Coherent destruction of tunneling in graphene irradiated by elliptically polarized lasers
Denis Gagnon, François Fillion-Gourdeau, Joey Dumont, Catherine Lefebvre, Steve MacLean
CCoherent destruction of tunneling in graphene irradiated by elliptically polarizedlasers
Denis Gagnon,
1, 2, ∗ Fran¸cois Fillion-Gourdeau,
1, 2
Joey Dumont, Catherine Lefebvre,
1, 2 and Steve MacLean
1, 2, † Universit´e du Qu´ebec, INRS– ´Energie, Mat´eriaux et T´el´ecommunications, Varennes, Qu´ebec, Canada, J3X 1S2 Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 (Dated: November 16, 2016)Photo-induced transition probabilities in graphene are studied theoretically from the viewpointof Floquet theory. Conduction band populations are computed for a strongly, periodically drivengraphene sheet under linear, circular, and elliptic polarization. Features of the momentum spectrumof excited quasi-particles can be directly related to the avoided crossing of the Floquet quasi-energylevels. In particular, the impact of the ellipticity and the strength of the laser excitation on theavoided crossing structure – and on the resulting transition probabilities – is studied. It is shown thatthe ellipticity provides an additional control parameter over the phenomenon of coherent destructionof tunneling in graphene, allowing one to selectively suppress multiphoton resonances.
I. INTRODUCTION
Graphene is a remarkable 2D material with propertiesparticularly suited to the development of electronic andoptoelectronic devices [1, 2], for instance ballistic transis-tors [3] and photo-detectors [4]. Besides this promise ofapplications, the Dirac band structure of graphene hasmotivated several studies of the fundamentals of light-matter interactions. In particular, a celebrated propertyof pristine graphene is its transformation in a Floquettopological insulator (FTI) upon irradiation with a cir-cularly polarized pulse [5]. In regular TIs, the band struc-ture is modified by spin-orbit interactions and edge statesare protected by time-reversal symmetry [6]. However, inthe case of graphene, a circularly polarized field breakstime-reversal symmetry and opens a gap at the Diracpoint, leading to a FTI state which cannot exist in the ab-sence of incident radiation [7]. This gives rise to Floquetedge states confined to the boundaries of graphene rib-bons, states with definite chirality and robustness againstdisorder [8].Another quantum optical phenomenon which has gar-nered some attention in recent work is the realiza-tion of Landau-Zener-St¨uckelberg interference (LZSI) ingraphene irradiated by a linearly polarized excitation.LZSI has been generically described for superconduct-ing qubits [9, 10], Majorana qubits [11], and also intheoretical [12–14] and experimental [15] works specificto graphene. In both physical realizations (qubits andgraphene), quantum interference stems from the peri-odic driving of the system through avoided crossings ofits adiabatic energy levels [10]. In the case of a circu-larly polarized excitation (which leads to the FTI statein graphene), LZSI is absent since the system no longergoes through the aforementioned avoided crossings.This article presents a theoretical study of the modifi-cation of Floquet quasi-energies and transition probabil- ∗ [email protected] † [email protected] ities in graphene when an incident monochromatic laseris tuned from a linear polarization to an elliptic polariza-tion. From a fundamental standpoint, the photo-inducedconduction band population calculated using Floqueteigenvectors is formally analogous to the production ofelectron-positron pairs from vacuum, as discussed in sev-eral recent articles [13, 16–19]. This population could,in principle, be probed using angle-resolved photoemis-sion spectroscopy (ARPES) on graphene [12, 20]. Themain goal of this article is to identify the various physi-cal features of the Floquet momentum maps that appearor disappear as the ellipticity parameter is varied, for in-stance coherent destruction of tunneling (CDT) [21–23].Our results show that the ellipticity of the polarizationprovides a control parameter over multiphoton processesin graphene. In particular, for some range of value of theellipticity of the incident field and for some momentumstates, electron-hole pair production can be suppressedfor several given field strengths. II. FLOQUET FORMULATION FORGRAPHENE
The Floquet formulation is an appropriate tool for thestudy of strongly, periodically driven quantum systems.It allows one to recast a time-dependent problem intoa time-independent one, thereby allowing the use of alltools specific to time-independent quantum mechanics,such as stationary perturbation theory [22, 24]. Severalarticles have been concerned with the application of Flo-quet theory to account for various physical phenomena ingraphene. These include the previously mentioned emer-gence of the FTI state under circularly polarized illumi-nation [5, 7], the prediction of ARPES results [20], themotion of Dirac points in the tight-binding model [25, 26],the photo-voltaic Hall effect [27], the presence of boundstates around adatoms [28], and quantum interference inthe case of linear polarization [13, 14]. In this section,we review the application of the Floquet formalism tographene, the ultimate goal being the computation of a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Figure 1. Graphene sheet irradiated by an elliptically polar-ized excitation. transition probabilities in momentum space for an ellip-tically polarized laser excitation.Consider Dirac fermions in a graphene mono-layer inthe presence of an elliptically polarized laser field, whichis uniform in space and normally incident to the grapheneplane (see Fig. 1). The fermion dynamics are governedby the following (2 ×
2) low energy Hamiltonian (we useunits such that (cid:126) = 1) H ξ ( t, p ) = ξv F σ · [ p + e A ( t )] , (1)where ξ = ± v F (cid:39) c/ σ = ( σ x , σ y ) is a vectorof Pauli matrices representing the sublattice pseudospin, p is the fermion momentum around the K ξ points and − e < A x ( t ) = A cos ωt,A y ( t ) = ηA sin ωt, (2)where η is the ellipticity of the excitation and A ≡ E /ω is the maximum amplitude of the vector potential. Asshown in Fig. 1, this vector potential describes a fieldprecessing around the z axis in an elliptic pattern, witha semi-major axis parallel in the x direction and a semi-minor axis in the y direction. A change of representationof the Dirac matrices can be performed via a unitarytransformation (which does not impact physical observ-ables) to obtain a Hamiltonian closer to the canonicalform of a strongly driven two-level atom [13]: U r ≡ e − iσ y π e − iσ x π . (3)Combining this transformation with the following changeof variables ε i ≡ − ξv F p i , (4a) A (cid:48) ≡ − ξv F eE /ω, (4b)leads to H ξ ( t, p ) = − (cid:18) ε x + A (cid:48) cos ωt ε y + ηA (cid:48) sin ωtε y + ηA (cid:48) sin ωt − ε x − A (cid:48) cos ωt (cid:19) . (5) One of the goals of this article is to explore the para-metric range 0 ≤ η ≤ η = 0), the Hamilto-nian in Eq. (5) is of the generic form describing a driventwo-level system, or a particle moving in a double-wellpotential [22]. The transverse momentum ε y plays therole of a coupling strength between the two basis states,or two wells [10, 24]. Thus, other quantum systems de-scribed by this Hamiltonian include atoms in intense laserfields [10], semiconductor superlattices [29–31], and su-perconducting qubits [23, 32], where ε y is generally called“tunnel splitting”. The terminology “transverse momen-tum” is also used to describe ε y , as it is proportionalto the quasi-particle momentum in the direction perpen-dicular to the (linearly polarized) electric field. In thecase of a circularly polarized excitation ( η = 1), thisdefinition is ambiguous, but we shall retain the termi-nology “transverse momentum” in the elliptic case withthe understanding that it is the momentum componentperpendicular to the semi-major axis of the polarizationellipse.The Fourier series of a periodic 2 × H ( t )reads (we use the notation of Son et al. [23] in the fol-lowing derivation) H ( t ) = ∞ (cid:88) n = −∞ H [ n ] e − inωt , (6)where n is an integer, the Fourier index. The Floquetstate nomenclature reads | νn (cid:105) = | ν (cid:105) ⊗ | n (cid:105) , (7)where ν is the system index which can take two values, α and β , labeling negative and positive energy eigenstatesof σ z , respectively. Switching to Fourier space and ap-plying the Floquet theorem yields (cid:88) µ = α,β (cid:88) m (cid:104) νn | H F | µm (cid:105) (cid:104) µm | q γl (cid:105) = q γl (cid:104) νn | q γl (cid:105) , (8)where q γl are the Floquet quasi-energies, | q γl (cid:105) are the Flo-quet eigenvectors, and H F is the Floquet Hamiltonian.This Floquet Hamiltonian has a block matrix structuregiven by (cid:104) µn | H F | νm (cid:105) = H [ n − m ] µν + nωδ µν δ nm . (9)Starting from (5), one can write the three non-vanishing2 × H [0] = − (cid:18) ε x ε y ε y − ε x (cid:19) , (10a) H [ ± = − A (cid:48) (cid:18) ± iη ± iη − (cid:19) . (10b)The transition probability between field-free eigen-states |−(cid:105) and | + (cid:105) (which have negative and positiveeigenenergies, respectively) can be written as a sum of k -photon transition probabilities [23]¯ P |−(cid:105)→| + (cid:105) = (cid:88) k (cid:88) γl |(cid:104) + , k | q γl (cid:105) (cid:104) q γl |− , (cid:105)| , (11)where |− , k (cid:105) = ε x + | ε |N | αk (cid:105) + ε y N | βk (cid:105) , (12) | + , k (cid:105) = − ε y N | αk (cid:105) + ε x + | ε |N | βk (cid:105) , (13) | ε | ≡ (cid:113) ε x + ε y , N ≡ (cid:113) ( ε x + | ε | ) + ε y . (14)There generally exists no closed form solution for thetransition probability, except for a special case of linearpolarization discussed below. The momentum-integrateddensity of photo-excited pairs can then be calculatedfrom the transition probability using the following def-inition [19] ¯ n = 1(2 π ) (cid:90) ¯ P |−(cid:105)→| + (cid:105) d p. (15) III. NUMERICAL RESULTS AND DISCUSSION
We now turn our attention to numerical calculations ofthe transition probability between the valence and con-duction band of graphene, Eq. (11), for several combi-nations of the coupled graphene-laser Hamiltonian pa-rameters. These parameters are the normalized energyscales ε x /ω , ε y /ω associated to the quasi-particle mo-mentum, A (cid:48) /ω associated to the applied field strength,and η . The eigenvalue problem, Eq. (8), can be solvednumerically by constructing a truncated version of theFloquet Hamiltonian and using standard linear algebraroutines. The eigenvectors can then be used to computethe time-averaged transition probability between the va-lence and conduction band in graphene, Eq. (11). In allnumerical calculations presented in this article, the Flo-quet Hamiltonian is truncated to 75 blocks to ensure anumerically converged solution, for a total matrix size of302 × A. Linear and circular polarization
Consider first the paradigmatic case of a linearly po-larized field incident on the graphene layer ( η = 0). Forthe purposes of this article, we use a normalized fieldstrength A (cid:48) /ω = 8 . E = 1 . × V/m oscillating at ε x / ω ε y / ω (a) 0.000.250.500 2 4 6 8 10 12 14 ε x / ω − − Q u a s i e n e r g y [ ω ] (b) Figure 2. (a) Transition probability between field-free eigen-states |−(cid:105) and | + (cid:105) computed via Floquet theory. The fieldis linearly polarized ( η = 0), with A (cid:48) /ω = 8 . ε x /ω = 0 and ε y /ω = 0. (b) Evolution of the Floquet quasi-energies for ε y /ω = 1, indicated by a white line on the top figure. The re-lation between the momentum space resonances’ widths andthe quasi-energy separation is apparent. For comparison, greylines indicate the quasi-energies for ε y /ω = 0. a frequency f = ω/ π = 10 . k -photonresonances) or of adjacent blocks (for odd k -photon res-onances). The quasi-energy gap is directly related to theresonance width: the closer the eigenstates anti-cross,the narrower the resonance. In the extreme case of zerogap, the resonances actually disappear, as can be seen forinstance at zero transverse momentum ε y /ω = 0. Notethat the maximal value of the transition probability is1 / f = 10 THz is ¯ n (cid:39) . × cm − .At small values of the transverse momentum, the pres-ence of Lorentzian shaped multiphoton resonances ismanifest. This can be interpreted using a closed-formsolution for the transition probability derived in Refs.[23, 32]. In the limit ε y (cid:28) ε x , | A (cid:48) ω | , and considering η = 0, the field-free eigenstates can be approximated bythose of σ z , that is | α (cid:105) and | β (cid:105) . Starting from Eq. (11),a leading order perturbation treatment leads to the fol-lowing analytic formula for the transition probability¯ P | α (cid:105)→| β (cid:105) = (cid:88) k
12 [ ε y J k ( A (cid:48) /ω )] [ ε y J k ( A (cid:48) /ω )] + [ kω − ε x ] , (16)where J k is the Bessel function of the first kind. Inother terms, for a linearly polarized laser excitation and asmall transverse momentum ε y , the time-averaged transi-tion probability can be expressed as the superposition ofLorentzian k -photon resonances. In this limit, the proba-bility assumes its maximal value of 1 / ε x = kω , and is suppressed at ε y = 0 (zero trans-verse momentum). The k -photon transition probabilityalso tends to zero when A (cid:48) /ω is equal to the n th zero j k,n of the Bessel function J k , a situation which corre-sponds to CDT [22, 23]. This explains why the k = 2and k = 5 multiphoton peaks are very narrow in Fig.2a, since A (cid:48) /ω = 8 .
413 approaches j , = 8 . j , = 8 . η = 1) are shown inFig. 3a. The momentum space pattern is now rotation-ally symmetric, and the circularly polarized excitationopens a gap in the quasi-energies at the Dirac point atsmall values of | ε | (see Fig. 3b). This photo-induced gapis consistent with the description of the quantum systemin terms of a FTI. Resonant multiphoton rings appear atfixed values of | ε | , the width of which is again directly re-lated to the quasi-energy gap. For large values of | ε | , therings become sharp and distinct, as they do in the caseof linear polarization ( η = 0). This can be explainedby the fact that the magnitude of the diagonal blocksof the Floquet Hamiltonian H [0] becomes much largerthan the magnitude of the field-dependent off-diagonalblocks H [ ± . In that case, the quasi-energy dispersionrelation approaches that of undressed, or diabatic, eigen-states (c.f. Figs. 2b and 3b for large ε x /ω ). The corre-sponding momentum integrated density for f = 10 THz ε x / ω ε y / ω (a) 0.000.250.500 2 4 6 8 10 12 14 ε x / ω − − Q u a s i e n e r g y [ ω ] (b) Figure 3. (a) Transition probability between field-free eigen-states |−(cid:105) and | + (cid:105) computed via Floquet theory. The field iscircularly polarized with ( η = 1), with A (cid:48) /ω = 8 . ε y /ω = 0. is ¯ n (cid:39) . × cm − . In the case of 0 < η < B. Elliptic polarization
Next, we study the effect of the ellipticity parameterof the incident field, η , on the Floquet transition proba-bility with an emphasis on the phenomenon of CDT [21].The condition for the occurrence of this phenomenon isa very close encounter of two quasi-energy levels, or inother words a nearly vanishing effective tunnel splitting.In that case, quantum tunneling is brought to an almostcomplete standstill and the transition probability is sup-pressed. This is due to the fact that the Rabi frequencyof population oscillations is directly proportional to theeffective tunnel splitting (or quasi-energy gap), and thusapproaches zero [21, 22]. Another way of interpreting thisresult is that as the quasi-energy gap reduces, the reso-nances narrow to the point of disappearing completely,as seen in the linearly polarized case described above for A ′ / ω η (a) 0.000.250.500 2 4 6 8 10 12 14 A ′ / ω ¯ P | − 〉 → | + 〉 (b)0 2 4 6 8 10 12 14 A ′ / ω Q u a s i e n e r g y [ ω ] (c) Figure 4. (a) Evolution of the transition probability betweenfield-free eigenstates |−(cid:105) and | + (cid:105) computed via Floquet theoryas a function of the field strength and ellipticity. This param-eter sweep is computed for a resonant value of ε x /ω = 3 and ε y /ω = 0 .
01. (b) Evolution of the transition probability for η = 0 .
1, indicated by a white line on the top figure. Dipsin the probability value are noticeable. (c) Evolution of theFloquet quasi-energies for η = 0 .
1. Black dashed lines in-dicate the expected position of the narrow avoided crossings(see Appendix). ε y /ω → ε x /ω = k where k is an integer. For illustration pur-poses, we use normalized momentum values of ε x /ω = 3and ε y /ω = 0 .
01. Since the system is resonant and thetransverse momentum value is small, Eq. (16) prescribesthat transition probability should take its maximal value of 1 / η = 0, except if the conditions for CDT aresatisfied. Regions where these conditions are satisfiedare visible in Fig. 4a in the form of five distinct dipsthat exist for relatively small values of η and given val-ues of the parameter A (cid:48) /ω . Looking at the specific caseof η = 0 .
1, results show that the transition probabilitycan be suppressed down to values of (cid:39) . A (cid:48) /ω (Fig. 4b). Interestingly, in-creasing the ellipticity allows one to increase the widthof the probability dips which are too narrow to be usefulat η = 0. Further increasing η shifts the dips towardssmaller field strengths, and they are destroyed for largevalues of η . The shift can be interpreted in terms of theAC Stark effect, whereas the destruction of the dips fora circular polarization is associated to the quasi-energygap opening caused by the breaking of time-reversal sym-metry. This result shows the potential for controlling theresponse of graphene-based optoelectronic devices by se-lectively suppressing multiphoton resonant peaks via theapplied field strength and ellipticity. This is in anal-ogy with the control of dynamical systems described bydriven double-well potentials, such as ammonia [22] orargon [33] molecules in a laser field.Once again, the position of the transition probabilitydips can be related to the Floquet quasi-energy diagramshown in Fig. 4c for η = 0 .
1. The locations of the dipscorrespond to very close avoided crossings of the Floqueteigenstates. For small values of the ellipticity and giventhe small value of the transverse momentum, their po-sition can be estimated as the zeros of the derivative ofthe Bessel function, that is A (cid:48) /ω = j (cid:48) ,n , as shown in Ap-pendix. We have checked numerically that this behavioris general for integer values of ε x /ω , that is whenever thecoupled laser-graphene system is tuned sufficiently closeto a multiphoton resonance at small transverse momen-tum.To conclude the discussion, let us mention that experi-ments to probe carrier dynamics in graphene could realis-tically be carried using laser excitations in the THz range,field strengths on the order of 10 V/m [34, 35], and tech-niques such as ARPES. The main impediment for therealization of these experiments lies in the fact that, ifelectron-electron interactions can not be neglected, thecarrier lifetime in pristine graphene is limited to ∼
10 fs[36]. A detailed discussion of the experimental feasibilityof momentum space experiments with graphene can befound in Ref. [13].
IV. SUMMARY
In this work, a detailed theoretical analysis of the prob-lem of electron-hole pair creation using graphene irra-diated by an elliptically polarized laser excitation waspresented. Concentrating on the CW regime, the photo-induced conduction band populations were numericallycomputed using Floquet theory. We started by present-ing the paradigmatic cases of linear and circular polar-ization, which are respectively related to Landau-Zener-St¨uckelberg interferometry and the emergence of a Flo-quet topological insulator phase. Most of the momen-tum space features can be explained in terms of theavoided crossing structure of the Floquet eigenstates. Re-sults for an elliptic polarization were subsequently ob-tained, and we showed that the ellipticity of the laserexcitation provides an additional control parameter overthe phenomenon of coherent destruction of tunneling ingraphene. In addition to being useful for the predictionof spectroscopy experiments, these results highlight thepossibility of using ellipticity as a further optimizationvariable to maximize or minimize electron-hole pair pro-duction in graphene. This optimization variable could beused in conjunction with pulse shaping [22], for example.
ACKNOWLEDGMENTS
Computations were made on the supercomputer
Mam-mouth from Universit´e de Sherbrooke, managed by Cal-cul Qubec and Compute Canada. The operation of thissupercomputer is funded by the Canada Foundation forInnovation (CFI), minist`ere de l’´Economie, de la Scienceet de l’Innovation du Qu´ebec (MESI) and the Fonds derecherche du Qu´ebec – Nature et technologies (FRQNT).
Appendix A: Appendix: Quasi-energy extrema
At small transverse momentum ( ε y /ω →
0) and for alinear polarization ( η = 0), it is possible to predict an-alytically the position of avoided crossings in the quasi-energies, as displayed in Fig. 4. Starting from the per-turbation approach presented by Son et al. , one obtainsthe following approximate formula for quasi-energies [23,Eqs. (26–27)] q ± = − kω ± (cid:112) λJ − k ( z ) + ( kω − ε x + 2 δ ) / , (A1)where z ≡ A (cid:48) /ω , λ ≡ − ε y / δ is a level shift termof order λ which also depends on z . Since the δ termis small, we take the derivative of q ± with respect to z by considering only the term proportional to λ under thesquare root: dq ± dz = ± λ J k ( z ) J (cid:48) k ( z ) (cid:112) λJ − k ( z ) + ( kω − ε x + 2 δ ) / . (A2)From this formula, one can clearly show that the locationof avoided crossings as z is varied are given by the Besselfunction zeros j k,n and j (cid:48) k,n , with the latter correspondingto the dashed lines in Fig. 4 for k = 3. [1] A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, andA. K. Geim, Rev. Mod. Phys. , 109 (2009).[2] M. Engel, M. Steiner, A. Lombardo, A. C. Ferrari, H. V.L¨ohneysen, P. Avouris, and R. Krupke, Nat. Commun. , 906 (2012).[3] A. K. Geim and K. S. Novoselov, Nat. Mater. , 183(2007).[4] D. Schall, D. Neumaier, M. Mohsin, B. Chmielak,J. Bolten, C. Porschatis, A. Prinzen, C. Matheisen,W. Kuebart, B. Junginger, W. Templ, A. L. Giesecke,and H. Kurz, ACS Photonics , 781 (2014).[5] J. Cayssol, B. D´ora, F. Simon, and R. Moessner, Phys.Status Solidi Rapid Res. Lett. , 101 (2013).[6] A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. ,021004 (2016).[7] G. Usaj, P. M. Perez-Piskunow, L. E. F. Foa Torres, andC. A. Balseiro, Phys. Rev. B , 115423 (2014).[8] P. M. Perez-Piskunow, G. Usaj, C. A. Balseiro, andL. E. F. Foa Torres, Phys. Rev. B , 121401 (2014).[9] A. V. Shytov, D. A. Ivanov, and M. V. Feigel’man, Eur.Phys. J. B , 263 (2003).[10] S. Shevchenko, S. Ashhab, and F. Nori, Phys. Rep. ,1 (2010).[11] Z. Wang, W.-C. Huang, Q.-F. Liang, and X. Hu,“Landau-Zener-St¨uckelberg Interferometry for MajoranaQubit,” (2016).[12] H. K. Kelardeh, V. Apalkov, and M. I. Stockman, Phys.Rev. B , 155434 (2016).[13] F. Fillion-Gourdeau, D. Gagnon, C. Lefebvre, andS. MacLean, Phys. Rev. B , 125423 (2016). [14] Y. I. Rodionov, K. I. Kugel, and F. Nori, Phys. Rev. B , 195108 (2016).[15] T. Higuchi, C. Heide, K. Ullmann, H. B. Weber, andP. Hommelhoff, “Light-field driven currents in graphene,”(2016).[16] D. Allor, T. D. Cohen, and D. A. McGady, Phys. Rev.D , 096009 (2008).[17] M. Lewkowicz, H. C. Kao, and B. Rosenstein, Phys. Rev.B , 035414 (2011).[18] B. D´ora and R. Moessner, Phys. Rev. B , 165431(2010).[19] F. Fillion-Gourdeau and S. MacLean, Phys. Rev. B ,035401 (2015).[20] M. A. Sentef, M. Claassen, A. Kemper, B. Moritz,T. Oka, J. Freericks, and T. Devereaux, Nat. Commun. , 7047 (2015).[21] F. Grossmann, T. Dittrich, P. Jung, and P. H¨anggi,Phys. Rev. Lett. , 516 (1991).[22] M. Grifoni and P. H¨anggi, Phys. Rep. , 229 (1998).[23] S.-K. Son, S. Han, and S.-I. Chu, Phys. Rev. A ,032301 (2009).[24] S. Ashhab, J. R. Johansson, A. M. Zagoskin, and F. Nori,Phys. Rev. A , 063414 (2007).[25] P. Delplace, ´A. G´omez-Le´on, and G. Platero, Phys. Rev.B , 245422 (2013).[26] P. Rodriguez-Lopez, J. J. Betouras, and S. E. Savel’ev,Phys. Rev. B , 155132 (2014).[27] T. Oka and H. Aoki, Phys. Rev. B , 081406 (2009).[28] D. A. Lovey, G. Usaj, L. E. F. Foa Torres, and C. A.Balseiro, Phys. Rev. B , 245434 (2016).[29] M. Holthaus, Phys. Rev. Lett. , 351 (1992). [30] J. Rotvig, A. P. Jauho, and H. Smith, Phys. Rev. Lett. , 1831 (1995).[31] G. Platero and R. Aguado, Phys. Rep. , Vol. 395 (2004)pp. 1–157.[32] W. D. Oliver, Y. Yu, J. C. Lee, K. K. Berggren, L. S.Levitov, and T. P. Orlando, Science , 1653 (2005).[33] E. Kierig, U. Schnorrberger, A. Schietinger, J. Tomkovic,and M. K. Oberthaler, Phys. Rev. Lett. , 190405 (2008).[34] X. Ropagnol, F. Blanchard, T. Ozaki, and M. Reid,Appl. Phys. Lett. , 161108 (2013).[35] H. A. Hafez, X. Chai, A. Ibrahim, S. Mondal,D. F´erachou, X. Ropagnol, and T. Ozaki, J. Opt.18