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Quantum Coherence Functions

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Quantum Coherence Functions 3, Coherent and Squeezed States 1. Coherent states 2. Squeezed states 3. Field Correlation Functions 4. Hanbury Brown and Twiss experiment 5. Photon Antibunching 6. Quantum Phenomena in Simple Nonlinear Optics Ref: Ch. 2, 4, 16 in ”Quantum Optics,” by M. Scully and M. Zubairy. Ch. 3, 4 in ”Mesoscopic Quantum Optics,” by Y. Yamamoto and A. Imamoglu. Ch. 6 in ”The Quantum Theory of Light,” by R. Loudon. Ch. 5, 7 in ”Introductory Quantum Optics,” by C. Gerry and P. Knight. Ch. 5, 8 in ”Quantum Optics,” by D. Wall and G. Milburn. IPT5340, Fall ’06 – p.1/85 Role of Quantum Optics photons occupy an electromagnetic mode, we will always refer to modes in quantum optics, typically a plane wave; the energy in a mode is not continuous but discrete in quanta of ~ω; the observables are just represented by probabilities as usual in quantum mechanics; there is a zero point energy inherent to each mode which is equivalent with fluctuations of the electromagnetic field in vacuum, due to uncertainty principle. IPT5340, Fall ’06 – p.2/85 Vacuum vacuum is not just nothing, it is full of energy. IPT5340, Fall ’06 – p.3/85 Vacuum spontaneous emission is actually stimulated by the vacuum fluctuation of the electromagnetic field, one can modify vacuum fluctuations by resonators and photonic crystals, atomic stability: the electron does not crash into the core due to vacuum fluctuation of the electromagnetic field, gravity is not a fundamental force but a side effect matter modifies the vacuum fluctuations, by Sakharov, Casimir effect: two charged metal plates repel each other until Casimir effect overcomes the repulsion, Lamb shift: the energy level difference between 2S1/2 and 2P1/2 in hydrogen. ... IPT5340, Fall ’06 – p.4/85 Casimir effect IPT5340, Fall ’06 – p.5/85 Uncertainty relation Non-commuting observable do not admit common eigenvectors. Non-commuting observables can not have definite values simultaneously. Simultaneous measurement of non-commuting observables to an arbitrary degree of accuracy is thus incompatible. variance: ∆Aˆ2 = Ψ (Aˆ Aˆ )2 Ψ = Ψ Aˆ2 Ψ Ψ Aˆ Ψ 2. h | −h i | i h | | i−h | | i 1 ∆A2∆B2 [ Fˆ 2 + Cˆ 2], ≥ 4 h i h i where [A,ˆ Bˆ]= iC,ˆ and Fˆ = AˆBˆ + BˆAˆ 2 Aˆ Bˆ . − h ih i Take the operators Aˆ =q ˆ (position) and Bˆ =p ˆ (momentum) for a free particle, ~2 [ˆq, pˆ]= i~ ∆ˆq2 ∆ˆp2 . →h ih i≥ 4 IPT5340, Fall ’06 – p.6/85 Uncertainty relation Schwarz inequality: φ φ ψ ψ φ ψ ψ φ . h | ih | i≥h | ih | i Equality holds if and only if the two states are linear dependent, ψ = λ φ , where λ | i | i is a complex number. uncertainty relation, 1 ∆A2∆B2 [ Fˆ 2 + Cˆ 2], ≥ 4 h i h i where [A,ˆ Bˆ]= iC,ˆ and Fˆ = AˆBˆ + BˆAˆ 2 Aˆ Bˆ . − h ih i the operator Fˆ is a measure of correlations between Aˆ andBˆ. define two states, ψ1 =[Aˆ Aˆ ] ψ , ψ2 =[Bˆ Bˆ ] ψ , | i −h i | i | i −h i | i the uncertainty product is minimum, i.e. ψ1 = iλ ψ2 , | i − | i [Aˆ + iλBˆ] ψ =[ Aˆ + iλ Bˆ ] ψ = z ψ . | i h i h i | i | i the state ψ is a minimum uncertainty state. | i IPT5340, Fall ’06 – p.7/85 Uncertainty relation if Re(λ) = 0, Aˆ + iλBˆ is a normal operator, which have orthonormal eigenstates. the variances, iλ i ∆Aˆ2 = [ Fˆ + i Cˆ ], ∆Bˆ2 = [ Fˆ i Cˆ ], − 2 h i h i − 2λ h i− h i set λ = λr + iλi, 2 1 2 1 2 ∆Aˆ = [λi Fˆ + λr Cˆ ], ∆Bˆ = ∆Aˆ , λi Cˆ λr Fˆ = 0. 2 h i h i λ 2 h i− h i | | if λ = 1, then ∆Aˆ2 =∆Bˆ2, equal variance minimum uncertainty states. | | if λ = 1 along with λ = 0, then ∆Aˆ2 =∆Bˆ2 and Fˆ = 0, uncorrelated equal | | i h i variance minimum uncertainty states. 2 λi 2 λ 2 1 if λr = 0, then Fˆ = Cˆ , ∆Aˆ = | | Cˆ , ∆Bˆ = Cˆ . 6 h i λr h i 2λr h i 2λr h i If Cˆ is a positive operator then the minimum uncertainty states exist only if λr > 0. IPT5340, Fall ’06 – p.8/85 Minimum Uncertainty State (ˆq qˆ ) ψ = iλ(ˆp pˆ ) ψ −h i | i − −h i | i 2r if we define λ = e− , then r r r r (e qˆ + ie− pˆ) ψ =(e qˆ + ie− pˆ ) ψ , | i h i h i | i the minimum uncertainty state is defined as an eigenstate of a non-Hermitian operator erqˆ + ie rpˆ with a c-number eigenvalue er qˆ + ie r pˆ . − h i − h i the variances of qˆ and pˆ are ~ ~ 2 2r 2 2r ∆ˆq = e− , ∆ˆp = e . h i 2 h i 2 here r is referred as the squeezing parameter. IPT5340, Fall ’06 – p.9/85 Quantization of EM fields ˆ ~ 1 the Hamiltonian for EM fields becomes: H = j ωj (ˆaj†aˆj + 2 ), the electric and magnetic fields become, P ~ω ˆ j 1/2 iωj t iωj t Ex(z,t) = ( ) [ˆaje− +a ˆj†e ] sin(kj z), ǫ0V Xj = cj [ˆa1j cos ωj t +a ˆ2j sin ωj t]uj (r), Xj IPT5340, Fall ’06 – p.10/85 Phase diagram for EM waves Electromagnetic waves can be represented by Eˆ(t)= E [Xˆ sin(ωt) Xˆ cos(ωt)] 0 1 − 2 where Xˆ1 = amplitude quadrature Xˆ2 = phase quadrature IPT5340, Fall ’06 – p.11/85 Quadrature operators the electric and magnetic fields become, ~ω ˆ j 1/2 iωj t iωj t Ex(z,t) = ( ) [ˆaje− +a ˆj†e ] sin(kj z), ǫ0V Xj = cj [ˆa1j cos ωj t +a ˆ2j sin ωj t]uj (r), Xj note that aˆ and aˆ† are not hermitian operators, but (ˆa†)† =a ˆ. 1 1 aˆ1 = (ˆa +a ˆ ) and aˆ2 = (ˆa aˆ ) are two Hermitian (quadrature) operators. 2 † 2i − † the commutation relation for aˆ and aˆ† is [ˆa, aˆ†] = 1, the commutation relation for and is i , aˆ aˆ† [ˆa1, aˆ2]= 2 and ∆ˆa2 ∆ˆa2 1 . h 1ih 2i≥ 16 IPT5340, Fall ’06 – p.12/85 Minimum Uncertainty State (ˆa1 aˆ1 ) ψ = iλ(ˆa2 aˆ2 ) ψ −h i | i − −h i | i 2r r r r r if we define λ = e , then (e aˆ1 + ie aˆ2) ψ =(e aˆ1 + ie aˆ2 ) ψ , − − | i h i − h i | i the minimum uncertainty state is defined as an eigenstate of a non-Hermitian operator eraˆ + ie raˆ with a c-number eigenvalue er aˆ + ie r aˆ . 1 − 2 h 1i − h 2i the variances of aˆ1 and aˆ2 are 2 1 2r 2 1 2r ∆ˆa = e− , ∆ˆa = e . h 1i 4 h 2i 4 here r is referred as the squeezing parameter. when r = 0, the two quadrature amplitudes have identical variances, 1 ∆ˆa2 = ∆ˆa2 = , h 1i h 2i 4 r r in this case, the non-Hermitian operator, e aˆ1 + ie− aˆ2 =a ˆ1 + iaˆ2 =a ˆ, and this minimum uncertainty state is termed a coherent state of the electromagnetic field, an eigenstate of the annihilation operator, aˆ α = α α . | i | i IPT5340, Fall ’06 – p.13/85 Coherent States r r in this case, the non-Hermitian operator, e aˆ1 + ie− aˆ2 =a ˆ1 + iaˆ2 =a ˆ, and this minimum uncertainty state is termed a coherent state of the electromagnetic field, an eigenstate of the annihilation operator, aˆ α = α α . | i | i expand the coherent states in the basis of number states, aˆn αn α = n n α = n 0 α = 0 α n , | i | ih | i | ih | √ | i √ h | i| i n n n! n n! X X X imposing the normalization condition, α α = 1, we obtain, h | i m n 2 (α∗) α α 2 1= α α = m n = e| | 0 α , h | i h | i √ √ |h | i| n m m! n! X X we have 1 2 n α ∞ α α = e− 2 | | n , | i √n! | i nX=0 IPT5340, Fall ’06 – p.14/85 Properties of Coherent States the coherent state can be expressed using the photon number eigenstates, 1 2 n α ∞ α α = e− 2 | | n , | i √n! | i nX=0 the probability of finding the photon number n for the coherent state obeys the Poisson distribution, 2 e α α 2n P (n) n α 2 = −| | | | , ≡ |h | i| n! the mean and variance of the photon number for the coherent state α are, | i nˆ = nP (n)= α 2, h i | | n X ∆ˆn2 = nˆ2 nˆ 2 = α 2 = nˆ , h i h i−h i | | h i IPT5340, Fall ’06 – p.15/85 Poisson distribution IPT5340, Fall ’06 – p.16/85 Photon number statistics For photons are independent of each other, the probability of occurrence of n photons, or photoelectrons in a time interval T is random. Divide T into N intervals, the probability to find one photon per interval is, p =n/N ¯ , the probability to find no photon per interval is, 1 p, − the probability to find n photons per interval is, N! n N n P (n)= p (1 p) − , n!(N n)! − − which is a binomial distribution. when N , → ∞ n¯nexp( n¯) P (n)= − , n! this is the Poisson distribution and the characteristics of coherent light. IPT5340, Fall ’06 – p.17/85 Real life Poisson distribution IPT5340, Fall ’06 – p.18/85 Displacement operator coherent states are generated by translating the vacuum state 0 to have a finite | i excitation amplitude α, 1 2 n 1 2 n α ∞ α α ∞ (αaˆ†) α = e− 2 | | n = e− 2 | | 0 , | i √ | i n! | i n=0 n! n=0 1 2 X X α αaˆ† = e− 2 | | e 0 , | i ∗ since aˆ 0 = 0, we have e α aˆ 0 = 0 and | i − | i 1 2 α αaˆ† α∗aˆ α = e− 2 | | e e− 0 , | i | i any two noncommuting operators Aˆ and Bˆ satisfy the Baker-Hausdorff relation, 1 Aˆ+Bˆ Aˆ Bˆ [A,ˆ Bˆ] e = e e e− 2 , provided [A,ˆ [A,ˆ Bˆ]]=0, using Aˆ = αaˆ , Bˆ = α aˆ, and [A,ˆ Bˆ]= α 2, we have, † − ∗ | | αaˆ† α∗aˆ α = Dˆ(α) 0 = e− − 0 , | i | i | i where Dˆ (α) is the displacement operator, which is physically realized by a classical oscillating current.
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