Semiclassical analysis of long-wavelength multiphoton processes: The Rydberg atom

Transcription

1 PHYSICAL REVIEW A 69, (2004) Semiclassical analysis of long-wavelength multiphoton processes: The Ryberg atom Luz V. Vela-Arevalo* an Ronal F. Fox Center for Nonlinear Sciences an School of Physics, Georgia Institute of Technology, Atlanta, Georgia , USA (Receive 12 December 2003; publishe 15 June 2004) We stuy the problem of multiphoton processes for intense, long-wavelength irraiation of atomic an molecular electrons. An exact, nonperturbative approach is applie to the stanar vector potential coupling Hamiltonian for a three-imensional hyrogenlike atom in a microwave fiel treate semiclassically. Multiphoton probability exchange is calculate in both the velocity an the length gauges, by applying the Goeppert- Mayer gauge transformation. The expansion of the time-epenent solution in terms of Floquet states elineates the mechanism of multiphoton transitions. A etaile analysis of the Floquet states an quasienergies as functions of the fiel parameters allows us to escribe the relation between avoie quasienergy crossings an multiphoton probability exchange. We formulate analytical expressions for the variation of quasienergies an Floquet states with respect to the fiel parameters, an emonstrate that avoie quasienergy crossings are accompanie by ramatic changes in the Floquet states. Analysis of the Floquet states, for small values of the fiel strength, yiels selection rules for the avoie quasienergy crossings. In the case of strong fiels, the simultaneous choice of frequency an strength of the fiel proucing an avoie crossing results in improve ionization probability. DOI: /PhysRevA I. INTRODUCTION Multiphoton processes involving atomic an molecular electrons have been wiely stuie both in experiments an in ifferent theoretical treatments, some of which are irectly relate to questions of quantum chaos [1 3]. The goal of this work is to stuy ynamical features of multiphoton transitions in the three-imensional hyrogenlike Ryberg atom in a semiclassical raiation fiel. The problem is stuie by irect integration of the system of orinary ifferential equations (ODE s) resulting from expaning the time-epenent solution in terms of a finite basis of unperturbe states. With this approach, we escribe the effect of the Goeppert-Mayer gauge transformation [4], that is, we calculate the wave function for the stanar vector potential coupling Hamiltonian (in the velocity gauge [5]) an for the electric ipole Hamiltonian (length gauge [5]). We compute time-evolving probabilities an their strong epenence on fiel parameters. We are focuse on fiel frequencies close to unperturbe resonances, which in general interfere with results obtaine from perturbative methos. A etaile analysis of the Floquet quasienergies an Floquet states for the Ryberg Hamiltonian allows us to escribe their role in multiphoton probability exchange. We provie analytical expressions of the variation of Floquet states an quasienergies as a function of the fiel parameters (strength an frequency). This analysis provies the basis for the unerstaning of the effects of avoie quasienergy crossings on probability exchange. We show that avoie quasienergy crossings are accompanie by ramatic changes in the components of the Floquet states, which may feature probability exchange. With this analysis we can preict transition rates an selection rules for multiphoton transitions. The analytic *Electronic aress: luzvela@cns.physics.gatech.eu Electronic aress: ron.fox@physics.gatech.eu PACS number(s): Rm, Wr, Sq an numerical techniques presente in this work are nonperturbative an can be extene to general Ryberg atoms. In the stuy of multiphoton processes ue to a raiation fiel, often perturbative methos have been attempte, which are base mainly on the reuction of the problem to a twolevel system (see, for instance, Refs. [5,6], an references therein). Naturally, low-orer transition probability is only justifie for the case of a weak fiel an for frequencies that are far from resonance, since the formulas have small enominators for resonant frequencies. The problem of intrashell ynamics has attracte attention since experiments in Ref. [2] suggest that intrashell ynamics is responsible for enhancement of ionization probability; in Ref. [7], intrashell ynamics is treate as two two-level problems. For the problem of a strong fiel, nonperturbative methos have been attempte, base on Floquet analysis introuce by Shirley [8]. With this approach, the Floquet states are obtaine in the Fourier omain, an the problem is reuce to an eigenvalue problem for a time-inepenent infinite-imensional matrix. Successful applications of this metho can be foun in Refs. [9,10]. In Ref. [11], there is a comparison between one- an two-imensional probability transitions of the Ryberg Hamiltonian. Also for the Ryberg problem, multiphoton transitions are stuie in Refs. [12,13] uner the influence of both a microwave an a static fiel. Their analysis is base on Floquet states obtaine by Blochinzew [14]. These Floquet states, however, are obtaine within an n shell of a Ryberg Hamiltonian uner a microwave fiel, in which noniagonal matrix elements between states with ifferent principal quantum number n have been neglecte. We are intereste in the Floquet analysis of the fully couple Hamiltonian, an we escribe the ynamics of the system in the time omain, that is, we provie the time evolution of the transition probabilities as a consequence of the expansion of the solution in terms of Floquet states. The raiation wavelength uner consieration in this work is much larger than the spatial extent of the electron /2004/69(6)/063409(13)/$ The American Physical Society

2 L. V. VELA-AREVALO AND R. F. FOX PHYSICAL REVIEW A 69, (2004) states even for high n. Therefore, the ipole approximation [15] applies an the fiel is inepenent of the spatial coorinates. We provie examples of multiphoton transitions in two regimes: transitions from the groun state to excite states of hyrogen an transitions between excite Ryberg states that are close to ionization. The same approach provie a negative proof for the existence of multiphoton transitions for the perioically riven simple harmonic oscillator [16]. We obtain analytical expressions for the variation of the quasienergies an Floquet states as functions of the fiel parameters. These expressions are analogs of some expressions obtaine in Ref. [17] in a ifferent context. When the fiel parameters vary, both crossings an avoie crossings of the quasienergies can be observe. Our analysis permits us to emonstrate that, when the quasienergies feature an avoie crossing, the Floquet states show ramatic changes. As a consequence, the time-epenent solution can be obtaine as the superposition of two or more Floquet states, which results in probability exchange between unperturbe states when the raiation fiel is present. When a crossing occurs, the Floquet states are not significantly affecte since the solution may be expane in terms of mainly one Floquet state, hence there is no transition probability. The ynamics close to avoie crossings was also stuie in Ref. [18]; however, our analytic results are ifferent an explain the numerical observations. For small values of the fiel strength, the Floquet states are very close to unperturbe states. This is use to obtain selection rules for the avoie quasienergy crossings. For a transition of N photons, an avoie crossing is possible if the states involve have a ifference in quantum number l that is even for N even or o for N o. We provie an expression for the rate of transition for these cases, which is inversely relate to the ifference in the quasienergies involve in the avoie crossing. As the fiel strength varies, the values of the fiel frequency proucing avoie crossings also vary (a consequence of the ac or ynamic Stark effect [6]). This shows that the tuning of the fiel parameters shoul be one simultaneously to achieve an optimal transition. The paper is organize as follows. In Sec. II we review the Hamiltonian of the Ryberg atom in a microwave fiel, in both the velocity an length gauges, which are relate by the Goeppert-Mayer gauge transformation. The solution is expane in terms of unperturbe states; the ynamics is then reuce to the auxiliary equations. Examples of multiphoton transitions from the groun state are presente in Sec. III; the time-evolving probabilities are obtaine in both velocity an length gauges by computing the effect of the Goeppert-Mayer gauge transformation. Floquet analysis of the auxiliary equations an the computation of Floquet states an quasienergies is escribe in Sec. IV; through examples, we escribe long-term ynamical features that can be obtaine from the expansion of the initial conition in terms of Floquet states at t = 0. We explore, analytically, the behavior of quasienergies an Floquet states close to avoie crossings as the parameters vary in Sec. V. The stuy of the Floquet states as a function of the parameters allows us to ientify selection rules for multiphoton transitions; this is escribe in etail in Sec. VI. The conclusions an iscussion is in Sec. VII. Finally, in Appenix A, we give heuristic reasons for the existence of multiphoton resonances by stuying the Laplace transform of the auxiliary equations. II. THE RYDBERG-ATOM HAMILTONIAN The Hamiltonian for the Ryberg atom in an intense semiclassical microwave raiation fiel is given by H = 1 + e 2 2m ep t c A e2 Z r. 1 The microwave fiel A with amplitue A 0 an frequency is given by A t = A 0 ˆ sint, in which ˆ is the polarization unit vector. The electron charge is e, e0. Applying the Goeppert-Mayer gauge transformation [4,5,19] r,t = ie c r A t, 2 the Hamiltonian (1) is relate to the electric ipole Hamiltonian given by H = H 0 + er E, 3 where H 0 is the unperturbe Hamiltonian for a hyrogenlike atom, H 0 = p2 e2 Z 2m e r, 4 an E = 1/cA /t is the electric fiel. The Hamiltonian (1) is in the so-calle velocity gauge [5]. The ipole Hamiltonian (3) is given in the so-calle length gauge [5]. The corresponing wave functions of Eq. (1) an of Eq. (3) are relate by r,t = exp ie c r A t r,t. 5 Let the time-epenent solution of Eq. (3) be expane as t = a k t k, 6 k where k are the unperturbe states of Eq. (4), i.e., H 0 k = E k k. Substituting Eq. (6) in the Schröinger equation, it can be shown that the coefficients a k t must satisfy auxiliary equations given by a k t = i E k a k + i j e c t A k r j a j. The etails of the erivation of the auxiliary equations can be foun in Ref. [16]. The erivation applies for any central

3 SEMICLASSICAL ANALYSIS OF LONG-WAVELENGTH PHYSICAL REVIEW A 69, (2004) force problem. For the calculations in this work, we consier the hyrogen atom Hamiltonian, that is, Z=1 in Eq. (4). For hyrogenlike atoms, the inices k an j in Eq. (8) are actually triples, corresponing to the quantum numbers n, l, m. The principal quantum number n is a positive integer; l = 0,...,n 1, an m = l,...,l. The energies are given by E k = E I /n 2, where E I =e 2 /2a 0 erg is the ionization energy an n is the principal quantum number of the corresponing state. The polarization irection of the microwave fiel is taken as the z axis, i.e., ˆ =kˆ. The vector potential is expresse as A t=a 0 kˆ sint. Normalizing coorinates an constants by atomic quantities, the auxiliary equations (8) are expresse in imensionless units as a k = i k a k + i cost k z j a j, t j where k = n = 1/2n 2 (n is the principal quantum number of the corresponing state), =E 0 / V/cm, an E 0 =A 0 /c. The matrix elements are known in close form [20]. The matrix elements k z j imply two selection rules: m = m an l = l ±1. 10 The auxiliary equations (9) form a nonautonomous, linear, time-perioic system of ODE s. In the Appenix we stuy the Laplace transform of the system to obtain riving fiel frequencies that may cause secularities, an it is immeiately obvious that an analytical solution is ifficult to achieve, even for a small number of states. Equations (9) form an infinite-imensional system. The analysis is one for a truncate system of equations or a finite basis of unperturbe states. The selection rules (10) imply that the quantum number m of the interacting states is fixe by the initial conition (initial state). The basis is forme by states in a finite set of principal quantum numbers n; the value of m fixes the minimum n to m+1. All couple l substates l=m,...,n 1 were inclue. The coupling term is larger between coefficients with the same principal quantum number n, an ecreases as the ifference n n increases. However, our computations show that the probability may sprea to many states, both within the same n an to states with ifferent n. Even small couplings can have a long-term effect, an therefore the integration must consier states within a large number of energy levels to avoi saturation. In our numerical experiments, the maximum n was etermine to have the higher states with negligible probability. Ionization probability is compute as the probability above an n cutoff value [3], which has to be smaller than the maximum n in the simulations. However, in strict terms, ionization probability must inclue the probability of continuum states, which are not consiere here. Incluing the continuum states as well as increasing the basis size to infinity may result in a ense spectrum [18]. The choice of, the frequency of the microwave fiel, provies the mechanism to effectively couple the initial state with some excite state. From the Laplace transform of these 9 FIG. 1. (Color online) Evolution of the square absolute value of some coefficients a k t for three ifferent values of the riving frequency: (a) = 3 1, (b) = 3 1 /2, an (c) = 3 1 /1+ 5/2. In all cases, the initial conition is a100 0=1 an zero for the rest. = There is probability exchange only in the resonant cases (a) an (b), between the groun state n=1 an excite states with n=3. The time t (in atomic unit) was normalize in each case by the value of the respective perio T =2/. equations (see the Appenix), we conclue that secularities arise with frequencies satisfying equations of the form j k ± N =0, where N is an integer. Numerical experiments showe that choosing satisfying (closely) this equation is key to prouce multiphoton transitions between the states with frequencies j an k, with N the number of photons involve in the transition. Base on Floquet analysis, we later establish that the values of proucing multiphoton transitions shoul be chosen simultaneously with the parameter, the strength of the fiel. can be use as a tuning parameter for the rate of transition. However, as we show in the following section, the epenence of the transition rate with respect to is rather complicate. III. MULTIPHOTON TRANSITIONS Three examples of solution of the auxiliary equations (9) are shown in Fig. 1. We took the groun state as the initial conition [a 100 0=1 an zero for all others], an obtaine probability exchange with excite states. We fixe the value of = = V/cm an evolve the system for three ifferent values of the coupling frequency: (a) = 3 1, (b) = 3 1 /2, an (c) = 3 1 /1 + 5/2. For the case (a), we observe Rabi-type oscillations between the states 1,0,0 an 3,1,0, alternating in probability to almost 100%; there is a small percentage of probability spreaing to other states. The first transition takes about 125 perios of the external fiel. Meanwhile for the case (b), two excite coefficients appear: the probability oscillates between the groun state, an the states 3,2,0 an 3,0,0. The first transition occurs after 250 fiel perios. On

4 L. V. VELA-AREVALO AND R. F. FOX PHYSICAL REVIEW A 69, (2004) FIG. 2. (Color online) Time evolution of the probabilities P k t [Eq. (11)]. The parameters are the same as in Fig. 1: (a) = 3 1, (b) = 3 1 /2, an (c) = 3 1 /1+ 5/2. Inall cases, = the other han, in the nonresonant case (c), the probability remains in the coefficient a 100 with small oscillations aroun it. There is no transition in this case. Goeppert-Mayer gauge transformation The solution of the auxiliary equations (9) provies the evolution of the system in the length gauge (3) in the form (6). In orer to obtain a time-epenent solution of the vector potential coupling Hamiltonian (1), we nee to apply the Goeppert-Mayer gauge transformation (2). Recall that the wave functions are relate by Eq. (5). Hence, in the velocity gauge, the time-epenent solution (in spherical imensionless coorinates) has the form t = exp i r cos sint a j t j. j The probability of fining the electron in an unperturbe state k at time t is given by where P k t = k t 2 = a j tm k,j t 2, j M k,j t = k exp i r cos sint j 0 11 e =20 i/r cos sint * k j r 2 sin r. 12 For the same parameters as in Fig. 1, the evolution of the probabilities is shown in Fig. 2. The computation of probability evolution requires the evaluation of the matrix elements (12) for all t0,t. (Note that the integral over the angle has been evaluate assuming that m=m.) For most cases, this ouble integral can only be evaluate numerically. In a few cases with low quantum FIG. 3. Some matrix elements M k,j t [Eq. (12)] for = 3 1 /2 an = numbers n an n the integral can be evaluate using symbolic computation software such as MATHEMATICA. But even the numerical integration is ifficult ue to the fast oscillations of the integran, as we observe in some attempts to use MATHEMATICA, which prouce inaccuracies. We trie to overcome this with our own FORTRAN coe using Gaussian integration over an Laguerre integration over r [21]. We sample time points in the interval 0,T/4, an by symmetry obtaine sample values of the matrix elements over one perio 0,T. The evaluation for any t0,t was then obtaine by interpolation. For the total integration time of 400T, we can use the perioicity of the matrix elements: M k,j t+t=m k,j t. Examples of some matrix elements (12) corresponing to low quantum numbers are shown in Fig. 3. In general, the matrix element is real if l l is even, an purely imaginary if the ifference in l is o. The parameters use in this figure are the same as in the example of Fig. 1(b): = 3 1 /2 an = We notice that for low principal quantum numbers the matrix elements M k,j are close to 1 for k = j an close to 0 otherwise. Having compute the coefficients a k t an the matrix elements M k,j t, we are able to obtain the time-epenent solution an probability evolution. From Fig. 3, we observe that k ta k t. Then, the evolution of probabilities (11) for this choice of parameters an initial conition is very similar to the evolution of the coefficients a k t. Compare Figs. 1 an 2. Figure 2(a) represents a 1-photon transition between the levels 1 an 3 with almost 100% of probability exchange; the probability oscillates in time, in a similar way of a Rabi oscillation; however, there is a small percentage of probability going to other states, which shows the effect of the multiple couple states. Meanwhile, (b) reflects a 2-photon transition between the groun state an 3,0,0 with probability of at most 30% an 3,2,0 with at most 60%. Again, the probability oscillates in time, after about 508 perios of the fiel the groun state has probability close to 100%. In these figures, very quick small oscillations make the lines look thick, as it is epicte in the inset of Fig. 2(c). Now, we stuy multiphoton transitions between states in the levels n=1 an n=8 as a function of the fiel strength

5 SEMICLASSICAL ANALYSIS OF LONG-WAVELENGTH PHYSICAL REVIEW A 69, (2004) FIG. 4. Time an initial phase average probabilities for 8 1 8=0 an varying, over a fixe time interval of 400T. In the lower panels we see the istribution of average probabilities among several states with quantum numbers n inicate. Darker areas correspon to larger probabilities. We integrate the auxiliary equations (9) for the fixe value of fiel frequency = 8 1 /8 an values of in the range an ( to V/cm). The initial conition was chosen with all the probability on the groun state, i.e., a 100 0=1 an all other coefficients zero. Equations (9) were integrate over a fixe interval of time of 400T, T=2/ an for ifferent initial phases. For each we compute the time an initial phase average of the probability in each state, that is, the average of a k t 2 over 400T. In the first panel of Fig. 4 we show the time an initial phase averages of a 100 2, a 870 2, an a as a function of. The graph in the secon panel shows the istribution of the average probability as a function of. The gray scale is the average probability for each state with quantum numbers n=1,6,7,8,9. In each n ban, the vertical axis correspons to the inex l of the coefficients a nl0 ; this is, l =0,...,n 1 (note that the vertical size increases for larger n, only because there are n levels represente in each ban). The arker gray zones correspon to greater values. We can observe in Fig. 4 that for values of aroun an 0.02, the probability of the groun state averages 0.5; there is an exchange of probability between the groun state an several states with quantum numbers n=7,8,9, an 10. This shows that although the fiel frequency is in resonance with states n=1 an n=8, the probability oes not exchange only between those states (the reuction to a twolevel system oes not hol), but it can isperse in neighboring states in same n shell an other n shells. It is also noticeable that the relation between an the transition probability is quite complicate, an by no means linear. In Fig. 5, we show the time evolution of some probabilities (11) for = 8 1 /8 an = We can see that, after 90 perios of the fiel, the probability of fining the electron in the groun state is zero. At that time, several states with n=7, n=8, an n=9 have nonzero probability; the state 9,2,0 is the one with largest probability at this point. In all these states, the probability is oscillating rapily, as we can observe in the insets of Fig. 5. We also note that the Goeppert-Mayer gauge transformation has important effects for the case of excite states. In FIG. 5. Probabilities P k t [Eq. (11)] for 8 1 8=0 an = The insets provie the fine etail of the quick oscillations. Fig. 6 we plot some matrix elements M k,j t that are neee to compute the probabilities [see Eqs. (11) an (12)]. Compare with the case of lower n s quantum numbers in Fig. 2. IV. FLOQUET ANALYSIS In this section, we calculate quantum Floquet states an quasienergies for the Ryberg Hamiltonian (1). One of the main avantages of the treatment iscusse in this work is that the Floquet analysis of the auxiliary equations (9) (a system of orinary ifferential equations) prouces Floquet states for the quantum problem. It can be shown [22] that the Floquet exponents of the auxiliary equations are the Floquet quasienergies of the quantum system. Also, the Floquet solutions a t yiel quantum Floquet states efine by the expansion (6) in terms of unperturbe states (7). Note that each Floquet solution a t is a vector of coefficients a k t. The auxiliary equations (9) for the coefficients a k t form a time-perioic system, with perio T=2/. Therefore, it is possible to fin a basis of solutions a mt, calle the Floquet solutions, of the form FIG. 6. Some matrix elements (12) for = 8 1 /8 an =

6 L. V. VELA-AREVALO AND R. F. FOX PHYSICAL REVIEW A 69, (2004) a mt = e i m t x m t, 13 where m is calle the Floquet exponent an x m t is a vector of T-perioic functions. Any vector solution at can be expresse as a superposition of Floquet solutions [23], such as at = c m a mt. 14 m The Floquet solutions yiel Floquet states for the Hamiltonian (1) of the form mt = exp i z sint k a k m t k, 15 where k are the unperturbe states (7). The Floquet exponents m are calle the quasienergies of the Hamiltonian (1). Therefore, any time-epenent solution of the Hamiltonian (1) can be expresse as a superposition of Floquet states, t = c m mt. 16 m The Floquet solutions a m are obtaine from the monoromy matrix. Let t,t 0 be the matrix of funamental solutions of the auxiliary equations (9) with t 0,t 0 =1, that is, any solution at with initial conition at 0 =a 0 is given by at=t,t 0 a 0. For simplicity, let t 0 =0. The monoromy matrix is T,0. The eigenvalues of the monoromy matrix are the Floquet multipliers. If m is a Floquet multiplier, the Floquet exponent m is efine by m =e imt. The Floquet exponent m is obtaine moulo, that is, m = i ln m /Tmo. The Floquet solution is given by a mt=t,0x m 0, where x m 0 is the eigenvector corresponing to the exponent m. Furthermore, a m has the form (13). Note, from expression (16), that the coefficients c m are constants, which are etermine by the initial conition. At t=0, the Floquet states (15) correspon to an expansion in terms of unperturbe states. The coefficients of this expansion are given by a m k 0. Now, from expression (13), these coefficients are precisely the components of the eigenvectors x m 0. We compute the monoromy matrix T,0 of the auxiliary equations (9) by irect numerical integration, that is, we integrate Eq. (9) taking as initial conitions each of the columns of the ientity matrix 1, over a time interval of one perio 0,T. Then, we are able to compute eigenvalues an eigenvectors with which we can construct the quantum Floquet states. FIG. 7. (Color online) First 15 Floquet quasienergies m for three ifferent frequencies of the riving fiel. The strength of the fiel is fixe, = The lines labele with the unperturbe energies n are plotte for reference. corresponing to 1, since 1 mo 3 [that is, 1 = 3 N, with N=1 for (a) an N=2 for (b)]. On the other han, for the nonresonant case (c), with = 3 1 /1+ 5/2 =0.2747, the Floquet quasienergy 1 is close to 1 mo = Let us iscuss with more etail the resonant case (a) in Fig. 7. Let the initial conition be the groun state, that is, at t=0, a 100 0=1, an the other coefficients zero. This initial conition is expane, almost exactly, in terms of only two eigenvectors: the eigenvectors corresponing to the quasienergies 3 an 6 (Fig. 7). Recall that, at t=0, the components of the eigenvectors are coefficients of the unperturbe states. The components of these eigenvectors are illustrate in Fig. 8(a). Therefore, the evolution of the coefficient a 100 can be approximate by a 100 tc 3 e i 3 t x t + c 6 e i 6 t x t, where x 3 t an x 6 t are vectors of T-perioic functions an only their first component k=1,0,0 appears. At multiples Probability exchange from superposition of Floquet solutions In Fig. 7, we plot the first 15 Floquet quasienergies for the same parameters previously stuie in the examples in Figs. 1 an 2. The labeling of the quasienergies is arbitrary; we calle them m for m=1,...,15 in increasing orer. We note that for the resonant cases (a) an (b) all the quasienergies are close to the unperturbe energies n = 1/2n 2. However, the quasienergies in the same n shell have split an the egeneracies isappeare. In the resonant cases, there are four quasienergies ( 3 to 6 ) which are close to 3. This aitional quasienergy splitting from 3 is the quasienergy FIG. 8. Square moulus of the components of eigenvectors corresponing to quasienergies (a) 3 an 6, with = 3 1 an (b) 3, 5, an 6 with frequency = 3 1 /2 [see Figs. 7(a) an 7(b)]. =

7 SEMICLASSICAL ANALYSIS OF LONG-WAVELENGTH PHYSICAL REVIEW A 69, (2004) of the perio, t=kt, the functions x 3 t an x 6 t are precisely the eigenvectors represente in Fig. 8(a). For this case, c 3 2 = an c 6 2 = When evaluate at multiples of the perio t=kt, a 100 KT 2 is the probability of fining the electron in the 1,0,0 state. That is, a 100 KT 2 =1,0,0 KT 2 since the Goeppert-Mayer gauge transformation is the ientity at those points. We can then fin that this probability evolves as a 100 KT 2 c 3 e ikt 3x c 6 e ikt 6x Expaning the previous expression, we get a 100 KT 2 c 3 x c 6 x coskt 6 3 Rec 3 c * 6 x x *. Then, this probability evolves as a Rabi-type oscillation. The K-epenent term has a minimum for K = Evaluating, we get K= This is the number of perios of the microwave fiel for which the probability will have a maximum transition; in other wors, this is half the perio of the Rabi oscillation. The approximation above gives a 100 KT 2 = coskt 6 3, then the maximum probability of the groun state after 1 perio can be calculate as 99.67%. The remaining fraction of probability spreas to other couple states. With the same argument we can obtain the probability a 310 KT 2 = coskt 6 3. The maximum probability for the state 3,1,0 is 98.04%. See Fig. 1(a). In general, if the initial conition can be expane in terms of mainly two Floquet states, the probabilities evolve as a Rabi-type flopping, an the quasienergies an Floquet states provie a very goo approximation to the rate of oscillation an probability exchange. Note, however, that although only two quasienergies an Floquet states participate in the expansions of probability evolution, the system is not equivalent to a two-level system. The Floquet analysis was one consiering all the couple states. Simplification to a two-level system coul not account for the fraction of probability that goes to other states. On the other han, for the resonant case (b) in Fig. 7, the initial conition a 100 =1 (an the rest zero) is expane in terms of mainly three eigenvectors, corresponing to the quasienergies 3, 6, an 5. The main coefficients in the Floquet expansion are c 3 2 =0.6566, c 6 2 =0.3411, c 5 2 = In Fig. 8(b), the components of the eigenvectors are represente. From the components of the eigenvectors, we see that three unperturbe states participate: 1,0,0, 3,0,0, an 3,2,0. This prouces the exchange of probability from the groun state 1,0,0 an those two excite states. Since two of the quasienergies 3 an 6 participate mostly in the expansion of the initial conition, we can still use expression (17) to approximate the rate of transition of the groun state as fiel perios; this is the point at which the groun state reaches the minimum probability of 9.95%. [See Fig. 1(b).] The previous analysis was repeate for several values of in the range an 0.1 (510 4 an V/cm). The quasienergies eviate more from the unperturbe energies as increases. This accounts for shorter perio of the probability oscillation [see expression (17)]. For larger, the expansion of the groun state (initial conition) in terms of Floquet states involves more significant coefficients c m. Also, the Floquet states are expresse in terms of more unperturbe states. This means that the probability spreas among more unperturbe states. In the following sections, we analyze the cases when the initial conition is an excite Ryberg state, with quantum number n60. In contrast with transitions involving the groun state an other low levels, where the energy gaps are well ifferentiate by n, in the case of n60 the energy gaps are smaller an closer; therefore the resonances ten to overlap. We observe that for excite levels, the Floquet analysis yiels a quite complicate behavior that is very sensitive to parameters. We present a etaile analysis of the variation of Floquet states an quasienergies with respect to parameters,. V. FLOQUET ANALYSIS AS A FUNCTION OF PARAMETERS The Hamiltonians for the Ryberg atom (1) an electric ipole (3) are relate by the Goeppert-Mayer gauge transformation (2). Note that, at times multiple of the perio, t = KT, the solutions for both Hamiltonians coincie [see expression (5)]. Particularly, the Floquet quasienergies also coincie, since they are compute as the eigenvalues of the monoromy matrix T,0 [see Sec. IV]. The Hamiltonian (3) is equivalent to the Floquet Hamiltonian in the extene phase space given by Kˆ = H 0 z cos + I, 18 where,i is an aitional pair of canonical conjugate variables, an quantum mechanically, I= i /. The Floquet Hamiltonian Kˆ can be iagonalize as Kˆ m = m m. 19 The eigenvalues m coincie with the Floquet quasienergies [8]. The eigenstates are the quantum Floquet states m in the extene phase space. A. Quasienergies an Floquet states as a function of We now consier variations with respect to the parameter, the strength of the microwave fiel. Taking the erivative with respect to of Eq. (19), an using the orthonormality of the basis m, we obtain (cf. Ref. [17]) n z cos m + n n m = m mn + m n m. Then, for n=m, we obtain

8 L. V. VELA-AREVALO AND R. F. FOX PHYSICAL REVIEW A 69, (2004) m = mz cos m. On the other han, for nm an n m, 20 n m = nz cos m. m n Therefore, applying completeness of the m basis, we obtain where m = n nz cos m + c m, 21 nm m n c = m m. However, a phase factor can be chosen such that c=0. We can see this in the following way. First, note that m m =1 implies m m + m m = c * + c =0; then c is purely imaginary. Now, for small, we have that m + = m + m + O 2. Since we can choose a phase factor of m + so that the prouct m + is real [15], we conclue that c must be zero. Substituting in Eq. (21), we obtain m = n nz cos m. 22 nm m n To obtain the secon erivative of the quasienergies m with respect to, from Eqs. (20) an (22) we have 2 m 2 = mz cos m = mz cos m m z cos m = nm n z cos m * n z cos m m n + m z cos n nz cos m nm m n =2 n z cos m 2. nm m n In numerical explorations, we observe avoie crossings of the quasienergies, together with ramatic changes in the Floquet states. Consier the case where two quasienergies m an n are very close; in the last erivation, we can approximate the sum with only the two closest quasienergies: 2 m 2 2 nz cos m 2 2 n m n 2. This implies that n + m behaves linearly with respect to aroun the avoie crossing. Close to an avoie crossing of m an n, we have nz cos m = nz cos m + n z cos m = pn p z cos n * p z cos m n p n z cos p pz cos m pm m p nz cos m m m n + n. In the last step, we kept only the terms for p=m in the first sum, an p=n in the secon sum, which are the most significant contributions close to the avoie crossing n m. Therefore, we obtain the following ifferential equation: nz cos m = nz cos m m n m n, which implies n z cos m m n = const. 23 The last ientity shows a common behavior at avoie crossings: if two quasienergies n an m are very close to each other, the matrix element n z cos m increases, in orer to preserve the prouct (23). As a consequence, from Eq. (22), const. m n m n 2. This implies ramatic changes in the Floquet states at avoie crossings when the parameter is varie. B. Quasienergies as a function of When the frequency fiel is varie, a similar analysis can be one. Proceeing in a similar way as for the variation with respect to, we can obtain an equation analog of Eq. (20): m = mi m, where I is the aitional action variable of the Floquet Hamiltonian (18). Also,

9 SEMICLASSICAL ANALYSIS OF LONG-WAVELENGTH PHYSICAL REVIEW A 69, (2004) FIG. 9. Floquet quasienergies m for varying. The frequency of the fiel is fixe = The range of the fiel strength is 0.5 V/cm 38.5 V/cm. The otte lines are the values of the unperturbe energies (moulo ). Avoie crossings prouce ramatic turns of the quasienergies. m = n ni m. nm m n Furthermore, it can be shown that m I m m n = const. These equations show that avoie crossings of the quasienergies as a function of are also accompanie by suen changes in the Floquet states. VI. SELECTION RULES FOR AVOIDED CROSSINGS The behavior of the quasienergies as a function of the fiel strength is illustrate in Fig. 9. For this computation, we consier excite states with quantum number m = 63. Since m is fixe, only states with principal quantum numbers n=64,65,... an l=63,...,n 1 interact [see Eq. (10)]. The choice of quantum number m = 63 prouces 45 interacting states for values of n = 64,..., 72. The fiel frequency is fixe = 65 64, an varies from to (0.05 to 38.5 V/cm). We observe that for small, the Floquet quasienergies split from the unperturbe energies (appearing in the figure as n mo ). This can be interprete as the ac or ynamic Stark shift [5,6].As increases, we observe apparent crossings an avoie crossings. There is no reason a priori to rule out crossings of the quasienergies in Eq. (19); egenerate Floquet states might exist for some values of. In our numerical experiments, we foun that some selection rules apply for these egeneracies, at least for small. This will be explaine in the next example. FIG. 10. (Color online) Floquet quasienergies m for varying. The frequency of the fiel is = We observe that some crossings occur as well as avoie crossings. See text for more etails. crossings an some avoie crossings. This can be appreciate in the secon an thir panel of Fig. 10. We can observe that the crossings an avoie crossings alternate. The explanation of this fact will be escribe below, from observing the behavior of the eigenvectors as a function of. Note also that, for this choice of, we have Recall that, at t=0, the Floquet states are obtaine as an expansion in terms of unperturbe states, with the coefficients given by the components of the eigenvectors [see Eqs. (13) an (15)]. We observe that for small there is one large eigenvector component close to 1, an the rest are close to 0; the Floquet states are very close to the unperturbe states. As increases, more components of the eigenvectors increase in value; hence, the Floquet states are expane in terms of more unperturbe states. When an avoie crossing occurs, the components of the eigenvectors also feature ramatic changes, as we showe analytically in Sec. V. In Fig. 11, we plot the square moulus of some components of three eigenvectors, corresponing to the quasiener- A. Quasienergy crossings, avoie crossings, the eigenvectors, an selection rules, as varies In Fig. 10, we present an example showing crossings an avoie crossings. For this case, the fiel frequency is = 67 64, an varies. A total of 45 states were integrate, for principal quantum numbers n=64,...,72 an m=63. The labeling of the quasienergies is arbitrary; in this case, we plotte 11 of the 45 compute quasienergies: 1, 2,..., 11. For small, we observe that the quasienergies corresponing to n = 65 an n = 72 get closer; proucing some FIG. 11. (Color online) Square moulus of some components of the eigenvectors x m k as a function of, corresponing to Floquet quasienergies 5, 6, an 7 in Fig. 10. The eigenvector components suffer ramatic changes at avoie crossings, the components showe are k=65,63,63, k=72,69,64, an k=72,67,

10 L. V. VELA-AREVALO AND R. F. FOX PHYSICAL REVIEW A 69, (2004) gies 7, 3, an 5 of Fig. 10; they are associate with quasienergies featuring avoie crossings. In each case, the components correspon to the unperturbe states 72,69,63, 65,63,63, an 72,67,63. Aroun the avoie crossing at = , we can see that the component k=65,63,63 of the eigenvector x 5 goes quickly to zero (see secon inset in the first panel). At that point, the same component of the eigenvector x 7 increases by the same amount (see inset in the thir panel). This behavior is reprouce in all components of the eigenvectors involve in avoie crossings. Exactly for a value of for which an avoie crossing occurs, the eigenvectors have two or more important components; these are the coefficients for the unperturbe states, an as a result, the Floquet states are a superposition of several unperturbe states. This results in probability exchange between the unperturbe states involve. On the other han, when a crossing occurs, the components of the eigenvectors o not reflect any suen change, neither o the Floquet states, an no probability transition takes place. As mentione above, for small values of, the Floquet states are still close to unperturbe states, so they can be easily ientifie with the n,l,m states. This leas to the following selection rule for this choice of : the quasienergies that have avoie crossings correspon to states for which the ifference in quantum number l is even; then probability exchange is possible only when l l is even. If the ifference in l is o, the quasienergies cross, an there is no probability exchange. This selection rule is relate to the particular resonance. In this case, we have that ; this avoie crossing involves a two-photon transition between states in n=65 an n=72. We note that this transition is slow, the perio of the Rabi oscillation, from Eq. (17), is / ,188 fiel perios. B. Transitions from the groun state as varies The selection rules also appear when we stuy probability transitions as a function of the fiel frequency. To illustrate this, first we stuy transitions from the groun state to low levels (small n). We show that crossings an avoie crossings of the quasienergies occur for values of satisfying closely a resonance equation of the form k 1 =N. When varies, keeping fixe, crossings an avoie crossings of the quasienergies are observe, as we show in Fig. 12 for low quantum numbers n=1,...,5 an m=0. In this example, the varying frequency is = 0 /, with 0 = 3 1, an the enominator varies continuously between 0.1 an 5. The fiel frequency is fixe as = The quasienergies are represente in the figure as a function of. We plotte, with otte lines, the curves 1 +N as well as the lines for the unperturbe energies 2,..., 5. The points where these lines an curves intersect, represent an unperturbe resonance (i.e., a resonance for the unperturbe energies) of the form k 1 =N. As we can expect, the crossings an avoie crossings of the quasienergies occur for values of close to an unperturbe resonance, i.e., k 1 /N. We observe that the quasienergies have split from the unperturbe energies, this is clearly seen in the secon panel of FIG. 12. (Color online) Floquet quasienergies varying the fiel frequency ; =0.006 fixe. The varying frequency is = 0 /, with 0 = 3 1. The states correspon to m=0 an principal quantum numbers n=1,2,...,5. The otte curves in the backgroun correspon to the resonances for the unperturbe energies. In the secon panel the splitting of the n=3 energy level is shown (ynamic Stark shift). Fig. 12, where a zoom of the quasienergies close to 3 is shown. In the inset, we can see one avoie crossing an several crossings. The avoie crossing occurs for 3 1 /3. Hence, this avoie crossing reflects a 3-photon transition between the groun state n=1 an a state in n=3. To stuy this transition, we compute the eigenvectors corresponing to the quasienergies in the avoie crossing, as a function of. Only two eigenvector components have high values, they are the coefficients of the unperturbe states 1,0,0 an 3,1,0. Our computations show that these eigenvector components change quickly aroun the avoie crossing. Therefore, for this value of, there is a 3-photon transition between the states 1,0,0 an 3,1,0. The transition is of the Rabi type. As before [see Eq. (17)], wecan calculate the perio of the Rabi oscillation, as / 9 8 =1602T. Selection rules. These computations confirm the selection rules obtaine in the preceing section. For these small values of, the Floquet states are close to the unperturbe states. Therefore, only one component of the eigenvectors is close to 1 an the rest are close to zero; this permits us to associate a label n,l,m to the Floquet states. Avoie crossings occur only when the ifference in l quantum number l l is even for N even (in the equation j k =N) or when l l is o for N o. An, since avoie crossings prouce ramatic exchange between the components of the Floquet states, probability exchange is possible only between states for which there is an avoie crossing. Therefore, these are the selection rules for probability transitions between unperturbe states. C. Transitions between excite states: The tuning of an When consiering excite state, the situation is more complicate ue to the small separation of the unperturbe energies. The resonant frequencies of the form = j k are small, then the quasienergies are very close together since

11 SEMICLASSICAL ANALYSIS OF LONG-WAVELENGTH PHYSICAL REVIEW A 69, (2004) FIG. 13. Floquet quasienergies varying the fiel frequency. =210 9 is fixe 10.3 V/cm. The varying frequency is = 0 /, with 0 = The states correspon to m=63 an principal quantum numbers n=64,65,...,80. they are obtaine moulo. Therefore, more crossings an avoie crossings occur. This can be observe in Fig. 13. For this case, we consiere states with quantum numbers n =64,...,80; m=63; an l=63,...,n 1. The varying frequency is = 0 /, with 0 = = , an the enominator is imensionless, taken between 1 an 3.2. The fiel strength parameter =210 9 is fixe 10.3 V/cm. The quasienergies are plotte as a function of the enominator. In otte lines, some unperturbe frequencies are shown (right vertical axis). For this kin of computation, some extra work is necessary in orer to obtain a goo representation of the quasienergies. Since the quasienergies are obtaine moulo, it is common to obtain iscontinuous curves that seem senseless. However, we can a multiples of as neee to obtain continuous curves. When this is one, we observe both crossings an avoie crossings, corresponing to all kins of resonance relations. The values of an can be both tune to achieve the esire transition an the rate of the Rabi oscillation. In Fig. 14, we observe that as varies, an the quasienergies feature an avoie crossing, also the coefficients c m in the expansion FIG. 15. (Color online) In the first panel, the quasienergy ifference 2 1 is plotte as a function of. Each curve correspons to a ifferent value of. The minimum in each curve is an avoie crossing relate to the 1-photon transition between n=64 an n=65. The avoie crossing for each occurs at a ifferent value of. These values of are epicte in the secon panel. (14) have quick changes. In the first panel, we mark in circles, squares, an iamons the three main quasienergies (in that orer) involve in the evolution of the initial state 64,63,63. The corresponing coefficients c m of the Floquet states are shown in the secon panel. Exactly at an avoie crossing, there are two (or at most three) significant coefficients c m. Then, for the choice of proucing an avoie crossing, the evolution of the initial state is a superposition of two or three Floquet states only. Our calculations show that the values of that prouce avoie crossings epen on the value of. In Fig. 15 the ifference between two quasienergies 1 an 2, relate to a 1-photon transition between n=64 an n=65, is plotte as a function of, an for ifferent values of. As increases, the ifference 2 1 also increases. For each fixe value of, we compute the value of proucing the minimum, i.e., the avoie crossing. As we can expect, the values of are close to = But, as increases, the value of proucing the avoie crossing also increases, as can be observe in the secon panel of Fig. 15. This computation also allows us to etermine the implications of the choice of parameters in probability transitions. For the parameters, in Fig. 15, an for 64,63,63 as the initial conition, the main Floquet states are expane in terms of the unperturbe states 64,63,63 an 65,64,63; but as increases, also the states 66,65,63, 67,66,63, etc., have significant coefficients. That is, if a new level n is involve, only the state with l=n 1 gains probability. Notice that this is the probability of a multiphoton transition to that state. As higher n levels have nonzero probability, there is higher probability of multiphoton ionization. VII. CONCLUSIONS FIG. 14. (Color online) Above, the three main Floquet quasienergies in the evolution of the state 64,63,63 are marke with circles, squares, an iamons (in that orer). At the bottom, the coefficients c m of the Floquet states [as in expansion (14)]. This was calculate for varying fiel frequency = 0 /, with 0 = = fixe 1.6 V/cm. The states correspon to m=63 an principal quantum numbers n=64,65,...,80. The problem of multiphoton transition of the Ryberg atom in a strong, long-wavelength microwave fiel is escribe with a nonperturbative approach, proviing a timeepenent solution of the Schröinger equation. Using the Goeppert-Mayer gauge transformation, the time-epenent solution is obtaine in both the length gauge (electric ipole Hamiltonian) an the velocity gauge (stanar vector potential coupling Hamiltonian)

12 L. V. VELA-AREVALO AND R. F. FOX PHYSICAL REVIEW A 69, (2004) The ynamics is reuce to a system of orinary ifferential equations for the coefficients a k t of the expansion of the time-epenent solution in terms of the unperturbe states (that is, the atomic states, when no raiation fiel is present). Coupling between states in ifferent levels, an between substates within an n level, results in spreaing of probability, for which many states nee to be inclue in the analysis. The auxiliary equations for the coefficients a k t form a linear, time-perioic system. The numerical integration of the auxiliary equations, together with the Goeppert-Mayer gauge transformation, provie exact time evolution of the transition probabilities between unperturbe states in the presence of the raiation fiel. The choice of fiel strength an frequency etermine the extent an rate of the transition. Multiphoton transitions can be observe, for k j /N, but the choice of epens on the value of. Note that, in previous work [16], the same approach was use to show that there are no multiphoton transitions of the simple harmonic oscillator in a raiation fiel. The introuction of Floquet analysis of the auxiliary equations (a system of time-perioic ODE s) prove to be an excellent tool to analyze transition probabilities. From the Floquet solutions an multipliers of the auxiliary equations, we obtain Floquet states an quasienergies of the quantum system. The expansion of the initial conition in terms of Floquet states (at t=0) is enough to know exactly the time evolution of probabilities. For cases when the initial conition is expane in terms of mainly two Floquet states, the probability transition is of Rabi type between two unperturbe states, an it is possible to obtain the rate of transition. We show, however, that other couple states also gain probability, therefore the strict reuction to a two-level system is not justifie. The Floquet analysis of transitions between excite Ryberg states n60 showe that the probability transitions are very sensitive to changes in the parameters an. We show, analytically an numerically, that the probability transitions are associate with multiple crossings an avoie crossings of the quasienergies, as the parameters vary. The expressions for the erivatives of the Floquet states an quasienergies with respect to the parameters an are erive analytically from the Floquet Hamiltonian, which permit us to show that avoie crossings are accompanie by suen changes of the Floquet states. Therefore, avoie crossings prouce probability transitions between the states involve. When a quasienergy crossing occurs (up to the numerical accuracy), the Floquet states o not show any suen change, an no probability transition takes place. We observe that the quasienergy crossings an avoie crossings are etermine by the unperturbe states involve, obeying selection rules for probability transitions. For small, if the ifference in quantum number l has the same parity as the N-multiphoton process, then there is an avoie crossing. That is, for a choice of k j /N, an avoie quasienergy crossing is prouce only when the corresponing Floquet states are expane in terms of states with l l o for N o or l l even for N even. Therefore, probability transition is only possible between unperturbe states that satisfy this selection rule. The value of that prouces an avoie crossing epens on the fiel strength. As increases, the value of at the avoie crossing also increases. For large, the choice of proucing an avoie crossing results in Floquet states that are not close anymore to unperturbe states; they have many components corresponing to unperturbe states with higher n an l=n 1. This means that the probability spreas to states of the form n,n 1,m, with larger n. As a consequence, there is an increase in multiphoton ionization probability. For large values of, more avoie quasienergy crossings appear an they may even overlap. Also, most Floquet states are expane in terms of many unperturbe states. The mixing of unperturbe states results in large iffusion of probability among many states. In this process, it is har to ifferentiate the exchange of probability in an n shell from the exchange between ifferent levels. In Ref. [2], some intriguing experiments showe that intrashell ynamics was responsible for enhancement of ionization probability. Some numerical an analytic work [7,12,13] has been one to stuy the effect of intrashell ynamics. However, our analysis shows that the intrashell ynamics cannot be isolate from transitions between ifferent levels, since these two processes occur together an for all neighboring states for strong enough fiels. ACKNOWLEDGMENT This work was supporte by National Science Founation Grant No. PHY APPENDIX: SECULARITIES OF THE AUXILIARY EQUATION Note that by efining b k =a k e i kt, we can write Eq. (9) as t b k = i Z kj e i k j +t + e i k j t b j, 2 j=1 where Z kj = k r cos j. Denoting the Laplace transform of b k t as Lb k =bˆ ks, an noting that L(/tb k )=sbˆ ks b k 0, the previous equation can be transforme to bˆ ks = 1 s b k0 + i Z kj bˆ js i k j + 2s j=1 + bˆ js i k j. The first iterate of this formula prouces