## Abstract

The state mixings necessary to mediate three new optical nonlinearities are shown to arise simultaneously and automatically in a 2-level atom with an ℓ = 0 ground state and an ℓ = 1 excited state that undergoes a sequence of electric and magnetic dipole-allowed transitions. The treatment is based on an extension of dressed state theory that includes quantized electric and magnetic field interactions. Magneto-electric rectification, transverse magnetization, and second-harmonic generation are shown to constitute a family of nonlinear effects that can take place regardless of whether inversion is a symmetry of the initial unperturbed system or not. Interactions driven jointly by the optical electric and magnetic fields produce dynamic symmetry-breaking that accounts for the frequency, the intensity dependence, and the polarization of induced magnetization in prior experiments. This strong field quantum model explains not only how a driven 2-level system may develop nonlinear dipole moments that are forbidden between or within its stationary states, but it also broadens the class of materials suitable for optical energy conversion applications and magnetic field generation with light so as to include all transparent dielectrics.

© 2014 Optical Society of America

## 1. Introduction

High-frequency magnetism has attracted increasing attention in the last few years due to its relevance to metamaterials, spintronics, quantum information, high-speed optical magnetic storage, and other topics. These fields have benefited from decades of advances in techniques to produce novel material properties reliant on magnetic response, such as negative permeability in structured materials [1], coherent optical spin control of semiconductor charge carriers [2] or luminescent centers [3], and ultrafast switching of magnetic domains [4]. In sharp contrast to this, the possibility of harnessing magnetic response in natural, unstructured, “non-magnetic” materials has received virtually no attention. This is understandable in light of traditional arguments suggesting that high frequency magnetism can be neglected in homogeneous media [5]. On the other hand, it is surprising in view of recent experimental findings that magneto-electric (M-E) interactions can reach intensities comparable to electric interactions in some transparent dielectric media [6–8]. In fact a trio of unanticipated M-E nonlinearities was discussed in earlier work, including magneto-electric rectification (charge separation), *transverse* magnetization, and second-harmonic generation. If high frequency magnetic response could be induced with moderate light intensities in optical materials regardless of their symmetry, it goes without saying that new horizons would emerge for materials and photonic technologies. Among the intriguing possibilities that have been discussed for this trio of new effects are the conversion of electromagnetic energy directly to electricity in “optical capacitors”, solar power generation without semiconductors, the achievement of negative permeability in natural materials, and the generation of large (oscillatory) magnetic fields without current-carrying coils [9–11].

Beginning in 2007, elastic scattering experiments [6–9] were performed at frequencies far off resonance in a variety of transparent dielectric compounds that revealed a second-order nonlinear optical process that produced magnetic dipole scattering nearly as intense as first-order electric polarization. Several classical analyses [7, 10] and a quantum treatment [11] of these unexpected findings followed to investigate how intense dipolar magnetic response might arise at optical frequencies. Traditional nonlinear optical theory, that assumes dissipation-free parametric interactions [12], dictates that magnetic dipole (MD) moments of atoms are at most small fractions of their electric dipole (ED) counterparts (unless the ED moment vanishes altogether), so the experimental results are difficult to understand in a conventional context. The traditional upper limit of the MD/ED intensity ratio is the square of the fine-structure constant, *α*^{2} ≐ (1/137)^{2}, and transverse magnetization would normally be ruled out altogether in transparent liquids where inversion symmetry is a property of the medium. Nevertheless, a very large magneto-electric susceptibility driven by the bilinear product of the optical electric and magnetic fields (*EH*) was inferred from the elastic scattering data of [7] in molecular liquids. Quantitative agreement was even achieved with a predicted maximum MD/ED intensity ratio of 1/4 at large detunings. However, the first quantum theory of transverse magneto-electric effects implicitly relied on rare electronic properties and handled the magnetic interaction as a weak perturbation. The system ground state was assumed to be of, or to acquire, the mixed angular momentum character necessary to mediate magnetic transitions between ground state sub-levels, whereas many atomic or molecular ground states are singlets that lack angular momentum altogether. Also, the intermediate state in the 2-photon interaction driven by *EH* was implicitly taken to be a superposition of the ground and excited states rather than an eigenstate of the unperturbed system. Because this earlier model did not begin with an explicit angular momentum structure and treated the magnetic interaction only perturbatively, it did not provide a universal model or a compelling proof that the trio of optically magnetic effects mentioned above could take place. The present work is intended to provide such a model.

In this paper it is shown using a simple, 2-level atom picture [13] that, remarkably, no specialized internal ground state structure is necessary to support the magneto-electric interactions of interest. This analysis establishes that an entire family of nonlinearities not predicted by conventional nonlinear optics can arise in any bound electron system driven by linearly-polarized light at non-relativistic intensities. The treatment quantizes both atom and field, takes both fields into account fully, and satisfies all quantum mechanical selection rules [14] while predicting three types of induced dipole moment in centrosymmetric media. For example, it demonstrates that the prohibition against a static electric dipole in eigenstates of systems with inversion symmetry is overturned when the system is driven simultaneously by optical *E* and *H* fields. Our formal procedure embeds the state-mixing caused by the electric and magnetic interactions into admixtures to the driven energy eigenstates and indicates that all bound electron systems support magneto-electric dynamics regardless of whether inversion symmetry is present or not.

The main result of our computations is an explanation of how second-order magneto-optical phenomena arise in a vastly expanded class of materials. The extension of these results to molecular systems with rotational degrees of freedom capable of enhancing the response by many orders of magnitude via 2-photon resonance is left to a forthcoming publication. The present work establishes the main features of dynamic magneto-electric phenomena in a general atomic model but does not go as far as analyzing the maximum achievable response in molecular materials for prospective applications to optical-to-electrical energy conversion, intense magnetic scattering, negative permeability, preservation of spin orientation in spin-based electronics, and the generation of spatially-programmable high-frequency magnetic fields [9, 11].

## 2. The atomic model

Consider a stationary two-level atom that is characterized by an electric dipole resonance at frequency *ω*_{0} as depicted in Fig. 1. The dynamics are assumed to begin in the atomic ground state and light of frequency *ω* is turned on adiabatically. Orbital angular momentum *ℓ* is assumed to be a good quantum number in the absence of light, and the ground (excited) state is taken to be *ℓ* = 0 (*ℓ* = 1). Hence the undriven system has inversion symmetry (centrosymmetry). The atomic nucleus is fixed in position at the origin (Born-Oppenheimer approximation) and the fields are assumed to be uniform over the region occupied by the atom (dipole approximation). The quantization axis is taken to lie along the electric field polarization *x̂*, perpendicular to the direction of light propagation (*ẑ*). In the excited state, the projections of angular momentum yield magnetic sub-state quantum numbers of *m* = {−1, 0, 1}. The ket notation |*α*, *ℓ*, *m*〉 will be used to denote the bare atomic states, where *α* specifies the principal quantum number (not to be confused with the fine-structure constant). The atomic basis states are assumed to consist of the set {|1, 0, 0〉, |2, 1, 0〉, |2, 1, 1〉, |2, 1, −1〉}. A single-mode optical field is specified by the Fock state |*n*〉, where *n* denotes the photon occupation number. The electric and magnetic field components correspond to one and the same mode. Four atom-field product states are chosen for the uncoupled basis, and a dressed state approach [14] is adopted to solve for the dynamics. As an extension to the usual treatment, both the electric and magnetic dipole interactions are incorporated in the system Hamiltonian from the outset. The secular equation is solved by diagonalization in the four state basis and new eigenenergies, eigenstates, and moments that develop within the driven system are determined at various frequencies.

First, a simplified notation is introduced for the uncoupled, |atom〉|field〉 product states

These states are eigenstates of the atom-field Hamiltonian given by*E*(for

_{i}*i*= 1, 2, 3, 4) defined by

*Ĥ*|

_{af}*i*〉 =

*E*|

_{i}*i*〉. The states specified by Eqs. (1)–(4) are the appropriate bare states for the present problem because they can be coupled by an absorptive electric interaction followed by a magnetic interaction that may be either absorptive or emissive. The eigenenergies are assumed to have the explicit values

_{21}≡ (

*E*

_{2}−

*E*

_{1})/

*ħ*=

*ω*

_{0}−

*ω*, Δ

_{31}≡ (

*E*

_{3}−

*E*

_{1})/

*ħ*=

*ω*

_{0}− 2

*ω*, Δ

_{41}≡ (

*E*

_{4}−

*E*

_{1})/

*ħ*=

*ω*

_{0}. For the purposes of the present paper, these energies are taken to be purely electronic in origin. Other excitations, for example molecular rotational degrees of freedom, are excluded, but will be examined in a future publication. In Eq. (5) above,

*σ̂*is a Pauli spin operator and

_{z}*â*

^{+}(

*â*

^{−}) is the raising (lowering) operator of the single mode field.

If one imagines starting from the ground state, the 2-photon (electric + magnetic) excitation of interest here proceeds in two steps, with the electric field acting first since the ground state lacks angular momentum. With the quantization axis parallel to the electric field direction *x̂*, the first step is the allowed ED transition from state |1〉 → |2〉. The second step is an MD transition either from |2〉 → |3〉 or from |2〉 → |4〉. The |2〉 → |3〉 transition is driven by field amplitude *H* and the |2〉 → |4〉 transition is driven by *H ^{*}*, without a change of principal quantum number in the uncoupled basis. These magnetic transitions involve the action of raising and lowering operators of the angular momentum, following the usual prescription

*L̂*

_{±}|

*α*,

*ℓ*,

*m*〉 =

*ħ*[

*ℓ*(

*ℓ*+ 1) −

*m*(

*m*± 1)]

^{1/2}|

*α*,

*ℓ*,

*m*± 1〉 as indicated below. The result of the sequence of ED and MD interactions is that all four atomic basis states are mixed into the new quasi-eigenvalues and quasi-eigenstates of the driven system (see Fig. 2).

The magneto-electric interaction Hamiltonian is introduced next in a fully-quantized form. Our procedure in this paper is to make the rotating wave approximation (RWA) for the electric interaction but not for the magnetic interaction. As a result, the electric interaction contains two co-rotating (energy-conserving) terms whereas the magnetic interaction contains four terms – two co-rotating and two counter-rotating. The interaction Hamiltonian is

*L̂′*

_{±}≡

*L̂*

_{±}/

*ħ*in Eq. (10) is defined for notational convenience so that the pre-factors

*ħg*and

*ħf*both have units of energy. Here

*g*=

*μ*

^{(}

^{e}^{)}

*ξ/ħ*and

*f*=

*μ*

^{(}

^{m}^{)}

*ξ/ħc*refer to the quantized field amplitude coefficients for mode volume

*V*, with

*ξ*= (

*ħω*/2

*ε*

_{0}

*V*)

^{1/2}. As a common point of reference, matrix elements

*μ*

^{(}

^{e}^{)}and

*μ*

^{(}

^{m}^{)}were arbitrarily chosen to be that of atomic hydrogen: ${\mu}^{(e)}=e{a}_{0}\sqrt{2}(128/243)$ on the first Lyman transition (

*ω*

_{0}= 1.55 × 10

^{16}s

^{−1}) and ${\mu}^{(m)}=(e\hslash /2{m}_{e})(1/\sqrt{2})$, where

*a*

_{0}is the Bohr radius and

*m*is the mass of an electron. The complete Hamiltonian is

_{e}All terms in Eq. (10) must be included in the current model because the magnetic transition caused by the incident light takes place in the excited state. The magnetic field components at frequency ±*ω* induce transitions between |2, 1, 0〉 and |2, 1, ±1〉 at zero frequency in an atomic model. Consequently, the 2-photon detuning is large; the magnetic response is far off resonance with respect to both the positive and negative components of the driving field. Hence both contribute similarly and must be taken into account.

Regrettably, the counter-rotating component of the interaction Hamiltonian generates new product states that are not included in the four-state basis given by Eqs. (1)–(4). One finds that a term such as *L̂′*_{+}*â*^{+} acting on |2, 1, 0〉|*n* − 1〉 generates |2, 1, 1〉|*n*〉, which is outside the initial basis. Thus as the interaction proceeds to higher order, additional states must be included to account completely for the dynamics. However, as we now proceed to show, the probability amplitudes associated with the additional basis states become negligible beyond a 6 × 6 description. The inclusion of counter-rotating terms from Eq. (10) is thereby found merely to double the various induced dipole moments.

In order to evaluate the relative importance of counter-rotating terms like *L̂′*_{+}*â*^{+} on the magnitude of second-order induced dipole moments, we proceed first by ignoring them in *Ĥ _{int}*. Then the problem is re-calculated in an eight-state basis that anticipates couplings beyond the initial four states. In the eight-state basis, one can evaluate effects that are third order in the optical interaction. With this approach, both the contributions of counter-rotating terms and the effects of expansion of the basis can be explicitly determined.

Following this strategy, we begin by dropping the non-secular terms in Eq. (10) and adopting the closed set of four basis states given by Eqs. (1)–(4). The eigenvalue equation

may then be written out explicitly in 4 × 4 matrix form as*n*this yields four “doubly-dressed” eigenvalues

*E*and eigenstates |

_{Di}*D*(

_{i}*n*)〉 distinguished by the index

*i*(where

*i*= 1, 2, 3, 4). The eigenstates are written as a linear combination of the basis states in accordance with

*a*|

_{i}^{2}+|

*b*|

_{i}^{2}+|

*c*|

_{i}^{2}+ |

*d*|

_{i}^{2}= 1. In this paper diagonalization has been performed numerically to calculate the dressed eigenstates, though analytic expressions are obtainable in principle for the dressed state coefficients by solving the quartic secular equation [15].

The energy level diagram for the doubly-dressed atom is shown in Fig. 2(a). Product states of the uncoupled basis have different photon occupation numbers, so when the states are mixed by the incident fields it is not surprising that *n* ceases to be a good quantum number. The quasi-states may nevertheless be distinguished by the number of photons associated with the first (ground state) term of the linear superposition of states in Eq. (14). Figure 2(b) then indicates that, within each 4-state manifold of the dressed atom, more values of *n* contribute to the state mixing than in conventional dressed state theory. Also, for low *n* values (and because *g* ≫ *f*) the doubly-dressed levels |*D*_{3}(*n*)〉 and |*D*_{4}(*n*)〉 shift very little with respect to their energies in the uncoupled basis. The splitting of dressed states |*D*_{1}(*n*)〉 and |*D*_{2}(*n*)〉 is given by the electric Rabi frequency
${\mathrm{\Omega}}_{g}\cong {({\mathrm{\Delta}}_{21}^{2}+4{\left|g\right|}^{2}n)}^{1/2}$, where Δ_{21} specifies the detuning on the 1-photon ED transition. Just as in traditional dressed atom theory, the separation of adjacent manifolds is the optical frequency *ω*. What is novel is the presence of the third and fourth basis state admixtures in each manifold, which alter not only the number of quasi-energy levels, but the system dynamics as well. Complete mixing of all electronic sub-states in the basis renders magneto-electric response “allowed”, thereby enabling longitudinal charge separation and the other unexpected magnetic effects depicted in Fig. 2(b) through dynamic symmetry-breaking, as detailed in the following section.

When the non-secular terms are included in Eq. (10), as ultimately they must be for consistency, doubly-dressed eigenstates that are accurate to any order of the optical interaction may be computed using progressively larger and larger sets of basis states. Fortunately, more than one application of the magnetic field has negligible effects at small 1-photon detunings. It is therefore sufficient to proceed next with a set of eight uncoupled product states to investigate corrections to the procedure above arising from non-RWA terms. One finds that an eight-state basis permits computation of second-order magneto-electric moments with negligible errors from higher-order effects. The eight states needed to formulate the theory comprehensively consist of the original four product states together with four more having the following photon numbers:

States |5〉 and |6〉 are generated by non-secular magnetic terms when*Ĥ*is applied to the initial basis states |1〉–|4〉. Thus the basis expands to six states. States |7〉 and |8〉 are generated by a second application of

_{int}*Ĥ*to states |1〉–|6〉. Again the basis expands to include two more states. This process may be repeated indefinitely to account for higher-order non-RWA couplings to the initial set of basis states given by Eqs. (1)–(4). As seen in the following section, however, extension of the basis to include states |7〉 and |8〉 already permits one to draw the conclusion that nonlinear contributions higher than second-order are negligible.

_{int}The new bare state energies in the expanded basis are

The corresponding eigenvalue equation for the eight-state basis is*i*= 1, 2,...,8 define the eight quasi-eigenstates of the driven system. As usual, the coefficients

*a*,

_{i}*b*,...,

_{i}*g*must obey the normalization condition

_{i}*g*or

_{i}*h*reflect system responses beyond second-order that originate from non-RWA interactions. These, and all other nonlinear responses that are higher than second-order, are shown to be very small in the next section.

_{i}## 3. Intensity and frequency dependence of magneto-electric moments

With the quasi-eigenstates of Eq. (14) in hand (together with those of Eq. (24)), various dipole moments that exist between the driven (dynamic) states of the system can readily be determined. However, before noting expressions for all three nonlinear moments of interest here, it is helpful to demonstrate that inversion symmetry of the charge distribution is lost in the driven system. This is a key feature of magneto-electric interactions that makes it possible for M-E nonlinearities to appear in centrosymmetric systems subjected to light.

As is well-known [16] a static electric dipole moment cannot exist in the initial ground eigenstate, |1〉, of a system with inversion symmetry as it is energetically indistinguishable in inverted and non-inverted states. On the other hand, in the driven centrosymmetric model considered here, finite dipole moments appear at three frequencies. Importantly, these include a static moment 〈*p _{ii}*(0)〉 because the system is mixed jointly by magnetic and electric components of light, as depicted for

*i*= 1 in Fig. 2(b). 〈

*p*(0)〉 is given in Eq. (27) and is shown below to be non-zero. Expressions for the other nonlinear moments are given in Eqs. (28)–(30).

_{ii}As an illustration for *i* = 1, the linear ED moment at the optical frequency in a system quantized along the *x*-axis, and designated in Fig. 2(b) by a solid straight arrow, is

*i*= 1) by

*ẑ*. The expression in Eq. (27) shows that the static ED moment along

*ẑ*is proportional to the matrix element $\u3008{\mu}^{(e)}\u3009=\u30082,1,-1\left|e\widehat{r}\right|1,0,0\u3009/\sqrt{2}$, which is not zero. Since the coefficients

*a*,

_{i}*b*,

_{i}*c*,

_{i}*d*are all non-zero, the result in Eq. (27) demonstrates that a static dipole can be sustained within any doubly-dressed state. Thus the inversion symmetry of the unperturbed atom has been lost as the result of dynamic symmetry-breaking by the 2-field optical interaction that creates the dressed states. This conclusion is upheld regardless of the size of the basis set.

_{i}The magnitude of the second-order polarization in Eq. (27) is proportional to the product of field amplitudes *EH ^{*}* through the coefficients

*a*and

_{i}*d*. The linear polarization on the other hand is proportional to

_{i}*E*alone. The various dependences of $\u3008{\widehat{p}}_{ii}^{(1)}(\omega )\u3009$, $\u3008{\widehat{p}}_{ii}^{(2)}(0)\u3009$, and other moments on photon number have been made evident in the plots of Fig. 3, where for the moment the counter-rotating terms have been omitted. Not shown in Fig. 3 are off-diagonal transition elements of the form 〈

*D*(

_{i}*n*)|

*p̂*(0)|

*D*(

_{j}*n′*)〉, 〈

*D*(

_{i}*n*)|

*p̂*(2

*ω*)|

*D*(

_{j}*n′*)〉, or 〈

*D*(

_{i}*n*)|

*m̂*(

*ω*)|

*D*(

_{j}*n′*)〉 where

*i*≠

*j*. Moments of this form with

*i*,

*j*= 3, 4 or 4, 3 are as large as those shown in Figs. 3(c) and 3(d) but have been omitted as repetitious. In Figs. 3(a) and 3(b), the rectification field $\u3008{\widehat{p}}_{ii}^{(2)}(0)\u3009$ is found to rise quadratically with respect to the input field amplitude (linearly in

*n*), whereas $\u3008{\widehat{p}}_{ii}^{(1)}(\omega )\u3009$ is linear in

*E*(square root dependence with respect to

*n*). This confirms that the rectification process is a second-order nonlinearity, though higher order field dependences can appear as dressed state index

*i*is varied by virtue of the fact that the dressed state formalism includes dynamic contributions originating from states other than the unperturbed ground state (Figs. 3(c) and 3(d)). Similar considerations apply to the second-harmonic and transverse magnetic moments that are simultaneously induced in the system. The expressions for these moments when the coefficients are taken to be real are (for

*i*= 1)

*l*) of Fig. 2(b) and Eq. (30) refers to the magnetic moment in the upper-half (subscript

*u*) of the diagram. The magnetic moments of Eqs. (29) and (30) oscillate at frequency

*ω*as implied by Fig. 2(b), yet they have quadratic field dependence as shown in Fig. 3(b). This emerges as a consequence of the magnetic transition dipole being induced by a magnetic field. The product ${a}_{1}{b}_{1}^{*}$ and its conjugate in the first and second terms of Eq. (29) and Eq. (30) accommodate the picture that the second-order moment requires the charge distribution to be in motion and the electric and magnetic interactions are considered sequential. An MD transition can then only be induced

*by a magnetic field*if the initial state not only has appropriate angular momentum (

*ℓ*= 1;

*m*= 0) but is also

*non-stationary*. Thus the charge distribution that initially occupies the ground state must be prepared by the electric field interaction in a superposition state on the |1〉 → |2〉 transition before the (otherwise linear) MD transition can be initiated. This simply reflects the well-known requirement that a magnetic field exerts force only on a charge in motion.

An alternative picture of the electric and magnetic interactions that produce magnetization ${m}_{ii}^{(2)}(\omega )$ is that they are simultaneous rather than sequential. When this picture is adopted, expressions for the magnetic moments change to ${\u3008{\widehat{m}}_{11}^{(2)}(\omega )\u3009}_{l}\propto ({a}_{1}{d}_{1}^{*}+c.c.)\widehat{y}$ and ${\u3008{\widehat{m}}_{11}^{(2)}(\omega )\u3009}_{u}\propto ({a}_{1}{c}_{1}^{*}+c.c.)\widehat{y}$ and the magnetic moments are considerably enhanced beyond the values predicted by Eqs. (29) and (30). However, a detailed treatment of this case is deferred to our forthcoming publication which addresses the question of how large the induced magnetic dipole moments can be.

The resonant behavior of the moments
${p}_{ii}^{(1)}(\omega )$,
${p}_{ii}^{(2)}(0)$,
${p}_{ii}^{(2)}(2\omega )$, and
${m}_{ii}^{(2)}(\omega )$ in Eqs. (26)–(30) are displayed in Figs. 4(a) through 4(f). Each of these curves is dotted outside the range [0.85*ω*_{0}, 1.15*ω*_{0}] where the RWA for the electric interaction is no longer valid. The purpose of these figures is only to display the presence and location of resonances. Due to the RWA, their magnitudes are not accurate over the entire range from zero to twice the optical resonant frequency. The curves shown for the electric moments have been calculated between dressed states with index *i* = 1, whereas the curves for the magnetic moments are for both *i* = 1 and *i* = 2. Both indices must be considered in the latter case in order to show a complete resonance profile for the magnetic effects. The “mirror symmetry” with respect to frequency *ω*_{0} of *i* = 1 and *i* = 2 curves in Figs. 4(d) and 4(e) is due to the anti-crossing of dressed state eigenenergies *E _{D}*

_{1}and

*E*

_{D}_{2}at

*ω*=

*ω*

_{0}. At this frequency the dressed states |

*D*

_{1}(

*n*)〉 and |

*D*

_{2}(

*n*)〉 exchange character, and the plots of magnetic moments for

*i*= 1 and

*i*= 2 switch character correspondingly (see for example Fig. 3(a) versus Fig. 3(b)). The main features to note are simply that there is a resonance at

*ω*

_{0}−

*ω*= 0 in all cases and one at

*ω*

_{0}− 2

*ω*= 0 in the magnetization and second-harmonic.

All three types of nonlinear moment display the same inverse proportionality with respect to the 1-photon detuning Δ_{21} that the linear moment,
${p}_{ii}^{(1)}(\omega )$, does. This reflects the fact that these nonlinear moments all depend on the first-order electric process which couples levels **|**1〉 and |2〉. A resonance governed by 2-photon detuning Δ_{31} is apparent in both the second-harmonic electric moment
${p}_{ii}^{(2)}(2\omega )$ and the magnetization
${m}_{ii}^{(2)}(\omega )$ for the upper magnetic transition depicted in Fig. 2(b) (see also Fig. 4(c) and Fig. 4(f)). The response peaks at *ω* = 0.5*ω*_{0}. Both the static electric moment
${p}_{ii}^{(2)}(0)$ and the magnetization for the lower magnetic transition
${m}_{ii}^{(2)}(\omega )$ have a 2-photon detuning of Δ_{41}. Since both Δ_{31} and Δ_{41} tend to be large in regions of transparency, the nonlinear moments are relatively small in the present model. For frequencies off resonance, the magnitudes of the nonlinear moments are generally smaller than the linear electric moment by factors of 10^{3} − 10^{8}. However if the optical field is taken to be close to the 2-photon resonance condition, Δ_{31} ≈ 0, then resonant enhancement of both the second-harmonic generation and the magnetization (on the upper transition) takes place. This is evident in Fig. 5 when the curves of
${p}_{11}^{(2)}(2\omega )$ and
${m}_{11}^{(4)}(\omega )/c$ for the upper transition are compared with the corresponding curves in Fig. 3(a). In fact,
${p}_{11}^{(2)}(2\omega )$ becomes comparable to the linear moment at moderate intensities under these conditions. Large enhancements of the rectification and magnetization on the lower transition arise in molecular systems in a way that makes them frequency-independent, but this topic is beyond the scope of the present model (see Section 4).

When counter-rotating terms are included in the calculations of the three nonlinear dipole moments, their magnitudes double, but little else changes. This can be seen by re-evaluating the nonlinear moments given by Eqs. (27)–(30) using the quasi-eigenstates in Eq. (24). Pedagogically it is useful to do this in two steps, first in a six-state basis and then in an eight-state basis. When states |5〉 and |6〉 are added to the basis, non-RWA contributions increase the moments over those given in Eqs. (27)–(30). These augmented dipole moments will be designated with a prime. As before, they are calculated for dressed state index *i* = 1. They are given by

*i*= 1 they are given by

*ω*.

The main contributions of non-RWA terms from Eqs. (31)–(34), as well as corrections to the magnetic moments from Eqs. (35) and (36), may be assessed by comparing the curves in Fig. 6 with those in Fig. 3. Figures 6(a) and 6(b) are identical to Figs. 3(a) and 3(b) for the rectification, second-harmonic, and magnetization. Consequently, the net moments are doubled. This is the result of including counter-rotating states |5〉 and |6〉 in the basis. Inclusion of states |7〉 and |8〉 in the basis only produces negligible corrections to the nonlinear magnetization, as is evident in the lower right corner of Figs. 6(a) and 6(b) where the curves barely make an appearance. As shown in Figs. 6(c) and 6(d), this conclusion is not altered for dynamics originating from states |7〉 or |8〉. Hence these results establish that the dominant magneto-electric nonlinearities are quadratic and that higher-order contributions to the phenomena analyzed here are negligible.

## 4. Discussion and conclusions

The main results of this paper are those in Figs. 3 and 6, and may be summarized as follows. By accounting fully for a sequence of electric and magnetic field interactions of light, it has been shown that dynamic symmetry-breaking takes place in a simple 2-level atomic model where inversion symmetry is initially present. This result is in accord with earlier calculations of classical trajectories of bound electron motion driven jointly by *E* and *H* fields [9], which showed that the electron distribution is not centered on the nucleus. Inversion ceases to be a symmetry of optically-driven charge systems when both an electric and a magnetic dipole interaction take place. New nonlinear processes are thereby enabled.

Using a simple quantum model, three second-order nonlinearities that are magneto-electric in origin are predicted. Their formation is enabled by the aforementioned loss of inversion symmetry. As an explicit example of this, the static second-order ED moment, ${p}_{ii}^{(2)}(0)$, was analyzed in a driven centrosymmetric system and found to be non-zero. Hence optically-induced charge separation [9] can take place regardless of the symmetry of the unperturbed medium. This has the effect of establishing a vast material class in which second-order magneto-electric charge separation and the other phenomena considered here can be observed. All transparent dielectric materials and semiconductors (in the forbidden energy gap region) should support the quadratic magneto-optic nonlinearities described in this paper.

Our results are in complete agreement with three key features of earlier experiments on magnetic dipole scattering in transparent dielectric media at intensities a factor of 10^{4} below white light generation threshold [6–8]. There, the intensity of elastically-scattered light polarized perpendicular to Rayleigh-scattered light in a non-chiral, non-polar liquid was found to be quadratic versus input intensity. The present analysis accounts for the occurrence of induced magnetization in centrosymmetric liquids at the incident frequency, its quadratic intensity dependence, and its transverse polarization (parallel to *H*). Note that the frequency of MD radiation predicted by this quantum model is in accord with a classical picture of circular motion of a bound charge driven at frequency *ω* along *x̂* by the electric force and at 2*ω* along *ẑ* by the Lorentz force imposed by *H*(*ω*). Such unusual, synchronized motion produces second-order response at frequency ±*ω*(= ∓*ω* ± 2*ω*), in addition to rectification (0 = ±*ω* ∓ *ω*) and second-harmonic generation (±2*ω* = ±*ω* ± *ω*).

Theory presented here does not attempt to explain the high intensity of magnetic scattering in non-magnetic molecular compounds reported in earlier experiments. Large 2-photon detuning factors limit the magnitudes of induced moments in the present atomic model. The smallness of these moments in turn derives from Eqs. (6)–(9) in which the bare state eigenvalues are taken to be the same for all the *ℓ* = 1 states. Such a choice forces the magnetic transitions to take place in the excited state far off resonance, in what might aptly be categorized as an “excited state” model of the magnetic dynamics. Much better agreement with experimental scattering intensities is found by treating the electric and magnetic interactions as phase-coherent and simultaneous, and by including rotational excitations in a molecular model. Treatment of such a “ground state” model is deferred to a future publication however.

Finally, it should be emphasized that in the present treatment, standard quantum mechanical selection rules are obeyed in all matrix elements that contribute to magneto-electric state mixing by the incident *E* and *H* fields or induced moments. These rules are the customary ones for ED and MD interactions, namely (Δ*ℓ _{ij}* = ±1; Δ

*m*= 0) and (Δ

_{ij}*ℓ*= 0; Δ

_{ij}*m*= ±1), respectively, when the subscripts refer to dressed state

_{ij}*components*. For example, the non-zero contribution to the magnetic moment between |

*D*

_{1}(

*n*)〉 and |

*D*

_{1}(

*n*+ 1)〉 is 〈

*m̂*〉 ∝ 〈

*n*|〈210|

*L̂*|21 − 1〉|

*n*〉, which obeys the rules Δ

*ℓ*

_{24}= 0 and Δ

*m*

_{24}= +1 for an MD transition. It is essential to recognize that the indices

*i*,

*j*= {1, 2, 3, 4} refer here to admixed components of the initial and final quasi-states of transitions in the problem, not the unmixed basis states. If specific reference is not made to these admixtures, confusion arises over how normally forbidden dipole moments can appear within or between energy levels of centrosymmetric media. In this paper we have shown that couplings mediated jointly by the

*E*and

*H*fields of light enable new moments to form

*only*in the driven system.

## Acknowledgments

This research was supported by grant FA9550_12_1_0119 from AFOSR and the MURI Center for Dynamic Magneto-Optics. The authors acknowledge useful discussion with P. Berman.

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