Unitary equilibrations: Probability distribution of the Loschmidt echo

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 81, Issue 2, 2010, Pages

  Campos Venuti, Lorenzo ^a   Zanardi, Paolo ^a,b  

^a ISI FOUNDATION   (Italy)

^b UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CLOSED QUANTUM SYSTEM; EQUILIBRIUM STATE; GAUSSIANS; INITIAL STATE; LARGE SYSTEM SIZE; LOSCHMIDT ECHOES; PURE STATE;

CRITICALITY (NUCLEAR FISSION); DISTRIBUTION FUNCTIONS; QUANTUM ELECTRONICS; QUANTUM OPTICS;

PROBABILITY;

EID: 76749093169     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.81.022113     Document Type: Article

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32. Here below we sketch why this is so. Let us suppose that ρ∞=limt→∞Ut(ρ0). Obviously ρ∞ is a fixed point for Ut, i.e., Ut(ρ∞)=Ut(limu→∞Uu(ρ0))= limu→∞Ut+u(ρ0)=limu→∞Uu(ρ0)=ρ∞. Using unitary invariance of the trace norm it follows that ∥Ut(ρ0)-ρ1= ∥Ut(ρ0-ρ∞)∥1=∥ρ0-ρ1 and therefore limt→Ut(ρ0)-ρ1=0 implies ρ0=ρ∞.
33. Notice that in the infinite-dimensional case discussed above Eq. (1) implies P(α)=δ(α-A∞). Indeed from Eq. (1) it follows for any continuous f that f[A(t)]̄=f(A∞). Whence P(α)= δ[α-A(t)]̄=1/2π∫eiλαe-iλA(t)̄= 1/2π∫eiλαe-iλA(t)̄=1/2π∫eiλα e-iλA∞=δ(α-A∞).
34. Ln(t)=ρψn,P(n)(ρψn)+ρψn,e-itH~(n)(ρψn) where H~(n)=(-P(n))H(n)(-P(n)). The time average of the second term is ρψn,FTn(ρψn)/T where FTn=(-iH~(n))-1[e-iTH~(n)-]. Since FTn is a bounded operator its expectation value divided by T goes to zero when T→∞.
35. In (ii) one has to measure H(n):=i=1n(i-1)H-(n-i) in ρψn.
36. A more precise approximation valid also in region where δh is large is given by L(t)=Πk>0(1+c0k+Xk(t))L̄(1+k>0(1+c0k)-1Xk(t)). The two expressions coincide when δh is small.
37. This is precisely Eq. (5.6) of [31] in the zero temperature limit.