^2 † 2 2 n 2 n 2 2 2 2 2 ^ p 1 2 ^2 ^ ^† ^ 1 ^ ^† ^† ^ ^ 1 ^ ^ ^ 1 ^ ^ ^ 1 d k 1/4 -1/2 (kx - Sqrt[2] α) ∞ ^ ∞ α -1/2 | α | ∞ α 1 ∞ -i/h pξ * 1 1 1 ∞ -i/h pξ k 1/2 -1/2 (k(x - 1/2 ξ) - Sqrt[2] α) -1/2 (k(x + 1/2 ξ) - Sqrt[2] α) 1 -(kx - Sqrt[2] α) - (p/h k) p 2 ^ ^ 1/4 RowBox[{Cell[TextData[{Coherent States and Phase Space, <br /><br />In one dimension the SHO Hamiltonian is written as<br /><br />, Cell[BoxData[H = --- + - MΩ x ]], <br /><br />The SHO Hamiltonian can also be written in terms of the creation and annihilation operators.<br /><br />, Cell[BoxData[H = hΩ(a a + -)]], <br /><br />Where the creation and annihilation operators acting on an eigenstate of the SHO Hamiltonian have the following properties:<br /><br />, Cell[BoxData[a | n > = Sqrt[n] | n >]], <br /><br />, Cell[BoxData[a | n > = Sqrt[n + 1] | n >]], <br /><br />, Cell[BoxData[a a | n > = n | n >]], <br /><br />The creation and annihilation operators can be expressed in terms of the position and momentum operators as follows:<br /><br />, Cell[BoxData[a = ------------------ (MΩ x - i p)]], <br />, Cell[BoxData[a = ------------------ (MΩ x + i p)]], <br /><br />The coherent states of the electromagnetic field were first investigated by Glauber. The coherent state is an eigenstate of the annihilation operator. The operator equation for a coherent state can be written:<br /><br />, Cell[BoxData[a ψ(x) - α ψ(x) = 0]], <br /><br />This equation can be expanded to yield:<br />, Cell[BoxData[------------------ (ih -- + MΩ x) ψ(x) - α ψ(x) = 0]], <br /><br />Solving the differential equation for ψ(x) yields the position state representation for the coherent state:<br /><br />, Cell[BoxData[ψ(x) = < x | α > = (------) e ]], <br /><br />The state vector for a coherent state can also be determined in another way as follows:<br /><br />, Cell[BoxData[| α > = ∑ c | n >]], <br /><br />, Cell[BoxData[a | α > = ∑ Sqrt[n] c | n - 1 > = α | α >]], <br /><br />Equating coefficients yields the following recursion:<br /><br />, Cell[BoxData[αc = Sqrt[n] c ]], <br /><br />Applying this equation recursively yields:<br /><br />, Cell[BoxData[--------- c = c ]], <br /><br />Subsituting this back into |α> and normalizing yields the following state vector for a coherent state:<br /><br />, Cell[BoxData[| α > = e ∑ --------- | n >]], <br /><br />The Wigner function for the phase space distribution of the coherent state can be determined by putting the state vector into the definition of the Wigner function. This is given by:<br /><br />, Cell[BoxData[W(x, p) = --------- ∫ e ψ (x - - ξ) ψ(x + - ξ) d ξ]], <br /><br />, Cell[BoxData[W (x, p) = --------- ∫ e (------) e e d ξ]], <br /><br />For α real, the integral yields:<br /><br />, Cell[BoxData[W (x, p) = ------- e ]], <br /><br />The Wigner function evolves according to classical dynamics (Liouville Equation) and the time varying Wigner function is determined by making the substitutions:<br /><br />, Cell[BoxData[W (x, p, t) = W (x -> cos(Ωt) x - sin(Ωt) ----, p -> sin(Ωt) k hx + cos(Ωt) p)]], <br /><br />The Mathematica notebook below shows a coherent state and it's evolution in phase space for different time intervals. The coherent state can also be squeezed by changing the squeezing parameter s. The squeezing operator , Cell[BoxData[S]], (s) operating on a state in the position representation is does the following:<br /><br />, Cell[BoxData[S(s) ψ(x) = s ψ(Sqrt[s] x)]], <br /><br />For a coherent state in phase space squeezing manifests itself as the generation of a cigar shaped state that evolves around in a similar fashion as the coherent state. Since the area is conserved in phase space, it maintains the area h. The minimized variance due to squeezing in one quadrature is also periodic as can be visualized in the following notebook. }]], }] 2 M 2 2 Sqrt[2 M h Ω] Sqrt[2 M h Ω] Sqrt[2 M h Ω] dx π n = 0 n n = 0 n n - 1 n Sqrt[n !] 0 n n = 0 Sqrt[n !] 2 πh -∞ 2 2 coh 2 πh -∞ π coh πh coh coh 2 h k

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k = 1 ; "Spring Constant for SHO" ; α = 3 ; "Displacement Parameter" ; h = 1 ; "Planck's Constant" ; Ω = .25 ; "Angular Frequency" ; "Time varying Wigner function for a coherent state with squeezing parameter s" ; W[x_, p_, t_] := 1/h e^(-(k * (Cos[Ω * t] * k * x - Sin[Ω * t] * p/(h * k^2)) - 2^(1/2) * α)^2 - 1/1 (Sin[Ω * t] * k * x + (Cos[Ω * t]) * p/(h * k))^2) ;

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"This plots a contour plot of the coherent state at four different times in phase space" ; A0 = Table[ContourPlot[Re[W[x, p, 3.1416 * t]], {x, -6, 6}, {p, -6, 6}, PlotRange -> {0, 1}, PlotPoints -> 100, AxesLabel -> {x, p, "W(x,p)"}, ColorFunction -> Hue], {t, 0, 8}] "This plots the coherent state at four different times in phase space" ; A1 = Table[Plot3D[Re[W[x, p, 3.1416 * t]], {x, -6, 6}, {p, -6, 6}, PlotRange -> {0, 1}, PlotPoints -> 90, Mesh -> False, AxesLabel -> {x, p, "W(x,p)"}], {t, 0, 8}]

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s = 3 ; "Squeezing Parameter" ; "Time varying Wigner function for a coherent state with squeezing parameter s" ; W2[x_, p_, t_] := 1/h e^(-(k * s * (Cos[Ω * t] * k * x - Sin[Ω * t] * p/(h * k^2)) - 2^(1/2) * α)^2 - 1/s (Sin[Ω * t] * k * x + (Cos[Ω * t]) * p/(h * k))^2) ; "This plots a contour plot of the coherent state at four different times in phase space" ; A2 = Table[ContourPlot[Re[W2[x, p, 3.1416 * t]], {x, -6, 6}, {p, -6, 6}, PlotRange -> {0, 1}, PlotPoints -> 100, AxesLabel -> {x, p, "W(x,p)"}, ColorFunction -> Hue], {t, 0, 8}] "This plots the coherent state at four different times in phase space" ; A3 = Table[Plot3D[Re[W2[x, p, 3.1416 * t]], {x, -6, 6}, {p, -6, 6}, PlotRange -> {0, 1}, PlotPoints -> 90, Mesh -> False, AxesLabel -> {x, p, "W(x,p)"}], {t, 0, 8}]

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Converted by Mathematica  (October 26, 2003)