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recently a new strategy for performing measurements on solid state ( josephson ) qubits was proposed which uses the entanglement of the qubit with states of a damped oscillator  @xcite , with this oscillator representing the plasma resonance of the josephson junction .
this system of a spin coupled to a damped harmonic oscillator ( see fig .  [
fig : system ] ) can be mapped to a standard model for dissipative quantum systems , namely the spin - boson model  @xcite .    .
]    here the spectral function governing the dynamics of the spin has a resonance peak .
such structured baths were also discussed in connection with electron transfer processes  @xcite .
we use the flow equation method introduced by wegner  @xcite to analyze the system shown in fig .
[ fig : system ] , consisting of a two - level system coupled to a harmonic oscillator @xmath0 , which is coupled to a bath of harmonic oscillators : @xmath1 with the spectral function @xmath2 . this system can be mapped to a spin - boson model  @xcite @xmath3 where the dynamics of the spin depends only on the spectral function @xmath4 given by @xmath5
using the flow equation technique we approximately diagonalize the hamiltonian @xmath6 [ eq.([eq : spin_boson ] ) ] by means of infinitesimal unitary transformations . the continuous sequence of unitary transformations @xmath7 is labelled by a flow parameter @xmath8 .
applying such a transformation to a given hamiltonian , this hamiltonian becomes a function of @xmath8 : @xmath9 . here
@xmath10 is the initial hamiltonian and @xmath11 is the final diagonal hamiltonian .
usually it is more convenient to work with a differential formulation @xmath12\quad\textrm{with}\quad \eta(l)=\frac{du(l)}{dl}u^{-1}(l).\ ] ] using the flow equation approach one can decouple system and bath by diagonalizing @xmath13  @xcite : @xmath14 here @xmath15 is the renormalized tunneling frequency . for the generator of the flow
we choose the ansatz  @xcite : @xmath16 @xmath17 the flow equations for the effective hamiltonian [ eq .  ( [ eq : effective_h ] ) ]
then take the following form : @xmath18 the unitary flow diagonalizing the hamiltonian generates a flow for @xmath19 which takes the structure @xmath20 where @xmath21 and @xmath22 obey the differential equations @xmath23 one can show that the function @xmath21 decays to zero as @xmath24 .
therefore the observable @xmath25 decays completely into bath operators  @xcite .
we integrated the flow equations numerically in order to calculate the fourier transform , @xmath26 , of the spin - spin correlation function @xmath27 @xmath28 can be used to calculate dephasing and relaxation times for measurements on qubits  @xcite .
[ fig : j+c](a ) shows @xmath29 and fig .
[ fig : j+c](b ) @xmath26 for different values of @xmath0 .
@xmath26 displays a double - peak structure , which can be understood from the term scheme shown in the inset .
the arrows indicate the transitions responsible for the peaks in @xmath30
. additional structure of @xmath30 due to higher order transitions in the term scheme is not seen in fig .
[ fig : j+c ] .
this is due to our ansatz for @xmath19 [ see eq.([eq : sigma_z ] ) ] , which does not include the corresponding higher order terms .
however , we do not expect the additional peaks to have much weight , as the sum rule  @xcite for the total spectral weight is fulfilled with an error of less than @xmath31 for all the plots in fig .
[ fig : j+c](b ) .
we leave the extension of the ansatz for @xmath19 for future work .
we discuss the dynamics of a spin coupled to a damped harmonic oscillator . 
 this system can be mapped to a spin - boson model with a structured bath , i.e. the spectral function of the bath has a resonance peak . 
 we diagonalize the model by means of infinitesimal unitary transformations ( _ flow equations _ ) , thereby decoupling the small quantum system from its environment , and calculate spin - spin correlation functions .    ,    
 ,    flow equations ; quantum dissipative systems , spin - boson , structured bath