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Why a norm preserving stochastic dynamics does not preserve the norm?

Posted 2 months ago
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For the following norm-preserving evolution of the quantum state in 2D Hilbert space: enter image description here with dW the usual Wiener increment, and |psi>={c1,c2}, I wrote this code:

collapseDynamics[\[Lambda]_, c10_, c20_] := 
       t] == -(\[Lambda]/
        2) \[DifferentialD]t (c1[t] - 2 c1[t] (c1[t]^2 - c2[t]^2) + 
         c1[t] (c1[t]^2 - c2[t]^2)^2) + \[DifferentialD]W[
         t] Sqrt[\[Lambda]] (c1[t] - 
         c1[t] (c1[t]^2 - c2[t]^2)), \[DifferentialD]c2[
       t] == -(\[Lambda]/
        2) \[DifferentialD]t (c2[t] + 2 c2[t] (c1[t]^2 - c2[t]^2) + 
         c2[t] (c1[t]^2 - c2[t]^2)^2) + \[DifferentialD]W[
         t] Sqrt[\[Lambda]] (-c2[t] - 
         c2[t] (c1[t]^2 - c2[t]^2))}, {c1[t], 
    c2[t]}, {{c1, c2}, {c10, c20}}, {t, 0}, 
   W \[Distributed] WienerProcess[]];

When I generate an evolution I noticed that the norm is not preserved:

data = RandomFunction[
   collapseDynamics[1, 1/Sqrt[2], -1/Sqrt[2]], {0., 1, 0.01}];
ListLinePlot[Total /@ ((data["States"][[1]])^2), PlotRange -> All]

Can anyone explain why the norm is not preserved? (attached you can find nb file)

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