## Dynamic symmetry-breaking in a simple quantum model of magneto-electric rectification, optical magnetization, and harmonic generation |

Optics Express, Vol. 22, Issue 3, pp. 2910-2924 (2014)

http://dx.doi.org/10.1364/OE.22.002910

Acrobat PDF (1655 KB)

### 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

1. J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. **108**, 097402 (2012). [CrossRef] [PubMed]

2. W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, “Room temperature coherent control of defect spin qubits in silicon carbide,” Nature **479**, 84–87 (2011). [CrossRef] [PubMed]

3. G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centers in diamond,” Nat. Phys. **7**, 789–793 (2011). [CrossRef]

4. C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and Th. Rasing, “All-optical magnetic recording with circularly polarized pulses,” Phys. Rev. Lett. **99**, 047601 (2007). [CrossRef]

6. S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: Molecular scattering in a dielectric liquid,” Phys. Rev. Lett. **98**, 093901 (2007). [CrossRef] [PubMed]

8. W. M. Fisher and S. C. Rand, “Dependence of optical magnetic response on molecular electronic structure,” J. Lumin. **129**, 1407–1409 (2009). [CrossRef]

*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

9. W. M. Fisher and S. C. Rand, “Optically-induced charge separation and terahertz emission in unbiased dielectrics,” J. Appl. Phys. **109**, 064903 (2011). [CrossRef]

11. S. C. Rand, “Quantum theory of coherent transverse optical magnetism,” J. Opt. Soc. Am. B **26**, B120–B129 (2009); Erratum J. Opt. Soc. Am. B **27**, 1983 (2010); Erratum J. Opt. Soc. Am. B **28**, 1792 (2011). [CrossRef]

6. S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: Molecular scattering in a dielectric liquid,” Phys. Rev. Lett. **98**, 093901 (2007). [CrossRef] [PubMed]

9. W. M. Fisher and S. C. Rand, “Optically-induced charge separation and terahertz emission in unbiased dielectrics,” J. Appl. Phys. **109**, 064903 (2011). [CrossRef]

7. S. C. Rand, W. M. Fisher, and S. L. Oliveira, “Optically-induced magnetization in homogeneous, undoped dielectric media,” J. Opt. Soc. Am. B **25**, 1106–1117 (2008). [CrossRef]

10. W. M. Fisher and S. C. Rand, “Light-induced dynamics in the Lorentz oscillator model with magnetic forces,” Phys. Rev. A **82**, 013802 (2010). [CrossRef]

11. S. C. Rand, “Quantum theory of coherent transverse optical magnetism,” J. Opt. Soc. Am. B **26**, B120–B129 (2009); Erratum J. Opt. Soc. Am. B **27**, 1983 (2010); Erratum J. Opt. Soc. Am. B **28**, 1792 (2011). [CrossRef]

12. P. S. Pershan, “Nonlinear optical properties of solids: Energy considerations,” Phys. Rev. **130**, 919–929 (1963). [CrossRef]

*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

7. S. C. Rand, W. M. Fisher, and S. L. Oliveira, “Optically-induced magnetization in homogeneous, undoped dielectric media,” J. Opt. Soc. Am. B **25**, 1106–1117 (2008). [CrossRef]

*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.

13. M. Frasca, “A modern review of the two-level approximation,” Annals of Physics **306**, 193–208 (2003). [CrossRef]

*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.

9. W. M. Fisher and S. C. Rand, “Optically-induced charge separation and terahertz emission in unbiased dielectrics,” J. Appl. Phys. **109**, 064903 (2011). [CrossRef]

11. S. C. Rand, “Quantum theory of coherent transverse optical magnetism,” J. Opt. Soc. Am. B **26**, B120–B129 (2009); Erratum J. Opt. Soc. Am. B **27**, 1983 (2010); Erratum J. Opt. Soc. Am. B **28**, 1792 (2011). [CrossRef]

## 2. The atomic model

*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.

*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 Several detunings have been introduced here that are important for later discussions: Δ

_{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.

*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).

*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:

*ω*

_{0}= 1.55 × 10

^{16}s

^{−1}) and

*a*

_{0}is the Bohr radius and

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

_{e}*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.

*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.

*L̂′*

_{+}

*â*

^{+}on the magnitude of second-order induced dipole moments, we proceed first by ignoring them in

*Ĥ*. 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.

_{int}*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 where the expansion coefficients obey the standard normalization condition |

*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].

*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

_{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.

*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}*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 and may be determined by diagonalizing the secular determinant of the eigenvalue equation. Nonlinear dipole moments containing

_{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

*p*(0)〉 because the system is mixed jointly by magnetic and electric components of light, as depicted for

_{ii}*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}*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 The second-order, static electric polarization in the doubly-dressed system, designated in Fig. 2(b) by a horizontal curved arrow, is given (again for

*i*= 1) by

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

*ẑ*is proportional to the matrix element

*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}*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

*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

*n*), whereas

*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) Eq. (29) refers to the magnetic moment in the lower-half (subscript

*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

*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.

*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.

_{21}that the linear moment,

**|**1〉 and |2〉. A resonance governed by 2-photon detuning Δ

_{31}is apparent in both the second-harmonic electric moment

*ω*= 0.5

*ω*

_{0}. Both the static electric moment

_{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

*i*= 1. They are given by When states |7〉 and |8〉 are added to the basis, the only changes that are possible are in the magnetic moments. These additional contributions will be designated with a double prime. Calculated for

*i*= 1 they are given by These corrections are fourth-order in the optical field to lowest order and oscillate at frequency

*ω*.

## 4. Discussion and conclusions

*E*and

*H*fields [9

**109**, 064903 (2011). [CrossRef]

**109**, 064903 (2011). [CrossRef]

^{4}below white light generation threshold [6

6. S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: Molecular scattering in a dielectric liquid,” Phys. Rev. Lett. **98**, 093901 (2007). [CrossRef] [PubMed]

8. W. M. Fisher and S. C. Rand, “Dependence of optical magnetic response on molecular electronic structure,” J. Lumin. **129**, 1407–1409 (2009). [CrossRef]

*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*).

*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.

*E*and

*H*fields or induced moments. These rules are the customary ones for ED and MD interactions, namely (Δ

*ℓ*= ±1; Δ

_{ij}*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

## References and links

1. | J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. |

2. | W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, “Room temperature coherent control of defect spin qubits in silicon carbide,” Nature |

3. | G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centers in diamond,” Nat. Phys. |

4. | C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and Th. Rasing, “All-optical magnetic recording with circularly polarized pulses,” Phys. Rev. Lett. |

5. | L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, |

6. | S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: Molecular scattering in a dielectric liquid,” Phys. Rev. Lett. |

7. | S. C. Rand, W. M. Fisher, and S. L. Oliveira, “Optically-induced magnetization in homogeneous, undoped dielectric media,” J. Opt. Soc. Am. B |

8. | W. M. Fisher and S. C. Rand, “Dependence of optical magnetic response on molecular electronic structure,” J. Lumin. |

9. | W. M. Fisher and S. C. Rand, “Optically-induced charge separation and terahertz emission in unbiased dielectrics,” J. Appl. Phys. |

10. | W. M. Fisher and S. C. Rand, “Light-induced dynamics in the Lorentz oscillator model with magnetic forces,” Phys. Rev. A |

11. | S. C. Rand, “Quantum theory of coherent transverse optical magnetism,” J. Opt. Soc. Am. B |

12. | P. S. Pershan, “Nonlinear optical properties of solids: Energy considerations,” Phys. Rev. |

13. | M. Frasca, “A modern review of the two-level approximation,” Annals of Physics |

14. | C. Cohen-Tannoudji, |

15. | M. Abramowitz and I. E. Stegun, |

16. | K. Konishi and G. Paffuti, |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 28, 2013

Revised Manuscript: January 6, 2014

Manuscript Accepted: January 22, 2014

Published: January 31, 2014

**Citation**

A. A. Fisher, E. F. Cloos, W. M. Fisher, and S. C. Rand, "Dynamic symmetry-breaking in a simple quantum model of magneto-electric rectification, optical magnetization, and harmonic generation," Opt. Express **22**, 2910-2924 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-2910

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### References

- J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012). [CrossRef] [PubMed]
- W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, D. D. Awschalom, “Room temperature coherent control of defect spin qubits in silicon carbide,” Nature 479, 84–87 (2011). [CrossRef] [PubMed]
- G. D. Fuchs, G. Burkard, P. V. Klimov, D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centers in diamond,” Nat. Phys. 7, 789–793 (2011). [CrossRef]
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