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Additional resources for Quantum optics : including noise reduction, trapped ions, quantum trajectories, and decoherence ; with 92 problems with hints for solutions
Example text
N! We can see that the phase ignorance washes out the off diagonal elements. 7. Define the characteristic function or “momentum generating function” (see also Chap. 7) as: ∞ CA (ξ) = Show that An = ( (iξ)n n A . n! n=0 ∂ n ) CA (ξ) |ξ=0 , ∂(iξ) CA (ξ) = T r(ρ exp(iξA)), CA (ξ) = P (A´/ρ) exp(iξA´)dA´, where A´is an eigenvalue of A and P (A´/ρ) the corresponding probability density. Hint: To prove the last property, use the second one for a continuous spectrum. 8. 7 be A = γa + γ ∗ a† . For a harmonic oscillator in a pure state α, show that 1 CA (ξ) = exp − ξ 2 γγ ∗ + i(α∗ γ ∗ + αγ) , 2 A = α∗ γ ∗ + αγ, 2 σA = A2 − A P (A´/ρ) = 2 =| γ |2 , 1 (A´− A )2 exp − 2 2 2σA 2πσA .
1 Squeezed States: General Properties and Detection We define the quadrature operators X and Y [1] X= a + a† = 2 ω q, 2 Y = a − a† = 2i 1 p. 2) ω em [X sin(ωt − k · r) − Y cos(ωt − k · r)] , 2εo v thus appearing as factor-operators in front of the sin and cos functions. 3) 1 . 11), one can write (ΔX)2 (ΔY )2 ≥ (ΔX)2 = 1 1 , (ΔY )2 = . 5) From the quadrature perspective, a coherent state is a minimum uncertainty state with equal fluctuations in both quadratures. 42 5. 7) Y )α = Im {α} . Pictorially, a coherent state can be represented, in the complex plane, as an error circle of diameter 12 and its center displaced by α (Fig.
28) are ∓R, where R≡ δ 2 + Ro2 , Eo dba | . 29) Normally, Ro is referred to as the Rabi frequency. 28) is Cb (t) = Ca (t) cos Rt 2 + iδ R sin Rt 2 o dab −i ER sin Rt 2 o dba −i ER sin Rt 2 Cb (0) cos Rt 2 − Ca (0) iδ R sin Rt 2 . 30) To take a simple example, if we start from the lower state [Cb (0) = 1], the transition probability for absorption is | Ca (t) |2 = | Rt Ro 2 | sin2 . 25) describes exactly the interaction between a two-level atom and the radiation field dCb = νCa (t) exp(−iωab t) cos ωt , dt dC i a = ν ∗ Cb (t) exp(iωab t) cos ωt , dt i where ν ≡ Eo dba .