March 15, 2015 / Vol. 40, No. 6 / OPTICS LETTERS
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Optical bistability of localized Josephson surface plasmons in cuprate superconductors Filippo Alpeggiani Dipartimento di Fisica and CNISM, Università di Pavia, via Bassi 6, 27100 Pavia, Italy ([email protected]) Received December 3, 2014; revised January 15, 2015; accepted January 16, 2015; posted January 22, 2015 (Doc. ID 228894); published March 3, 2015 Microparticles made of high-T c cuprate superconductors are characterized by localized plasmonic excitations known as Josephson surface plasmons, whose electromagnetic response is intrinsically nonlinear, giving rise to yet unexplored optical phenomena. In this work bistability effects in the near-resonance excitation of Josephson surface plasmons of dipolar symmetry are investigated for spheroidal superconducting particles. The threshold for the incident intensity is estimated, and experimental probing strategies are discussed. The system can be of interest in view of terahertz light switching and detection. © 2015 Optical Society of America OCIS codes: (190.1450) Bistability; (300.6495) Spectroscopy, terahertz. http://dx.doi.org/10.1364/OL.40.000867
High-T c cuprate superconductors have attracted much attention as potential negative-index-metamaterials [1] and terahertz nonlinear media [2–6]. In the superconducting (SC) phase their c axis optical response is related to the sine-Gordon equation for which various nonlinear phenomena, such as bistability and soliton propagation, have long been studied [7,8]. Recently, optical excitation of solitons in a cuprate superconductor has been experimentally demonstrated [9], testifying for a growing interest in probing nonlinear optical effects in cuprates, in view of applications in terahertz plasmonics and information processing [4]. In [2], optical bistability in the Fabry–Perot resonances of a cuprate SC slab has been predicted for sufficiently high incident fields. In this work, we study bistability effects in the presence of a different feedback mechanism, i.e., the localized plasmonic excitations of cuprate SC particles, known as Josephson surface plasmons (JSPs), whose frequencies can be obtained from a modal expansion of the electric field, as shown in [10] for spherical particles in the quasi-static approximation. These modes are analogous to localized surface plasmons of metallic nanoparticles [11]. Here, we generalize the formalism of [10] to prolate and oblate spheroidal particles of several aspect ratios, which are meant to simulate experimentally realizable geometries like pillars and disks, and we apply it to the study of optical bistability effects. Both geometries could be obtained by patterning a superconductor film, e.g., by focused ion beam lithography [12]. We consider monolayered cuprate superconductors, such as La2−x Srx CuO4 , whose structure can be modeled as a stack of conductive CuO2 planes along the c axis (which we take directed as zˆ) alternating with insulating block layers, as sketched in Figs. 1(a) and 1(b). In the SC phase, each pair of consecutive planes acts as a nanoscale Josephson junction and supports the tunneling current, J n1;n t −J 0 sinφn1;n t;
[4,10]. In the long wavelength approximation, we introduce the continuous phase field, φr; t Reφω r exp−iωt, where φω r is related to the complex z component of the electric field as [10] φω r −2iesE ω;z r∕ℏω
(2)
[s is the interplane separation, see Fig. 1(a)]. In the linear regime, the electromagnetic response of the condensate is modeled by the diagonal dielectric tensor [1,10] ↔ ε ω diagε∥ ω; ε∥ ω; εzz ω, where εzz (neglecting dissipation) is obtained from the linearized version of Eq. (1) in the Drude form, ω2J 2J 0 es εzz ω ε∞ − ≐ε∞ 1 − 2 : ω ℏε0 ω2
(3)
The c axis Josephson plasma frequency ωJ is typically in the THz range, much lower than the plasma frequency for in-plane motion (of the order of an eV). Observing that, neglecting higher-order harmonics, sinφr; t
2J 1 jφω rj φr; t Ocos3ωt; …; jφω rj
(4)
(1)
with φn1;n being the gauge-invariant phase difference of the condensate order parameter between two consecutive copper planes and J 0 being the critical current 0146-9592/15/060867-04$15.00/0
Fig. 1. (a) La2−x Srx CuO4 crystalline structure along the ac plane, the unit cell is delimited by the black contour; (b) corresponding electrodynamical model, as referred to in the text; and (c) prolate spheroidal superconducting particle. © 2015 Optical Society of America
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(J n denotes a Bessel function of the first kind), Eq. (3) can be modified to include nonlinear effects in the form 2ω2 J jφ rj ω2 ω2 ε0zz 1 − J21 ω ≃ 1 − J2 J2 jφω rj2 …: ε∞ ω jφω rj ω 8ω (5) We notice that, to a first approximation, the correction for εzz has the same quadratic dependence on the field as the optical Kerr effect. For instance, in the case of La2−x Srx CuO4 around the Josephson frequency (νJ ≈ 2 THz, s 0.66 nm, and ε∞ ≈ 16), we can estimate a second-order nonlinear refractive index n2 ≈ 4e2 s2 ∕ε0 cℏ2 ω2J ≈ 10−7 cm2 ∕W, which is several orders of magnitude larger than that of common nonlinear materials, pinpointing the interest in high-T c superconductors as potential nonlinear materials in the THz range. In addition, the nonlinearity described here is an intrinsic phenomenon that is directly related to phase coherence in the superconductor, i.e., to the macroscopic emergence of a quantum mechanical effect. Multilayered superconductors, characterized by at least two different intrinsic junctions in the unit cell, present a more complex response due to the presence of transverse optical modes [13] and coupling with phonons [14], and they are not explicitly treated in the present work, although similar phenomena are expected to occur. As shown in Fig. 1(c), we suppose that the cuprate is spatially confined in the form of a particle, smaller than the wavelength but still much larger than the coherence length ξ (for La2−x Srx CuO4 , ξab ≈ 3 nm), in order to minimize quantum size and fluctuation effects that are known to occur in conventional SC nanoparticles with size comparable to ξ [15]. The resonance frequencies ω0 and the (complex) field profiles E0 r of JSPs can be calculated from a modal expansion of the electric field in the linear regime [10]. In the quasi-static approximation, the total energy contained in each mode is proportional to the quantity [16] Z U tot
↔
d3 rE0 r
∂ω ε ω; r E0 r: ∂ω
(6)
Nonlinearity affects the resonance frequency of plasmonic modes. In particular, applying perturbation theory for plasmonic systems [17] and Eq. (5), we find that the resonance frequency of the JSP shifts to ω00 ω0 Δω0 ;
(7)
with m 0 and l 1) in the quasistatic approximation, which we call dipolar JSPs (D-JSPs). These modes are characterized by the important property that the field inside the particle is homogeneous and polarized along zˆ, and we will denote its amplitude as E 0;int . Similarly to the case of metallic nanoparticles [18], in the quasi-static approximation the unperturbed resonance frequency ω0 of D-JSPs does not depend on the size, but only on the aspect ratio r through the geometrical parameter L3 , according to the relation ω20 ω2J
ε∞ L3 ; 1 ε∞ − 1L3
(9)
which is shown in Fig. 2. Analytical expressions for L3 are given in [19]. For spheres, e.g., L3 1∕3 and ω0 ωJ ε∞ ∕2 ε∞ 1∕2 (for La2−x Srx CuO4 , ν0 ≃ 1.9 THz). Notice that ω0 redshifts with increasing the aspect ratio. We consider an external excitation by a plane wave with frequency ω and field magnitude E ext linearly polarized along zˆ . Our results are generalized to other configurations by considering only the z component of the incident field. Near resonance, the intensity of the internal field is approximated by a Lorentzian function of the (perturbed) excitation frequency, E 20;int
ω20 κ 2 E2 ; ω − ω00 2 γ 2 ∕4 ext
(10)
where the parameter γ takes into account homogeneous broadening of the plasmonic mode, which is mainly of nonradiative origin and can be significant even for highT c superconductors, due to losses by noncondensed carries. For D-JSPs, the coefficient κ 2 has the value κ 2 1∕f41 ε∞ − 1L3 2 g;
(11)
also represented in Fig. 2. The intensity-dependent shift of the plasmonic frequencies in Eq. (8) is the physical origin of bistability in Josephson plasmons. The situation is similar to optical bistability in dielectric spheres [20]. In particular, for D-JSPs the correction in Eq. (8) is evaluated as Δω0 J 1 jφ0 j 1 jφ j2 − ≈ − 0 Ojφ0 j4 ; ω0 16 jφ0 j 2
(12)
independently of the aspect ratio. This property crucially depends on the fact that in these modes the electric field
where the first-order frequency correction is given by Δω0 ε ∞ ω0 U tot
Z Vp
d3 r
ω2J 2J 1 jφ0 rj − 1 jE 0;z rj2 ; jφ0 rj ω20
(8)
with V p being the particle volume and φ0 r being related to E 0;z r through Eq. (2) with ω ω0 . We consider SC particles of spheroidal shape, with varying the aspect ratio r height∕diameter. In particular, we focus on plasmons with dipolar symmetry and out-of-plane polarization (in the notation of [10], modes
Fig. 2. Resonance frequency ω0 (normalized to ωJ , solid curve) and coupling factor κ2 (dashed curve) as a function of the aspect ratio for dipolar surface plasmons in spheroidal superconducting particles.
March 15, 2015 / Vol. 40, No. 6 / OPTICS LETTERS
is homogeneous inside the particle. Notice that the shift is negative in sign. From Eq. (2), we derive that the natural unit for the electric field is E 0 ℏω0 ∕2es. By replacing Eq. (12) into Eq. (10) and defining the dimensionless quantities W jφ0 j2
E 20;int E 20
;
κ2 E 2ext;z I ; E 20
ω − ω0 Δ ; ω0
and γ¯ γ∕ω0 1∕Q (Q is the resonance Q-factor), Eq. (10) can be rewritten in the form p p2 1 1 I W Δ − J 1 W ∕ W γ¯ 2 ; (13) 2 4 which is solved for the internal intensity W . It represents the fundamental equation for the study of bistability effects in the present system. As shown in Fig. 3(a), bistability is achieved for values of the incident intensity I above a certain threshold, in a range of negative values for the detuning parameter Δ (highlighted in the figure). Correspondingly, the internal intensity W plotted as a function of the incident intensity I (for a fixed detuning Δ in the bistability range) shows the hysteresis behavior of Fig. 3(b), with sharp transitions between high- and low-intensity states (as usual, a third intermediate state is unstable [2]). The hysteresis phenomenon derives from positive and negative feedback mechanisms between mode pulling [Eq. (12)] and coupling of external radiation to the Josephson plasmon [Eq. (10)]. The bistability threshold for the incident intensity can be estimated by expanding J 1 in series and approximating Eq. (13) with the cubic equation I W Δ W ∕162 γ¯ 2 ∕4;
(14)
which is similar to that of dielectric microspheres [20]. The threshold for entering the bistability regime corresponds to the condition
p Δth − 3γ¯ ∕2 and
p I th 16¯γ 3 ∕ 3 3 ;
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(15)
which implies E 2ext;th
16¯γ 3 2 64 1 ε∞ − 1L3 2 2 p p E0: E 0 Q3 3κ2 3 3 3
(16)
The threshold crucially depends on the resonance width γ¯ or, equivalently, on the resonance Q-factor of surface plasmons. Unfortunately, very few experimental data exist on the properties of JSPs in confined geometry. Extinction spectroscopy of powdered samples within the so-called sphere resonance method [21,22] revealed broad resonances, which we may expect to be at least partially of inhomogenous origin, due to size and shape variability of the particles. Nevertheless, even in the conservative assumption of γ¯ ≈ 0.1, i.e., Q ≈ 10, for La2−x Srx CuO4 we can estimate the threshold E ext;th ≈ 40 kV cm−1 for a spherical particle and E ext;th ≈ 9 kV cm−1 for a prolate spheroid with r 5. These values are within the reach of current technology, being of the same order of magnitude of the THz fields employed in the experiment described in [9] to generate solitons in cuprate superconductors. A further reduction of the threshold can be obtained by tailoring the geometry of the samples. In particular, as shown by Fig. 2, prolate spheroids with high aspect ratios are favored by both lower resonance frequencies and larger coupling factors κ 2 . Of course, different growth techniques must be chosen according to the desired aspect ratio. In any case, the use of single-crystal samples is highly convenient in order to maximize the effect. Bistability effects can be detected, for instance, with time resolved scattering or extinction experiments with a pulsed incident field. In the linear approximation, D-JSPs are characterized by the dipole moment [19], Z p ε0 εzz ω0 − 1
Vp
d3 rE0 r −ε0
Vp zˆ E : L3 0;int
(17)
Then the far-field intensity of scattered radiation at point R; θ reads SR; θ
Fig. 3. Solutions of the bistability Eq. (13) (¯γ 0.1): (a) the adimensional internal intensity W as a function of detuning Δ for some values of the incident intensity I (bistability region is highlighted); (b) W as a function of I for Δ −0.15.
ε0 sin θ 2 V 2p 4 2 ω E 0;int ; 2c3 4π L23 R2
(18)
i.e., it is proportional to the internal intensity. When the temporal width of the incident pulse T satisfies the condition γT ≫ 1, the quasi-steady-state approximation holds, and the intensity of the incident field can be calculated by solving the steady-state Eq. (13) for each value of the slowly-varying envelope of the incident intensity. We notice that the condition is much less strict for plasmonic systems than dielectric ones, since resonance lifetimes are shorter (Q ∼ 10). The time-dependent profile of the scattered intensity is shown in Fig. 4 for some values of the detuning parameter Δ. When the frequency of incident radiation is in the bistability region, the profile of the scattered field is strongly modified with respect to the incident one, showing an asymmetric shape with a sharp
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Fig. 4. Envelope of the scattered intensity as a function of time for different values of the detuning Δ. The incident intensity has a slowly-varying profile ∝ exp−t∕T2 shown in the background, with a maximum corresponding to the value I 0.023 for the parameter I defined in the text.
transition to a high-intensity state. This can be understood by following the hysteresis cycle in Fig. 3(b) for an increasing and then decreasing incident intensity. Particle heating due to incident radiation could be significant in the case of cw irradiance and it could drive the system out of the SC state. However, we believe that this is not the case of a pulsed excitation. For instance, the authors of [9] irradiated a cuprate sample with 25-ps pulses of THz light with peak fields of 10 kV cm−1 and reported minimal heating effects. Indeed, SC particles should be favored as they can better dissipate heat when cooled in a cryogenic system due to higher surface-tovolume ratios than bulk samples. In conclusion, we have theoretically investigated optical bistability in Josephson surface plasmons of spheroidal high- T c superconducting particles. Bistability originates from phase coherence in the superconducting order parameter through a nonlinear term in the tunneling current equation and it can be probed by terahertz time-resolved spectroscopy. The threshold for the incident intensity to enter the bistable regime critically depends on the third power of the resonance Q-factor. This phenomenon has potential application in the field of terahertz plasmonics, e.g., for ultrafast optical switching. Moreover, from a more fundamental point of view, nonlinear optical experiments could help in studying the condensate dynamics and clarifying some aspects of the mechanism of high- T c superconductivity. The author is grateful to Lucio Claudio Andreani for a critical reading of the manuscript and to Marco Liscidini for useful discussions.
References 1. A. L. Rakhmanov, V. A. Yampol’skii, J. A. Fan, F. Capasso, and F. Nori, Phys. Rev. B 81, 075101 (2010). 2. V. Yampol’skii, S. Savel’ev, A. Rakhmanov, and F. Nori, Phys. Rev. B 78, 024511 (2008). 3. S. S. Apostolov, Z. A. Maizelis, M. A. Sorokina, V. A. Yampol’skii, and F. Nori, Phys. Rev. B 82, 144521 (2010). 4. S. Savel’ev, V. A. Yampol’skii, A. L. Rakhmanov, and F. Nori, Rep. Prog. Phys. 73, 026501 (2010). 5. A. Wall-Clarke and S. Savel’ev, Phys. Rev. B 85, 214521 (2012). 6. T. N. Rokhmanova, S. S. Apostolov, Z. A. Maizelis, V. A. Yampol’skii, and F. Nori, Phys. Rev. B 88, 014506 (2013). 7. A. Barone, F. Esposito, C. J. Magee, and A. C. Scott, Riv. Nuovo Cimento 1, 227 (1971). 8. R. Khomeriki and J. Leon, Phys. Rev. E 71, 056620 (2005). 9. A. Dienst, E. Casandruc, D. Fausti, L. Zhang, M. Eckstein, M. Hoffmann, V. Khanna, N. Dean, M. Gensch, S. Winnerl, W. Seidel, S. Pyon, T. Takayama, H. Takagi, and A. Cavalleri, Nat. Mater. 12, 535 (2013). 10. F. Alpeggiani and L. C. Andreani, Phys. Rev. B 88, 174513 (2013). 11. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007). 12. A. Palau, C. Monton, V. Rouco, X. Obradors, and T. Puig, Phys. Rev. B 85, 012502 (2012). 13. D. van der Marel and A. A. Tsvetkov, Phys. Rev. B 64, 024530 (2001). 14. Y. Hirata, K. M. Kojima, M. Ishikado, S. Uchida, A. Iyo, H. Eisaki, and S. Tajima, Phys. Rev. B 85, 054501 (2012). 15. E. Bernardi, A. Lascialfari, A. Rigamonti, L. Romanò, V. Iannotti, G. Ausanio, and C. Luponio, Phys. Rev. B 74, 134509 (2006), and references therein. 16. L. D. Landau, E. M. Lifshits, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984). 17. A. Raman and S. Fan, Phys. Rev. B 83, 205131 (2011). 18. A. Moussiaux, A. Ronveaux, and A. Lucas, Can. J. Phys. 55, 1423 (1977). 19. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957). 20. M. M. Mazumder, S. C. Hill, D. Q. Chowdhury, and R. K. Chang, J. Opt. Soc. Am. B 12, 297 (1995). 21. T. W. Noh, S. G. Kaplan, and A. J. Sievers, Phys. Rev. B 41, 307 (1990). 22. H. Shibata and T. Yamada, Phys. Rev. B 54, 7500 (1996).
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Optical bistability of localized Josephson surface plasmons in cuprate superconductors Filippo Alpeggiani Dipartimento di Fisica and CNISM, Università di Pavia, via Bassi 6, 27100 Pavia, Italy ([email protected]) Received December 3, 2014; revised January 15, 2015; accepted January 16, 2015; posted January 22, 2015 (Doc. ID 228894); published March 3, 2015 Microparticles made of high-T c cuprate superconductors are characterized by localized plasmonic excitations known as Josephson surface plasmons, whose electromagnetic response is intrinsically nonlinear, giving rise to yet unexplored optical phenomena. In this work bistability effects in the near-resonance excitation of Josephson surface plasmons of dipolar symmetry are investigated for spheroidal superconducting particles. The threshold for the incident intensity is estimated, and experimental probing strategies are discussed. The system can be of interest in view of terahertz light switching and detection. © 2015 Optical Society of America OCIS codes: (190.1450) Bistability; (300.6495) Spectroscopy, terahertz. http://dx.doi.org/10.1364/OL.40.000867
High-T c cuprate superconductors have attracted much attention as potential negative-index-metamaterials [1] and terahertz nonlinear media [2–6]. In the superconducting (SC) phase their c axis optical response is related to the sine-Gordon equation for which various nonlinear phenomena, such as bistability and soliton propagation, have long been studied [7,8]. Recently, optical excitation of solitons in a cuprate superconductor has been experimentally demonstrated [9], testifying for a growing interest in probing nonlinear optical effects in cuprates, in view of applications in terahertz plasmonics and information processing [4]. In [2], optical bistability in the Fabry–Perot resonances of a cuprate SC slab has been predicted for sufficiently high incident fields. In this work, we study bistability effects in the presence of a different feedback mechanism, i.e., the localized plasmonic excitations of cuprate SC particles, known as Josephson surface plasmons (JSPs), whose frequencies can be obtained from a modal expansion of the electric field, as shown in [10] for spherical particles in the quasi-static approximation. These modes are analogous to localized surface plasmons of metallic nanoparticles [11]. Here, we generalize the formalism of [10] to prolate and oblate spheroidal particles of several aspect ratios, which are meant to simulate experimentally realizable geometries like pillars and disks, and we apply it to the study of optical bistability effects. Both geometries could be obtained by patterning a superconductor film, e.g., by focused ion beam lithography [12]. We consider monolayered cuprate superconductors, such as La2−x Srx CuO4 , whose structure can be modeled as a stack of conductive CuO2 planes along the c axis (which we take directed as zˆ) alternating with insulating block layers, as sketched in Figs. 1(a) and 1(b). In the SC phase, each pair of consecutive planes acts as a nanoscale Josephson junction and supports the tunneling current, J n1;n t −J 0 sinφn1;n t;
[4,10]. In the long wavelength approximation, we introduce the continuous phase field, φr; t Reφω r exp−iωt, where φω r is related to the complex z component of the electric field as [10] φω r −2iesE ω;z r∕ℏω
(2)
[s is the interplane separation, see Fig. 1(a)]. In the linear regime, the electromagnetic response of the condensate is modeled by the diagonal dielectric tensor [1,10] ↔ ε ω diagε∥ ω; ε∥ ω; εzz ω, where εzz (neglecting dissipation) is obtained from the linearized version of Eq. (1) in the Drude form, ω2J 2J 0 es εzz ω ε∞ − ≐ε∞ 1 − 2 : ω ℏε0 ω2
(3)
The c axis Josephson plasma frequency ωJ is typically in the THz range, much lower than the plasma frequency for in-plane motion (of the order of an eV). Observing that, neglecting higher-order harmonics, sinφr; t
2J 1 jφω rj φr; t Ocos3ωt; …; jφω rj
(4)
(1)
with φn1;n being the gauge-invariant phase difference of the condensate order parameter between two consecutive copper planes and J 0 being the critical current 0146-9592/15/060867-04$15.00/0
Fig. 1. (a) La2−x Srx CuO4 crystalline structure along the ac plane, the unit cell is delimited by the black contour; (b) corresponding electrodynamical model, as referred to in the text; and (c) prolate spheroidal superconducting particle. © 2015 Optical Society of America
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(J n denotes a Bessel function of the first kind), Eq. (3) can be modified to include nonlinear effects in the form 2ω2 J jφ rj ω2 ω2 ε0zz 1 − J21 ω ≃ 1 − J2 J2 jφω rj2 …: ε∞ ω jφω rj ω 8ω (5) We notice that, to a first approximation, the correction for εzz has the same quadratic dependence on the field as the optical Kerr effect. For instance, in the case of La2−x Srx CuO4 around the Josephson frequency (νJ ≈ 2 THz, s 0.66 nm, and ε∞ ≈ 16), we can estimate a second-order nonlinear refractive index n2 ≈ 4e2 s2 ∕ε0 cℏ2 ω2J ≈ 10−7 cm2 ∕W, which is several orders of magnitude larger than that of common nonlinear materials, pinpointing the interest in high-T c superconductors as potential nonlinear materials in the THz range. In addition, the nonlinearity described here is an intrinsic phenomenon that is directly related to phase coherence in the superconductor, i.e., to the macroscopic emergence of a quantum mechanical effect. Multilayered superconductors, characterized by at least two different intrinsic junctions in the unit cell, present a more complex response due to the presence of transverse optical modes [13] and coupling with phonons [14], and they are not explicitly treated in the present work, although similar phenomena are expected to occur. As shown in Fig. 1(c), we suppose that the cuprate is spatially confined in the form of a particle, smaller than the wavelength but still much larger than the coherence length ξ (for La2−x Srx CuO4 , ξab ≈ 3 nm), in order to minimize quantum size and fluctuation effects that are known to occur in conventional SC nanoparticles with size comparable to ξ [15]. The resonance frequencies ω0 and the (complex) field profiles E0 r of JSPs can be calculated from a modal expansion of the electric field in the linear regime [10]. In the quasi-static approximation, the total energy contained in each mode is proportional to the quantity [16] Z U tot
↔
d3 rE0 r
∂ω ε ω; r E0 r: ∂ω
(6)
Nonlinearity affects the resonance frequency of plasmonic modes. In particular, applying perturbation theory for plasmonic systems [17] and Eq. (5), we find that the resonance frequency of the JSP shifts to ω00 ω0 Δω0 ;
(7)
with m 0 and l 1) in the quasistatic approximation, which we call dipolar JSPs (D-JSPs). These modes are characterized by the important property that the field inside the particle is homogeneous and polarized along zˆ, and we will denote its amplitude as E 0;int . Similarly to the case of metallic nanoparticles [18], in the quasi-static approximation the unperturbed resonance frequency ω0 of D-JSPs does not depend on the size, but only on the aspect ratio r through the geometrical parameter L3 , according to the relation ω20 ω2J
ε∞ L3 ; 1 ε∞ − 1L3
(9)
which is shown in Fig. 2. Analytical expressions for L3 are given in [19]. For spheres, e.g., L3 1∕3 and ω0 ωJ ε∞ ∕2 ε∞ 1∕2 (for La2−x Srx CuO4 , ν0 ≃ 1.9 THz). Notice that ω0 redshifts with increasing the aspect ratio. We consider an external excitation by a plane wave with frequency ω and field magnitude E ext linearly polarized along zˆ . Our results are generalized to other configurations by considering only the z component of the incident field. Near resonance, the intensity of the internal field is approximated by a Lorentzian function of the (perturbed) excitation frequency, E 20;int
ω20 κ 2 E2 ; ω − ω00 2 γ 2 ∕4 ext
(10)
where the parameter γ takes into account homogeneous broadening of the plasmonic mode, which is mainly of nonradiative origin and can be significant even for highT c superconductors, due to losses by noncondensed carries. For D-JSPs, the coefficient κ 2 has the value κ 2 1∕f41 ε∞ − 1L3 2 g;
(11)
also represented in Fig. 2. The intensity-dependent shift of the plasmonic frequencies in Eq. (8) is the physical origin of bistability in Josephson plasmons. The situation is similar to optical bistability in dielectric spheres [20]. In particular, for D-JSPs the correction in Eq. (8) is evaluated as Δω0 J 1 jφ0 j 1 jφ j2 − ≈ − 0 Ojφ0 j4 ; ω0 16 jφ0 j 2
(12)
independently of the aspect ratio. This property crucially depends on the fact that in these modes the electric field
where the first-order frequency correction is given by Δω0 ε ∞ ω0 U tot
Z Vp
d3 r
ω2J 2J 1 jφ0 rj − 1 jE 0;z rj2 ; jφ0 rj ω20
(8)
with V p being the particle volume and φ0 r being related to E 0;z r through Eq. (2) with ω ω0 . We consider SC particles of spheroidal shape, with varying the aspect ratio r height∕diameter. In particular, we focus on plasmons with dipolar symmetry and out-of-plane polarization (in the notation of [10], modes
Fig. 2. Resonance frequency ω0 (normalized to ωJ , solid curve) and coupling factor κ2 (dashed curve) as a function of the aspect ratio for dipolar surface plasmons in spheroidal superconducting particles.
March 15, 2015 / Vol. 40, No. 6 / OPTICS LETTERS
is homogeneous inside the particle. Notice that the shift is negative in sign. From Eq. (2), we derive that the natural unit for the electric field is E 0 ℏω0 ∕2es. By replacing Eq. (12) into Eq. (10) and defining the dimensionless quantities W jφ0 j2
E 20;int E 20
;
κ2 E 2ext;z I ; E 20
ω − ω0 Δ ; ω0
and γ¯ γ∕ω0 1∕Q (Q is the resonance Q-factor), Eq. (10) can be rewritten in the form p p2 1 1 I W Δ − J 1 W ∕ W γ¯ 2 ; (13) 2 4 which is solved for the internal intensity W . It represents the fundamental equation for the study of bistability effects in the present system. As shown in Fig. 3(a), bistability is achieved for values of the incident intensity I above a certain threshold, in a range of negative values for the detuning parameter Δ (highlighted in the figure). Correspondingly, the internal intensity W plotted as a function of the incident intensity I (for a fixed detuning Δ in the bistability range) shows the hysteresis behavior of Fig. 3(b), with sharp transitions between high- and low-intensity states (as usual, a third intermediate state is unstable [2]). The hysteresis phenomenon derives from positive and negative feedback mechanisms between mode pulling [Eq. (12)] and coupling of external radiation to the Josephson plasmon [Eq. (10)]. The bistability threshold for the incident intensity can be estimated by expanding J 1 in series and approximating Eq. (13) with the cubic equation I W Δ W ∕162 γ¯ 2 ∕4;
(14)
which is similar to that of dielectric microspheres [20]. The threshold for entering the bistability regime corresponds to the condition
p Δth − 3γ¯ ∕2 and
p I th 16¯γ 3 ∕ 3 3 ;
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(15)
which implies E 2ext;th
16¯γ 3 2 64 1 ε∞ − 1L3 2 2 p p E0: E 0 Q3 3κ2 3 3 3
(16)
The threshold crucially depends on the resonance width γ¯ or, equivalently, on the resonance Q-factor of surface plasmons. Unfortunately, very few experimental data exist on the properties of JSPs in confined geometry. Extinction spectroscopy of powdered samples within the so-called sphere resonance method [21,22] revealed broad resonances, which we may expect to be at least partially of inhomogenous origin, due to size and shape variability of the particles. Nevertheless, even in the conservative assumption of γ¯ ≈ 0.1, i.e., Q ≈ 10, for La2−x Srx CuO4 we can estimate the threshold E ext;th ≈ 40 kV cm−1 for a spherical particle and E ext;th ≈ 9 kV cm−1 for a prolate spheroid with r 5. These values are within the reach of current technology, being of the same order of magnitude of the THz fields employed in the experiment described in [9] to generate solitons in cuprate superconductors. A further reduction of the threshold can be obtained by tailoring the geometry of the samples. In particular, as shown by Fig. 2, prolate spheroids with high aspect ratios are favored by both lower resonance frequencies and larger coupling factors κ 2 . Of course, different growth techniques must be chosen according to the desired aspect ratio. In any case, the use of single-crystal samples is highly convenient in order to maximize the effect. Bistability effects can be detected, for instance, with time resolved scattering or extinction experiments with a pulsed incident field. In the linear approximation, D-JSPs are characterized by the dipole moment [19], Z p ε0 εzz ω0 − 1
Vp
d3 rE0 r −ε0
Vp zˆ E : L3 0;int
(17)
Then the far-field intensity of scattered radiation at point R; θ reads SR; θ
Fig. 3. Solutions of the bistability Eq. (13) (¯γ 0.1): (a) the adimensional internal intensity W as a function of detuning Δ for some values of the incident intensity I (bistability region is highlighted); (b) W as a function of I for Δ −0.15.
ε0 sin θ 2 V 2p 4 2 ω E 0;int ; 2c3 4π L23 R2
(18)
i.e., it is proportional to the internal intensity. When the temporal width of the incident pulse T satisfies the condition γT ≫ 1, the quasi-steady-state approximation holds, and the intensity of the incident field can be calculated by solving the steady-state Eq. (13) for each value of the slowly-varying envelope of the incident intensity. We notice that the condition is much less strict for plasmonic systems than dielectric ones, since resonance lifetimes are shorter (Q ∼ 10). The time-dependent profile of the scattered intensity is shown in Fig. 4 for some values of the detuning parameter Δ. When the frequency of incident radiation is in the bistability region, the profile of the scattered field is strongly modified with respect to the incident one, showing an asymmetric shape with a sharp
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Fig. 4. Envelope of the scattered intensity as a function of time for different values of the detuning Δ. The incident intensity has a slowly-varying profile ∝ exp−t∕T2 shown in the background, with a maximum corresponding to the value I 0.023 for the parameter I defined in the text.
transition to a high-intensity state. This can be understood by following the hysteresis cycle in Fig. 3(b) for an increasing and then decreasing incident intensity. Particle heating due to incident radiation could be significant in the case of cw irradiance and it could drive the system out of the SC state. However, we believe that this is not the case of a pulsed excitation. For instance, the authors of [9] irradiated a cuprate sample with 25-ps pulses of THz light with peak fields of 10 kV cm−1 and reported minimal heating effects. Indeed, SC particles should be favored as they can better dissipate heat when cooled in a cryogenic system due to higher surface-tovolume ratios than bulk samples. In conclusion, we have theoretically investigated optical bistability in Josephson surface plasmons of spheroidal high- T c superconducting particles. Bistability originates from phase coherence in the superconducting order parameter through a nonlinear term in the tunneling current equation and it can be probed by terahertz time-resolved spectroscopy. The threshold for the incident intensity to enter the bistable regime critically depends on the third power of the resonance Q-factor. This phenomenon has potential application in the field of terahertz plasmonics, e.g., for ultrafast optical switching. Moreover, from a more fundamental point of view, nonlinear optical experiments could help in studying the condensate dynamics and clarifying some aspects of the mechanism of high- T c superconductivity. The author is grateful to Lucio Claudio Andreani for a critical reading of the manuscript and to Marco Liscidini for useful discussions.
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