Mixed gauge in strong laser-matter interaction
We show that the description of laser-matter interaction in length gauge at short short and in velocity gauge at longer distances allows for compact physical modeling in terms of field free states, rapidly convergent numerical approximation, and efficient absorption of outgoing flux. The mathematical and numerical framework for using mixed gauge in practice is introduced. We calculate photoelectron spectra generated by a laser field at wavelengths of 400800 nm from single-electron systems and from the helium atom and hydrogen molecule. We assess the accuracy of coupled channels calculations by comparison to full two-electron solutions of the time-dependent Schrödinger equation and find substantial advantages of mixed over velocity and length gauges.
The choice of gauge in the interaction of strong, long wave-length fields with atoms and molecules affects the physical modeling Bauer et al. (2005), perturbative expansions as the -matrix series Faisal (2007); Vanne and Saenz (2009), as well as the efficiency of numerical solutions Cormier and Lambropoulos (1996). For systems where field quantization can be neglected and the field appears only as a time- and space-dependent external parameter, the wavefunctions in all gauges are unitarily related by time- and space-dependent multiplicative phases. An extensive discussion of gauge transformations in the context of strong field phenomena can be found in Bandrauk et al. (2013). When approximations are made, the unitary equivalence of the wave-functions and the corresponding time-dependent Schrödinger equation (TDSE) is lost. An important example is the strong field approximation, where the system is assumed to either remain in the field-free initial state or move exclusively under the influence of the laser field: the function representing the field-free initial state depends on gauge. A similar situation arises, when a series expansion is truncated to a finite number of terms, as in an -matrix expansion: the physical meaning of any finite number of terms is different in different gauges. Also the discretization errors in a numerical calculation are gauge dependent. In particular, multiplication by a space-dependent phase changes the smoothness of the solution. As a result, numerical accuracy and convergence are depend on gauge.
Mathematical and numerical aspects of using general gauges were adressed in Refs. Burke et al. (1991); Dorr et al. (1992); Robicheaux et al. (1995) in the context of Floquet theory and the time dependent Schrödinger equation, where various options for mixing different gauges were discussed. In Refs. Burke et al. (1991); Dorr et al. (1992), mixing length, velocity, acceleration gauge in the R-matrix Floquet method was achieved by the introduction of Bloch operators at the boundaries between the gauges. In Ref. Robicheaux et al. (1995), it was pointed out that alternatively the transition between regions can be taken to be differentiably smooth, which also allows application to the time-dependent Schrödinger equation (TDSE).
Here we will show that physical modeling on the one hand and efficient numerical solution on the other hand impose conflicting requirements on the choice between the standard length and velocity gauges. We introduce the mathematical and numerical techniques for resolving this conflict by using general gauges. We restrict our discussion to gauge transformations in the strict sense, i.e. local phase multiplications, which does not include the acceleration “gauge”, as it involves a time-dependent coordinate transformation. Numerical performance of the various gauges is compared on a one-dimensional model system. We show that, also with discontinuous transition between gauges, there is no need for the explicit inclusion of the -like Bloch operators. We demonstrate validity and accuracy of mixed length and velocity gauge calculations in three dimensions by comparing to accurate velocity gauge results for the hydrogen atom at 800 nm wavelength. Finally, we combine local length gauge with asymptotic velocity gauge to compute photoelectron spectra of and at a laser wavelength of 400 nm. Efficiency and accuracy of the approach is shown by comparing to complete numerical solutions of the two-electron problem. We find that mixed gauge allows low-dimensional approximations, while in velocity gauge we achieve convergence only when we allow essentially complete two-electron dynamics. We will conclude that few-body dynamics in the realm of bound states is more efficiently represented in length gauge, while the 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 with light at wavelength down to the extreme ultraviolet one employs the dipole approximation, i.e. one neglects the variation of the field across the extension of the system . In length gauge, the interaction of a charge with the dipole field is
while in velocity gauge it is
Here and below we use atomic units with , electron mass , and electron charge , unless indicated otherwise. In these two gauges the dependence of the dipole interaction operators on is particularly simple and wavefunctions are unitarily related by
The transformation from length to velocity gauge is a special case of the general gauge transformation, namely multiplication by a space- and time-dependent phase
As is unitary, it leaves the system’s dynamics unaffected, if operators and the time-derivative are transformed as
The above relations are valid for general that are differentiable w.r.t. . If is twice differentiable in space, the gauge transforms of momentum operator and Laplacian are
We see, in particular, that a gauge transform introduces a time- and space-dependent momentum boost .
A standard TDSE transforms as
By explicitly writing the dot-product in the kinetic energy we emphasize that does not commute with space-dependent and space derivatives of appear in the Hamiltonian. The velocity gauge interaction Eq. (2) requires spatially uniform . More generally, any time-dependence of the potential energy can be transformed into a time- and space-dependent momentum by defining
ii.1 Discontinuous gauge transformation
The local phase multiplication need not be continuously differentiable or even continuous in space. One only must make sure that the gauge tranformed differential operators are defined on functions from a suitable domain . With discontinuous , formally, -like operators appear in Eq. (7). must be adjusted to compensate for those terms. The very simple, mathematically correct solution is to choose , i.e. functions of the form
where the are differentiable.
With at ranges and for large , one can switch from lenght to mixed gauge. Clearly, the particular form of the transition and the corresponding modulations of the wavefunction do not bear any physical meaning. Still, accurate modeling of the transition is needed to correctly connect the length to the velocity gauge part of the solution. In the transition region one needs to densly sample the solution, which increase the number of expansion coefficients. In many cases, this will also increase the stiffness of the time propagation equations and further raise the penalty for a smooth transition.
When or any of its derivates is discontinuous, the discretization in the vicinity of the discontinuity must be adjusted appropriately. Because of the lack of differentiability, any higher order finite difference scheme or approximations by analytic basis functions will fail to improve the approximation or may even lead to artefacts. The general solution for this problem is to explicitly build the known non-analytic behavior of the solution into the discretization. With the finite element basis set used below, this is particularly simple, as well-defined discontinuities can be imposed easily. We will also demonstrate below that a spatially abrupt transition does not increase stiffness and allows calculations with similar efficiency as in uniform velocity gauge.
ii.2 Gauge in the strong field approximation (SFA)
When we describe a physical process in terms of a few quantum mechanical states it is implied that the system does not essentially evolve beyond those states. The “strong field approximation” (SFA) is a simple model of this kind, which plays a prominent role in 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 atomic binding forces.
The SFA must be reformulated appropriately depending on the gauge on chooses. Let be the initial state in absence of the field. The physical picture above implies that the velocity distribution of the initial state remains essentially unchanged also in presence of the field. However, using the same function for all gauges, effectively leads to a set of different models with different, time-dependent velocity distributions for different gauges. In length gauge the operator has the meaning of a velocity of the electron and the velocity distribution is independent of the field:
In contrast, in velocity gauge, the velocity distribution varies with time as
The difference becomes noticeable when the variation of is not negligible compared to the width of the momentum distribution. This is typically the case in strong field phenomena. Findings that SFA in length gauge better approximates the exact solution in cases where the picture remains suitable at all Awasthi et al. (2008); Busuladzic and Milosevic (2010) are consistent with this reasoning.
ii.3 Single active electron (SAE) approximation
The gauge-dependent meaning of eigenstates has important consequences for the numerical approximation of few-electron systems. The functions corresponding to few-electron bound states have their intended physical meaning only in length gauge. In velocity gauge, the same functions correspond to time-varying velocity distributions. The problem affects the “single active electron” (SAE) approximation, where one lets one “active” electrons freely react to the laser field, but freezes all other electrons in their field-free states. Below we will demonstrate that this ansatz generates artefacts in velocity gauge.
As a simple illustration of the problem let us consider two non-interacting electrons with the Hamiltonian . The ansatz for the solution is exact, if the functions and are unrestricted. Matrix elements of the Hamiltonian are
In an exact calculation, the exchange terms in the second line vanish, if initially, as the unitary time evolution maintains orthogonality. However, if we restrict the time evolution of one of the functions, say , orthogonality is violated and unphysical exchange terms appear in the Hamiltonian matrix as the system evolves. Their size depends on the extend to which orthogonality is lost. If e.g. 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 . However, depending on the size of , in velocity gauge this does not hold and the exchange terms become sizable.
With interacting electrons, the direct term (Hartree potential) of electron-electron interactions is unaffected, as it only depends on the gauge-invariant electron density. In the exchange terms, however, the frozen orbitals with their length gauge meaning are inconsistently combined with the velocity gauge functions of the active electron.
The same gauge dependence appears also when the non-active electrons are not frozen in their initial states, but restricted in their freedom to evolve. We will demonstrate the superiority of length gauge for and with limited freedom for the non-active electron.
ii.4 Gauge in numerical solutions
While length gauge lends itself to intuitive interpretation and modeling, velocity gauge performs better in numerical calculations Cormier and Lambropoulos (1996). Fewer discretization coefficients can be used and the stiffness of the equations is reduced. This is due to the dynamics of free electrons in the field. From Eq. (7) one sees that for a free electron () the velocity gauge canonical momentum is conserved. In contrast, in length gauge, momenta are boosted by , reflecting the actual acceleration of the electron in the field. As large momenta correspond to short range modulations of the solution, length gauge requires finer spatial resolution than velocity gauge. This modulation affects numerical efficiency, when the variation of 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 infinite range exterior complex scaling (irECS) Tao and Scrinzi (2012) for absorption at the box boundaries. This method is highly efficient and free of artefacts, but it cannot be applied for systems with length gauge asymptotics, as clearly observed in simulations Scrinzi (2010). An intuitive explanation of this fact can be found in McCurdy et al. (2004) and the wider mathematical background is laid out in Reed and Simon (1982).
ii.5 Mixed gauge
The conflicting requirements on gauge can be resolved by observing that bound states are, by definition, confined to moderate distances, whereas the effect of phase modulation is important for free electrons, usually far from the bound states. Using length gauge within the reach of bound states and velocity gauge otherwise largely unites the advantage of both gauges: locally, the system can be modeled intuitively, while at the same time maintaining efficient numerical spatial discretization and asymptotics suitable for absorption.
Iii Implementations and examples
iii.1 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
In absence of the field, the ground state energy is exactly .
Spatial discretization is by a high order finite element basis, which is described in detail in Ref. Scrinzi (2010). Apart from being numerically robust, the basis is flexible, which allows, in particular, easy implementation of the discontinuity Eq. (9).
In general, basis functions used for spatially discretizing the TDSE do not need to be twice differentiable. Rather, they can have discontinuous first derivatives. Although the second derivative is not defined as an operator on the Hilbert space, matrix elements can be calculated correctly by using the symmetrized form
This relaxed condition on the differentiability is explicitly used with finite element bases, where usually first derivatives are discontinuous at the boundaries between elements.
leads to the system of ordinary differential equations for the expansion coefficients ,
with the matrices
For time-integration, we use the classical 4th order explicit Runge-Kutta solver. As an explicit method it is easy to apply, but it is also susceptible to the stiffness of the system of equations (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 of the different gauges. In the one-dimensional case we use a simulation box large enough such that reflections at the boundary remain well below the error level.
iii.1.1 Mixed gauge implementations (1d)
Matrix elements of the kinetic energy are always computed in the explicitly hermitian form
which also avoids the calculation of spatial derivatives of . Also, for non-differentiable , no -like operators appear.
For differentiable functions we have domains and no adjustments need to be made for the basis functions. To avoid loss of numerical approximation order, one must make sure that is smooth to the same derivative order as the the numerical approximation. We will refer to this case as “smooth switching”.
With a discontinuous change of gauge
the Hamiltonian is
where we denote the standard velocity gauge Hamiltonian as . There appear time-dependent discontinuities at the “gauge radius”
In a finite element basis such discontinuities can be imposed explicitly. Among others it leads to a time-dependent overlap matrix, whose inverse at each time-step can be obtained at low computational cost by low-rank updates. The technical details on this will be presented elsewhere.
One can avoid discontinuities at by “continuous gauge switching”
with the Hamiltonian
Note that is continuous, but not differentiable at , which leads to discontinuous first derivatives in the solution. As discussed above, a finite element basis admits discontinuities of the derivatives and there is no penalty in the numerical approximation order, if one ensures that fall onto element boundaries.
The and formulations are nearly equivalent in their numerical behavior, as
The respective solutions differ only by the phases :
Depending on , the time-dependence of this phase is slow compared to phase-oscillations due to high energy content of the solution and does not change stiffness for numerical integration.
In the following, we compare mixed gauge in the three forms given here with pure length and velocity gauge calculations.
iii.1.2 Comparison of the gauges
We compare electron densities at the end of the laser pulse. Size of the spatial discretization and the number of time-steps are adjusted to reach the same local error in all gauges relative to a fully converged density . For suppressing spurious spikes at near-zeros of the density, we include some averaging into the definition of the error:
We use a single cycle 800 nm pulse with -shape and peak intensity , which leads to about 25% ionization of this 1-d system. The -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 linear coefficients (150 elements) and about time steps for accuracy . Length gauge has the largest discretization with and , amounting to an overall increase in computation time of almost a factor 2. The larger number of time-steps arises because the explicit propagation scheme is sensitive to the stiffness of the equations, which can grow . The actual increase of times steps does not exactly reflect this behavior, as a finer discretization is used in the inner region to obtain comparably accurate initial states in all calculations. Stiffness from this discretization is always present in the calculations. For the mixed gauge, we use continuous switching Eq. (24) with . With this small length gauge section, the same discretization as in the velocity gauge can be used with and .
We also investigated the effect of a smooth transition between the gauge over an interval of size , using
where is a 3rd order polynomial smoothly connecting the length with the velocity gauge region.
For a smoothing interval , we need a rather dense discretization by 18th order polynomials 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 time steps. With smoothing , stiffness is reduced and , which still exceeds by 50% the number of time steps with continuous, but non-differentiable transition.
The dependence on is not surprising: the correction terms to the kinetic energy involve derivatives of , 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 a rather strong change in temporal and spatial behavior of the solution, which necessitates the 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, when using the discontinuous Hamiltonian , Eq. (21), or be left to be adjusted numerically with the continuous Hamiltonian , Eq. (24). We conclude that, wherever technically possible, a sudden transition is to be preferred.
iii.2 Mixed gauge for the Hydrogen atom
The length gauge Hamiltonian for the hydrogen atom in a laser field is
The velocity gauge Hamiltonian is
In three dimensions, problem size grows rapidly and truncation of the simulation volume is advisable. As the error free absorbing boundary method irECS Tao and Scrinzi (2012) is incompatible with length gauge calculations, in this section we only compare velocity to mixed gauge calculations. Following the findings of the one dimensional calculation, we use continuous gauge switching for its numerically efficiency and moderate programming effort. In three dimensions, it can be defined as
denoting . The corresponding Hamiltonian is
The gradient of the angle-dependent phase introduces an extra quadrupole type coupling:
asymptotically coincides with standard velocity gauge as tends to . In an expansion into spherical harmonics, the quadrupole terms introduce additional non-zeros into the Hamiltonian matrix, which increase the operations count for applying the Hamiltonian by 60%.
For the numerical solution we use polar coordinates with a finite element basis on the radial coordinate and spherical harmonics for the angular dependence. A discussion of the basis can be found in Tao and Scrinzi (2012). We assume linear polarization and fix the magnetic quantum number at . We use a -shaped pulse with 3 optical cycles FWHM at central wavelength and and peak intensity , which leads to about 16% ionization.
We compare the errors of the different gauges in the angle-integrated electron density at the end of the pulse and in the photoelectron spectra. The spectra are computed by the tSURFF method described in Refs. Tao and Scrinzi (2012); Scrinzi (2012). Errors are again defined relative to a fully converged velocity gauge calculation.
On the radial coordinate we use 5 finite elements of order 16 up to radius in all gauges. Beyond that, the solution is absorbed by irECS. The stronger phase oscillations of the length gauge solution requires more angular momenta compared to velocity gauge Cormier and Lambropoulos (1996).
Figure 2 shows the relative errors in of a velocity gauge calculation with angular momenta with mixed gauge calculations at two different gauge radii , and . As expected, the mixed gauge calculation needs higher as increases.
The same general error behavior of the different gauges is also found in the photoelectron spectra, Figure 3. Here, the requires even more angular momenta . This may be due to the particular sensitivity of photoelectron spectra to the wavefunction at the radius where the surface flux is picked up and integrated, in the present case at .
iii.3 Helium atom and molecule
The length gauge Hamiltonian for a two-electron problem interacting with a dipole laser field is
This includes the molecule at fixed internuclear distance and the helium atom . We assume linear polarization in -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 this calculation will be reported elsewhere Zielinski et al. (). Photoelectron spectra for the various ionic channels were computed using the 2e form of tSURFF (see Scrinzi (2012)). As 2e calculations are very challenging at long wavelength, we use a 3-cycle pulse at somewhat shorter wavelength of and an intensity of only . To facilitate the extraction of photoelectron momenta, all potentials where smoothly turned off beyond distances as described in Ref. Scrinzi (2012).
We compare the 2-electron calculation with a coupled channels computation using the expansion
which includes the field-free neutral ground state and the ionic states multiplied by the same single-electron basis functions as for the hydrogen atom. Anti-symmetrization is indicated by . The neutral ground state as well as the ionic states were obtained from the COLUMBUS quantum chemistry package Lischka et al. (2011). Calculations were performed in velocity and mixed gauge (continuous switching), as described for the hydrogen atom. Details of the coupled channels method will be reported reported elsewhere Majety and Scrinzi ().
By the arguments above, in the coupled channels basis we expect the mixed-gauge calculation to converge better than the velocity gauge calculation: the COLUMBUS wavefunctions and have their intended physical meaning only in length gauge. Figures 4 and 5 confirm this expectation.
For helium, Figure 4, the velocity gauge 2e calculation agrees well with the mixed gauge coupled channels calculation using the neutral and only the 1s ionic state. With the 5 ionic states with principal quantum number the error is for a large part of the spectrum up to 1 au. In contrast, the single-ion velocity gauge calculation is far off. It does somewhat approach the 2e result when the number of ionic states is increased to include the ionic states up to . In velocity gauge, convergence could not be achieved for two reasons. One reason is a technical limitation of the coupled channels code, which uses Gaussian basis functions that do not properly represent the higher ionic states. The second reason is more fundamental: in velocity gauge the ionic cores transiently contain significant continuum contributions, which are not included in our ionic basis by construction.
The error pattern is similar in , but the accuracy of all calculations is poorer, Fig. 5: 2e and coupled channel mixed gauge calculations qualitatively agree already when only the ionic ground state is included. With the lowest 6 ionic and states the two spectra differ by . Remarkably, the height of a small resonant peak at is faithfully reproduced in mixed gauge with 6 ionic states. The resonance can be tentatively assigned to the near degerate second and third doubly excited states of at the internuclear equlibrium distance of 1.4 au (see Ref. Sanchez and Martin (1997)). As a note of caution, the single center expansion used in the 2e code converges only slowly for and cannot be taken as an absolute reference. The coupled channels velocity gauge calculation is off by almost two orders of magnitude when only a single ionic state is included. With 6 ionic states it compares to the full 2e on a similar level as the mixed gauge. However, in velocity gauge the resonance is not reproduced correctly.
For both systems, analogous results were found at shorter wavelength down to . At even shorter wavelength and realistic laser intensities, gauge questions are less important as the magnitude of .
In summary, we have shown that a transition between gauges within the same calculation bears substantial advantages and requires only moderate implementation effort. For low-dimensional problems, the advantage can be technical, such as reducing the size of the spatial discretization and the equations’ stiffness. We have shown that with a suitably chosen basis a sudden, non-differentiable transition from length to velocity gauge is preferable over a differentiably smooth transition in terms of both, simplicity of implementation and numerical efficiency.
Mixed gauge opens the route to a highly efficient, coupled channels type description of laser-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 only in length gauge the field free ionic eigenfunctions retain their physical meaning in presence of a strong pulse. In contrast, in velocity gauge the same functions represent a momentum-boosted system with unphysical dynamics. Therefore typical physical models suggest the use 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 functions: mixed gauge calculations converge with very few channels. Most dramatically, the single-ionization spectrum of helium was calculated to accuracy using only the ionic ground state channel. In contrast, in velocity gauge the single channel result is by nearly two 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 discretization size.
Convergence with only the field-free neutral and very few ionic states can justify a posteriori wide-spread modeling of laser-atom interactions in terms of such states. It also supports the view that length gauge is the natural choice for this type of models. The convergence behavior of mixed gauge calculations — possibly contrasted with pure velocity gauge calculations — may help to judge the validity of these important models in more complex few-electron systems.
V.P.M. is a fellow of the EU Marie Curie ITN “CORINF”, A.Z. acknowledges support by the DFG through the excellence cluster “Munich Center for Advanced Photonics (MAP)”, and by the Austrian Science Foundation project ViCoM (F41). A.S. gratefully acknowledges partial support by the National Science Foundation under Grant No. NSF PHY11-25915.
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