Atoms and Molecules in Intense Laser Fields: Gauge Invariance of Theory and Models

Atoms and Molecules in Intense Laser Fields: Gauge Invariance of Theory and Models

A. D. Bandrauk, F. Fillion-Gourdeau and E. Lorin Laboratoire de chimie théorique, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, Canada, J1K 2R1 School of Mathematics Statistics, Carleton University, Canada, K1S 5B6 Centre de Recherches Mathématiques, Université de Montréal, Montréal, Canada, H3T 1J4
Abstract

Gauge invariance was discovered in the development of classical electromagnetism and was required when the latter was formulated in terms of the scalar and vector potentials. It is now considered to be a fundamental principle of nature, stating that different forms of these potentials yield the same physical description: they describe the same electromagnetic field as long as they are related to each other by gauge transformations. Gauge invariance can also be included into the quantum description of matter interacting with an electromagnetic field by assuming that the wave function transforms under a given local unitary transformation. The result of this procedure is a quantum theory describing the coupling of electrons, nuclei and photons. Therefore, it is a very important concept: it is used in almost every fields of physics and it has been generalized to describe electroweak and strong interactions in the standard model of particles. A review of quantum mechanical gauge invariance and general unitary transformations is presented for atoms and molecules in interaction with intense short laser pulses, spanning the perturbative to highly nonlinear nonperturbative interaction regimes. Various unitary transformations for single spinless particle Time Dependent Schrödinger Equations, TDSE, are shown to correspond to different time-dependent Hamiltonians and wave functions. Accuracy of approximation methods involved in solutions of TDSE’s such as perturbation theory and popular numerical methods depend on gauge or representation choices which can be more convenient due to faster convergence criteria. We focus on three main representations: length and velocity gauges, in addition to the acceleration form which is not a gauge, to describe perturbative and nonperturbative radiative interactions. Numerical schemes for solving TDSE’s in different representations are also discussed. A final brief discussion is presented of these issues for the relativistic Time Dependent Dirac Equation, TDDE, for future super-intense laser field problems.

: J. Phys. B: At. Mol. Phys.

1 Introduction

Advances in current laser technology allow experimentalists to access new laser sources for probing and controlling molecular structure, function and dynamics on the natural time scale of atomic (nuclear) motion, the femtosecond ( fs s) [1, 2] and electron motion on attosecond ( s) time scale [3, 4]. A new regime of nonlinear nonperturbative laser-matter interaction is leading to new physical phenomena such as nonlinear photoelectron spectra called Above Threshold Ionization (ATI) [5, 6, 7, 8] and high order harmonic generation (HHG) [9, 10]. In these two new examples of nonperturbative electron response, ionized electron trajectories are completely controlled by the electric laser field [3]. The atomic unit of laser intensity W cm corresponds to the atomic unit (a.u.) of electric field Vcm, the field at the atomic radius nm of the 1 hydrogen () atom orbit. For an intensity Wcm a.u. currently used in many high intensity experiments, the electron ponderomotive energy at wavelength nm ( a.u.), (a.u.) a.u. eV, exceeds the ionization potential a.u. = eV of the ground state atom. The corresponding maximum field induced excursion of the electron is a.u. nm, where the Coulomb potential becomes negligible and a field dressed electron description becomes appropriate. This new regime of nonperturbative radiative interactions has motivated the development of simple but highly predictive physical models, such as the recollision model in atoms [11, 12, 13] or molecules [8], and the strong field approximation (SFA) [3, 5, 7] which have become standard models for the advancement and development of this new field of nonlinear physics.

The theoretical description of perturbative and nonperturbative radiative interactions relies on one of the most important concepts of physics: gauge invariance. Gauge invariance was discovered in the development of classical electromagnetism when the latter was formulated in terms of scalar and vectorial potentials [14]. It is now considered to be a fundamental principle of nature, stating that different forms of potentials yield the same physical description, i.e. they describe the same electromagnetic fields as long as they are related to each other by gauge transformations [15]. In the quantum description of matter interacting with electromagnetic fields, gauge invariance is obtained by transforming wave functions under local unitary transformations, resulting in different Hamiltonians in the corresponding TDSE’s. However unitary transformations can also give rise to “representations” which are not considered as “gauge” transformations (the latter are obtained rigorously only from transformations of classical Lagrangians leaving the dynamics invariant [2, 15]). According to the gauge principle (this will be described in more details in latter sections), all physical observables are gauge invariant. However, their calculations usually involve the evaluation of gauge dependent quantities: for instance, the expression of the electron wave function depends on the gauge chosen. When the exact analytical solution of the TDSE is known, it is possible to move from one gauge to the other by a gauge transformation (see [16] for instance), and then, all gauges will yield the same physical result. In this case, the gauge choice is simply a matter of convenience: it may be easier to solve the TDSE in one specific gauge than in others. However, when approximations of gauge dependent quantities are involved (for instance, when using numerical methods or perturbation theory), the gauge invariance of physical observables may be lost as the error induced by the approximation scheme may not transform in the same way as the full solution. This can be illustrated as follows. Let us consider a single electron in interaction with an electromagnetic field expressed in two different gauges such that our system can be described equivalently by the following wave functions:

(1)
(2)

where is the general exact solution of the TDSE, coupled to the electromagnetic field expressed in gauge 1, while is given in gauge 2. The two wave functions are related by a gauge transformation as , with the following calculated observables:

(3)

provided that the observable transform as , which can be verified explicitly in most cases. The last equation (3) expresses the essence of the gauge principle, i.e. that physical observables are independent of the gauge chosen. In Eqs. (1) and (2), are approximate wave functions, obtained from a solution of the TDSE using approximation methods. Thus, are the remainder or error of this method. Typically, it will be given by for numerical methods (where is the grid size and ) or for perturbation theory (where is a small parameter). If the approximate wave functions obey the same gauge transformation as the full solution, that is , the approximation of the observable will also be gauge invariant:

(4)

However, this is generally not the case because and usually transform differently in different gauges

(5)
(6)

but

(7)

and thus . Therefore, we have lost gauge invariance by approximating the wave function. As the approximate wave functions get closer to the exact solution, the observables calculated in two different gauges converge towards each other

(8)

and of course, gauge invariance is recovered in the limit of the exact solution when .

The previous discussion illustrates the fact that approximating a gauge independent quantity (an observable) by implementing an approximation of a gauge dependent quantity (the wave function) may destroy the gauge independence of the former: the breaking of gauge invariance is an artifact of the approximation method. In this paper, we will compare the calculation of many observables in different gauges and show that certain gauge choices have better convergence properties towards the exact solution. One should always try to choose the gauge which gives the best approximation of the physical quantity under consideration. This has been studied extensively using either analytical or numerical approximations. For instance, it was remarked by Lamb in his celebrated study of the Hydrogen atom fine structure, that the theoretical results obtained in the length and velocity gauges (defined later) differ in perturbation theory [17]. Moreover, the result of the length gauge calculation was in better agreement with experiments, suggesting that this gauge would be more “fundamental”. This apparent paradox was resolved latter when it was observed that the “naive” perturbation theory is not gauge invariant and that special care is required to calculate observables in a gauge independent way. Many papers have attempted to clarify this issue [18, 19, 20, 21, 21, 22, 23, 15, 24]. The equivalence of the velocity and length gauge was also demonstrated in specific calculations for multiphotons transition probabilities [25] and induced polarization [26]. Other calculations have shown the equivalence of results in different gauge choices [27, 28]. Given the status and importance of this issue, we will attempt to formulate a consistent gauge invariant perturbation theory using general arguments in the following sections. Concerning numerical calculation and gauge independence, it is now generally admitted that certain gauge choices are better than others for the solution of the TDSE. For instance, it was concluded that the velocity gauge is more appropriate for calculations in dynamical laser-matter interactions [29]. Other calculations where gauge choices are compared can be found in [30] for high harmonic generation, HHG. It should be stressed here that these comparisons should be performed with great care because the structure of the mathematical equation may change under a gauge transformation and the resulting TDSE may require a different numerical scheme. This issue will be discussed in details in this work, along with the description of current numerical methods.

Throughout this work, a single particle system is mainly considered but extension to -particle problems exists naturally in the literature, and can often be easily deduced. Recent work has examined gauge invariance of coupled electron-nucleus dynamics using the time-dependent Hartree-Fock equation (TDHF) in [31] and the Time-dependent Kohn-Sham equation (TDKS) density functional theory [32] and also nonadiabatic molecular dynamics [33]. However, they are much more involved technically and are outside the scope of our discussion.

It is also assumed that the electromagnetic field is not quantized and treated as a classical field. This approximation is valid when the number of photons is large (such as in a macroscopic laser field) such that quantum fluctuations can be neglected [15]. Concluding whether this approximation is true or not for the field induced by the particle is a non-trivial task and certainly depends on the dynamic of the system. Also, from the practical point of view, this approach simplifies the calculations significantly, which is certainly the main reason why it has been so popular among practionners. It is important to note that using a classical theory spontaneous emission is not taken into account but must be introduced by “hand” as in HHG.

This article is separated as follows. In Section 2, the classical and quantum dynamics of particles coupled to an electromagnetic field is treated, with a special emphasis on gauge transformations. It will be shown how to obtain the TDSE from the classical dynamics of a scalar particle and how it interacts with the electromagnetic field. More generally, the gauge principle, which allows to derive the interaction of matter with force carriers will be described. In Section 3, the dipole approximation is detailed along with the regime where it can be used. The gauges commonly used in laser-matter interaction are reported in Section 4. More precisely, we will consider the length, the velocity and the acceleration gauges. The transformations allowing to change the gauge representation are also described. Section 5 contains details on analytical approximations used to evaluate quantum transition amplitudes. For instance, a general description of perturbation theory and the strong field approximation, with respect to gauge transformation, is included. The gauge invariance of these techniques is also shown in some specific examples. Section 6 contains a discussion of numerical methods for the solution of TDSE’s. It is emphasized that the mathematical structure of the TDSE depends on the gauge and thus, the numerical scheme used to solve the equation should be chosen by taking this fact into account. Finally, gauge transformations in relativistic quantum mechanics, which is relevant for ultra-intense laser field, is considered in Section 7.

2 Classical and quantum dynamics of particles coupled to an electromagnetic field

In this section, we recall Maxwell’s equations and the notion of gauge transformations and gauge choice, as well as some basic informations about the Hamiltonian and Lagrangian formulations of classical mechanics. The latter will then be applied to the case of a single particle interacting with a classical electromagnetic field. The corresponding quantum dynamics will be derived at the end of this section using the usual canonical quantization scheme. This allows to obtain a TDSE describing the quantum dynamics of a particle coupled to an electromagnetic field: this equation is the basis of this paper. Finally, we will discuss the derivation of the TDSE from the point of view of the Gauge principle and show how a general symmetry principle can be used to obtain an interaction of matter with an electromagnetic field. This non-exhaustive survey will allow us to set the framework. This first part is a summary of some key sections of [15] (in particular we use similar notations).

2.1 Maxwell’s equations

The electromagnetic field can be characterized by two physical quantities: the electric field and the magnetic field , where is a four-vector. These are vector fields which exist only in the presence of electric charges: an electric field is produced by a stationary charge while the magnetic field is induced by charges in motion (currents). Their dynamics and the relation between them is governed by Maxwell’s equations. Denoting the electromagnetic field by , , microscopic Maxwell’s equations in non-Gaussian units are written as:

(9)

where is the charge density, is the current density, is the vacuum permittivity and is the velocity of light. When we are considering a system of point-like particles of charges located at at time , the charge density is given by [15]

(10)

and the current density by

(11)

Of course, there exists a generalization of these last two formulas to the case of a continuous distribution of charges [14], but it is not required in this paper. The particle positions are determined from the Newton-Lorentz equations, which describes the non-relativistic dynamics of point-particles immersed in an electromagnetic field. Thus, the classical particles trajectories are solution of:

(12)

for , and where we consider the electromagnetic field as external. Here, are the particle velocities, is the initial time and are the initial particle positions and velocities, respectively.

From the divergence free equation implying the non-existence of magnetic monopoles, and the Maxwell-Faraday equation, we deduce the existence of vector and scalar functions and such that

(13)

where is the vector electric potential and the scalar electric potential. Maxwell’s equations can be rewritten in terms of and as a second order (in time and space) wave equation:

(14)

This system of equations is equivalent in principle to Maxwell’s equations: it should give the same electromagnetic field . However, we will see in the following that this is not quite the case because the potentials are not defined uniquely and thus, we need another condition to obtain the appropriate dynamics. This can be understood in the following way. Let us consider a regular scalar function . It can be easily shown from (13) that the electromagnetic field is unchanged (using the fact that ) by the so-called gauge transformation:

(15)

As a consequence , is not uniquely defined by the potentials: there exists an infinite number of potentials related by gauge transformations that yield the same electromagnetic field. An additional condition, called a gauge condition, allows to fix these superfluous “degrees of freedom” and to determine a unique definition of potentials: this procedure is called gauge fixing111It is interesting here to note that it took almost a century to deduce this condition [34].. There exists in principle an infinite number of these conditions but from a practical point of view, only a certain number of gauge choices are generally utilized because the resulting equations are simpler or have certain interesting properties. In the framework of electrodynamics coupled to non-relativistic classical particles, the most popular choices are the Lorentz and Coulomb gauges:

  • Lorentz Gauge:
    Consists of imposing the condition

    (16)

    and Maxwell’s equations become

    (17)

    This gauge has the merit of being manifestly covariant, which is a very useful property when one is interested in symmetries of Maxwell’s equations or in the covariant perturbation theory. It is to be noted that scalar , and vector potentials are decoupled in this gauge, as well as charge and current.

  • Coulomb Gauge or minimal coupling:
    Consists of imposing the condition , and as consequence Maxwell’s equations are rewritten

    (18)

There exist many other possibilities, such as the light-cone gauge, temporal gauge, axial gauge, Fock-Schwinger gauge and Poincaré gauge (for an exhaustive enumeration and definitions, see [15, 34]). Also, it is interesting to note that the null divergence of the vector potential has interesting connections to the incompressibilty condition in fluid dynamics. This is discussed in more details in Section 6.

2.2 Lagrangian and Hamiltonian formulation

This section is not an exhaustive presentation of the notion of Lagrangian and Hamiltonian operators. However simple facts are important to recall, in particular the least action principle, which lies as the basis of classical and quantum mechanics. Two cases will be treated: the discrete and the continuum cases. The former is used for the description of point-like particles. The latter is important when one is interested in the dynamics of a field variable such as the electromagnetic field, the velocity field in fluid dynamics and even the quantum wave function.

2.2.1 Discrete case: particle-like systems

We denote by the set of trajectories of particles of mass . In the Lagrangian formalism the search of trajectories is equivalent to solving an extremum problem for the action , between times and (see [35, 36]):

(19)
(20)

where is called the Lagrangian, with the kinetic energy and the potential energy. Above and in the following, the notation denotes a time derivative of . It should be noted here that the Lagrangian depends only on the variables , their time derivative (velocities) and possibly on time, but not on the acceleration , which is not included. The variables are the dynamical variables and they completely specify the state of a classical system (this is the reason why the Lagrangian does not depend on higher time derivatives). The explicit time dependence of the Lagrangian is included to describe external forces acting on the dynamical system under consideration. In this latter case, it can be shown that the energy is not conserved.

The equations of motion are then obtained from the least action principle, which states that the particle paths minimize the action, that is: where is the functional differential:

(21)

Assuming that the coordinate variation vanishes at , the Euler-Lagrange equations can be obtained [35], for all :

(22)

These equations allow us to obtain a description local in time (equation of motion) from a global in time principle (least action principle).

Another important quantity can be obtained from the Lagrangian: the conjugate momentum. It is given, for the th particle, by

(23)

It should also be noted that the conjugate momentum will be especially important when the theory is quantized and it will happen that is not always equal to .

The last equation suggests that it is possible to obtain a description of the dynamics in terms of momenta and coordinates , instead of the velocity used in the Lagrangian formulation. This is the Hamiltonian formulation, which is related to the Lagrangian by a Legendre transformation:

(24)

Thus, the Hamiltonian represents the total energy (kinetic + potential energy) of the system. From the above equation it follows that if the Lagrangian is time-independent, the Hamiltonian or the total energy is conserved and thus, is a constant of motion.

2.2.2 Continuous case: classical field theory

In this section, we recall some basic facts about Hamiltonian and Lagrangian operators in classical field theory. The main difference with the preceding section is the fact that now, the object under study are fields, that is quantities which take a value at each point of space time. The latter come in various form depending on their transformation properties: scalar fields, vector fields or tensor fields for example. Also, they can be used to describe many physical entities such as the electromagnetic field, flow velocity in fluid mechanics, temperature distribution in a material, etc. Their dynamics can also be formulated in terms of Lagrangian and Hamiltonian mechanics, which is the subject of this section.

First, let us consider the dynamical variables given by and , that is the field under consideration. Here the index is an integer which denotes one of the field vectorial components, that is:

(25)

Also, the argument is such that has a value over and . Finally, the index denotes a derivative with respect to a space-time coordinate (when considering the Minkowski metric or in other words, when looking at relativistic systems, the usual notation is to have where is a Lorentz index).

Now, since the Lagrangian is a function of the dynamical variables, we can write the field action as

(26)

where is a Lagrangian density. This action is a scalar and is a generalization to the continuous case of the action for the discrete case. It can actually be derived from the discrete case in the limit of an infinite number of degrees of freedom and by assuming local interactions. The least action principle is unchanged in this procedure and can still be used to compute the equations of motion. The latter states that the field minimizes the action such that where . In other words, the action is stationnary under the perturbation . So under this infinitesimal variation, the action changes according to

(27)

Of course, by requiring stationarity of the action, we get the usual Euler-Lagrange equation of motion:

(28)

Here, Einstein notation convention is assumed, that is repeated indices are summed.

The conjugate momenta and the Hamiltonian density can also be defined in a similar way as in the discrete case. They are given by

(29)
(30)

Finally, it is often convenient to use the Lagrangian of a continuous system obtained from the Lagrangian density as . This notation will be used frequently in the following sections.

2.2.3 Symmetry transformations

In both the discrete and the continuous cases, the physical system may have symmetries, that is a set of transformation which leave the dynamics invariant. In the Lagrangian formulation, it can be shown that under such symmetry transformation, the Lagrangian is unchanged, up to a divergence term. Symmetries are very important in all areas of physics because they are related to conserved quantities via Noether’s theorem.

First, we will look at the possible transformations that can be implemented on the Lagrangian. These exist in two varieties (in the discrete case, only the first one can be implemented):

  1. Transformations on coordinates:
    Examples of these are the Lorentz and Galilean transformations, which include translations and rotations.

  2. Transformation on the fields:
    Examples of these are the phase transformation of the wave function ( ). Note that in the following, we will consider a set of invertible transformations which depend only on the field itself (not its derivative).

Mathematically, a general way of writing these two transformations is to define a linear mapping as:

(31)

where is a continuous parameter (which may depend on spacetime) such that when , the transformation is the identity. Then, it can be shown that a condition to confirm that is a symmetry transformation (for the continuous case) is that

(32)

where is the Jacobian of the coordinate transformation. In the discrete case, one finds that this symmetry condition is written as

(33)

where are arbitrary functions. If these conditions are fulfilled, the transformed and the initial Lagrangian will lead to the same dynamics (the equation of motion will have the same form).

2.3 Lagrangian of the electromagnetic field

In this section, we will look at the Lagrangian density describing the electromagnetic field. It is convenient for this discussion to use the manifestly covariant formulation of electrodynamics. In this case, we define the four vector potential and current as

(34)
(35)

These quantities are contravariant tensors which are related to their covariant counterparts by , where is the Minkowski metric. It is also convenient to define the antisymmetric field strength tensor as

(36)

Then, within this formulation, Maxwell’s equations can be written in a very compact and manifestly covariant (invariant under Lorentz transformations) form:

(37)

Also, the conservation of current is written as

(38)

and finally, the gauge transformation is

(39)

We would like to stress that the covariant formulation is equivalent to the set of Maxwell’s equations presented before. However, it is independent of the choice of referential frame: the latter are related by Lorentz (or Poincaré transformations).

From these equations, it is possible to obtain the corresponding Lagrangian density for the electromagnetism sector. It is given by

(40)
(41)

It is a straightforward calculation to show that the Euler-Lagrange equation for this Lagrangian density is given by Maxwell’s equations.

The coupling of the electromagnetic field to matter is done through the source terms which contain the charge density and current. In a classical setting, this would be related to the charge position and thus, another term should be included in the Lagrangian to take care of the particles dynamics. This is done in section 2.4. In the quantum setting however, the tensor is related to the wave function of the particle under study (in our case, a single electron). This is described is Section 2.6.

2.4 Lagrangian of a particle in an electromagnetic field

The Lagrangian for a system of non-relativistic free particles is given by . To introduce its coupling with an electromagnetic field, we can add the electromagnetic field Lagrangian and we get

(42)
(44)
(45)

where the current and density in the interaction terms of are given by Eq. (11) and (10), respectively. The Euler-Lagrange equation of motion are given by Maxwell’s equations and by the Newton-Lorentz equation as the dynamical variables are .

We now discuss the effect of a gauge transformation via (15), on this Lagrangian where we set , that is for the single particle case. Replacing by , leads to a new Lagrangian :

(46)

Due to charge conservation

(47)

and the divergence theorem (), we easily deduce that

(48)

The new and old Lagrangians are then equivalent as they obey the symmetry condition in Eq. (33). This transformation is called a gauge transformation of the first kind, according to Pauli [37]. The previous Lagrangian was written in a general way such that gauge invariance is explicitly satisfied. However, it contains redundant degrees of freedom and as discussed earlier, this can be discarded by the gauge fixing procedure. Thus, it is possible to write Lagrangians for specific gauge choices. For instance, in the Coulomb gauge ( with ), the Lagrangian (45) can be written:

(49)

and the corresponding Hamiltonian, is then given by (using (24))

(50)

There are a few interesting remarks to make about this Lagrangian:

  • It is not gauge invariant, rather, it is obtained from the gauge invariant Lagrangian by gauge fixing. Thus, it is valid only for the Coulomb gauge choice.

  • The electromagnetic field dynamics does not depend on the scalar potential . The gauge fixing procedure () allowed to eliminate this degree of freedom.

  • The Coulomb potential of the particle appears naturally from the term involving the charge density [15]. Thus, in this gauge, the field of the particle is simply given by the usual Coulomb law.

  • It is not manifestly covariant because the gauge condition is not invariant under Lorentz transformation.

For these three reasons, the Coulomb gauge is a very popular choice to describe laser-matter interactions.

We would like to conclude this section by considering the special case of a single particle () subject to an external electromagnetic field where we assume that the field is not part of the dynamical system. This approximation is often used to simplify the calculations. The simplified Lagrangian becomes

(51)

where represent the potential of an external electromagnetic field . In this model, the Euler-Lagrange equation are given by the Newton-Lorentz equation for the particle and there is no backreaction of the particle on the field. The conjugate momentum is while the Hamiltonian is

(52)

This Hamiltonian is then used in the special case where it is assumed that the backreaction on the electromagnetic field is negligible. Throughout this paper, we will refer to this case as the external field approximation (minimal gauge).

2.5 Quantization

The equations of motion obtained in the last section can be quantized in the usual canonical quantization where essentially the position and conjugate momentum become operators. This method attempts to quantize a classical system while keeping its main properties such as symmetries. We would like to stress again that in this work, we are not quantizing the electromagnetic field: only the single particle described by the Newton-Lorentz equation will be quantized. The canonical quantization states that the classical dynamical variables, that is the position and conjugate momentum , becomes operators with the following commutation relations:

(53)
(54)
(55)

Throughout this work, we will work in the “position representation” where is a vector in the Hilbert space describing the quantum state of our system. The position operator has the property that . From this result and the commutation relation, it is straightforward to obtain that (in the free case).

A general state is obtained by the linear superposition . The wave function is a projection of such a state on the position state, that is . The dynamics of the wave function is given by the TDSE:

(56)

where is the Hamiltonian operator for the system under consideration. Projecting this equation on position space, we get the usual TDSE in coordinate space:

(57)

where the quantum Hamiltonian is obtained from the classical Hamiltonian by using the following prescription: . This procedure will be used in the rest of this paper to obtain wave equations describing the quantum dynamics of a single particle interacting with different electromagnetic fields.

2.6 Time dependent Schrödinger equation coupled to an electromagnetic field and the gauge principle

In this section, we consider the gauge invariance of the TDSE coupled to an electromagnetic field from the Lagrangian viewpoint and the gauge principle. It should be noted here that just like the electromagnetic field, the wave function is also a field (a complex scalar field) in the sense that it is a function that is defined over all . Therefore, its dynamics can be understood from a variational principle analogous to the treatment of the electromagnetic field. This will be presented in the first section. Finally, the coupling of the two along with gauge invariance and physical consequences will be proven.

2.6.1 Schrödinger scalar field

The free Schrödinger equation for a particle of mass is given by

(58)

Then, it can be shown that the free Schrödinger equation is given by the Euler-Lagrange equation of the following Lagrangian density:

(59)

It should be noted that since the field is complex, both and should be considered as dynamical variables [15], leading to two different Euler-Lagrange equations which are complex conjugate of each other. Also, the Schrödinger Lagrangian is not invariant under Lorentz transformations because of course, it describes a non-relativistic particle. It is however invariant under the Galilean transformations which relates different referential frames in the non-relativistic setting. There exists relativistic generalizations of the Schrödinger wave equation, such as the Klein-Gordon (spin-0) and the Dirac (spin-1/2) equations, which are invariant under Lorentz transformations.

2.6.2 Minimal coupling prescription and gauge invariance

Now, we would like to couple the “matter field” described by the Schrödinger Lagrangian with the electromagnetic field. This can be achieved by imposing a symmetry on the Lagrangian . Let us assume that the theory describing our system is invariant under local phase transformations ( symmetry), that is:

(60)

such that . However, an explicit calculation shows that . Rather, we have

(61)

The only way to cancel the extra terms in Eq. (61) is to add a new field with appropriate transformations. This new field is the electromagnetic field and it is added via the minimal coupling prescription, that is partial derivatives are replaced by:

(62)

This is complemented by adding the kinetic term such that we obtain a dynamical field. Then, the Lagrangian for the coupled system becomes

(63)
(64)

This represents physically the dynamics of the matter field with its “own” electromagnetic field, that is the magnetic field generated by the particle itself. However, it is also possible to add an external electromagnetic field by letting . The corresponding Euler-Lagrange equation then becomes

(65)

where

(66)

Before continuing further, we summarize the above derivations. We started with a Lagrangian describing matter: the Schrödinger Lagrangian in Eq. (59). Then, we imposed a local symmetry (note that symmetry with a space-independent phase parameter is related to charge conservation) to this Lagrangian. The consequence of this is that we had to add a new field, the electromagnetic gauge field, such that the symmetry is obeyed. This field obeys the gauge transformation properties. This whole procedure is an example of the gauge principle which is used in many field of physics to obtain interaction terms between matter and force carriers. For example, the theory of Strong and Weak nuclear interaction are based upon this very principle, albeit on its non-abelian generalization (the symmetry groups are local and in these cases). Therefore, it is generally believed that gauge symmetries are one of the fundamental organizing principles of nature.

To summarize, the gauge transformation can be written as

(67)

where is an arbitrary function. This was shown to be a symmetry transformation because it obeys Eq. (32).

2.7 Gauge invariance, unitary transformation and quantum observables

The goal of this section is to show that although gauge transformations are unitary, not all unitary transformations correspond to a gauge transformation. This may occur when the transformation parameter depends on dynamical variables such as , etc. Also, the fact that transition matrix elements are invariant under any unitary transformations is discussed. This is crucial for our analysis because it states that physical observables are the same, for any unitary transformations of the wave function. The corollary to this is that observables are invariant under gauge transformations.

To describe these results, we will start by looking at the invariance of quantum observables under unitary transformations. Let us consider a transformation operator defined by

(68)

where can be any space-time dependent operator. Of course, this transformation is unitary: . The operator acts on the wave function as , while the wave functions obey the following TDSE’s:

(69)

It can be verified by using these TDSE’s, along with the definition of unitary transformations, that the Hamiltonians in the two representations are related by

(70)

Note here that this is derived by assuming that . The average energy is then given by

(71)

Therefore, the average energy is not invariant under a general unitary transformation if the last term of the last equation is non-zero. However, it can be easily deduced that physical observables, which are given in terms of transition amplitudes, are invariant under unitary transformations, even if the Hamiltonian does not. This result can be obtained as follows.

Denoting the evolution operator for (where stands for the time-ordering operator) see for instance [38], such that for ,

(72)

Then, we have

(73)

Thus, we can deduce that the evolution operator transforms as

(74)

under a unitary transformation. As a consequence, the transition amplitudes from to obey:

(75)

This equation states that the transition amplitudes are equal and are invariant under a unitary transformation. This is a very important result because most physical observables can be obtained from the transition amplitudes and consequently, these observables are also invariant under unitary transformations. For instance, in HHG experiments, one can measure the amplitude and phase of each harmonic electric field, and these are related to photon emission transition moments, which can result in direct tomography of wave functions [9, 39].

We can now specialize the general unitary transformations to the special case of gauge transformations. Then, the unitary transformation takes the form:

(76)

In this case, and commute and the transformation of the Hamiltonian is given by

(77)

It was shown in previous sections that under this transformation, the Lagrangian of a particle interacting with electromagnetic radiation transforms into an equivalent Lagrangian via

(78)

where we set . This is the more general unitary transformation that yields a Lagrangian respecting the symmetry condition in Eq. (33). Note that the function is independent of , or , to avoid any dependence in , or in the Lagrangian as these quantities are not required to specify the dynamics of the system. The consequence of adding these terms is that the new Lagrangian would not describe a physical system obeying Hamiltonian mechanics. Thus, certain unitary transformations change the classical dynamics of the system, such as

(79)

although they leave transition amplitudes invariant. These are not gauge tranformations because they may change the form of the Schrödinger equation. An example of these are the Kramers-Henneberger (also called Bloch-Nordsieck [40]) transformations, presented below, which allow to obtain the acceleration gauge from the length gauge.

3 Long wavelength or dipole approximation

In this section, we describe the long wavelength approximation which is relevant when the wavelength of the electromagnetic field is much larger than the dimension of the system , that is . In that case, the spatial variation of the electromagnetic field over the size of the quantum system is very small and thus, it can be neglected. This approximation is often used in laser-matter interaction as it simplifies the calculations significantly.

More precisely, we consider a quantum system centered in and an electric field with a space-time dependence given by . Here, we assume that the -axis of the coordinate system is in the direction of the wave propagation and thus, we define the wave number . This form of the electric field is relevant as is a solution of a wave equation, which has plane wave solutions with this space-time dependence (for instance ). A general solution can be written as the linear combination of plane waves. Then, making a Taylor expansion as in [41],

(80)

This last equation can be re-written in two different but equivalent ways:

(81)
(82)

The first one allows us to understand the limit of the long wavelength approximation. Indeed, from this equation, we obtain the change in the electric field over the size of the system: