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Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Laroze, D.; Rivera, R. On the transition probability for a quantum dot in a time dependent magnetic field Revista Mexicana de Física, vol. 53, núm. 7, diciembre, 2007, pp. 112-116 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57036163025 How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative REVISTA MEXICANA DE FÍSICA S 53 (7) 112–116 DICIEMBRE 2007 On the transition probability for a quantum dot in a time dependent magnetic field D. Laroze∗ and R. Rivera Instituto de Fı́sica, Pontificia Universidad Católica de Valparaı́so, Casilla 4059, Valparaı́so, Chile Recibido el 30 de noviembre de 2006; aceptado el 8 de octubre de 2007 In this work we analyze in detail the dynamical behavior of a quantum dot in presence of a uniform time-dependent magnetic field, in the effective mass approximation. An exact solution for the time-evolved wave function is obtained when the initial state is a particular Fock– Darwin state. Moreover, we calculate the transition probability from an initial Fock-Darwin state to a given final Fock-Darwin state for the case of a magnetic field that changes linearly in time, flipping its direction. Keywords: Quantum Dot; Exact Solution; Time dependent Problems. En este trabajo analizamos en detalle, en la aproximación de masa efectiva, la dinámica de un electrón en un punto cuántico en presencia de un campo magnético uniforme dependiente del tiempo. Se encuentra una solución exacta para la evolución temporal de la función de onda cuando el estado inicial es un estado de Fock-Darwin. Se calcula también la probabilidad de transición entre estados de Fock-Darwin especı́ficos cuando el campo magnético invierte su dirección cambiando linealmente en el tiempo. Descriptores: Cuánticos; soluciones exactas; problemas dependientes del tiempo. PACS: 73.63.Kv, 73.21.La, 73.21.–b 1. Introduction Quantum dots (QD) are zero dimensional objects constructed by patterning and epitaxial growth techniques in semiconductor heterostructures [1]. An important characteristic of these systems is that the phase coherent length of the electron wave functions exceeds the size of the dots, and consequently, their motion is considered ballistic. There exists a vast amount of literature discussing diverse physical aspects of these systems, like spectroscopy, nonlinear optics, magnetotransport, many-body effects, etc [2]. In many theoretical studies of the QD, the electrons are considered independent, and their energy level structure is determined accordingly. In addition, it is usually assumed that the lateral con?ning potential has a parabolic shape. Apart from the obvious mathematical simplicity brought about by this assumption, self-consistent calculations, as well as experimental observations, have provided a support to the validity of this approach [3]. On the other hand, the dynamics of electrons in magnetic fields has played a fundamental role in physics and its application to technology in a wide spectrum of topics, such as the Aharonov – Bohm effect [4–7], the bidimensional electron gas at the interface of semiconductor heterostructures [8], and electromagnetic lenses with time-dependent magnetic fields [9–12]. All these situations are interesting by themselves, and present rather complex mathematical structures; therefore, exact analytic solutions have been found only in a few special cases [13]. Therefore, the study of electrons in a QD under the presence of the magnetic field is important and in order. In the present work, we study the dynamics of electrons of a quantum dot interacting with a homogeneous timedependent magnetic field together with the corresponding in- duced electric field, assuming a parabolic dot con?ning potential in the effective mass approximation. We find an exact solution for the corresponding propagator of the Schrödinger equation. An analytical expression for the time-evolved wave function is found when the initial state is a Fock-Darwin state. We also calculate the transition probability to a given FockDarwin state as a consequence of the interaction with the electromagnetic field. The arrangement of the paper is as follows: In Sec. 2, the theoretical model is presented. In Sec. 3, the application is performed. Finally, the conclusions are presented in Sec. 4. 2. Theoretical Model In this section we will develop the quantum mechanical formalism to describe the dynamics of electrons in a quantum dot interacting with a homogeneous time-dependent magnetic field, B (t), and its corresponding induced electric field. Since the Hamiltonian is time-dependent, energy will not be conserved and there will not be an energy spectrum. Therefore, the problem must be approached by directly solving the time-dependent Schrödinger equation: ½ e i2 1 h P+ A +U 2m c ¾ Φ (r, t) = i~ ∂ Φ (r, t) , ∂t (1) where m is the electron effective mass, q = −e is the electron charge, c is the speed of light and the vector potential is chosen to be A = −r × B/2. In order to analyze a QD phenomenon, we will model the dot by a parabolic potential 1/2 whose strength is related to the dot size ld = (~/m$0 ) , where $0 is the corresponding oscillator frequency. There- 113 ON THE TRANSITION PROBABILITY FOR A QUANTUM DOT IN A TIME DEPENDENT MAGNETIC FIELD fore, the potential, U , can be written as: 1 m$02 r2 (2) 2 The temporal evolution of the system from an initial instant t0 to a final instant tf is given by Z Φ (rf , tf ) = d2 ri G (rf , tf , ri , t0 ) Φ0 (ri , t0 ) , (3) U (r) = where G (rf , tf , ri , t0 ) is the propagator and Φ0 (ri , t0 ) is the initial wave function. We will assume that the magnetic field has the form B (t) = B0 f (t) ẑ; Hence, the transversal propagator, can be cast in the form: m G (rf , tf , ri , t0 ) = 2πi~µ1 (tf ) ½ im £ × exp µ̇1 (tf ) r2f + µ2 (tf ) r2i 2~µ1 (tf ) ¾ ¤ − 2rf · R (θ (tf )) ri (4) where · R (θ (t, t0 )) = cos θ (t, t0 ) − sin θ (t, t0 ) sin θ (t, t0 ) cos θ (t, t0 ) ¸ (5) Rt with θ (t, t0 ) = t0 ω (t0 ) dt0 and where µ1,2 (t) denote two linear independent solutions of the equation ¸ · 2 d + Ω (t) µ (t) = 0 (6) dt2 2 with Ω (t) = ω (t) + $02 , such that ω (t) = (eB0 /(2mc)) f (t) = ωL f (t) . It is convenient to choose µ1 (t) having dimensions of time and µ2 (t) being dimensionless, and to choose the initial conditions µ1 (t0 ) = µ̇2 (t0 ) = 0, µ2 (t0 ) = µ̇1 (t0 ) = 1. We note that, if ωL = 0 or $0= 0 the proper limiting behaviors are recovered [16–17]. Also, we remark that the properties of the system generally follow from the behavior of the solutions of Eq. (6). A particular case of this theoretical model will be analyzed in the next section. 3. Application In this section we will focus in a particular shape of the time dependence of the magnetic field. We will consider a magnetic field that changes linearly in time, flipping its direction in a given time interval [−τ, τ ]. The z-component of the magnetic field is−B0 for t ≤ −τ . During the interval under consideration, it linearly increases in time as B (t) = B0 t/τ , and fort ≥ τ it becomes +B0 . In order to calculate the corresponding propagator, we must find the general solution of Eq. (6), that is: ³ ´ 2 2 µ̈ (t) + $02 + ωL (t/τ ) µ (t) = 0 (7) with −τ ≤ t ≤ τ . Let (ζ1 , ζ2 ) be two linearly independent solutions of Eq. (7) satisfying the initial conditions ζ1 (−τ ) = ζ̇2 (−τ ) = 0 and ζ2 (−τ ) = ζ̇1 (−τ ) = 1. We can easily express these solutions in terms of the solutions of Eq. (7) for the initial conditions µ1 (0) = µ̇2 (0) = 0, µ2 (0) = µ̇1 (0) = 1. The result is: ζ1 (t) = µ2 (τ ) µ1 (t) + µ1 (τ ) µ2 (t) (8) ζ2 (t) = µ̇2 (τ ) µ1 (t) + µ̇1 (τ ) µ2 (t) (9) where ¡ ¢ µ1 (t) = t exp −iωL t2 /2τ ¡ ¢ × 1 F1 i$02 τ /(4ωL )+3/4, 3/2, iωL t2 /τ ¡ ¢ µ2 (t) = exp −iωL t2 /2τ ¡ ¢ × 1 F1 i$02 τ /(4ωL )+1/4, 1/2, iωL t2 /τ (10) (11) 1 F1 (a, b, z)being the usual Hypergeometric function; also, using the series expansion of this function it can be easily verified that the functions µ1,2 are real, and it is trivially proven that the functions {µ1 , µ2 }are odd and even functions of t, respectively. Figure 1 shows the time dependence of functions {ζ1 , ζ2 }in the interval −τ < t < τ for different values of the dimensionless ratio $0 /ωL , where we have chosen ωL =1 and τ = 10. Both functions have an oscillating behavior whose period decreases as the ratio $0 /ωL increases. Now let us elucidate the structure of the propagator G (rf , τ, ri , −τ ). Firstly, we choose θ (−τ ) = 0, hence θ (τ ) = 0; and using the parity of the functions {µ1 , µ2 } we find that ζ1 (τ ) = 2µ1 (τ ) µ2 (τ ) , ζ2 (τ ) = 2µ̇1 (τ ) µ2 (τ ) − 1, ζ̇1 (τ ) = ζ2 (τ ) and ζ̇2 (τ ) = 2µ̇1 (τ ) µ̇2 (τ ). Therefore, Eq. (4) becomes m 4πi~µ1 (τ ) µ2 (τ ) ¶¸ µ · im ζ2 (τ ) 2 (12) r2f + r2i − rf · ri × exp 4~µ1 (τ ) µ2 (τ ) ζ2 (τ ) G (rf , τ, ri , −τ ) = In what follows we will analyze the temporal evolution of the QD wave function. The wave function at t = −τ can be expressed as a superposition of Fock–Darwin states [16–17]: X Cl,n exp (iEl,n τ /~) ψl,n (ri ) (13) Φ (ri , −τ ) = l,n where the Cl,n are constants, µ ¶ q 2 El,n =~$0 (2l+1+ |n|) 1+ (ωL /$0 ) +n (ωL /$0 ) , Rev. Mex. Fı́s. S 53 (7) (2007) 112–116 114 D. LAROZE AND R. RIVERA The integral in Eq. (21) converges if A − Re [Z] > 0, and in such case its value is: and ³q ´ ψl,n (ri ) = pl|n| /πλ2ω · ¸ q 2 2 2 × exp inφi − mri $0 + ωL /2~ × (ri /λω ) |n| |n| ³ Ll Z∞ |n|+1 dri ri 2 ´ (ri /λω ) ¡ ¢ exp −Z (τ ) ri2 Jn (s (τ ) rri ) 0 (14) 1 = 2Z (τ ) where pl|n| = l!/(l + |n|)!, µ ¶1/2 q 2 λω = (~/(m$0 ))/ 1 + (ωL /$0 ) , µ ¶n α (τ ) r ζ2 (τ ) Z (τ ) " µ ¶2 # 1 α (τ ) r × exp − , (22) Z (τ ) ζ2 (τ ) |n| Ll (x) are the generalized Laguerre polynomials, and {ri , φi } are the polar coordinates of ri . For simplicity, we will analyze the case l = 0. In such scenario the initial wave function can be cast as £ ¤ |n| Φ0,n (ri , −τ ) = C̃0,n (τ ) exp inφi − Ari2 ri (15) p 2 /2~ and where A = m $02 + ωL where, for simplicity, we have assumed that n > 0. Therefore, Eq. (16) can be expressed as: C̃0,n (τ ) Φ (r, τ ) = i × exp with C0 a normalization constant. Therefore, using Eqs. (3), (12) and (15), the time-evolved wave function is given by C̃0,n (τ ) α (τ ) πiζ2 (τ ) £ ¤ × exp iα (τ ) r2 ηn (r, ri ; τ ) (16) where α (τ ) = mζ2 (τ )/(4~µ1 (τ ) µ2 (τ )) and Z £ ¤ |n| ηn (r, ri ; τ ) = d2 ri exp inφi − Z (τ ) ri2 ri × exp [−isr · ri ] (17) with s (τ ) = 2α (τ )/ζ2 (τ ) and Z (τ ) = A − iα (τ ). Let us first calculate the angular integral: dφi exp [inφi − isrri cos (φi − φ)] Iφ = 1 iα (τ ) − Z (τ ) ¶n+1 µ r|n| α (τ ) ζ2 (τ ) # ¶2 ) r 2 (23) Equation (23) is the exact expression for the time evolved wave function when the initial wave function is a particular Fock-Darwin state. We note that it has a similar structure to the initial wave function, the dissimilarity appearing in the coefficients and in a numerical phase. Now let us calculate the probability transition to different Fock-Darwin states due to the particular time dependence of the magnetic field. Let us consider the states Φ0,k (−τ ) and Φ0,n (τ ). The transition amplitude is: Z h Φ (−τ )| Φ (τ )i = Z2π α (τ ) ζ2 (τ ) Z (τ ) × exp (in (φ − π/2)) "( C̃0,n (τ ) = C0 exp (iτ E0,n /~) Φ0,n (r, τ ) = µ d2 r Φ∗0,k (r, −τ ) Φ0,n (r, τ ) (24) (18) 0 Using the integral representation of the Bessel function [18]: Jn (ξ) = 1 2π 2π+a Z exp [i (nx − ξ sin x)] dx (19) Equation (24) can be written as Z2π h Φ (−τ )| Φ (τ )i = Xn,k (τ ) dφ exp (iφ (n − k)) 0 a Z∞ Equation (18) can be written as × Iφ = 2π exp (in (φ − π/2)) Jn (s (τ ) rri ) (20) Thus, we finally obtain the following expression for Eq. (11) Z∞ |n|+1 ηn (r, ri ; τ ) = 2π exp (in (φ − π/2)) dri ri ¡ × exp −Z (τ ) ri2 ¢ 0 Jn (s (τ ) rri ) , £ ¤ drr|n|+|k|+1 exp −∆ (τ ) r2 (25) 0 with ∗ (τ ) C̃0,n (τ ) C̃0,k Xn,k (τ ) = i (21) Rev. Mex. Fı́s. S 53 (7) (2007) 112–116 µ α (τ ) ζ2 (τ ) Z (τ ) ¶n+1 × exp (−inπ/2) (26) ON THE TRANSITION PROBABILITY FOR A QUANTUM DOT IN A TIME DEPENDENT MAGNETIC FIELD and ∆ (τ ) = Z (τ ) + 1 Z (τ ) µ α (τ ) ζ2 (τ ) ¶2 (27) Since Z2π dφ exp (iφ (n − k)) = 2πδn,k (28) 115 the ratio $0 /ωL for the n = 1case. We note that it has a nontrivial structure, with an oscillatory dependence on the length of the time interval τ , and a peak as a function of $0 /ωL . As a consequence of these behaviors, the transition probability presents a line of maximums in certain regions of the parameter space; besides, for a fixed τ , the probability transition decays asymptotically for large $0 /ωL , away from the line of maximums. 0 where δn,k is the Kronecker delta, the only possible transition is that which conserves the z-component of the angular momentum. Hence, Eq. (25) is reduced to h Φ (−τ )| Φ (τ )i = 2πXn,n (τ ) Z∞ × ¤ £ drr2|n|+1 exp −∆ (τ ) r2 (29) 0 Since ½ α2 Re Z + 2 ζ2 Z ¾ =A+ α2 A > 0, ζ22 (α2 + A2 ) the integral converges. Its value is Z∞ £ ¤ drr2|n|+1 exp −∆ (τ ) r2 = 0 |n|! |n|+1 2 (∆ (τ )) . (30) Therefore, h Φ (−τ )| Φ (τ )i = ³ ¶n+1 ¯2 µ 2π ¯¯ m ¯ ¯C̃0,n (τ )¯ i 4~µ1 (τ ) µ2 (τ ) |n|! exp (−inπ/2) 2 Z (τ ) + (α (τ )/ζ2 (τ )) 2 Conclusions The dynamics of electrons in a quantum dot interacting with a uniform time-dependent magnetic field has been analyzed when the magnetic field changes linearly in time, flipping its direction in a given time interval. An exact analytical solution for the time-evolved wave function has been obtained when the initial state corresponds to a specific Fock-Darwin level. The classical dynamics of this system has been found; it presents a non trivial oscillatory behavior through its dependence on the hypergeometric functions in Eq. (12). Both the amplitude and the period of the oscillation decrease as the strength of the dot potential, and therefore its confining properties, increases. The evolved wave function has a structure similar to that of the initial state, but with different timedependent coefficients. In addition, the probability transition has been calculated for these states; we found that this probability is appreciably different from zero only in some regions of the parameter space. Acknowledgments ´|n|+1 (31) The transition probability is of course proportional to 2 |h Φ (−τ )| Φ (τ )i| . Figure 2 shows the un-normalized transition probability as a function of τ (in arbitrary units) and ∗ 4. . Corresponding Author: Tel. +56-32-273136; Fax +56-32273529; e-mail [email protected]. 1. U. Woggon, Optical Properties of Semiconductor Quantum Dots, Springer Tracts in Modern Physics, Vol. 136, (Springer, Berlin, 1997). 2. 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