# Damped Quantum Harmonic Oscillator

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```					Quantum Matter                                                                   Winter Semester 2009/10
u
LMU M¨nchen                                                                               due: 18.12.2009
Prof. Jan von Delft, Dr. Andreas Weichselbaum                                          Problem Set Nr. 6

Damped Quantum Harmonic Oscillator

Consider the Caldeira-Leggett model to describe the eﬀect of a quantum bath on a quantum
harmonic oscillator,

ˆ  p2
ˆ  1                                 p2
ˆi  1                ˆ
ci x 2
H=    + mΩ2 x2 +
ˆ                                     2
+ mi ωi (ˆi −
q          2
)      ,            (1)
2m 2                     i
2mi 2             mi ωi

and choose the spectral function

c2
i
J(ω) = π                       δ(ω − ωi )                             (2)
i
2mi ωi

to be ohmic, J(ω) = mγω.
ˆ        ˆ                     ˆ
By setting up the Heisenberg equations of motion for x(t) and qi (t), and solving for qi (t)
ˆ
as function of x(t), it can be shown (see lecture notes) that the operator x(t) satisﬁes a
Langevin equation of the following form:
∞
¨
mx + mΩ2 x + m
ˆ       ˆ                                   ˙       ˆ
dt γ(t − t )x(t ) = ξ(t) .
ˆ                                 (3)
0

ˆ
The damping kernel γ(t) and “stochastic force” ξ(t) are expressed in terms of the bath
operators as

1         c2
i
γ(t) =                  2
cos(ωi t) ,                                          (4)
m    i
mi ωi

ˆ                                            ˆ
pi (0)
ξ(t) =               ˆ
ci δ qi (0) cos(ωi t) +             sin(ωi t) ,                (5)
i
mi ωi

where δ qi (0) = qi (0) − mci 2 x(0) are the deviations of the bath oscillators from their equilib-
ˆ        ˆ         i ωi
ˆ
rium positions at time t = 0 .

1. Ohmic damping
Show that the last term on the left-hand side of the Langevin equation (3) can be simpliﬁed
to the form of velocity-proportional damping, mγ x.˙
ˆ
2                                                                                   Problem Set Nr. 6

2. Noise correlator
Show that the correlator of the stochastic force has the form

ˆ ˆ                     dω
ξ(t)ξ(0)   T   =           J(ω) nB (ω)eiωt + [nB (ω) + 1]e−iωt ,                    (6)
π

where nB (ωi ) = ˆ†ˆi T = [eβωi − 1]−1 is the Bose distribution function at temperature
bi b
kB T = 1/β. [Hint: see the lecture notes from 30.11.09 on the noise spectral function of a
quantum harmonic oscillator.]

3. Noise spectral function
Show that the corresponding noise spectral function (the Fourier transform of Eq.(6)) has
the form:
Sξξ [ω] = 2 ωγ θ(−ω)[−ωnB (−ω)] + θ(ω)[ω(nB (ω) + 1)]                 (7)
Show that in the classical limit kB T                ω, this reduces to classical white noise, with
Sξξ [ω] = 2mγkB T .

4. Spectral function of damped harmonic oscillator
The spectral function of a damped harmonic oscillator subject to a Langevin force can be
expressed as (see lecture notes from 11.12.09)

Sxx [ω] = |A(ω)|2 Sξξ [ω] ,                                     (8)

where for Ω    γ the function A(ω) can be approximated as the sum of two Lorenzians,

π                                                                   γ/(2π)
|A(ω)|2 =               δγ (ω + Ω) + δγ (ω − Ω) ,            where δγ (ω) =                 .    (9)
2m2 Ω2 γ                                                          ω2   + (γ/2)2

Show that the spectral function for the damped oscillator consists of two peaks of diﬀerent
heights,
Sxx [ω] 2πx2 ZPF δγ (ω + Ω)nB (Ω) + δγ (ω − Ω)[nB (Ω) + 1] .            (10)

[Here x2 = /(2mΩ).]
ZPF

5. Amplitude of thermal ﬂuctuations
Show (by integrating Sxx [ω]) that the amplitude of thermal ﬂuctuations of the damped
quantum harmonic oscillator is:

Ω
x2   T   =         coth           .                             (11)
2mΩ          2kB T

Does this reduce to expected value in the classical limit?

```
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