position and momentum operators. – ∼ real and imaginary parts of a^ Coherent states are minimum uncertainty states Visualising states in quadratures Quadrature operators: time dependence Time evolution from Heisenberg equation ^ 1 −i!t y i!t X1(t) = p (ae^ + a^ e ) 2 ^ −i −i!t y i!t X2(t) = p (ae^ − a^ e ); 2 ) ^ ^ ^ X1(t) = X1(0) cos(!t) + X2(0) sin(!t) ^ ^ ^ X2(t) = −X1(0) sin(!t) + X2(0) cos(!t): 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 jλi, p 1 ∗ hX1i = p (λ + λ ) = 2<(λ) 2 p −i ∗ hX2i = p (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 ≥ jcj=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 quantum 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(i!t) but quantum E !Ee(a^ + a^y): Generation of coherent states (non-exam) Physical origin of coherent states For one mode of the field we now have something like y i!t −i!t H0 + (Ee:P0)(a^ + a^ )(e + e ) Suppose at time t = 0 field is in the ground state j0i 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 ! Switch to “interaction picture” operators a^ ! ae^ −i!t . State vector obeys dj i i = H^ j i; ~ dt int where ^ y Hint ≈ (Ee:P0)(a^ + a^ ): Integrating iEe:P − 0 t(a^+a^y) j i = e ~ j0i ∗ y itEe:P ≡ e−α a^+αa^ j0i α = − 0 ~ Generation of coherent states (non-exam) Physical origin of coherent states iEe:P − 0 t(a^+a^y) j i = e ~ j0i ∗ y itEe:P ≡ e−α a^+αa^ j0i α = − 0 ~ Gives : 2 − jαj αay ) j i = e 2 e j0i: = 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 a^. 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 lasers.
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