Density matrix theory is presented to explain recent experimental observations of intense optically induced magnetism due to a “mixed” type of nonlinearity proportional to the product of the electric- and magnetic-field strengths of light. Two previously unknown quadratic optical effects are predicted—namely, transverse optical magnetization and magnetic charge separation—and quantitative agreement is obtained with experimental results regarding the former of these. The mechanistic origin of a third quadratic nonlinearity, namely, magneto-electric second-harmonic generation, which is familiar on a phenomenological basis in classical nonlinear optics, is also examined. Transverse optical magnetism is shown to enable large permeability changes at optical frequencies accompanied by magnetic dispersion near resonances. This phenomenon provides for all-optical generation of magnetic moments, large transverse magnetic fields, static electric dipoles, and terahertz radiation in (unbiased) transparent homogeneous dielectrics or semiconductors. Intriguing possibilities for applications are considered, including magneto-electric refractive index modification, optical electric power generation, and spin control.
© 2009 Optical Society of America
CorrectionsStephen C. Rand, "Quantum theory of coherent transverse optical magnetism: erratum 2," J. Opt. Soc. Am. B 28, 1792-1792 (2011)
Throughout the long history of nonlinear optics, an enormous number of phenomena have been discovered in which electromagnetic waves combine to alter the properties of materials in complex ways. Nonlinear interactions proportional to various powers of the incident electric fields (, , etc.) have contributed greatly to the emergence of entire scientific fields such as ultrafast photochemistry, quantum optics, holography, quantum information, spintronics, and laser cooling. Although nonlinear optics is now a mature topic, the possibility of having the magnetic field of light work in concert with the electric component to yield a nonlinear magnetic response proportional to the product of electric and magnetic field strengths has gone unnoticed. Longitudinal magneto-optic effects such as the Faraday and inverse Faraday effect are well documented, but transverse magneto-electric effects in which both contributing fields oscillate at the optical frequency are unknown. In the past, the strength of magnetic dipole interactions was invariably thought to be limited to a small fraction of that of electric dipole interactions. Following a recent discovery , however, new geometries have become available for all-optical interactions and magnetic device applications that provide unexpectedly direct ways of controlling spins and creating magnetic moments in ordinary matter. An entirely new class of nonlinear optical interactions is emerging that holds promise for unanticipated electromagnetic technologies based on natural materials. This article formulates the quantum theory of this class of effects against the backdrop of recent observations of transverse optical magnetism.
Recently published experiments [1, 2, 3] and classical analysis  have revealed a quadratic mechanism whereby intense magneto-electric response can be obtained in (nominally nonmagnetic) dielectric media. Magnetic dipole emission nearly as intense as the electric polarization has been reported at infrared and visible wavelengths in transparent dielectric liquids (, , and ), far from any electronic resonances. Although the possibilities for magneto-electric material modification with light are expected to be more dramatic on resonance, where a quantum mechanical analysis is required, no quantum treatment of this problem has appeared. Here an analysis is made of the parametric origin of intense magnetic dipole emission and magnetization induced by linearly polarized optical waves in bound electron systems using the density matrix. This theory is in quantitative accord with experiments performed to date at nonrelativistic intensities in several insulators. Furthermore, it predicts the possibility of magneto-electric modification of the index of refraction in low-loss natural materials, and introduces a novel approach to generate intense magnetic fields that may be of use in many disciplines that rely on spin physics.
Since the time of Maxwell , it has been widely accepted that the effects of the magnetic component H of light can be ignored compared to effects of the electric component E at ordinary intensities. Though magnetic phenomena exhibit close parallels to electric behavior at relativistic intensities, strong magnetic response is absent in the optical range under moderate irradiation conditions. This has been attributed in the past to the comparative weakness of the (optical) magnetic Lorentz force. Arguments have been made suggesting that, unlike the electric permittivity ε, the magnetic permeability μ loses even its physical meaning at optical frequencies . Consequently, the many possibilities available through nonlinear optics for controlling the propagation of electromagnetic waves by altering the permittivity with external fields were thought not to extend to the permeability . In natural materials, where μ typically has the value characteristic of vacuum, it appeared that arbitrary control over the spatial properties of waves that depend in principle on both ε and μ was impossible. Until now, controlled variation of magnetic permeability even at relatively low frequencies in the gigahertz range has been realizable only in artificial materials called metamaterials . Experimental research on transformation optics [7, 8], which calls for controlled spatial variations of ε and μ together with low losses at much higher frequencies, therefore faces even greater challenges. In this paper, however, a new mechanism is analyzed that is capable of generating intense magnetic fields at single points, or along surfaces or in macroscopic volumes, and allows precise control of magnetic dispersion in low-loss, homogeneous dielectrics. The magnetic fields and dipole moments resulting from transverse optical magnetism are derived in subsequent sections, and then their possible relevance is considered to next-generation magnetic memories , spintronics , control of spins in Bose–Einstein condensates , atom optics , quantum information science , and electric power generation.
Research performed in transparent dielectrics (water, , and benzene) at optical intensities as low as , fully ten orders of magnitude below the relativistic optics threshold of , has now provided experimental evidence of a second-order magneto-electric optical process capable of generating intense magnetic dipole emission [1, 2, 3]. The measured magnetic susceptibility can be as large as half the electric dipole susceptibility, even far from any electronic resonances. Classical theory  that accounts for these surprising effects through the leading term of the multipole expansion, and predicts many new magneto-optical phenomena, agrees with these observations. However a quantum mechanical analysis of transverse optical magnetism has not yet been provided.
The main purpose of this paper is, therefore, to formulate a quantum mechanical theory of the intense transverse magnetic dipole moments and static electric dipole moments that form in bound electron systems as the result of irradiation with coherent light of moderate intensity. A radiant magnetization that is driven simultaneously by the optical electric and magnetic fields is first calculated analytically using density matrix analysis of 2-level atoms or molecules. The calculation shows that directly driven magnetic resonance is possible, and by considering the induced magnetic dispersion near an electronic resonance, predictions are made of intensity-dependent, magnetic modifications of the refractive index in dielectric media. We show that it is theoretically possible to attain negative permeability in natural materials that are completely homogeneous, and that moderately intense light can be used to program the spatial distribution of . We further deduce that static charge separation takes place under the same conditions, and we mention briefly the prospect of electric power generation in transparent insulators.
2. DENSITY MATRIX ANALYSIS
We begin by considering a system of identical 2-level atoms or molecules with a resonance frequency subjected to an electromagnetic plane wave of frequency ω that propagates in the positive z direction. The light is linearly polarized along and detuned from resonance by . Population dynamics and coherences are found using the density matrix equation of motion:
Substitution of Eqs. (4, 5, 6, 7, 8) into Eq. (3) furnishes the irreducible form of the interaction Hamiltonian. The use of Eqs. (2, 3) in Eq. (1) then permits solution of the density matrix . The main purpose of this paper is to predict the temporal behavior of induced electric and magnetic dipole moments in the medium by calculating the expectation values and , respectively.
3. MATRIX ELEMENTS OF TRANSVERSE MAGNETIC MOMENTS
Before proceeding to solve the equation of motion for the density matrix, a discussion of the calculation of expectation values is warranted due to the transverse nature of the magnetic dipole moment to be considered here. Individual light quanta carry spin angular momentum with projections on the axis of propagation given by , , or 0, depending on whether their state is left-circular, right-circular, or linear polarization respectively. In linear single-field MD interactions, the angular momentum needed to create or destroy a magnetic dipole moment is therefore provided by an appropriate state of circularly polarized incident light. However, in what follows, the induced angular momentum is not along , which denotes the propagation axis, but along the optical B field (chosen here to point along the transverse direction). The induced transverse moment, therefore, has no projection on the axis of propagation and can be generated without the transfer of any angular momentum from the optical field. For the quadratic interaction of interest here, it will be shown that when the angular momentum of the light field is zero (the case of linearly polarized light), a large magnetic moment that oscillates at the optical frequency itself can be induced perpendicular to . The time average value of this orbital angular momentum is zero so that angular momentum is conserved overall. However, this process gives rise to intense, radiant magnetic dipole fields.
To facilitate a comparison of longitudinal and transverse magnetic moments, it is convenient to consider two coordinate systems in which the polar axis is either parallel or perpendicular to the wavevector . First we discuss longitudinal magnetic moments by considering the coordinate system in which the polar and quantization axes are parallel to , and the azimuthal angle φ is measured with respect to the x axis. This is the geometry of conventional magnetic dipole transitions. The electric dipole transition moment on a single atom is10) is, in general, a sum over all states of the system. Here however, we assume that the dynamics are dominated by only two states. State 1 is the ground state and state 2 denotes the particular excited state that has minimum detuning from the incident light frequency and opposite parity with respect to state 1. The quantities and are the probability amplitudes of state 1 and state 2, respectively.
The magnetic dipole transition moment for a one-photon interaction connecting states 1 and 2 is
At low intensities the magnetic moment in Eq. (12) is negligible for linear polarization, because the linear momentum of the electron is very nearly parallel to its displacement. The Lorentz force is negligible compared to the force of the electric field at nonrelativistic intensities. The cross product in the integrand of Eq. (11) and the associated angular momentum are therefore nearly zero. Only an electric dipole oriented along is induced. In the case of circular polarization, the electron follows the electric field adiabatically, circulating around the propagation axis, inducing a steady magnetic moment oriented along the z axis . This motion mediates the inverse Faraday effect caused by circularly polarized light , which is not of interest in this paper. Consequently, in one-photon, electric-field-mediated interactions, only the angular momentum carried by the field , where z is the quantization axis, can be transferred to the atom and the initial and final states and must differ in magnetic quantum number m accordingly .
In the case of an interaction mediated jointly by the E and B components of a linearly polarized field, the orientation of the magnetic moment is along the laboratory y axis, and its calculation is significantly different because two driving forces contribute to the motion. The Lorentz force causes the linear momentum induced by the electric field E to acquire a small transverse component that is azimuthal with respect to the B component of the optical field. We therefore introduce new source coordinates with a polar axis along B . As before E defines the axis. The third basis vector is , and is considered to be measured from the axis. In this primed coordinate system the linear momentum may be written as . The expression for the magnetic moment in Eq. (11) becomes13) is implicitly written in the rotating frame where the rapid time dependence is associated with the azimuthal momentum and the magnetic moment points in the expected direction anti-parallel to the B field (along ).
Under the action of the forces due to orthogonal fields E and B, the time dependence of radial and azimuthal motions may differ. Hence we assume the wavefunction is separable according to and introduce separate c coefficients for the radial and angular parts of the wavefunction as follows: and . Correspondingly, we define electric and magnetic density sub-matrices by and in the laboratory reference frame. The time development of is determined by E, and that of is determined by B. Both fields oscillate at the optical frequency, so by invoking the slowly varying envelope approximation (SVEA) the two submatrices can be written in the lab frame as
In terms of these quantities, the expression for the transverse magnetic moment in Eq. (13) becomes16), when the direction of the magnetic field is fixed along , the expectation value for the transverse magnetic moment is given by
Note that the expectation value of the magnetic moment in Eq. (16) is second order in the wavefunction as expected, not fourth order. The full density matrix is just the product of the submatrices and , given explicitly by
4. STEADY-STATE SOLUTION OF THE DENSITY MATRIX
To write the total magneto-optical interaction consistent with any particular choice of reference frame requires some care. The reason for this is that interactions driven by involve motional effects of two orthogonal fields, one of which is a polar vector and the other of which is axial. The rotating frame of linear optical interactions governed by alone, for example, has an axis along , whereas the magnetic moment induced by the combined action of and involves currents circulating about the axis. In this paper the joint effect of the electric and magnetic interactions will ultimately be described in the lab frame of reference, but to provide perspective on the kinematics, use will be made of a sequence of three reference frames. The calculation of dynamics begins in the rotating frame, is next transformed to the ordinary lab frame, and finally ends up in a z-adjusted lab frame.
Customarily, the rotating wave approximation is introduced in optical analysis to solve for system dynamics. One result of this is that in the frame co-rotating with a circular component of , the induced electric dipole is a constant. For this reason the electric dipole interaction is written as , where only the field varies with time. However, in the same reference frame, the magnetic moment oscillates at the optical frequency. This is implied by Eqs. (15, 17) where magnetic charge oscillation varies rapidly with time as whereas the electric coherence is only slowly varying. To include electric and magnetic interactions in an atom-field Hamiltonian referenced to a single frame, this must be taken into account.
The interaction Hamiltonian in the rotating frame has the form19). The electric and magnetic transition matrix elements of V are therefore
The charge oscillations induced by an electric field acting alone follow the time dependence of E. Hence the electric-field-induced coherence has the form in the lab frame or in the rotating frame, as previously indicated in Eq. (14). Charge oscillations that are jointly driven by electric and magnetic forces similarly follow the time dependence of the applied fields, and this gives rise to three distinct frequencies of oscillation in the lab frame, namely, 0 and , because there are combination terms in the product of the driving fields E and B.18) is therefore expected to take the form23) have the same time dependence as those in the magnetic interaction Hamiltonian of Eq. (21) and agree with the lab frame product of the magnetic and electric submatrices given by Eqs. (14, 15).
Equation (1) may now be solved directly for steady-state solutions by setting . We treat the electric interaction exactly, by applying it as a strong field in zeroth order . The magnetic dipole interaction is then applied as a perturbation in concert with the electric dipole interaction in first order , and the equation of motion is solved for the submatrix coherences by collecting terms at each frequency.
This procedure yields first-order results for the coherences, which are25) is designated by . The detunings in the resonant denominators are defined by and . In obtaining Eq. (25) the magnetic interaction has been treated as a perturbation, so the population difference equals the initial value, which may be assumed to correspond to the ground state . Population saturation effects due to the electric interaction are nevertheless taken into account when the electric field interaction is applied a second time to obtain the first-order result. The population difference that appears in Eq. (24) is then given by
5. CALCULATION OF TRANSVERSE OPTICAL MAGNETIZATION
The steady-state solution for the magnetization —the same macroscopic magnetization that appears in the constitutive relation associated with Maxwell’s equations—is given in the lab frame by
With the results of Eqs. (24, 25) in hand, we now specialize to the case of linear polarization. Upon substitution of the coherences (24, 25) into Eq. (27), the magnetization of Eq. (27) yields the result
The magnetization in Eq. (28) has the general form29) oscillates at the fundamental frequency ω not .
Before we can determine the dimensionless ratio R of magnetic to electric-dipole moments as a function of incident field strength, we must account for the axial versus polar nature of MD and ED moments. An adjustment is needed to account for the fact that of all the electrons that can be set in motion by the electric field to produce polarization P within a given volume, at most one half can be deflected to contribute to a magnetic moment M in the same volume . For a given number of charges per unit volume, the amplitude of the oscillatory magnetization must therefore be corrected by another factor of 2 before direct comparison with the amplitude of electric polarization is possible.
This correction is equivalent to a transformation that rescales the laboratory z coordinate, since oscillatory motion in an arc about B resolves itself differently on the Cartesian x and z axes (See  for further discussion). Circular arc motion projected onto reverses twice as often as the same motion projected on . As a result, the amplitude of magnetic charge oscillations projected onto the propagation axis must be halved for comparison with the amplitude of electric dipole oscillation measured along . The halving of the z amplitude may be taken into account with the substitution in Eq. (30). We also note that the second term in Eq. (30) is much smaller than the first due to the factor in the denominator. To an excellent approximation our expression for the radiant magnetization at the optical frequency therefore reduces to32) cannot exceed unity. Also, the maximum value of the off-diagonal matrix element is (see Appendix A). So the ratio R has a maximum value that is32), it is important to note that the ratio of magnetic to electric susceptibility can attain the maximum value given by Eq. (33) at nonrelativistic intensities. This may be demonstrated by direct numerical integration of the equations of motion  and is the subject of a forthcoming publication . Ultrafast growth (on a timescale [1, 3]) of magnetic response takes place via energy transfer from electric field-induced linear motion along x, to the azimuthal motion initiated by the magnetic field along , and is due to the phenomenon of parametric resonance .
On the basis of Eqs. (29, 30, 31, 32, 33), the development to this stage can be summarized in a few points. The radiant magnetic emission intensity is predicted to be quadratic with respect to the input intensity. It may be enhanced by electronic resonance at and is governed secondarily by a parametric detuning factor . It can grow to a value of, at most, one fourth that of the electric dipole emission intensity. These findings are in excellent agreement with experimental results [1, 2, 3] at intensities ten orders of magnitude below the relativistic threshold.
To calculate the magnetic susceptibility, and to compare it with the electric susceptibility, we now make use of Eq. (31).10, 24). This yields34, 37), local field renormalization has been ignored. The ratio of magnetic and electric susceptibilities obtained from these results is1 . The matrix element in the numerator of Eq. (38) reflects transformation of the magnetic susceptibility as a rotation about the y axis. According to Eq. (34), the magnetic susceptibility is also proportional to the electric dipole transition moment. Hence the matrix elements and must both be nonzero for optical magnetization to be allowed, and the electric field amplitude must be large for it to be intense.
Some further comments about selection rules are in order. Explicit evaluation of the magnetic matrix element between states of well-defined total initial and final angular momentum and , respectively, using the Wigner-Eckart theorem , yields39) indicates explicitly that magnetic interactions induced by circularly polarized components of the B field exchange spin angular momentum of with the atom. By contrast, linearly polarized fields exchange no spin angular momentum with the atom. Nevertheless, at moderate intensities, the combined action of linearly polarized E and B fields can drive the formation of a parametrically enhanced, oscillating transverse orbital angular momentum as specified by Eq. (31). For this to happen, the reduced matrix elements of and x must be simultaneously nonzero, and the selection rules and must be satisfied.
6. SECOND-HARMONIC AND DC ELECTRIC DIPOLE PROCESSES
Electric dipole moments can also be generated by the joint action of optical E and B fields. Two additional processes emerge from this analysis by considering expectation values of the electric dipole operator in combination with the magneto-electric coherences developed in Eqs. (24, 25). One process yields a radiant polarization at the second-harmonic frequency, and the other produces a static electric dipole in the direction of propagation of light.
We now consider electric dipole moments that develop perpendicular to and . A z-directed, magnetically induced electric dipole moment is clearly distinct from either the linear electric dipole induced along or the nonlinear magnetic dipole induced along . Its macroscopic polarization is calculated using24, 25, 41) into Eq. (40), and specializing again to the case of linear input polarization, one finds in the Cartesian lab frame where the charge oscillation along z is at a doubled frequency (i.e., ) that31), the second-harmonic signal is longitudinally polarized and lacks the parametric resonance factor , so it is expected to produce only weak emission perpendicular to the pump wave. The second term is a zero-frequency term that predicts a static charge separation induced by light in dielectric media illuminated by moderately intense coherent light. Since it originates from the oscillatory coherence in Eq. (23) and contains the same parametric denominator as the magnetization in Eq. (31), its magnitude is expected to be strongly enhanced. In ultrashort pulse interactions, this effect will therefore generate intense longitudinally polarized terahertz radiation, although conventional phase-matching of the output will not be possible.
These same optical effects were predicted previously using steady-state analysis of a classical model of electron motion subject to external electric and magnetic forcing fields and Hooke’s law restoring forces . The present density matrix treatment has the merit of identifying the relative intensities, detuning dependences, emission frequencies, selection rules, directionality and multipole character of these effects in an independent, systematic way that requires no interpretation and is valid near resonances.
7. MAGNETIC DISPERSION
As one example of an application of these findings, we now briefly discuss the dispersion of the magnetic dipole response calculated in Eqs. (28, 38) and its connection with refractive index behavior. The refractive index of a medium is determined by the relative permittivity and permeability according to . Ordinarily the permeability of dielectrics is very close to the vacuum value at all frequencies, and because it is constant it does not contribute to dispersion of the index. However, through the intensity dependence of the magnetic susceptibility , the refractive index itself becomes intensity-dependent. can therefore be modified using induced magnetic dispersion, particularly near electronic resonances .
and have opposite signs and a small phase shift as illustrated in Fig. 1. Consequently, the main effect of magnetic dispersion is to reduce the refractive index. In addition, for a sufficiently sharp electronic resonance at , there is a frequency range on the red (long wavelength) side of resonance where the optically induced permeability may acquire negative values.
In the figure, the dashed curves illustrate induced magnetic susceptibilities of different magnitudes, near a resonance at . The linewidth-to-frequency ratio is arbitrarily taken to be . The plasma frequency is assumed to be , close to the value for silica. The ratio of magnetic-to-electric susceptibility has a value determined by the incident intensity and lies between zero and . R is negligible at low intensities, grows linearly at intermediate intensities, and eventually reaches at a “saturation” intensity that is material-dependent . Above the magnetization ceases to grow quadratically, becoming merely proportional to the incident intensity rather than to its square. In this “saturation” regime, the magnetic susceptibility therefore maintains a fixed proportionality with respect to the electric susceptibility.
The main results of this paper are contained in Eqs. (31, 33, 42). The first of these predicts that light of frequency ω induces a coherent oscillatory magnetization that radiates at frequency ω in bound electron systems. This dynamic magnetization M arises via a quadratic nonlinearity  driven by the product of the optical field amplitudes and can be comparable in magni tude to the electric polarization P at high but subrelativistic intensities. Equation (33) shows that the maximum ratio of magnetic-to-electric susceptibility is , without resorting to classical arguments based on geometric considerations or Maxwell’s equations. Thus the maximum intensity of magnetic dipole emission from insulators, which depends on the square of the susceptibility, is predicted to be one fourth that from electric dipole polarization, which is in excellent quantitative agreement with experiments to date. The coherence established by this type of interaction also gives rise to two other nonlinear optical effects. The first term in Eq. (42) describes magnetically induced second-harmonic generation, in which the emitted electric field lies along , parallel to the propagation of the pump light. The second term is static charge separation. The orientation of the static electric dipole is again along . Finally, we note that, according to Eqs. (31, 36), the calculated electric and magnetic susceptibilities have opposite signs across electronic resonances, indicating that the induced magnetic response is diamagnetic. As a result, the magnetic dispersion that accompanies transverse optical magnetization near resonances provides a unique new method of modifying the refractive index of unstructured dielectrics.
The enhancement of magnetic response by a factor close to the speed of light as described in this work is due to parametric resonance. This phenomenon manifests itself in Eqs. (31, 42) through the doubled-frequency detuning parameter , but its impact only becomes apparent in numerical simulations . The result of this dynamic enhancement is that magnetic dipole response can be nearly as intense as the electric response at intensities far below the relativistic threshold  due to transfer of energy between the motions associated with the E and B components of the light field. Physically, this process requires both an electric dipole transition and a magnetic dipole transition to be driven on the atom simultaneously by linearly polarized incident light. It is therefore not surprising that the transverse MD and ED selection rules require that rotations about y transform the same way as translations along x, and the MD and ED transition moments must be nonzero for the same initial and final states. This requirement is met for states of different parity in many crystallographic point groups.
In systems that satisfy these symmetry requirements, an important role can be anticipated for the manipulation of orbital and spin magnetism by optical waves. In a manner quite different from earlier experiments that demonstrated ultrafast control over magnetic moments without applying static magnetic fields [14, 22, 23, 24], the present results explain how transverse coherent magnetization can generate magnetic moments that are eight to ten orders of magnitude larger than expected, and internal magnetic fields that are in the Tesla range, by a new process at subrelativistic intensities. Since optical modulators can provide real-time control of the intensity and frequency of light within large volumes of dielectric material, this finding may enable programmable transformation optics, prolongation of coherence times for spintronic circuitry, ultrafast reading and writing of magnetic memories, and a new family of magnetic sensors or imagers. The capability of creating large magnetic moments or torques at an optical focus in dense systems of bound electrons, including condensates, should provide new modalities of spin control for research. Short-period magnetic field distributions produced by intense standing waves of light may be useful for atom optics or free electron lasers operating at short wavelengths. The availability of programmable distributions of intense magnetic fields could be used to improve the fidelity of compact quantum information systems based on spin qubits. Furthermore, these results have significant implications for high-field and plasma science, as well as laser fusion, since they make it clear that in ultrafast pulse interactions magnetic dynamics cannot be ignored at pre-pulse intensities even ten orders of magnitude below the relativistic threshold on very short timescales. Finally, the existence of a mechanism whereby light causes static charge separation in nonconducting media has been outlined. This opens the door to both capacitive and inductive electric power generation using coherent light.
Here it is shown that the maximum value of an off-diagonal density matrix element in a 2-level system is one-half. Consider a system described by the wavefunctionA.9) into Eq. (A.6) yields , or
The author gratefully acknowledges funding by the National Science Foundation (DMR-0502715 and CISE-0531086) and assistance in preparation of this paper by William Fisher and Yuwei Li. The author thanks the following individuals and institutions for hospitality during the course of this research: C. deAraujo at Universidade Federale de Pernambuco, O. Svelto at Politecnico di Milano, W. Sibbett at the University of St. Andrews, and C. Mirasso at the Universitat de les illes Balears.
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