Resonance wave functions located at the Stark saddle point
aa r X i v : . [ phy s i c s . a t o m - ph ] M a y Resonance wave functions located at the Stark saddle point
Holger Cartarius, ∗ J¨org Main, Thorsten Losch, and G¨unter Wunner
Institut f¨ur Theoretische Physik 1, Universit¨at Stuttgart, 70550 Stuttgart, Germany (Dated: November 1, 2018)We calculate quantum mechanically exact wave functions of resonances in spectra of the hydro-gen atom in crossed external fields and prove the existence of long-lived decaying quantum stateslocalized at the Stark saddle point. A spectrum of ground and excited states reproducing the nodalpatterns expected from simple quadratic and cubic expansions of the potential in the vicinity of thesaddle point can be identified. The results demonstrate the presence of resonances in the vicinity ofthe saddle predicted by simple approximations.
PACS numbers: 32.60.+i, 32.80.Fb, 82.20.Db
I. INTRODUCTION
The hydrogen atom in crossed static electric and mag-netic fields is an important example for a quantum sys-tem accessible both in experiments and numerical calcu-lations. Adding external fields to the Coulomb poten-tial of the hydrogen atom opens the possibility for wavefunctions to be localized far away from the nucleus. Theexistence of such localized states in quantum systems hasattracted the attention of both theoretical [1–4] and ex-perimental [5, 6] investigations over a long period of time.First investigations considered a gauge dependent variantof the system, in which the paramagnetic term was omit-ted [1, 2, 5] due to its smallness as compared with thediamagnetic term at large field strengths. The resultingsum potential possesses a minimum away from the nu-cleus in which localized states can exist. However, if onetakes into account all terms of the magnetic field, onecan immediately see that the outer potential minimumdoes not exist in the real physical system, rather, onefinds classical electron motion close to the Stark saddlepoint which is bound in two directions and unbound inthe third direction. In fact, localized resonance quan-tum states in the vicinity of the saddle point have beenpredicted by Clark et al. [4] as quantized version of thisquasi-bound motion.The question whether or not resonances located at theStark saddle point do exist has also a crucial meaningfor the ionization mechanism. Classical electron pathsdescribing an ionization must pass the vicinity of theStark saddle. This process was investigated, e.g., usingthe transition state theory [7–11]. The transition statetheory is applicable to many dynamical systems whichevolve from an initial to a final state and has its origin inthe calculation of chemical reaction rates. The concept isbased on classical trajectories which describe a reactionby passing a boundary in phase space, viz. the transitionstate, which must be crossed by all trajectories connect-ing the initial (“reactants”) and final (“products”) side.For the hydrogen atom in crossed fields the theory postu- ∗ Electronic address: [email protected] lates classical orbits confined in the vicinity of the Starksaddle point [7].The work by Clark et al. [4] can be regarded as a firststep in discussing the ionization mechanism of the hydro-gen atom in crossed electric and magnetic fields in thiscontext. The quasi-bound states confined to the vicin-ity of the saddle found by Clark et al. coincide with anapproximation to the Hamiltonian in the framework ofthe transition state theory, however, only an algorith-mic procedure based on a normal-form representation ofa power-series expansion of the Hamiltonian allowed foridentifying the transition state [7, 10, 11]. Even thoughthe transition state for the hydrogen atom in crossedfields has been found, the question of whether or notthe classical trajectories in its vicinity leave a signaturein the exact quantum spectrum remained unanswered fora long time. Finally, clear evidence for signatures in theenergies, i.e., the eigenvalues of the quantum resonances,was found [12].It is the purpose of this paper to demonstrate thatthere is, indeed, a relationship between the exact quan-tum resonances and the quantized energy levels of theelectron motion near the transition state due to a defi-nite spatial restriction of the resonances’ probability den-sity to a small region around the saddle. We calculatethe position space representation of the correspondingwave functions and are able to show that they are clearlylocated in a vicinity of the Stark saddle point. From aquadratic approximation of the potential around the sad-dle point one expects a harmonic oscillator spectrum ofenergy levels and wave functions. We even find nodalpatterns in the exact quantum states which correspondto “excited” states of the quadratic approximation, andthus demonstrate the usefulness of the simple approxi-mation and the existence of the resonances predicted bya classical treatment of the system [4, 7].In Sec. II we introduce the system as well as the second-and third-order power-series expansions of the potentialin the vicinity of the Stark saddle point. Position spacerepresentations of exact quantum wave functions whichprove the existence of resonances located closely at theposition of the saddle are presented in Sec. III. Conclu-sions are drawn in Sec. IV.
II. HAMILTONIAN AND APPROXIMATIONSIN THE VICINITY OF THE STARK SADDLEPOINTA. Hamiltonian and exact quantum calculations
In Hartree units the Hamiltonian of a hydrogen atom incrossed external fields with an electric field f orientatedalong the x -axis and a magnetic field along the z -axisrepresented by the vector potential A has the form H = 12 ( p + A ) − r + f x. (1)The parity with respect to reflections at the ( z = 0)-planeis a good quantum number and allows considering stateswith even and odd z -parity separately.It is a common and efficient method to rewrite theSchr¨odinger equation for numerical calculations in di-lated semiparabolic coordinates [12, 13], µ = 1 b √ r + z, ν = 1 b √ r − z, ϕ = arctan yx , (2)which yields (cid:26) ∆ µ + ∆ ν − (cid:0) µ + ν (cid:1) + b γ (cid:0) µ + ν (cid:1) i ∂∂ϕ − b γ µ ν (cid:0) µ + ν (cid:1) − b f µν (cid:0) µ + ν (cid:1) cos ϕ (cid:27) ψ = (cid:8) − b + λ (cid:0) µ + ν (cid:1)(cid:9) ψ (3)with ∆ ̺ = 1 ̺ ∂∂̺ ̺ ∂∂̺ + 1 ̺ ∂ ∂ϕ , ̺ ∈ { µ, ν } , (4)and the generalized eigenvalues λ = − (1 + 2 b E ), whichare related to the energies E of the quantum states. Thecalculation of the resonances is done with a diagonaliza-tion of a matrix representation of the Schr¨odinger equa-tion (3) and the complex rotation method [14–16]. Thenecessary complex scaling of the coordinates r is intro-duced via the complex convergence parameter b = | b | e i ϑ/ . (5)Resonances appear as complex eigenvalues E , where thereal part of E represents the energy and the imaginarypart is related to the width Γ = − E ).An adequate complete basis for the matrix representa-tion of the Schr¨odinger equation (3) is given by | n µ , n ν , m i = | n µ , m i ⊗ | n ν , m i , (6)where | n ̺ m i are the eigenstates of the two-dimensionalharmonic oscillator. The position space representation indilated semiparabolic coordinates has the form ψ n µ n ν m ( µ, ν, ϕ ) = s [( n µ − | m | ) / n µ + | m | ) / n ν − | m | ) / n ν + | m | ) / × r π f n µ m ( µ ) f n ν m ( ν )e i mϕ (7a) with the “radial” wave functions f nm ( ̺ ) = e − ̺ / ̺ | m | L | m | ( n −| m | ) / ( ̺ ) (7b)expressed in terms of Laguerre polynomials L αn ( x ). Dueto the complex scaling µ and ν become complex coor-dinates. In properly complex conjugated wave functionsobtained with the complex rotation method one has tobear in mind that only the intrinsically complex partshave to be conjugated [16, 17], i.e., in the complex con-jugated version ψ ∗ n µ n ν m of the wave function (7a) e i mϕ is replaced with e − i mϕ and the “radial” parts f n µ m ( µ ), f n ν m ( ν ), which become complex only by the complexscaling, are not conjugated.It is very effective to perform the matrix setup of theSchr¨odinger equation (3) completely algebraically by ex-pressing Eq. (3) in terms of harmonic oscillator creationand annihilation operators. Consequently, the expansioncoefficients c in µ n ν m of the eigenstates obtained in a ma-trix diagonalizationΨ i ( µ, ν, ϕ ) = X n µ ,n ν ,m c in µ n ν m ψ n µ n ν m ( µ, ν, ϕ ) , (8a)Ψ ∗ i ( µ, ν, ϕ ) = X n µ ,n ν ,m c in µ n ν m ψ ∗ n µ n ν m ( µ, ν, ϕ ) (8b)belong to position space wave functions (7a), which arenormalized eigenstates of two coupled two-dimensionalharmonic oscillators, ∞ Z d µ ∞ Z d ν π Z d ϕ µν ψ ∗ n µ n ν m ( µ, ν, ϕ ) ψ n ′ µ n ′ ν m ′ ( µ, ν, ϕ )= δ n µ n ′ µ δ n ν n ′ ν δ mm ′ , (9)but are not orthogonal and not normalized in the physicalposition space r = ( x, y, z ) T . The eigenstates (8a) and(8b) of the Schr¨odinger equation (3), however, can benormalized in such a way that Z d r Ψ ∗ i ( µ, ν, ϕ )Ψ j ( µ, ν, ϕ )= b ∞ Z d µ ∞ Z d ν π Z d ϕ µν ( µ + ν ) Ψ ∗ i Ψ j = δ ij . (10)Since the proper complex conjugation of the wave func-tions effects only intrinsically complex parts, their squaremoduli must be replaced with Ψ ∗ j Ψ j as introduced in Eqs.(8a) and (8b) to obtain a measure for the probabilitydensity. For wave functions of decaying states this prod-uct will be complex, and we will visualize the modulus | ψ ∗ n µ n ν m ψ n µ n ν m | in Sec. III to show where the resonancewave functions are located. -100 -50 0 50 100-100-50 0 50 100-0.15-0.10-0.05 0.00 0.05 V ( ξ , y ,0) [a.u.](b) ξ [a.u.] y [a.u.] V ( ξ , y ,0) [a.u.] -100 -50 0 50 100-100-50 0 50 100-0.15-0.10-0.05 0.00 0.05 V ( ξ , y ,0) [a.u.](a) ξ [a.u.] y [a.u.] V ( ξ , y ,0) [a.u.] FIG. 1: (Color online) Potential of the hydrogen atom inan external electric field (dashed red grids in (a) and (b))in comparison with the second-order (a) and third-order (b)power-series expansions around the saddle in the ( ξ , y )-planewith ξ = x − x s . Both approximations reproduce the potentialthe vicinity of the saddle at the origin correctly, however, bothare only valid very close to the saddle point. B. Approximations in the vicinity of the saddlepoint
To investigate resonances located at the saddle point,the electric potential of the combined contributions of thenucleus and the external electric field, V f = − r + f x, (11)is expanded into a series up to third-order. As wasshown in Ref. [12], already the second-order approxima-tion leads to a good one-to-one correspondence with ener-gies of exact resonances of the full Hamiltonian (1). Here,we additionally consider an approximation including allthird-order terms to estimate the quality of the approxi-mation carried out in the vicinity of the saddle point bycomparing the second- and third-order results. The sad-dle point has the coordinates r s = ( − / √ f , , T andits energy has the value V f ( r s ) = − √ f . Using the co-ordinate shift ξ = x − x s a power-series expansion around r s yields V f ( r ) = − p f − p f ξ + 12 p f (cid:0) y + z (cid:1) − f ξ + 32 f ξ ( y + z ) + O(( r − r s ) ) . (12)In Fig. 1 the second- and third-order power-series expan-sions around the saddle point and the full electric po-tential (11) are compared in the ( ξ , y )-plane. The saddle structure is clearly visible and the figure demonstratesthat both approximations are only valid very close to thesaddle point, and thus can describe resonances correctlyonly in its vicinity.The approximated Hamiltonian close to r s reads H = 12 (cid:0) p ξ + p y + p z (cid:1) − γyp ξ + 12 γ y − p f + 12 p f (cid:0) y + z − ξ (cid:1) − f ξ + 32 f ξ ( y + z ) + O(( r − r s ) ) , (13)where the gauge A = ( − γy, ,
0) was used as in Refs.[4, 7, 12], since it leads to a simple structure of the terms.Introducing complex scaling parameters s i via ξ = Q ξ /s ξ , p ξ = s ξ P ξ , (14a) y = Q y /s y , p y = s y P y , (14b) z = Q z /s z , p z = s z P z (14c)we calculate the resonances of the Hamiltonian with thecomplex rotation method. To do so, a matrix represen-tation with a basis of products | n ξ , n y , n z i = | n ξ i ⊗ | n y i ⊗ | n z i (15)of one-dimensional harmonic oscillator states | n i i is builtup and diagonalized.In the much simpler quadratic approximation, i.e.,in the case in which the two third-order terms V = − f ξ + 3 f ξ ( y + z ) / E n z ,n ,n = − p f + ω z (cid:18) n z + 12 (cid:19) + ω (cid:18) n + 12 (cid:19) + ω (cid:18) n + 12 (cid:19) , (16)where all frequencies ω i can be calculated analytically[4, 7, 12]. It includes one inverted oscillator, which leadsto a purely imaginary frequency ω , and thus representsthe resonance character of the states. The frequency ω z belongs to the z motion of the electron, whereas ω and ω belong to new variables introduced via a canonicaltransformation [7] to separate the x and y motions. III. RESULTS AND DISCUSSION
In Ref. [12] it was shown that the energies of some ofthe resonances calculated in the quadratic approximationaround the saddle point agree very well with the exactquantum energies over a large range in the parameterspace. The region considered was defined by lines, i.e.,one-dimensional objects, in the two-dimensional param-eter space to obtain clearly identifiable results. In this -0.030-0.028-0.026-0.024-0.022-0.020-0.018-0.016-0.0140.0 0.2 0.4 0.6 0.8 1.0 R e ( E ) [ a . u . ] α (n z =0, n =0)(2, 0)(0, 1)(4, 0)(0, 2)(a) A1A2A3A4A5 C1C2EF-0.030-0.028-0.026-0.024-0.022-0.020-0.018-0.016-0.0140.0 0.2 0.4 0.6 0.8 1.0 R e ( E ) [ a . u . ] α (n z =1, n =0)(3,0)(1,1)(b) BDG FIG. 2: (Color online) Comparison of the resonance energies(real parts of the complex energy eigenvalues) obtained in ex-act solutions of the full Hamiltonian (1) (red dotted lines)and the second- (solid blue lines) and third-order (filled greensquares) approximations. The results are shown for even (a)and odd (b) z parity separately. Position space represen-tations of the exact quantum resonances labeled by capitalletters are shown in Figs. 3 to 6. In (a) and (b) the dashedgrey lines represent the saddle point energy. paper we chose one of these lines, viz. γ = 0 . × α, (17a) f = 0 . × α, (17b)0 < α < , (17c)on which all parameters used in what follows are located.Figure 2 shows how the real parts of the resonance ener-gies behave on this line. The red dotted lines representthe exact quantum solution of the full Hamiltonian (1),the solid blue lines and the filled green squares denotethe second- and third-order energies, respectively, andthe saddle point energy is marked by the dashed greylines. Results are shown for even [Fig. 2(a)] and odd[Fig. 2(b)] z parity.As was already pointed out in Ref. [12] some of thesecond-order resonances are traced by the exact solu-tions of (1). Here we see that this is in particular true forall resonances of the second-order power-series expansion(solid blue lines in Fig. 2), which have a clear correspon-dence in the third-order results (filled green squares in FIG. 3: (Color online) Density plots of the resonance labeledA1 in Fig. 2 in the planes z = 0, y = 0, and x = x s . Thesolid magenta lines mark the saddle point. One can clearlysee that the probability density of the resonance is restrictedto a close vicinity around the saddle point. Fig. 2), i.e., which have almost the same energies in bothapproximations for all parameters α . In other words, ap-proximated resonances in the vicinity of the Stark saddlepoint whose energies seem already to be converged inthe two lowest-order power-series expansions have alsoa counterpart in the exact spectrum of the full quan-tum system described by the Hamiltonian (1). In thesecases the simple power-series expansions seem to pro-vide a good approximation for resonances located at thesaddle. Due to the very good agreement of the second-order and the exact quantum results in a large regionof the parameter space it was already concluded in Ref.[12] that the corresponding exact crossed-fields hydrogenatom resonances must be closely related with localizedstates predicted by Clark et al.In this paper we show using the position space repre-sentation of the exact quantum wave functions that theyare clearly centered at the Stark saddle point in all cases,in which the good agreement in the energies describedabove appears. The best correspondence is expected forthe lowest transition state resonance with quantum num-bers n z = 0, n = 0 in the second-order approximationand for the largest field strengths. The position spacewave function of this resonance for α = 1, i.e., γ = 0 . f = 0 . z = 0, y = 0, and x = x s . All three sections showthat the probability density of the resonance is centeredat and restricted to a close vicinity of the saddle point,which is marked by the solid magenta lines. The trans-formation of the quantum resonances corresponding tothe same second-order line for decreasing field strengths FIG. 4: (Color online) Density plots of the resonances labeledA2, A3, A4, and A5 in Fig. 2 in the z = 0 plane. For decreas-ing field strengths one can observe that the resonances aredecreasingly centered at the saddle point. can be observed in Fig. 4, where examples for α = 0 . α = 0 . α = 0 .
3, and α = 0 . z = 0plane. Between α = 1 (resonance labeled A1 in Fig.2) and α = 0 . α = 0 . α = 0 . α = 0 . n z quantum number. In the second-order approx-imation these energies correspond to excited harmonicoscillator states in z direction. The resonance labeled Brepresents the quantum number n z = 1, i.e., the first ex-cited state. One nodal line at z = 0 is expected in the sec-tion and can be observed in Fig. 5. One may argue that anodal plane at z = 0 is not surprising for a state with odd z parity, however, further excitations whose nodal pat-terns reproduce the expectations correctly can be found.Comparing the second-order power-series result for quan-tum numbers n z = 2, n = 0 and its third-order counter-part in Fig. 2(a) we find two possibilities to assign exactquantum resonances. Indeed, we find the two states la-beled C1 and C2 whose probability densities are locatedvery close to the saddle point. Both of these resonancesshow the two nodal lines expected for n z = 2 and demon- FIG. 5: (Color online) Density plots of the resonances B, C1,C2, and D (cf. Fig. 2) in the z = 0 plane. The resonances canbe assigned to second-order approximations with n = 0 and n z = 0. The nodal patterns expected for excited states in z direction can be observed. strate that a rich spectrum of states dominated by thelocal shape of the potential around the Stark saddle pointis present. The resonances C1 and C2 have almost thesame real part of the energy but differ significantly intheir imaginary parts. One can assume that these reso-nances can be assigned to two second-order states with n z = 2, n = 0 and different n values, which do notinfluence the real energy in the second-order approxima-tion. A definite answer, however, is not possible becausethe excellent agreement in the real parts of the energies ofthe approximated states with the exact quantum resultsof the full Hamiltonian (1) does not hold for the imag-inary parts [12]. The imaginary parts of the energiesobtained with the second- and third-order power-seriesexpansions of the potential do not agree as well as theirreal parts and cannot be considered converged in thesesimple approximations. Their quality does not allow fora comparison with their numerically exact counterparts.Note that the exact computations with the complex ro-tation method provide accurate results for both the realand imaginary parts of the energies. Finally one can evenobserve the resonance labeled D with three nodal lines asone would expect for a n z = 3 state, and indeed, the res-onance energy can be mapped to that of the second-orderstate n z = 3, n = 0. The line belonging to resonance n z = 4, n = 0 in Fig. 2(a) does not seem to have acorrespondence in the third-order approximation and itwas not possible to find exact quantum resonances withthe proper nodal structure.Further examples including excitations n = 0 canbe found in Fig. 6. Due to the energy diagram in Fig.2(a) resonance E can be assigned to quantum numbers n z = 0, n = 1. The position space representation inFig. 6 reveals a complicated shape of the probability den-sity, which is not very surprising as the excitation in n belongs to a coordinate whose origin is in a canonical FIG. 6: (Color online) Density plots of the resonances E, F,and G (cf. Fig. 2) corresponding to states of the quadraticapproximation excited in n . Complicated nodal patterns arefound. transformation including the positions x , y as well as themomenta p x and p y . However, the resonance is clearlylocated close to the position of the saddle. This is alsotrue for resonance F, which is assigned to the second-order quantum numbers n z = 0, n = 2 and shows evena more complicated nodal pattern in position space. Res-onance G is connected with an excitation in both n z and n ( n z = 1, n = 1) and exhibits a nodal plane at z = 0due to the respective quantum number n z = 1. IV. CONCLUSION
By determining the position space representation ofquantum mechanically exact wave functions we haveproved the existence of resonances located in a closevicinity of the Stark saddle point which were predictedby simple classical approximations as, e.g., the transitionstate theory. The results show that near the saddle onecan find a spectrum of several states belonging to thelocal neighborhood of the saddle and ignoring the globalstructure of the Coulomb potential. It reproduces the expectations of a simple quadratic power-series expan-sion around the saddle in the energies as well as in theshape of the wave functions. In particular, one can, inthe quadratic approximation, separate three one dimen-sional harmonic oscillators one of which is connected withthe z direction. In this spatial direction we found nodalplanes of resonances which have the structure of groundand excited states from n z = 0 up to n z = 3. Com-plicated nodal patterns are found for excitations in thequantum number n not directly connected to a spatialcoordinate.The comparison of the two approximations of the reso-nance energy eigenvalues (cf. Fig. 2) showed that some ofthe second-order energies have a distinct correspondencein the third-order approximation while some have not.In all cases in which an assignment of the results of bothapproximations is possible we were also able to detectan exact quantum resonance with a strong localizationat the saddle, whereas such a strong connection to thesaddle could not be observed for exact resonances not be-longing to a pair of second- and third-order resonances.Despite this clear result it must be noted that, as wasalready pointed out in a previous publication [12], thesimple power-series expansions are not capable of repro-ducing the resonance widths (or imaginary parts of theenergy eigenvalues) correctly. We have demonstrated,however, that a convergence of the approximated reso-nance energies (or real parts) is already a strong signa-ture of the existence of exact quantum states centered atthe saddle.The results prove that resonances located at an outersaddle predicted in a variety of theoretical work [1, 2, 4,5, 7] and supported by experimental results [6] do existin the Coulomb potential superimposed by two externalfields. The correspondence already appears for very sim-ple second- and third-order power-series expansions ofthe potential at the saddle point. For future work com-parisons with more thorough approximations applicableto the problem, e.g., the normal form expansion devel-oped for identifying the classical transition state in high-dimensional systems [7, 10, 11] or its quantum analogue[11, 18] will be of high value. [1] S. K. Bhattacharya and A. R. P. Rau, Phys. Rev. A ,2315 (1982).[2] J. C. Gay, L. R. Pendrill, and B. Cagnac, Phys. Lett. A , 315 (1979).[3] A. R. P. Rau and L. Zhang, Phys. Rev. A , 6342 (1990).[4] C. W. Clark, E. Korevaar, and M. G. Littman, Phys.Rev. Lett. , 320 (1985).[5] M. Fauth, H. Walther, and E. Werner, Z. Phys. D , 293(1987).[6] G. Raithel, M. Fauth, and H. Walther, Phys. Rev. A ,419 (1993).[7] T. Uzer, C. Jaff´e, J. Palaci´an, P. Yanguas, and S. Wig- gins, Nonlinearity , 957 (2002).[8] C. Jaff´e, D. Farrelly, and T. Uzer, Phys. Rev. Lett. ,610 (2000).[9] C. Jaff´e, D. Farrelly, and T. Uzer, Phys. Rev. A , 3833(1999).[10] S. Wiggins, L. Wiesenfeld, C. Jaff´e, and T. Uzer, Phys.Rev. Lett. , 5478 (2001).[11] H. Waalkens, R. Schubert, and S. Wiggins, Nonlinearity , R1 (2008).[12] H. Cartarius, J. Main, and G. Wunner, Phys. Rev. A ,033412 (2009).[13] J. Main and G. Wunner, J. Phys. B , 2835 (1994). [14] W. P. Reinhardt, Ann. Rev. Phys. Chem. , 223 (1982).[15] D. Delande, A. Bommier, and J. C. Gay, Phys. Rev. Lett. , 141 (1991).[16] N. Moiseyev, Phys. Rep. , 212 (1998).[17] T. N. Rescigno and V. McKoy, Phys. Rev. A , 522 (1975).[18] R. Schubert, H. Waalkens, and S. Wiggins, Phys. Rev.Lett.96