Spin-spin correlation functions for a spin-boson model with a structured bath
SilviaKleff∗, StefanKehrein1, and Janvon Delft
Lehrstuhl f¨ur Theoretische Festk¨orperphysik, Ludwig-Maximilians Universit¨at, Theresienstr.37, 80333 M¨unchen, Germany
1Theoretische Physik III – Elektronische Korrelationen und Magnetismus, Universit¨at Augsburg, 86135 Augsburg, Germany (Received August 19, 2002)
KEYWORDS: flow equations, quantum dissipative systems, spin-boson, structured bath, qubits, adiabatic renor- malization
Recently a new strategy for performing measurements on solid state (Josephson) qubits was proposed, in which the qubit is coupled to the measurement device through a single damped harmonic oscillator.1) Due to this cou- pling, the measurement comes arbitrarily close to a von- Neumann measurement. This system was also discussed in connection with electron transfer processes2) and de- coherence control of two-level atoms in lossy cavities.3)
In this short note we discuss the dynamics of the sys- tem shown in Fig. 1, namely a two-level system coupled to an harmonic oscillator, which is coupled to a bath of harmonic oscillators. This system can be mapped to a standard model for dissipative quantum systems, namely thespin-boson model.2) In this case the spectral function governing the dynamics of the spin will have a resonance peak. We diagonalize the model by means of infinitesimal unitary transformations (flow equations),4) thereby decoupling the small quantum system from its environment. We calculate the renormalized tunneling matrix element for different coupling strengths and com- pare our results with an adiabatic renormalization cal- culation. The renormalization of the tunneling matrix element plays an important role for performing quantum measurements on qubits as it tells how close a measure- ment comes to a von-Neumann measurement.1) We also calculate spin-spin correlation functions for several in- structive parameter choices. Spin-spin correlation func- tions can be used to calculate dephasing and relaxation times for measurements on qubits.1)
The system shown in Fig. 1, namely H˜ =−∆0
2 σx+ ΩB†B+g(B†+B)σz+
k
˜ ωk˜b†k˜bk
+ (B†+B)
k
κk(˜b†k+ ˜bk) + (B†+B)2
k
κ2k
˜ ωk, with the spectral functionJ(ω)≡
kκ2kδ(ω−ω˜k) = Γω can be mapped to a spin-boson model2)
H=−∆0 2 σx+1
2σz
k
λk(b†k+bk) +
k
ωkb†kbk, (1) where the dynamics of the spin depends only on the spec-
∗email: kleff@theorie.physik.uni-muenchen.de
Ω
+ −
1 2
0
bath
Fig. 1. A two-level system is coupled to anharmonic oscillator with frequency Ω, which is coupled to a bath of harmonic oscil- lators.
tral functionJ(ω)≡
kλ2kδ(ω−ωk) given by
J(ω) = 2αωΩ4
(Ω2−ω2)2+ (2πΓωΩ)2 withα= 8Γg2 Ω2 . (2) Using the flow equation technique we approximately diagonalize the Hamiltonian H[Eq.(1)] by means of in- finitesimal unitary transformationsU(l):
H(l) =U(l)HU†(l), (3) wherelis the so-called flow parameter. HereH(l= 0) = H is the initial Hamiltonian and H(l = ∞) is the final diagonal Hamiltonian. In a differential formulation
dH(l)
dl = [η(l),H(l)] with η(l) = dU(l)
dl U−1(l). (4) Using the flow equation approach one can decouple sys- tem and bath by diagonalizingH(l= 0):4, 5)
H(l=∞) =−∆∞
2 σx+
k
ωkb†kbk. (5) Here ∆∞ is the renormalized tunneling frequency. For the generator of the flow we choose the Ansatz:5)
η =
k
iσy∆(bk+b†k) +σzωk(bk−b†k) λk
2
∆−ωk
∆ +ωk
+∆ 2
q,k
λkλqI(ωk, ωq, l)(bk+b†k)(bq−b†q), (6)
with I(ωk, ωq, l) = ωq ω2k−ωq2
ωk−∆
ωk+ ∆ +ωq−∆ ωq+ ∆
. The flow equations for the effective Hamiltonian [Eq. (5)]
then take the following form:
∂J(ω, l)
∂l =−2(ω−∆)2J(ω, l) (7)
Quantum Transport and Quantum Coherence (Localisation 2002) J. Phys Soc. Jpn. Vol. (2003) Suppl. A pp. 161–162
© 2003 The Physical Society of Japan
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0 0.02 0.04 0.06 0.08 0.1 α
0 0.2 0.4 0.6 0.8 1
∆∞/∆0
flow equations
adiabatic renormalization
0 0.05 0.1
1 1.2 1.4
Fig. 2. The renormalized tunneling frequency ∆∞as function ofα calculated using the flow equation approach and adiabatic renor- malization(withp= 1) for ∆0= 1.0, Γ = 0.02 and ∆0/Ω = 0.1.
The inset shows the flow equation result for ∆0= 1.0, Γ = 0.06 and ∆0/Ω = 1.1. The maximumα corresponds to a coupling between two-level system and harmonic oscillator ofg≈7.9/0.4.
+ 2∆J(ω, l)
dωJ(ω, l)I(ω, ω, l), d∆
dl =−∆
dωJ(ω, l)ω−∆
ω+ ∆. (8)
The unitary flow diagonalizing the Hamiltonian gener- ates a flow for the spin which takes the structure
σz(l) =h(l)σz+σx
k
χk(l)(bk+b†k), (9) whereh(l) andχk(l) obey the differential equations
dh
dl =−∆
k
λkχkωk−∆
ωk+ ∆, (10)
dχk
dl = ∆hλkωk−∆ ωk+ ∆+
q
χqλkλq∆I(ωk, ωq, l). (11) One can show that the function h(l) decays to zero as l → ∞. Therefore the observable σz decays completely into bath operators.5) We integrated the flow equations numerically in order to calculate the renormalized tun- neling matrix element, ∆∞, for different values ofα. In Fig. 2 we compare our result with anadiabatic renormal- ization6) calculation for the spectral function of Eq.(2):
∆∞
∆0 =
p2∆2∞−ω22 p2∆2∞
− αΩ2
8πΓ√π2Γ2−1ω12
2 (12)
×
p2∆2∞−ω21 p2∆2∞
− αΩ2
8πΓ√π2Γ2−1ω12
1
with
ω1/22 = Ω2
1−(2πΓ)2/2±2πΓ π2Γ2−1
, (13) where p is an unspecified constant. The main plot in Fig. 2 shows ∆∞/∆0 for ∆0 Ω. In this limit both methods yield qualitatively the same dependence. For
∆0 > Ω (see inset of Fig. 2) the flow equation result shows the expected level repulsion, namely ∆∞ > ∆0 forα >0. For conceptual reasons, adiabatic renormal-
ization can not work in this limit. We also calculated
0 100 200
300 10000 × J(ω)/∆0 C(ω)
0 0.2 0.4 0.6 0.8 1 1.2 ω/∆0
0 20 40 60
80 100 × J(ω)/∆0 C(ω)
(a)
(b) ω0,+
ω1,+
ω2,+
ω0,- ω1,- ω2,-
+ -
1 2
∆0 > Ω (c)
1 2 2
Fig. 3. (a) C(ω) for ∆0 = 1.0, Ω = 0.8, Γ = 1.6 an dg2 = 0.005 (corresponds to α = 0.1). (b) C(ω) for ∆0 = 1.0, Ω = 0.8, Γ = 0.01 andg= 0.1 (corresponds toα= 0.00125). (c) Term scheme of a two-level system coupled to anharmonic oscillator for ∆0>Ω.
the Fourier transform,C(ω), of thespin-spin correlation function
C(t)≡ 1
2 σz(t)σz(0) +σz(0)σz(t). (14) Fig. 3 showsC(ω) for two instructive limits, namely (a) Γ≈∆0≈Ω,g Γ: Due to the small coupling be- tween two-level system and harmonic oscillator, we ex- pect the tunneling matrix element not to be altered very much, ∆∞≈∆0, andC(ω) is expected to show a single peak at ∆∞.
(b) Ω≈∆0,g(Ω, ∆0), Γg: In this case we expect a double peak structure, which can be understood from the term scheme shown in Fig. 3(c). A second order per- turbation calculation for the coupled two-level-harmonic oscillator system yields the following two frequencies for the peak position of C(ω), corresponding to transitions 1 and 2 in Fig. 3(c)
ω1,+−ω0,+= Ω−g22∆0/(∆20−Ω2) ω0,−−ω0,+= ∆0+g22∆0/(∆20−Ω2).
This is consistent with Fig. 3(b).
In summary, we investigated the renormalization of the tunneling matrix element and calculated spin-spin correlation functions for the system depicted in Fig. 1, using a flow equation approach.
The authors would like to thank F. Wilhelm for help- ful discussions. S. Kehrein acknowledges support by the SFB 484 of the Deutsche Forschungsgemeinschaft.
1) F.K. Wilhelm, preprint.
2) A. Garget al., J. Chem. Phys.83, 3391 (1985).
3) M. Thorwartet al., J. mod. Optics47, 2905 (2000).
4) F. Wegner, Ann. Phys.3, 77 (1994).
5) S. Kehrein and A. Mielke, Ann. Phys.6, 90 (1997).
6) A.J. Leggettet al., Rev. Mod. Phys.59, 1 (1987).
162 J. Phys. Soc. Jpn. Vol. 72 (2003) Supplement A S. Kleffet al.
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