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3, Coherent and Squeezed States 1. Coherent States 2. Squeezed States 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. 1/50 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. 2/50 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. 3/50 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. 4/50 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) = ( ) [ˆaj e− +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. 5/50 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. 6/50 Quadrature operators the electric and magnetic fields become, ~ω ˆ j 1/2 iωj t iωj t Ex(z,t) = ( ) [ˆaj e− +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. 7/50 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. 8/50 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. 9/50 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. 10/50 Poisson distribution IPT5340, Fall ’06 – p. 11/50 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. 12/50 Real life Poisson distribution IPT5340, Fall ’06 – p. 13/50 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. IPT5340, Fall ’06 – p. 14/50 Displacement operator the coherent state is the displaced form of the harmonic oscillator ground state, 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, the displacement operator Dˆ(α) is a unitary operator, i.e. 1 Dˆ †(α)= Dˆ ( α)=[Dˆ (α)]− , − Dˆ (α) acts as a displacement operator upon the amplitudes aˆ and aˆ†, i.e. 1 Dˆ − (α)ˆaDˆ (α) =a ˆ + α, 1 Dˆ − (α)ˆa†Dˆ (α) =a ˆ† + α∗, IPT5340, Fall ’06 – p. 15/50 Radiation from a classical current the Hamiltonian (p A) that describes the interaction between the field and the · current is given by V = J(r, t) Aˆ(r, t)d3r, · Z where J(r, t) is the classical current and Aˆ(r, t) is quantized vector potential, 1 iωkt+ik r Aˆ(r, t)= i Ekaˆke− · + H.c., − ωk Xk the interaction picture Schrödinger equation obeys, d i Ψ(t) = V Ψ(t) , dt | i − ~ | i the solution is Ψ(t) = exp[α aˆ α aˆ ] 0 , where | i k k † − k∗ k | ik 1 t iωt′ ik r αk = Ek dt′ drJ(r, t)e − · , ~ωk 0 Q R R this state of radiation field is called a coherent state, α = (αaˆ† α∗aˆ) 0 .
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