Quantum Dynamics of a Harmonic Oscillator in a Defomed Bath in the Presence of Lamb Shift

DOI 10.1007/s10773-012-1188-5

Quantum Dynamics of a Harmonic Oscillator in a Defomed Bath in the Presence of Lamb Shift

M. Daeimohamad·M. Mohammadi

Received: 14 February 2012 / Accepted: 18 April 2012

© Springer Science+Business Media, LLC 2012

Abstract In this paper, we investigate the dissipative quantum dynamics of a harmonic os- cillator in the presence a deformed bath by considering the Lamb shift term. The deformed bath is modelled by a collection of deformed quantum harmonic oscillators as a generaliza- tion of Hopfield model. The Langevin equation for both the photon number and the fluctua- tion spectrum under the Weisskopf–Winger approximation are obtained and discussed.

Keywords Lamb shift·Dissipation bath·Langevin equation·Fluctuation spectrum

1 Introduction

The Lamb shift (LS) is one of the most important quantum electrodynamics effects in atom Physics and quantum optics [1]. The energy level shift in hydrogen due to the virtual photon processes, measured first by Willis lamb, stimulated the study of the renormalized quan- tum field theory and confirmed the existence of the quantum vacuum. It was realized early, through the work of Bethe [2], that most of the LS can be explained within nonrelativistic quantum electrodynamics. There are a number of approaches to the calculation of the LS.

One such approach is due to Feynman [3,4] and is beautifully reviewed by Milonni [5]. In this approach, it is argued that the presence of an atom inside a box leads to a change in the resonant frequencies fromωktoωk/n(ωk), where is the refractive index atωk. This lead to a change in the zero-point energy due to the presence of the atom, and the calculated change of the energy corresponds to the LS.

This motivates us to consider a situation where the refractive indexn(ωk)can be con- trolled by an external driving field, and hence we can coherently control the LS. Such a

M. Daeimohamad

Department of Physics, Najafabad Branch, Islamic Azad University, Najafabad, Isfahan, Iran e-mail:Daeimohamad@phys.ui.ac.ir

M. Mohammadi (

⁾

Department of Physics, Shahreza Branch, Islamic Azad University, Shahreza, Isfahan, Iran e-mail:majid471702@yahoo.com

situation can, for example be realized in a coherently driven system such as in electromag- netically induced transparency [6,7]. Coherent atomic effect are a hot area of research in quantum optics and have led to a number of interesting and counterintuitive phenomena, such as correlated emission laser [8,9], lasing without inversion [10–13], and suppression of atomic decay by spontaneous emission [14]. One of the important problems in physics is the investigation of a system coupled to its environment which in its simplest from is the standard paradigm for quantum theory of Brownian motion [15–27]. The success of quantum theory of Brownian motion can be seen in various areas such as quantum optics, transport processes, coherence effects and macroscopic quantum tunneling, electron trans- fer in large molecules, thermal activation processes in chemical reaction, etc. and each one forms a large body of current literature. In the present work we consider a quantum har- monic oscillator, Which can be a one mode of quantized electromagnetic field, in a medium which is described by a bosonic heat bath in the presence of the LS. The layout of the paper is as follows. In Sect.2, we solve the Heisenberg equation of motion for a harmonic oscil- lator under the Weisskopff–Wigner approximation (WWA) in the presence of LS term. In Sect.3, we obtain the fluctuation spectrum in the presence of LS term. In Sect.4, we obtain Langevin equation for photon number by considering the LS term.

2 Langevin Equation in a Deformed Medium with the LS

In this section we solve the Heisenberg equation of motion for a damped harmonic oscillator considering the WWA. These allow us to obtain the Langevin equation in the presence of a deformed medium. A quantum damped harmonic oscillator is described by the Hamiltonian.

Hˆ_T = ˆH0+ ˆH_B+ ˆHint

=ω0aˆ⁺aˆ+

j

ω_j 2

Bˆ_jBˆ_j⁺+ ˆB_j⁺Bˆ_j

+

j

k_jBˆ_jaˆ⁺+k_j^∗Bˆ_j⁺aˆ

. (1)

The first term(Hˆ0)is the Hamiltonian of the harmonic oscillator, the second term(Hˆ_B)is the Hamiltonian of the deformed medium or heat-bath which is considered as a combination of deformed harmonic oscillator described by annihilation(Bˆ_j)and creation(Bˆ_j⁺)bosonic operators which can be considered as a deformed version of Hopfield model, and the third term(Hˆint)is the interaction between the oscillator and its environment. The algebra of the usual operatorsaˆandaˆ⁺is the Weyl–Heisenberg algebra

[ ˆN ,a] = −ˆˆ a, N ,ˆ aˆ⁺

= ˆa⁺, a,ˆ aˆ⁺

=1,

(2)

whereNˆ = ˆa⁺aˆ is the number operator and the Hamiltonian describing the oscillator is defined by

Hˆ=ω0

Nˆ+1 2

. (3)

Here we have omitted the constant^ω₂⁰ form the total Hamiltonian. For anyj, the deformed operatorBˆj,Bˆ_j⁺are defined by their nondeformed partnersbˆj,bˆ_j⁺respectively as follows

Bˆj= ˆbjf (Nˆj)=f (Nˆj+1)bˆj,

Bˆ_j⁺=f (Nˆj)bˆ⁺_j = ˆb⁺_jf (Nˆj+1), (4)

whereNˆj= ˆb⁺_jbˆjandf (Nˆj)is the deformation operator and if for eachjwe setf (Nˆj)≡1, we recover the usual definition of a heat bath. The deformed bosonic of the medium fulfill the following deformed Weyl–Heisenberg algebra

[ ˆNj,Bˆj] = − ˆBj, Nˆj,Bˆ_j⁺

= ˆB_j⁺, Bˆj,Bˆ_j⁺

=(Nˆj+1)f²(Nˆj+1)− ˆNjf²(Nˆj).

(5) In the Heisenberg picture we have

daˆ

dt = −iω0aˆ−i

j

kjBˆj, (6)

and

dBˆj

dt =−iBˆjΩˆj

2 −iΩˆjBˆj

2 −ik^∗_jΩˆjaˆ ωj

, (7)

where

Ωˆj=

(Nˆj+1)f²(Nˆj+1)− ˆNjf²(Nˆj)

ωj. (8)

SinceΩˆj is an operator depending onNˆj nonlinearly, the analytic solution of (6) and (7) will be impossible for an arbitrary deformation function. Therefore we simplify the problem by replying the operatorΩˆjwith its classical value ˆΩjin the presence of the bath. For this purpose let us assume that the bath has a Maxwell–Boltzmann distribution, in this case we have

ˆΩj =Tr ρ_B^TΩˆj

=Tr 1

Ze^−βHˆBΩˆj

= 1 Z

∞ nj=0

nj|e^{−βjωj2 (BˆjBˆj++ ˆBj+Bˆj)}Ωˆj|nj

=ωj

Z ∞ n=0

e^{−βωj2 [(n+1)f2(n+1)+nf2(n)]}

(n+1)f²(n+1)+nf²(n)

, (9)

whereZ=Tr(e^−βHˆB)is the partition function of the deformed bath.

Now (7) becomes

dBˆj

dt = −i ˆΩj ˆBj−ik_j^∗ ˆΩjˆa ωj

, (10)

with the following solution

Bˆj(t )=e^{−i ˆΩjt}−ik_j ^∗ t

0

a t

e^{i ˆΩj(t−t )}dt, (11) where we letk_j ^∗=^{k∗j ˆ}_ω^Ω_j^j. By considering this recent solution, for (6) we find

daˆ

dt = −iω0aˆ−

j

k_j² _t

0

a t

e^{i ˆΩj(t−t )}dt+Ga, (12) where we have defined

Ga= −i

j

kBˆj(0)e^{−i ˆΩjt}. (13)

In order to removing the high-frequency behavior (12) let us define the new operatorAˆas A(t )ˆ = ˆa(t )e^iω0t, (14) and (12) reduce to

dAˆ

dt = −

j

k_j² _t

0

dtA t

exp i

ˆΩj −ω0

t−t

+GA, (15) where

G_A= −i

j

k_jBˆ_j(0)exp

−i

ˆΩ_j −ω0

t

. (16)

Using the WWA in the presence the LS term we have [28]

ˆ

a(t )=u(t )a(0)ˆ +

j

vj(t )Bj(0)=e^−iω0tA(t ),ˆ (17) where we have defined

u(t )=exp− 1

2γ+i(ω0+ω)

t, (18)

v_j(t )=−k_je^{−i ˆΩjt}[1−expi( ˆΩj −ω0−ω)t e^−γ2t]

[ω0− ˆΩj +ω−i^γ₂] , (19)

γ =2πg(ω0)k(ω0)², (20)

ω= −

g( ˆΩj)k( ˆΩj)²d ˆΩj ˆΩj −ω0

. (21)

Therefore,

dAˆ dt = −

γ 2 +iω

Aˆ+GA(t ), (22)

whereωis the LS term. The operatorGAis a random or noise operator, the term−(^γ₂ + iω)Aˆ is responsible for a drift motion in the presence of LS termω. Since the noise operatorG_Ais a linear combination in bosonic operatorBˆ_j so its reservoir average is zero.

GA(t )

=Tr_B

ρ_B^TGA(t )

=0, (23)

where TrBmeans taking trace over the reservoir degree of freedom. Therefore, from (21) we find

d dt

A(t )ˆ

B= − γ

2 +iωA(t )ˆ

B, (24)

with the solution

A(t )ˆ

B=e^{−(γ2+iω)t}a(0).ˆ (25) Note that ˆA(0)B= ˆa(0)B≡ ˆa(0). The time evolution of the harmonic oscillator number operatoraˆ⁺aˆin the presence of LS term is found as

d

dtaˆ⁺aˆ= 1 i

aˆ⁺a,ˆ Hˆ_T

=i

j

k_j ^∗Bˆ_j⁺aˆ−i

j

k_jBˆ_jaˆ⁺, (26) and using (11) it reduces to

d

dtaˆ⁺aˆ= −

j

k_j² t 0

aˆ⁺ t

e^{−i ˆΩj(t−t )}a(t )ˆ + ˆa⁺(t )a t

e^{i ˆΩj(t−t )}

dt+Ga⁺a

= − γ

2 +iω

a⁺a− γ

2 −iω

a⁺a+Ga⁺a

= −γ a⁺a+G_a+a, (27)

where we have used the WWA and defined Ga⁺a=i

j

k_j ^∗Bˆ_j⁺(0)e^{i ˆΩjt}a(t )−ik_jaˆ⁺(t )Bˆj(0)e^{−i ˆΩjt}

. (28)

If we insert the solution (17),Ga⁺abecomes G_a+a=i

j

k ^∗_j e^{i( ˆΩi−ω0)t}e^−(γ2+iω)a(0)ˆ

+

j,k

k_j ^∗k_kBˆ_j⁺(0)e^{i( ˆΩj−ω0)t}Bˆk(0)[e^{−i( ˆΩk−ω0)t}−e^{−(γ2+iω)t}]

γ

2−i( ˆΩk −ω0−ω) . (29) Using the equation

Tr_B

ρ_B^TBˆ_j⁺(0)

=Tr_B

ρ_B^TBˆj(0)

=Bˆ_j⁺(0)

B=Bˆj(0)

B=0, TrB

Bˆ_j⁺(0)ρ_B^TBˆk(0)

=Bˆ_j⁺(0)Bˆk(0)

B=δj kn¯j, (30)

where

ˆ nj=ωj

Z ∞ n=0

e^{−βωj2 [(n+1)f2(n+1)+nf2(n)]}nf²(n), (31) Z=

∞ n=0

e^{−βωj2 [(n+1)f2(n+1)+nf2(n)]}. (32) Now we find from (29)

Ga⁺a(t )

B=

j

k_j²

1−e^{[i( ˆΩj−ω0−ω)−γ2]t}

γ

2−i[ ˆΩj −ω0−ω]+c.c

=

j

|k_j|²n¯j{γ−γ e^−(γ2)tCos( ˆΩj −ω0−ω)t}

[(^γ₂)²+(Ωj −ω0−ω)²]

+

j

|kj|²n¯j{2( ˆΩj −ω0−ω)t e^(−γ2)tSin( ˆΩj −ω0−ω)t}

[(^γ₂)²+(Ωj −ω0−ω)²] . (33) Since|k_j|²n¯jis slowly varying and the summand is so strongly peak ˆΩj =ω0+ω=ω₀, we may convert the sum to an integral and remove the slowly varying factors. This gives

Ga⁺a =k ω₀²g

ω₀

¯ n

ω₀

∞

−∞dx{γ−γ e^−γ2tCos(xt )+2xe^−γ2tSin(xt )}

(^γ₂)²+x² , (34) wherex= ˆΩj −ω0−ωanddx=d ˆΩjand we assumed that the reservoir modes are closely spaced withg(ωj)dωj the number of modes betweenωj andωj+dωj. Using the following definite integral

_∞

−∞

dx

(^γ₂)²+x² =2π γ , _∞

−∞

Cos(xt ) (^γ₂)²+x² =2π

γ e^−(γ2)|t|, _∞

−∞

xSin(x|t|) (^γ₂)²+x² =

π e^−(γ2)|t| t=0,

0 t=0.

(35)

Equation (34) reduces to

G_a+aB=γn,¯ (36) where we used the relation (20). Now by averaging both sides of (27) and using (36) we obtain

d dt

aˆ⁺a

B= −γ ˆ a⁺a

B+γn,¯ (37)

with the solution

aˆ⁺(t )a(t )ˆ

=e^{−γ t}aˆ⁺(0)a(0)ˆ + ¯n

1−e^{−γ t}

. (38)

Now let us rewrite (27) and include the reservoir average ofG_a+a, sinceAˆ⁺Aˆ= ˆa⁺aˆ we have

d

dtAˆ⁺Aˆ= −γAˆ⁺Aˆ+γn¯+G_A+A, (39) where

GA⁺A=Ga⁺a− Ga⁺aB=Ga⁺a−γn.¯ (40) Note thatGA⁺A(t )B=0, so the (39) lead to same (37).

The Langevin has zero thermal average and the remaining terms (39) give a thermally average drift. Therefore, we see that the result is formally identical with the dissipative quantum dynamics of a harmonic oscillator in a deformed bath in the presence of LS term.

3 Spectra in the Presence of LS

Once we have the approximate solution of the Heisenberg equations, we may calculate the various spectra. The fluctuation spectrum is given by

_∞

−∞e^−iωt ˆ

a⁺(t )a(0)ˆ dt=

_∞

−∞e^{−i(ω−ω0)t}Aˆ⁺(t )A(0)ˆ

, (41)

where we used (14) and its adjoints. This spectrum is just the Fourier transform of correlation function

kA⁺A(t )≡Aˆ⁺(t )A(0)ˆ

=TrB,SAˆ⁺(t )ˆa(0)ρ(0), (42) where the initial density operator is

ˆ

ρ(0)= ˆρS(0)⊗ρ_B^T =ρˆS(0)⊗e^−βHB

Tr_B(e^−βHB) , (43)

whereρˆs(0)is the initial density matrix of the oscillator andρˆ_B^T is the initial density matrix of the deformed reservoir. We assume ρˆ_B^T has a Maxwell–Boltzmann distribution. If use adjoint of (11) we have for the two-time correlation function in the presence of LS

KA⁺A(t )=Tr_B,SρˆS(0)ρˆ_B^T e^{−(γ2−iω)t}aˆ⁺(0)+ t

0

e^{(γ2−iω)(t−t )}G⁺_A t

dt

ˆ a(0)

=Tr_BρˆS(0)aˆ⁺(0)a(0)eˆ ^{−(γ2−iω)t} +TrS

ρS(0)

_t

0

e^{(γ2−iω)(t−t )} G⁺_A

t

Bdta(0)ˆ

, (44)

or

K_A+A(t )=e^{−(γ2−iω)|t|} ˆ

a⁺(0)a(0)ˆ

, (45)

where we used the adjoint (23) and have let aˆ⁺(0)a(0)ˆ

=TrSρS(0)aˆ⁺(0)a(0).ˆ (46) We have used the absolute value oftsince for a stationary process [29]

K(t )=K(−t ).

From (41), the fluctuation spectrum in the presence of LS reduces to γˆa⁺(0)a(0)ˆ

(ω−ω0−ω)²+(^γ₂)², (47) which is Lorentzian centered atω=ω0+ωwith half-width ^γ₂. If we assume att=0 the cavity is in thermal equilibrium with the reservoir, we have

¯ n=

ˆ

a⁺(0)ˆa(0)

= 1

e

ω0 KB T −1

. (48)

Therefore, the fluctuation spectrum in the presence of LS term is given by

¯ nγ

[ω−ω0−ω]²+(^γ₂)². (49) We therefore see that the only effect ofωis to change slightly the cavity resonant fre- quencyω0.

4 Langevin Equation for Photon Number in the Presence of LS

We have already obtained the Langevin equation of motion for the photon number (39).

For obtaining more insight into the nature of the Markov approximation and formulating the Langevin method for more general systems, let us obtain the Langevin equation for the photon number by another method.

If we use the Langevin equation (22) and its adjoint, we obtain d

dtA⁺A= − γ

2 −iω

A⁺A− γ

2 −iω

A⁺A+A⁺GA+AG⁺_A

= −γ A⁺A+A⁺GA+AG⁺_A. (50) We need reservoir average to give us the drift motion

d dt

A⁺(t )A(t )

B= −γ

A⁺(t )A(t )

B+

A⁺(t )GA(t )

B+

GA⁺(t )A(t )

B. (51)

We have already evaluated[A⁺(t )GA(t )+GA⁺(t )A(t )]B=γn. We used the solution for¯ A(t )andA⁺(t ). The method we now use relies more directly on the Markov approximation

and dose not require knowledge of the solution forA(t )andA⁺(t ). Consequently, it is more general.

We begin by writing the identity Ga⁺a(t )

=

A⁺(t )GA(t )+GA⁺(t )A(t )

B

≡ A⁺(t0)+ _t

t₀

dA⁺ ds ds

GA(t )

B

+

G_A+(t ) A(t0)+ _t

t₀

dsdA ds

B

, (52) wheret > t0andγ⁻¹t−tcτc. Clearly,

A⁺(t0)GA(t )

B=0, GA⁺(t )A(t0)

B=0. (53)

There timeτcis called reservoir correlation time. This result must be true under the Markov approximation for the system operator and reservoir Langevin force were correlated over this time interval the system would develop memory. Since we happen to know the solution forA⁺(t0), let us verify that (53) is indeed true before proceeding. We have by (16) and the adjoint of (11) that fort > t_c>0

A⁺(t0)GA(t )

B

= −i

e^{−(γ2−iω)t}a(0)ˆ +i

j

k_j ^∗b_j⁺(0)[e^{i( ˆΩj−ω0)t0}−e^{−(γ2−iω)t0}] [^γ₂+i( ˆΩj −ω0−ω)]

×

k

k_k ^∗bˆ_k(0)e^{i( ˆΩj−ω0)t}

=

j

k_j²n¯_j [e^{i( ˆΩj−ω0−ω)t0}−e^−γ2t0]

[^γ₂+i( ˆΩj −ω0−ω)]e^{−i( ˆΩj−ω0−ω)t}. (54) The integrand is now highly peak ˆΩj =ω0+ωso that we may without serious error let the lower limit extent to−∞and remove the slowly varying termsg|k|²n¯from the integral.

If we use (20) and (30), we obtain A⁺(t0)GA(t )

B=γn¯ 2π

_∞

−∞

[e^{−ix(t−t0)}−e^−(γ2)t0e^−ixt]

γ

2+ix dx, (55)

where we letx= ˆΩj −ω0−ω. Then since forα >0 _∞

−∞

e^−iαx

γ

2+ixdx=0, (56)

(53) follows directly. A similar argument verifies the second relation of (53). Again (53) is a direct consequence of the Markov approximation and is not peculiar to the damped oscillator. Accordingly, (52) reduce to

Ga⁺a(t )

B= _t

t₀

dA⁺ ds GA(t )

B

+

G⁺_A(t )dA ds

B

ds. (57)

If we next use (15) and (16), we obtainG_a+a(t )Bin the presence of LS G_a+a(t )

B= _t

t₀

ds −

γ 2 −iω

A⁺(s)+G⁺_A(s)

+G⁺_A(t ) − γ

2 +iω

A(s)+GA(s)

. (58)

By the Markov approximation again sincet > s, we have let A⁺(s)GA(t )

B=0, GA⁺(t )A(s)

B=0. (59)

Consider next the cross-correlation function defined by KA⁺A(t1−t2)=

GA⁺(t1)GA(t2)

B

=

j,k

k_j ^∗k_k ^∗Bˆ_j⁺(0)Bˆk(0)

Be^{i[( ˆΩj−ω0)t1−( ˆΩk−ω0)t2]}

=

j

k_j²n¯je^{i( ˆΩj−ω0)(t1−t2)}

= _∞

0

g

ˆΩjk

ˆΩj²n¯ ˆΩj

e^{i( ˆΩj−ω0)(t1−t2)}d ˆΩj. (60) The integrand is now highly peaked at ˆΩj =ω0so that we may without serious error let the lower limit extend−∞and removeg|k|²n¯from the integral. If we use (20), we have

K_A+A(t1−t2)=

G_A+(t1)GA(t2)

B

=gk²n¯ _∞

−∞e^{i( ˆΩj−ω0)(t1−t2)}=γnδ(t¯ 1−t2). (61) Therefore, we see that over the intervalt−t0

GA⁺(s)GA(t )

B=

GA⁺(t )GA(s)

B=γnδ(t¯ −s). (62)

Thus (57) reduces to

G_a+a(t )

B= _t

t₀

ds2γnδ(t¯ −s)=γn.¯ (63) Therefore, (51) becomes

d dt

A⁺A

B= −γ A⁺A+γn¯+GA⁺A. (64) We may take as our Langevin equation

d

dtA⁺A= −γ A⁺A+γn¯+GA⁺A, (65) where the Langevin force is

Ga⁺a(t )=A⁺(t0)Gp(t )+G⁺_A(t )A(t ), (66) and has the property that

GA⁺A(t )

B=0. (67)

Again the drift motion is explicitly in clouded and a random force is added to retain the correct quantum fluctuation. The first in (64) is the rate of loss of photon into the reservoir while the second gives the rate at which photons enter from the reservoir. The force cause fluctuation from the mean photon number.

5 Summary and Conclusions

In summary, we studied the dissipative quantum dynamics of a harmonic oscillator in the presence LS term under the WWA. By solving the Heisenberg equation of motion, we found the Langevin equation for photon number. By using the correlation function we obtain fluctuation spectrum in the presence of LS. Then by considering the LS, we obtained the Langevin equation for photon number under the Markov approximation.

Acknowledgements The authors wish to thank the Office of Research of Islamic Azad University, Na- jafabad Branch, for their support.

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