Cent. Eur. J. Phys. • 10(4) • 2012 • 983-988DOI: 10.2478/s11534-012-0064-5

Central European Journal of Physics

Rabi type oscillations in damped two-dimensionalsingle electron quantum dots

Research Article

Madhuri Mukhopadhyay1, Ram Kuntal Hazra2, Manas Ghosh3, Samaresh Mukherjee4, Shankar P.Bhattacharyya4∗

1 Kemicentrum, Lunds Universitet, Getingevägen 60, Lund S-22241, Sweden

2 Department of Chemistry, North Campus, University of Delhi, Delhi 110007, India

3 Department of Chemistry, Physical Chemistry Section, Visva Bharati University, Santiniketan, Birbhum 731 235, West Bengal,India

4 Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India

Received 16 March 2012; accepted 16 April 2012

Abstract: We present a quantized model of a harmonically confined dot atom with inherent damping in the presenceof a transverse magnetic field. The model leads to a non-Hermitian Hamiltonian in coordinate space. Wehave analytically studied the effects of damping on Rabi type oscillations of the system. The model explainsthe decoherence of Rabi oscillations in a Josephson Junction.

PACS (2008): 78.67.-n, 78.67.Hc, 78.67.De

Keywords: damped quantum dot • quantization of damping • Rabi oscillation© Versita Sp. z o.o.

1. Introduction

The phenomenon of Rabi oscillations [1] is one of the fun-damental observations in light matter interaction that oc-curs coherently and nonlinearly [2] and which has no clas-sical analogue. The generation of coherent superpositionof quantum states using ultra short laser pulses and thesubsequent decoherence due to some inherent damping orinteraction with the environment is of great interest espe-cially in semiconductor quantum dots due to the prospectof future applications [3–5] in quantum information pro-∗E-mail: pcspb@iacs.res.in

cessing and making novel laser devices [6]. The miniatur-ization of semiconductor devices reaches the bottom of theavenue with the advent of so called low dimensional struc-tures such as quantum dots (QD). Rabi oscillations usingexcitons in single quantum dots [7–10] have been stud-ied successfully by different groups in the past few years[11–14]. Control of the decoherence of Rabi oscillationsin quantum dots, the mechanism of which is still a mat-ter of investigation, has attracted wide attention [15, 16].There has been rapid progress in experimental control ofdephasing of coherent states in quantum dots [17]. On thecontrary, theoretical studies on quantum dots leading tothe dynamics in the presence of damping are scarce. Todate theoretical studies on quantum dots have been donemainly on the basis of damping that has been introduced983

Rabi type oscillations in damped two-dimensional single electron quantum dots

phenomenologically. To the best of our knowledge, noquantum theoretical model of the dot has been developedwith the inherent damping incorporated in the model.We develop a model of a damped quantum dot which goesbeyond the phenomenological description used so far. Themodel, we believe provides some insight into the Rabi dy-namics. Our analytical results lead to an understanding ofthe experimental observation of decoherence in Josepshonjunctions [18].2. ModelWe show, in what follows, that the damped one electrondot can be described by the eigenstates of a quantumHamiltonian (H) that is non Hermitian. The artificialatom that we have modeled is composed of a single elec-tron confined in two dimension by harmonic potential withsome inherent damping and a homogeneous magnetic fieldapplied normal to the confinement plane. Let us start withthe classical equation of motion of the damped harmonicoscillator which reads

me−→r + γ−→r + k−→r = 0, (1)

where k is the harmonic force constant and γ is the damp-ing constant and me is the oscillator mass. The systemdescribed by Eq. (1) is known to have a time-dependentLagrangian and Hamiltonian [19–21]. There have beenmany attempts to quantize the damped linear oscillator[22–24] but a completely satisfactory solution has beenelusive. The stumbling block has been the lack of atime-independent Hamiltonian formalism. Recently, how-ever such a formalism has been proposed making definiteprogress [25, 26]. We proposed a different strategy thatbrings in a non-Hermitian Hamiltonian formalism. FromEq. (1) we start by noting that it is immediately possibleto write down the Euler-Lagrange equation for the dissi-pative system by defining a velocity dependent force Fgenand settingddt

(∂L∂r

)− ∂L∂r = Fgen, (2)

where Fgen is defined as the negative derivative ofRayleigh dissipative function (f ) with respect to r [19].

Fgen = − ∂∂r (f ), (3)

f is determined by the damping constant γ and the velocity(r) as follows:f = 12γr2, (4)

Equations (3) and (4) suggest that the time-dependentdamping force (Fd) is linearly related to the velocity:Fd = −γ−→r , (5)

With Eq. (3) the Euler-Lagrange Eq. (2) now readsddt

(∂L∂r

)− ∂L∂r + ∂f

∂r = 0, (6)Eq. (6) requires that the Lagrangian L is chosen as

L = 12mer2 + γrr − 12kr2, (7)Clearly the Lagrangian of Eq. (7) is consistent with theequation of motion of the damped harmonic oscillator Eq.(1). The momentum p = ∂L

∂r the modified momentum forthe damped harmonic oscillator becomesp = me

−→r + γ−→r , (8)Let the damped oscillator have a charge ’q’ and let itexperience an electric field (E) and a transverse magneticfield (B). The Lorentz force acting on it isF = q

[E + 1

c (v × B)]=q [−∇φ − 1c

(∂A∂t

)+ 1c (v × B)] ,(9)where E = −∇φ (φ =scalar potential) and B = ∇× A(A= Vector potential). The electric and magnetic fieldsbring in additional terms in the Lagrangian (L = L, say)where

L = 12mer2 + γrr − qφ + qc−→A −→r , (10)

where qφ = 12kr2, scalar potential. The modified momen-tum (p) for the system (described by L) isp = me

−→r + γ−→r + qc−→A . (11)

The modified momentum p leads to the Hamiltonian (H)of the system represented by a single carrier electron ina damped quantum dot as follows:H= 12me

[(me−→r +γ−→r + q

c−→A )·(me

−→r + γ−→r + qc−→A )]+qφ.(12)Taking the cyclotron frequency ωc = qB

mec , the confinementpotential qφ = 12meω20(x2 + y2) and replacing the classi-cal operators by their respective quantum analogues, the984

Madhuri Mukhopadhyay et al.

quantum mechanical Hamiltonian of the system in Carte-sian coordinates becomesH = − ~22me

(∂2∂x2 + ∂2

∂y2)− i~γme

(1 + x ∂∂x + y ∂∂y

)− i~ωc2

(−y ∂

∂x + x ∂∂y

)+ γ22me(x2 + y2)

+ me8 ω2c (x2 + y2) + 12meω20(x2 + y2). (13)

Transforming from Cartesian to polar coordinates theHamiltonian changes toH = − ~22me

(∂2∂r2 + 1

r∂∂r + 1

r2 ∂2∂φ2

)− i~γ

me

(1 + r ∂∂r

)− i~ωc2

(∂∂φ

)+ Ω2dr2,(14)

where Ω2d = 12me

[ω2c4 + γ2

m2e + ω20]. H is manifestly non-Hermitian. H may be thought of as defining a set of eigen-states ψn,l(r, φ) with complex energy En,l, if we assumethat H obeys the energy eigenvalue equation

Hψn,l(r, φ) = En,lψn,l(r, φ). (15)A straight forward series solution of Eq. (15) (Appendix A)leads to the quantized energy eigenvalues of the dampeddot:En,l = ωcl2 + (2n+ |l|+ 1)Ω− iγ(2n+ |l|+ 1)Ω, (16)

where n and l are principal, and angular momentum quan-tum numbers, respectively and Ω2 = [ω2c4 + γ2

m2e + ω20]. Theenergy is clearly complex and the imaginary part of it isrelated to the dissipating energy which is given by

Γn,l = −γ(2n+ |l|+ 1)Ω (17)Thus, starting from the classical equation of motion of thedamped harmonic oscillator quantization has been carriedout through a Lagrange-Hamiltonian formalism, where theHamiltonian is non-Hermitian [27, 28] as expected for anon conservative system [29, 30]. We have described thesystem in terms of real positional coordinates in contrastwith attempts to handle the problem in terms of a complexcoordinate [27].Thus, proceeding with the assumption that the system de-scribed by the non-Hermitian Hamiltonian of Eq. (13) sat-isfies the time-independent Schrodinger equation Hψ =

Eψ [27], we have obtained all the quasi energy eigen-states.ψn,l(r, φ) = C2√π eiφe− Ω2r22 rlL|l|n , (18)

where L|l|n is the Laguerre series and C is the normal-ization constant. In the absence of damping these statesmerge into Fock-Darwin energy spectrum [31, 32], whilethe presence of damping makes the energy levels quasistationary. The important outcome is that for a knownω0 and ωc comparison of the energy separation betweentwo states as observed from experiment and obtained fromthe expressions with and without damping can lead to therealization of the intrinsic damping coefficient of a dot sys-tem. For damped dot system the energy states are shiftedfrom the energy levels without damping and the shifts aremore pronounced for stronger damping whereas for greatereffective mass of the carrier electron the effect of dampingis somewhat quenched. Since the non-Hermitian Hamilto-nian obtained for the damped dot has complex eigenvaluesthat correlate with the energy eigenvalues of the dot inthe limit of zero damping, it could be interesting to in-vestigate the dynamics of the damped dot in response toperturbation by laser light.3. Dynamics of damped quantumdotLet us consider the time-dependent Schrodinger equationfor the complex energy eigen states of H;

i~∂Ψn,l(r, t)∂t = (ER − iΓ)n,lΨn,l(r, t),

The corresponding wave function is decaying and theprobability P(r,t) is proportional to |ψn,l(r, 0)|2e −2Γt~ . Theexponential function accounts for the exponential fall-off ofthe amplitude with time, the first factor being the the am-plitude of the initial state which is now damped. The in-trinsic life time τn,l = 1Γ n,l of these quasi-stationary statesare therefore determined by the damping coefficient andthe quantum number characterizing the states.We now consider the two energy levels (g and e) of thedamped quantum dot system, the two states, designatedas ψg and ψe are assumed to be well separated from allother states. The system interacts with a laser of fre-quency ωL and ωa ≡ ωeg is the resonance frequency(Fig. 1). The effect of perturbation produced by the lasercan be treated semiclassically using the eigenfunctions ofthe damped dot Ψn,l as the zeroth order wave function.

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Rabi type oscillations in damped two-dimensional single electron quantum dots

Figure 1. Energy level diagram for a two-level system showing de-cay rates for the ground and excited states (Γg and Γe),respectively. The difference in frequency between the ex-cited and ground state is denoted by ωeg. The Rabi os-cillation ΩR is introduced by an external laser frequencyωL.

The perturbed Hamiltonian is partitioned into H0 and V ,where the unperturbed dot Hamiltonian H0 is given by theEq. (13) and the perturbation in the dipole approximation:−→V = e.rE0 cos(ωLt).We may now consider the semiclassical perturbation treat-ment based on the damped wave functions of the dampeddot already obtained. The time-dependent Schrodingerequation for the perturbed system is

|Ψ(r, t)〉 = − i~H(r, t)|Ψ(r, t)〉 (19)

while the solution is (k = e, g)Ψ(r, t) =∑

k

Ck (t)ψk (r)e−iωRk te−γωIk t . (20)Projecting on to the states |e〉 and |g〉 and integrating overspatial coordinates in each case we arrive at the equationsgoverning the time development of the amplitudes (Cg, andCe);iCg = Cg(t)(ωRg − iγωIg) + Ce(t)ΩR

−→V (t)e−γωIate−iωRa t (21)iCe = Ce(t)(ωRe − iγωIe) + Cg(t)ΩR

−→V (t)eγωIate−iωRa t , (22)where the Rabi frequency is defined as ΩR = eE0

~ 〈e|r|g〉and the dipole approximation has been used. Introduc-ing the transformations Cg = [Cge(iωRg +γωIg)t ] and Ce =[Cee(iωRe +γωIe)t ] we obtainiCg = CeΩR

−→E (t)e(−iωRa−γωIa)t (23)iCe = CgΩR

−→E (t)e(iωRa +γωIa)t , (24)where −→E (t) = eiωLt+e−iωLt2 , ωL+ωa = ω+, and ωL−ωa = δ ,the detuning frequency.

Invoking the rotating wave approximation we obtaini ˙Cg = ΩR2 Cee−γω

Iateiδt (25)

i ˙Ce = ΩR2 CgeγωIate−iδt . (26)

Eqs. (26) and (27) can be uncoupled by the standardroute, leading to¨Cg − (γωIa − iδ) ˙Cg + Ω2

R4 Cg = 0 (27)¨Ce + (γωIa − iδ) ˙Ce + Ω2

R4 Ce = 0. (28)Taking the initial condition that Cg(0) = 1, Ce(0) = 0 weobtain the solutions

Cg = e− ∆2 t[cos ΩRdt2 + i ∆ΩRd

sin ΩRdt2] (29)

Ce = e ∆2 t[i ΩRΩRd

sin ΩRdt2]. (30)

where ∆ = γωIa − iδ and ΩRd =√Ω2R − (γωIa − iδ)2.Hence,

Ce = e−γ( ωIg+ωIe2 )te−i δ2 te−iωRe t[i ΩRΩRd

sin ΩRdt2]. (31)

Hence, the excited state population Pe = |Ce|2 is givenby|Ce|2 = e−γ(ωIg+ωIe)t

[ ΩR2ΩRd2 sin2 ΩRdt2

]. (32)

The result shows that the contribution of a given state tothe evolving wave function (Ψ) of the system at a par-ticular time is given in terms of the damping coefficientand the sum of the energies of the two levels coupled bylaser light. The coherent temporal oscillations of the pop-ulation in the excited state obtained above matches withthe experimental observations made by Yu et al [18] in aJosephson phase qubit. The observed oscillatory behaviorof the decaying amplitude reported by them is success-fully explained by our model based on the descriptionof a damped quantum dot by a non-Hermitian Hamilto-nian in real Coordinates. The probability of being in thestate ’k ’ (e or g) at any given time is therefore given byPk (t) = |Ck (t)|2. For the excited state e, Fig. 2 showsthe nature of the time-dependence of Pe. As expected itis coherently oscillatory and exponentially damped. Wenote that the Eq. (32) was earlier developed by Yu et al.[18] as the asymptotic limit of solution of the appropriateLiouville equation for the density operator in the rotatingwave approximation, and used to interpret their experi-mental observation. We have arrived at the same resultsbased on our non-Hermitian Hamiltonian.

986

Madhuri Mukhopadhyay et al.

Fig. 2

Figure 2. Plot of total population of the excited states versus realtime (in a.u) with me =1 a.u., ωc = 10−3 a.u., ω0 = 0.0141a.u., ΩR = 0.01 a.u., (ωg+ωe) = 0.04 a.u., and γ = 0.0001a.u.

4. ConclusionsIn summary, the proposed model describes correctly theeffects of inherent damping in a quantum dot. The amountof dissipating energy in a particular state in a quantum dotis naturally related to the damping coefficient. The deco-herence of Rabi oscillations shows that the rate of deco-herence is exponentially related not only to the dampingcoefficient but also to the energy separation between thetwo levels. Again one interesting point is that the in-herent life times of all the different states is predictableassuming that the life time of any one particular state areknown from experiment. We also note that the temporalcoherent oscillation of population in Josephson junctions iscorrectly explained by the present model. We expect thatthe results obtained could have important implications intechnological applications of quantum dot nanomaterials.AppendixIn atomic units the Hamiltonian of Eq. (14) reads

H = − 12me

(∂2∂r2 + 1

r∂∂r + 1

r2 ∂2∂φ2

)− iγ

me

(1 + r ∂∂r

)− iωc2

(∂∂φ

)+ Ω2dr2, (33)

where Ω2d = 12me

[ω2c4 + γ2

m2e + ω20]. Substituting f (r, φ) =

e(ilφ)Ψ(r)√2π and multiplying both sides by √2πe−ilφr 12 leadsto radial Schrodinger equation,− 12me

[d2dr2 + 14r2 − l2

r2 + Ω2dr2] f (r)

+ lwc2 − iγme

f (r) + rf ′(r)− 12 f (r)

= Ef (r) (34)

or [d2dr2 + ( 14 − l2) 1

r2 −Ω′2d r2 −meωcl

+2iγ( 12 + r ∂∂r ) + 2Eme

]f (r) = 0, (35)

where Ω′2d = m2e(ω2

c4 + γ2m2e +ω20). Substituting r = x√Ω′d theradial function f (r) changes to g(x).[

d2dx2 + ( 14 − l2) 1

x2 −me(ωc l−2E)Ω′d+2i γΩ′d ( 12 + x d

dx)+ 2Eme

]g(x) = 0. (36)

Asymptotic analysis leads to g(x) = g0(x)V (x)g∞(x)with g0(x) = e−x2/2, g∞(x) = x 12 +|l|, and g(x) =e−x2/2V (x)x 12 +|l|. V (x) = ∑

j bjx j satisfies Laguerre se-ries. Again taking z = x2 the function V (x) changes tofunction q(z), such that q(z) = ∑k akzk satisfies the La-guerre series.

[z d

2dz2 + (l+ 1− z) ddz −

l+ 12 +me

(ωcl− 2E)4Ω′d

+2i γ4(1 + +2z ddz + l− z

)]q(z) = 0,(37)

where E is complex (= ER − iΓ). The Lageurre seriessatisfies Eq. (36). Accordingly the total wave functionreadsψn,l(r, φ) = C2√π eiφe− Ω2

dr22 rl

n∑0 bnr2n. (38)

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