Mixed gauge in strong laser-matter interaction
aa r X i v : . [ phy s i c s . c o m p - ph ] A ug Mixed gauge in strong laser-matter interaction
Vinay Pramod Majety, Alejandro Zielinski, and Armin Scrinzi ∗ Physics Department, Ludwig Maximilians Universit¨at, D-80333 Munich, Germany (Dated: February 15, 2018)We show that the description of laser-matter interaction in length gauge at shortshort and in velocity gauge at longer distances allows for compact physical model-ing in terms of field free states, rapidly convergent numerical approximation, andefficient absorption of outgoing flux. The mathematical and numerical frameworkfor using mixed gauge in practice is introduced. We calculate photoelectron spectragenerated by a laser field at wavelengths of 400 ∼
800 nm from single-electron sys-tems and from the helium atom and hydrogen molecule. We assess the accuracyof coupled channels calculations by comparison to full two-electron solutions of thetime-dependent Schr¨odinger equation and find substantial advantages of mixed overvelocity and length gauges.
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I. INTRODUCTION
The choice of gauge in the interaction of strong, long wave-length fields with atoms andmolecules affects the physical modeling [1], perturbative expansions as the S -matrix series[2, 3], as well as the efficiency of numerical solutions [4]. For systems where field quantizationcan be neglected and the field appears only as a time- and space-dependent external pa-rameter, the wavefunctions in all gauges are unitarily related by time- and space-dependentmultiplicative phases. An extensive discussion of gauge transformations in the context ofstrong field phenomena can be found in [5]. When approximations are made, the unitaryequivalence of the wave-functions and the corresponding time-dependent Schr¨odinger equa-tion (TDSE) is lost. An important example is the strong field approximation, where thesystem is assumed to either remain in the field-free initial state or move exclusively under ∗ Electronic address: [email protected] the influence of the laser field: the function representing the field-free initial state dependson gauge. A similar situation arises, when a series expansion is truncated to a finite numberof terms, as in an S -matrix expansion: the physical meaning of any finite number of termsis different in different gauges. Also the discretization errors in a numerical calculationare gauge dependent. In particular, multiplication by a space-dependent phase changes thesmoothness of the solution. As a result, numerical accuracy and convergence are depend ongauge.Mathematical and numerical aspects of using general gauges were adressed in Refs. [6–8] inthe context of Floquet theory and the time dependent Schr¨odinger equation, where variousoptions for mixing different gauges were discussed. In Refs. [6, 7], mixing length, velocity,acceleration gauge in the R-matrix Floquet method was achieved by the introduction ofBloch operators at the boundaries between the gauges. In Ref. [8], it was pointed out thatalternatively the transition between regions can be taken to be differentiably smooth, whichalso allows application to the time-dependent Schr¨odinger equation (TDSE).Here we will show that physical modeling on the one hand and efficient numerical solutionon the other hand impose conflicting requirements on the choice between the standard lengthand velocity gauges. We introduce the mathematical and numerical techniques for resolvingthis conflict by using general gauges. We restrict our discussion to gauge transformationsin the strict sense, i.e. local phase multiplications, which does not include the acceleration“gauge”, as it involves a time-dependent coordinate transformation. Numerical performanceof the various gauges is compared on a one-dimensional model system. We show that, alsowith discontinuous transition between gauges, there is no need for the explicit inclusionof the δ -like Bloch operators. We demonstrate validity and accuracy of mixed length andvelocity gauge calculations in three dimensions by comparing to accurate velocity gaugeresults for the hydrogen atom at 800 nm wavelength. Finally, we combine local length gaugewith asymptotic velocity gauge to compute photoelectron spectra of He and H at a laserwavelength of 400 nm. Efficiency and accuracy of the approach is shown by comparingto complete numerical solutions of the two-electron problem. We find that mixed gaugeallows low-dimensional approximations, while in velocity gauge we achieve convergence onlywhen we allow essentially complete two-electron dynamics. We will conclude that few-bodydynamics in the realm of bound states is more efficiently represented in length gauge, whilethe long-range representation of the solution prefers the velocity gauge. II. LENGTH, VELOCITY, AND GENERAL GAUGES
In the interaction of small systems of sizes of . . nm with light at wavelength down tothe extreme ultraviolet λ & nm one employs the dipole approximation, i.e. one neglectsthe variation of the field across the extension of the system ~ E ( ~r, t ) ≈ ~ E ( t ). In length gauge,the interaction of a charge q with the dipole field is I L ( t ) = q ~ E ( t ) · ~r, (1)while in velocity gauge it is I V ( t ) = − ~A ( t ) · ~p + 12 | ~A ( t ) | , ~A ( t ) := Z t −∞ q ~ E ( τ ) dτ. (2)Here and below we use atomic units with ~ = 1, electron mass m e = 1, and electron charge e = −
1, unless indicated otherwise. In these two gauges the dependence of the dipoleinteraction operators on ~r is particularly simple and wavefunctions are unitarily related byΨ V ( ~r, t ) = e i ~A ( t ) · ~r Ψ L ( ~r, t ) . (3)The transformation from length to velocity gauge is a special case of the general gaugetransformation, namely multiplication by a space- and time-dependent phaseΨ g = U g Ψ , U g := e ig ( ~r,t ) . (4)As U g is unitary, it leaves the system’s dynamics unaffected, if operators and the time-derivative are transformed as b O → O g = U g OU ∗ g , ddt → ddt + U g ˙ U ∗ g = ddt − i ˙ g. (5)The above relations are valid for general g ( ~r, t ) that are differentiable w.r.t. t . If g is twicedifferentiable in space, the gauge transforms of momentum operator and Laplacian are ~p = − i ~ ∇ → ~p g = − i ~ ∇ − ~B, ∆ → ∆ g = [ − i ~ ∇ − ~B ] , ~B := ~ ∇ g. (6)We see, in particular, that a gauge transform introduces a time- and space-dependent mo-mentum boost ~B ( ~r, t ).A standard TDSE transforms as i ddt Ψ = (cid:20) ~p V + q ~ E ( t ) · ~r (cid:21) Ψ → i ddt Ψ g = " ( ~p − ~B ) · ( ~p − ~B )2 + V + q ~ E ( t ) · ~r − ˙ g Ψ g ( ~r, t ) . (7)By explicitly writing the dot-product in the kinetic energy we emphasize that ~p does notcommute with space-dependent ~B ( ~r, t ) and space derivatives of ~B appear in the Hamiltonian.The velocity gauge interaction Eq. (2) requires spatially uniform ~B ( ~r, t ) ≡ ~A ( t ). Moregenerally, any time-dependence of the potential energy V ( ~r, t ) can be transformed into atime- and space-dependent momentum by defining g V ( ~r, t ) = Z t V ( ~r, t ′ ) dt ′ . (8) A. Discontinuous gauge transformation
The local phase multiplication need not be continuously differentiable or even continuousin space. One only must make sure that the gauge tranformed differential operators ~ ∇ g aredefined on functions χ from a suitable domain D ( ~ ∇ g ). With discontinuous g , formally, δ -likeoperators appear in Eq. (7). D ( ~ ∇ g ) must be adjusted to compensate for those terms. Thevery simple, mathematically correct solution is to choose D ( ~ ∇ g ) = U g D ( ~ ∇ ), i.e. functionsof the form χ ( ~r, t ) = U g ( ~r, t ) ϕ ( ~r ) , ϕ ∈ D ( ~ ∇ ) , (9)where the ϕ ( ~r ) ∈ D ( ~ ∇ ) are differentiable.With g ( ~r, t ) = 0 at ranges | ~r | < R g and g ( ~r, t ) → ~A ( t ) · ~r for large | ~r | , one can switch fromlenght to mixed gauge. Clearly, the particular form of the transition and the correspondingmodulations of the wavefunction do not bear any physical meaning. Still, accurate modelingof the transition is needed to correctly connect the length to the velocity gauge part of thesolution. In the transition region one needs to densly sample the solution, which increasethe number of expansion coefficients. In many cases, this will also increase the stiffness ofthe time propagation equations and further raise the penalty for a smooth transition.When g or any of its derivates is discontinuous, the discretization in the vicinity of thediscontinuity must be adjusted appropriately. Because of the lack of differentiability, anyhigher order finite difference scheme or approximations by analytic basis functions will failto improve the approximation or may even lead to artefacts. The general solution for thisproblem is to explicitly build the known non-analytic behavior of the solution into thediscretization. With the finite element basis set used below, this is particularly simple, aswell-defined discontinuities can be imposed easily. We will also demonstrate below that aspatially abrupt transition does not increase stiffness and allows calculations with similarefficiency as in uniform velocity gauge. B. Gauge in the strong field approximation (SFA)
When we describe a physical process in terms of a few quantum mechanical states itis implied that the system does not essentially evolve beyond those states. The “strongfield approximation” (SFA) is a simple model of this kind, which plays a prominent rolein strong field physics. One assumes that an electron either resides in its initial state or,after ionization, moves as a free particle in the field, whose effect largely exceeds the atomicbinding forces.The SFA must be reformulated appropriately depending on the gauge on chooses. LetΦ ( ~r ) be the initial state in absence of the field. The physical picture above implies thatthe velocity distribution of the initial state remains essentially unchanged also in presenceof the field. However, using the same function Φ for all gauges, effectively leads to a setof different models with different, time-dependent velocity distributions for different gauges.In length gauge the operator − im e ~ ∇ has the meaning of a velocity of the electron and thevelocity distribution is independent of the field: n L ( ~p, t ) ≡ n ( ~p ) = | ˜Φ ( ~p ) | . (10)In contrast, in velocity gauge, the velocity distribution varies with time as n V ( ~p, t ) = 1 m e | ˜Φ [ ~p + ~A ( t )] | . (11)The difference becomes noticeable when the variation of ~A ( t ) is not negligible compared tothe width of the momentum distribution. This is typically the case in strong field phenom-ena. Findings that SFA in length gauge better approximates the exact solution in caseswhere the picture remains suitable at all [9, 10] are consistent with this reasoning. C. Single active electron (SAE) approximation
The gauge-dependent meaning of eigenstates has important consequences for the numericalapproximation of few-electron systems. The functions corresponding to few-electron boundstates have their intended physical meaning only in length gauge. In velocity gauge, thesame functions correspond to time-varying velocity distributions. The problem affects the“single active electron” (SAE) approximation, where one lets one “active” electrons freelyreact to the laser field, but freezes all other electrons in their field-free states. Below we willdemonstrate that this ansatz generates artefacts in velocity gauge.As a simple illustration of the problem let us consider two non-interacting electronswith the Hamiltonian H ( x, y ) = h ( x ) + h ( y ). The ansatz for the solution Φ( x, y, t ) = ϕ ( x, t ) χ ( y, t ) − χ ( x, t ) ϕ ( y, t ) is exact, if the functions ϕ and χ are unrestricted. Matrixelements of the Hamiltonian are h Φ | H | Φ i = h ϕ | h | ϕ ih χ | χ i + h ϕ | ϕ ih χ | h | χ i (12) −h ϕ | h | χ ih χ | ϕ i − h ϕ | χ ih χ | h | ϕ i . (13)In an exact calculation, the exchange terms in the second line vanish, if h ϕ | χ i = 0 initially,as the unitary time evolution maintains orthogonality. However, if we restrict the timeevolution of one of the functions, say ϕ , orthogonality is violated and unphysical exchangeterms appear in the Hamiltonian matrix as the system evolves. Their size depends on theextend to which orthogonality is lost. If e.g. ϕ remains very close to its field-free state(e.g. if it is closely bound), then in length gauge the time evolution is well approximated as ϕ ( t ) ≈ ϕ (0). However, depending on the size of ~A ( t ), in velocity gauge this does not holdand the exchange terms become sizable.With interacting electrons, the direct term (Hartree potential) of electron-electron inter-actions is unaffected, as it only depends on the gauge-invariant electron density. In theexchange terms, however, the frozen orbitals with their length gauge meaning are inconsis-tently combined with the velocity gauge functions of the active electron.The same gauge dependence appears also when the non-active electrons are not frozenin their initial states, but restricted in their freedom to evolve. We will demonstrate thesuperiority of length gauge for He and H with limited freedom for the non-active electron. D. Gauge in numerical solutions
While length gauge lends itself to intuitive interpretation and modeling, velocity gaugeperforms better in numerical calculations [4]. Fewer discretization coefficients can be usedand the stiffness of the equations is reduced. This is due to the dynamics of free electronsin the field. From Eq. (7) one sees that for a free electron ( V = 0) the velocity gaugecanonical momentum ~p = − i ~ ∇ is conserved. In contrast, in length gauge, momenta areboosted by ~A ( t ), reflecting the actual acceleration of the electron in the field. As largemomenta correspond to short range modulations of the solution, length gauge requires finerspatial resolution than velocity gauge. This modulation affects numerical efficiency, whenthe variation of ~A ( t ) is comparable or exceeds the momenta occurring in the field-free system.We will illustrate this below on one- and three-dimensional examples.A second important reason for velocity gauge in numerical simulation is the use of infiniterange exterior complex scaling (irECS) [11] for absorption at the box boundaries. Thismethod is highly efficient and free of artefacts, but it cannot be applied for systems withlength gauge asymptotics, as clearly observed in simulations [12]. An intuitive explanationof this fact can be found in [13] and the wider mathematical background is laid out in [14]. E. Mixed gauge
The conflicting requirements on gauge can be resolved by observing that bound statesare, by definition, confined to moderate distances, whereas the effect of phase modulation isimportant for free electrons, usually far from the bound states. Using length gauge withinthe reach of bound states and velocity gauge otherwise largely unites the advantage of bothgauges: locally, the system can be modeled intuitively, while at the same time maintainingefficient numerical spatial discretization and asymptotics suitable for absorption.
III. IMPLEMENTATIONS AND EXAMPLESA. TDSE in one dimension
We use a basic model for discussing the various options for implementing mixed gauges.We solve the TDSE with the length gauge Hamiltonian H L ( t ) = − ∂ x − √ x + 2 − x E ( t ) . (14)In absence of the field, the ground state energy is exactly − . | j i used for spatially discretizing the TDSE do not need to betwice differentiable. Rather, they can have discontinuous first derivatives. Although thesecond derivative is not defined as an operator on the Hilbert space, matrix elements can becalculated correctly by using the symmetrized form h k | − ∂ x | j i := h ∂ x k | ∂ x j i . (15)This relaxed condition on the differentiability is explicitly used with finite element bases,where usually first derivatives are discontinuous at the boundaries between elements.The ansatz | Ψ( x, t ) i ≈ N X j =1 | j i c j ( t ) (16)leads to the system of ordinary differential equations for the expansion coefficients ~c , ( ~c ) j = c j i ddt b S~c ( t ) = b H ( t ) ~c ( t ) (17)with the matrices b H kj ( t ) = h k | H ( t ) | j i , b S kj = h k | j i . (18)For time-integration, we use the classical 4th order explicit Runge-Kutta solver. As anexplicit method it is easy to apply, but it is also susceptible to the stiffness of the system ofequations (17). This is a realistic setting, as in many practical implementations explicit time-integrators are used. For the present purpose, it clearly exposes the numerical properties ofthe different gauges. In the one-dimensional case we use a simulation box large enough suchthat reflections at the boundary remain well below the error level.
1. Mixed gauge implementations (1d)
Matrix elements of the kinetic energy are always computed in the explicitly hermitianform h k | [ − i∂ x − B ] | j i = h ∂ x k | ∂ x j i − i h ∂ x k | Bj i + i h Bk | ∂ x j i + h k | B | j i , (19)which also avoids the calculation of spatial derivatives of B ( x, t ). Also, for non-differentiable B , no δ -like operators appear.For differentiable functions B ( x, t ) we have domains D (∆ g ) = D (∆) and no adjustmentsneed to be made for the basis functions. To avoid loss of numerical approximation order,one must make sure that B ( x, t ) is smooth to the same derivative order as the the numericalapproximation. We will refer to this case as “smooth switching”.With a discontinuous change of gauge U D = | x | < R g e iA ( t ) x for | x | > R g (20)the Hamiltonian is H D ( t ) = H L ( t ) for | x | < R g H V ( t ) for | x | > R g , (21)where we denote the standard velocity gauge Hamiltonian as H V ( t ). There appear time-dependent discontinuities at the “gauge radius” R g Ψ D ( ± R g + ǫ ) = e iA ( t ) R g Ψ D ( ± R g − ǫ ) . (22)In a finite element basis such discontinuities can be imposed explicitly. Among others it leadsto a time-dependent overlap matrix, whose inverse at each time-step can be obtained at lowcomputational cost by low-rank updates. The technical details on this will be presentedelsewhere.One can avoid discontinuities at R g by “continuous gauge switching” U C = | x | < R g e iA ( t )( x ∓ R g ) for x ≷ ± R g (23)with the Hamiltonian H C ( t ) = H L ( t ) for | x | < R g H V ( t ) ± q E ( t ) R g for x ≷ ± R g . (24)Note that U C is continuous, but not differentiable at R g , which leads to discontinuous firstderivatives in the solution. As discussed above, a finite element basis admits discontinuitiesof the derivatives and there is no penalty in the numerical approximation order, if oneensures that x = ± R g fall onto element boundaries.0The U D and U C formulations are nearly equivalent in their numerical behavior, as U C = U U D , U = | x | < R g e ∓ iA ( t ) R g for x ≷ ± R g . (25)The respective solutions differ only by the phases e ∓ iA ( t ) R g :Ψ C = U Ψ D . (26)Depending on R , the time-dependence of this phase is slow compared to phase-oscillationsdue to high energy content of the solution and does not change stiffness for numericalintegration.In the following, we compare mixed gauge in the three forms given here with pure lengthand velocity gauge calculations.
2. Comparison of the gauges
We compare electron densities n ( x ) at the end of the laser pulse. Size of the spatialdiscretization and the number of time-steps are adjusted to reach the same local error ǫ ( x )in all gauges relative to a fully converged density n ( x ). For suppressing spurious spikes atnear-zeros of the density, we include some averaging into the definition of the error: ǫ ( x ) = 2∆ x | n ( x ) − n ( x ) | / Z ∆ x − ∆ x dx ′ n ( x ′ ) (27)with ∆ x = 1.We use a single cycle 800 nm pulse with cos -shape and peak intensity 2 × W/cm ,which leads to about 25% ionization of this 1-d system. The x -axis is confined to [-1000,1000]with Dirichlet boundary conditions, discretized by finite elements of polynomial order 20.Figure 1 shows results in the different gauges. Velocity gauge requires N ≈ T ≈ ǫ ( x ) . − .Length gauge has the largest discretization with N ≈ T ≈ ∝ N . The actual increase of times steps does not exactly reflect thisbehavior, as a finer discretization is used in the inner region to obtain comparably accurate1initial states in all calculations. Stiffness from this discretization is always present in thecalculations. For the mixed gauge, we use continuous switching Eq. (24) with R g = 5. Withthis small length gauge section, the same discretization as in the velocity gauge can be usedwith N ≈ T ≈ S , using g ( x, t ) = | x | < R g xs ( x ) A ( t ) for | x | ∈ [ R g , R g + S ] xA ( t ) for | x | > R g + S (28) -4 -3 -2 -1 E l e c t r o n d e n s i t y n ( x ) −40 −30 −20 −10 0 10 20 30 40 x (a.u.) -5 -4 -3 R e l a t i v e e rr o r FIG. 1: Electron density of the one-dimensional model system at the end of a single-cycle pulse(see text for exact pulse definition. Upper panel: fully converged velocity gauge calculation withsimulation box size [-1000,1000], finite element order 20, N ≈ T = 12 × time steps. Lower panel: Relative errors Eq. (27) in various gauges. Velocity gauge, N ≈ . × (red line), length gauge, N ≈ . × (green), mixed gauge, R g = 5, N ≈ . × (blue). Errors of velocity and mixed gauge nearly coincide. s ( x ) is a 3rd order polynomial smoothly connecting the length with the velocity gaugeregion.For a smoothing interval S = 5, we need a rather dense discretization by 18th orderpolynomials on the small interval to maintain the spatial discretization error of ≈ − .While this leads only to a minor increase in the total number of discretization coefficients,it significantly increases the stiffness of the equations requiring T = 1 . × time steps.With smoothing S = 10, stiffness is reduced and T ≈ , which still exceeds by 50% thenumber of time steps with continuous, but non-differentiable transition.The dependence on S is not surprising: the correction terms to the kinetic energy involvederivatives of s ( x ), which grow inversely proportional to the size of the transition region,leading to large matrix elements. Thinking in terms of the solution, we need to follow arather strong change in temporal and spatial behavior of the solution, which necessitatesthe dense grid. With the sudden transition, this change is reduced to a single discontinuity,whose behavior we know analytically. It can either be build explictly into the solution, whenusing the discontinuous Hamiltonian H D ( t ), Eq. (21), or be left to be adjusted numericallywith the continuous Hamiltonian H C ( t ), Eq. (24). We conclude that, wherever technicallypossible, a sudden transition is to be preferred. B. Mixed gauge for the Hydrogen atom
The length gauge Hamiltonian for the hydrogen atom in a laser field is H L ( t ) = −
12 ∆ − r − ~ E ( t ) · ~r. (29)The velocity gauge Hamiltonian is H V ( t ) = −
12 [ − i ~ ∇ − ~A ( t )] − r . (30)In three dimensions, problem size grows rapidly and truncation of the simulation volume isadvisable. As the error free absorbing boundary method irECS [11] is incompatible withlength gauge calculations, in this section we only compare velocity to mixed gauge calcula-tions. Following the findings of the one dimensional calculation, we use continuous gaugeswitching for its numerically efficiency and moderate programming effort. In three dimen-3sions, it can be defined as U C = r < R g exp h i ~A ( t ) · ˆ r ( r − R g ) i for r > R g , (31)denoting ˆ r := ~r/r . The corresponding Hamiltonian is H C ( t ) = H L ( t ) for r < R g h − i ~ ∇ − ~B ( ~r, t ) i − r − ~ E ( t ) · ˆ rR g for r > R g . (32)The gradient of the angle-dependent phase introduces an extra quadrupole type coupling: ~B ( ~r, t ) = ~ ∇ h ~A ( t ) · ˆ r ( r − R g ) i = ~A ( t ) (cid:20) − R g r (cid:21) + h ~A ( t ) · ˆ r i ˆ rR g r . (33) H C ( t ) asymptotically coincides with standard velocity gauge as | ~r | tends to ∞ . In an ex-pansion into spherical harmonics, the quadrupole terms introduce additional non-zeros intothe Hamiltonian matrix, which increase the operations count for applying the Hamiltonianby ∼
1. Comparisons
For the numerical solution we use polar coordinates with a finite element basis on theradial coordinate and spherical harmonics for the angular dependence. A discussion of thebasis can be found in [11]. We assume linear polarization and fix the magnetic quantumnumber at m ≡
0. We use a cos -shaped pulse with 3 optical cycles FWHM at centralwavelength λ = 800 nm and and peak intensity 2 × W/cm , which leads to about 16%ionization.We compare the errors of the different gauges in the angle-integrated electron density n ( r )at the end of the pulse and in the photoelectron spectra. The spectra are computed bythe tSURFF method described in Refs. [11, 15]. Errors are again defined relative to a fullyconverged velocity gauge calculation.On the radial coordinate we use 5 finite elements of order 16 up to radius R = 25 in allgauges. Beyond that, the solution is absorbed by irECS. The stronger phase oscillations ofthe length gauge solution requires more angular momenta compared to velocity gauge [4].Figure 2 shows the relative errors in n ( r ) of a velocity gauge calculation with L max = 22angular momenta with mixed gauge calculations at two different gauge radii R g = 5 , L max =430, and R g = 20 , L max = 35. As expected, the mixed gauge calculation needs higher L max as R g increases.The same general error behavior of the different gauges is also found in the photoelectronspectra, Figure 3. Here, the R g = 20 requires even more angular momenta L max = 40. Thismay be due to the particular sensitivity of photoelectron spectra to the wavefunction at theradius where the surface flux is picked up and integrated, in the present case at r = 25. -4 -3 -2 -1 E l e c t r o n d e n s i t y n ( r ) r (a.u.) -5 -4 -3 R e l a t i v e e rr o r FIG. 2: Velocity vs. mixed gauge for the hydrogen atom in three dimensions. Upper panel:electron-density up to the absorption radius R = 25, fully converged calculation. Lower panel:Relative errors Eq. (27) compared with a fully converged calculation. Red: velocity gauge L max =21, green: mixed gauge, R g = 5 , L max = 30, blue: mixed gauge at R g = 20 , L max = 35. Radialdiscretization by N = 80 functions. -7 -5 -3 -1 Sp e c t r a l d e n s i t y σ ( E ) ( a . u . ) Energy E (a.u.) -4 -3 -2 -1 R e l a t i v e e rr o r FIG. 3: Photoelectron spectrum (upper panel). For pulse parameters see text. Lower panel:relative errors. Red: velocity gauge L max = 21, green: mixed gauge, R g = 5 , L max = 30, blue:mixed gauge at R g = 20 , L max = 40. Radial discretization as in Fig. 2. C. Helium atom and H molecule The length gauge Hamiltonian for a two-electron problem interacting with a dipole laserfield is H L ( t ) = X k =1 , " −
12 ∆ k − | ~r k − ~R/ | − | ~r k + ~R/ | − ~ E ( t ) · ~r k + 1 | ~r − ~r | . (34)This includes the H molecule at fixed internuclear distance ~R = (0 , , . au ) and the heliumatom | ~R | = 0. We assume linear polarization in z -direction.We compare total photoelectron spectra. As a reference, we solved the two-electron (2e)TDSE fully numerically in velocity gauge using a single-center expansion. Details of thiscalculation will be reported elsewhere [16]. Photoelectron spectra for the various ionicchannels were computed using the 2e form of tSURFF (see [15]). As 2e calculations are very6challenging at long wavelength, we use a 3-cycle pulse at somewhat shorter wavelength of λ = 400 nm and an intensity of only 1 × W/cm . To facilitate the extraction of photo-electron momenta, all potentials where smoothly turned off beyond distances | ~r i | > a.u. as described in Ref. [15].We compare the 2-electron calculation with a coupled channels computation using theexpansion | Ψ( ~r , ~r , t ) i = | i c ( t ) + X I,j A [ | I i| j i ] c Ij ( t ) , (35)which includes the field-free neutral ground state | i and the ionic states | I i multiplied bythe same single-electron basis functions | j i as for the hydrogen atom. Anti-symmetrizationis indicated by A [ . . . ]. The neutral ground state | i as well as the ionic states | I i wereobtained from the COLUMBUS quantum chemistry package [17]. Calculations were per-formed in velocity and mixed gauge (continuous switching), as described for the hydrogenatom. Details of the coupled channels method will be reported reported elsewhere [18].By the arguments above, in the coupled channels basis we expect the mixed-gauge calcula-tion to converge better than the velocity gauge calculation: the COLUMBUS wavefunctions | i and | I i have their intended physical meaning only in length gauge. Figures 4 and 5confirm this expectation.For helium, Figure 4, the velocity gauge 2e calculation agrees well with the mixed gaugecoupled channels calculation using the neutral and only the 1s ionic state. With the 5 ionicstates with principal quantum number n ≤ ∼
2% for a large part of thespectrum up to 1 au. In contrast, the single-ion velocity gauge calculation is far off. It doessomewhat approach the 2e result when the number of ionic states is increased to includethe ionic states up to n = 3. In velocity gauge, convergence could not be achieved fortwo reasons. One reason is a technical limitation of the coupled channels code, which usesGaussian basis functions that do not properly represent the higher ionic states. The secondreason is more fundamental: in velocity gauge the ionic cores transiently contain significantcontinuum contributions, which are not included in our ionic basis by construction.The error pattern is similar in H , but the accuracy of all calculations is poorer, Fig. 5: 2eand coupled channel mixed gauge calculations qualitatively agree already when only the σ g ionic ground state is included. With the lowest 6 ionic π and σ states the two spectra differ by . ∼ . au is faithfully reproduced7 -10 -9 -8 -7 -6 -5 -4 Sp e c t r a l d e n s i t y σ ( E ) ( a . u . ) Energy E (a.u) -3 -2 -1 R e l a t i v e e rr o r FIG. 4: Photoelectron spectrum of helium at 400 nm. Upper panel, thick line: full 2e calculationin velocity gauge, thin lines coupled channels plus the neutral ground state: include n=1 ionic state(red), n ≤ n ≤ n ≤ in mixed gauge with 6 ionic states. The resonance can be tentatively assigned to the neardegerate second and third Σ + u doubly excited states of H at the internuclear equlibriumdistance of 1.4 au (see Ref. [19]). As a note of caution, the single center expansion usedin the 2e code converges only slowly for H and cannot be taken as an absolute reference.The coupled channels velocity gauge calculation is off by almost two orders of magnitudewhen only a single ionic state is included. With 6 ionic states it compares to the full 2e on asimilar level as the mixed gauge. However, in velocity gauge the resonance is not reproducedcorrectly.For both systems, analogous results were found at shorter wavelength down to λ = 200 nm .8 -5 -4 -3 -2 -1 Sp e c t r a l d e n s i t y σ ( E ) ( a . u . ) Energy E (a.u) -3 -2 -1 R e l a t i v e e rr o r FIG. 5: Photoelectron spectrum of H at 400 nm. Upper panel, thick line: full 2e calculationin velocity gauge, thin lines coupled channels plus the neutral ground state: include only the σ g ionic ground state (red), include the 6 lowest σ and π ionic states (green). Lower panel: relativedifference between mixed gauge and the 2e calculation. Include only σ g ionic ground state (blue)and include the 6 lowest σ and π ionic states (magenta). The dashed line marks the resonanceposition. At even shorter wavelength and realistic laser intensities, gauge questions are less importantas the magnitude of | ~A ( t ) | ∝ λ . IV. CONCLUSIONS
In summary, we have shown that a transition between gauges within the same calculationbears substantial advantages and requires only moderate implementation effort. For low-dimensional problems, the advantage can be technical, such as reducing the size of the spatialdiscretization and the equations’ stiffness. We have shown that with a suitably chosen basis9a sudden, non-differentiable transition from length to velocity gauge is preferable over adifferentiably smooth transition in terms of both, simplicity of implementation and numericalefficiency.Mixed gauge opens the route to a highly efficient, coupled channels type description oflaser-matter interaction. As the meaning of the individual channel functions is gauge-dependent, a finite set of channels leads to gauge-dependent results. We argued that onlyin length gauge the field free ionic eigenfunctions retain their physical meaning in presenceof a strong pulse. In contrast, in velocity gauge the same functions represent a momentum-boosted system with unphysical dynamics. Therefore typical physical models suggest theuse of length gauge. This was clearly demonstrated by a mixed gauge calculation of two-electron systems, where the length gauge region was chosen to cover the ionic channel func-tions: mixed gauge calculations converge with very few channels. Most dramatically, thesingle-ionization spectrum of helium was calculated to .
10% accuracy using only the ionicground state channel. In contrast, in velocity gauge the single channel result is by nearlytwo orders of magnitude away and convergence could not be achieved with up to 9 channels.In pure length gauge a computation is out of reach because of the required discretizationsize.Convergence with only the field-free neutral and very few ionic states can justify a posteri-ori wide-spread modeling of laser-atom interactions in terms of such states. It also supportsthe view that length gauge is the natural choice for this type of models. The convergencebehavior of mixed gauge calculations — possibly contrasted with pure velocity gauge cal-culations — may help to judge the validity of these important models in more complexfew-electron systems.
Acknowledgement
V.P.M. is a fellow of the EU Marie Curie ITN “CORINF”, A.Z. acknowledges support bythe DFG through the excellence cluster “Munich Center for Advanced Photonics (MAP)”,and by the Austrian Science Foundation project ViCoM (F41). A.S. gratefully acknowledges0partial support by the National Science Foundation under Grant No. NSF PHY11-25915. [1] D. Bauer, D. B. Milosevic, and W. Becker, Phys. Rev. A , 23415 (2005).[2] F. H. M. Faisal, Phys. Rev. A , 063412 (2007).[3] Y. V. Vanne and A. Saenz, Phys. Rev. A , 023421 (2009).[4] E. Cormier and P. Lambropoulos, Journal of Physics B: Atomic, Molecular and Optical Physics , 1667 (1996).[5] A. D. Bandrauk, F. Fillion-Gourdeau, and E. Lorin, Journal of Physics B: Atomic, Molecularand Optical Physics , 153001 (2013).[6] P. G. Burke, P. Francken, and C. J. Joachain, Journal of Physics B: Atomic, Molecular andOptical Physics , 761 (1991).[7] M. Dorr, M. Terao-Dunseath, J. Purvis, C. J. Noble, P. G. Burke, and C. J. Joachain, Journalof Physics B: Atomic, Molecular and Optical Physics , 2809 (1992).[8] F. Robicheaux, C. T. Chen, P. Gavras, and M. S. Pindzola, Journal of Physics B: Atomic,Molecular and Optical Physics , 3047 (1995).[9] M. Awasthi, Y. V. Vanne, A. Saenz, and P. Decleva, Physical Review A , 1 (2008).[10] M. Busuladzic and D. B. Milosevic, Phys. Rev. A (2010).[11] L. Tao and A. Scrinzi, New Journal of Physics , 013021 (2012).[12] A. Scrinzi, Phys. Rev. A , 053845 (2010).[13] C. McCurdy, M. Baertschy, and T. Rescigno, J. Phys. B , R137 (2004).[14] M. Reed and B. Simon, Methods of Modern Mathematical Physics , Vol. IV (Academic Press,New York, 1982) p. 183.[15] A. Scrinzi, New Journal of Physics , 085008 (2012).[16] A. Zielinski, V. P. Majety, and S. A, “A general solver for the time-dependent schr¨odingerequation of one and two particle systems,” (unpublished).[17] H. Lischka, T. Mueller, P. G. Szalay, I. Shavitt, R. M. Pitzer, and R. Shepard, Wiley Inter-discip. Rev.-Comp. Mol. Science , 191 (2011).[18] V. P. Majety and A. Scrinzi, “Photo electron spectra of H , N e , and N ,” (unpublished).[19] I. Sanchez and F. Martin, The Journal of Chemical Physics106