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Quantum Vacuum Properties of the Intersubband Cavity Polariton Field Cristiano Ciuti, Gérald Bastard, Iacopo Carusotto

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Quantum Vacuum Properties of the Intersubband Cavity Polariton Field Cristiano Ciuti, Gérald Bastard, Iacopo Carusotto Quantum vacuum properties of the intersubband cavity polariton field Cristiano Ciuti, Gérald Bastard, Iacopo Carusotto To cite this version: Cristiano Ciuti, Gérald Bastard, Iacopo Carusotto. Quantum vacuum properties of the intersubband cavity polariton field. Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2005, 72, pp.115303. hal-00004617v2 HAL Id: hal-00004617 https://hal.archives-ouvertes.fr/hal-00004617v2 Submitted on 8 Jul 2005 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Quantum vacuum properties of the intersubband cavity polariton field 1, 1 2 Cristiano Ciuti, ∗ G´erald Bastard, and Iacopo Carusotto 1Laboratoire Pierre Aigrain, Ecole Normale Sup´erieure, 24, rue Lhomond, 75005 Paris, France 2CRS BEC-INFM and Dipartimento di Fisica, Universit`adi Trento, I-38050 Povo, Italy (Dated: July 8, 2005) We present a quantum description of a planar microcavity photon mode strongly coupled to a semiconductor intersubband transition in presence of a two-dimensional electron gas. We show that, in this kind of system, the vacuum Rabi frequency ΩR can be a significant fraction of the inter- subband transition frequency ω12. This regime of ultra-strong light-matter coupling is enhanced for long wavelength transitions, because for a given doping density, effective mass and number of quantum wells, the ratio ΩR/ω12 increases as the square root of the intersubband emission wave- length. We characterize the quantum properties of the ground state (a two-mode squeezed vacuum), which can be tuned in-situ by changing the value of ΩR, e.g., through an electrostatic gate. We finally point out how the tunability of the polariton quantum vacuum can be exploited to generate correlated photon pairs out of the vacuum via quantum electrodynamics phenomena reminiscent of the dynamical Casimir effect. In the last decade, the study of intersubband elec- a significant fraction of the intersubband transition (in 1 8 tronic transitions in semiconductor quantum wells has the pioneering experiments by Dini et al. , 2¯hΩR = 14 enjoyed a considerable success, leading to remarkable meV compared to ¯hω12 = 140 meV). Furthermore, re- opto-electronic devices such as the quantum cascade cent experiments have also demonstrated the possibility lasers2,3,4. In contrast to the more conventional inter- of a dramatic tuning of the strong light-matter coupling band transitions between conduction and valence bands, through application of a gate voltage11 which is able to the frequency of intersubband transitions is not deter- deplete the density of the two-dimensional electron gas. mined by the energy gap of the semiconductor material Although the quest for quantum optical squeezing ef- system used, but rather can be chosen via the thickness of fects in the emission from atoms strongly coupled to a the quantum wells in the active region, providing tunable cavity mode has been an active field of research12, all sys- sources emitting in the mid and far infrared. tems realized up to now show a vacuum Rabi frequency One of the most fascinating aspects of light-matter ΩR much smaller than the frequency of the optical tran- interaction is the so-called strong light-matter coupling sition. In this parameter regime, the relative importance regime, which is achieved when a cavity mode is reso- of the anti-resonant terms in the light-matter coupling nant with an electronic transition of frequency ω12, and is small and, as far as no strong driving field is present, the so-called vacuum Rabi frequency ΩR exceeds the cav- they can be safely neglected under the so-called rotating- ity mode and electronic transition linewidths. The strong wave approximation. In the presence of a strong driving coupling regime has been first observed in the late ’80s field, however, anti-resonant terms are known to play a using atoms in metallic cavities5,6, and a few years later significant role, giving, e.g., the so-called Bloch-Siegert in solid-state systems using excitonic transitions in quan- shift in magnetic resonance experiments13, or determin- tum wells embedded in semiconductor microcavities7. In ing the quantum statistical properties of the emission this regime, the normal modes of the system consist of from dressed-state lasers14. linear superpositions of electronic and photonic excita- A few theoretical studies have pointed out the intrin- tions, which, in the case of semiconductor materials, are sic non-classical properties of exciton-polaritons in solid- the so-called polaritons. In both these systems, the vac- state systems15,16,17, but the small value of the ratio uum Rabi frequency ΩR does not exceed a very small ΩR/ωexc, typically less than 0.01, has so far prevented the fraction of the transition frequency ω12. observation of quantum effects due to the anti-resonant Recently, Dini et al.8 have reported the first demon- terms of the light-matter coupling. All the squeezing stration of strong coupling regime between a cavity pho- experiments that have been performed so far in fact re- ton mode and a mid-infrared intersubband transition, in quired the presence of a strong coherent optical pump agreement with earlier semiclassical theoretical predic- beam in order to inject polaritons and take advantage of tions by Liu9. The dielectric Fabry-Perot structure real- nonlinear polariton parametric processes18,19,20,21,22. ized by Dini et al.8 consists of a modulation doped mul- In this paper, we show that in the case of intersub- tiple quantum well structure embedded in a microcavity, band cavity polaritons, it is instead possible to achieve ccsd-00004617, version 2 - 8 Jul 2005 whose mirrors work thanks to the principle of total in- an unprecedented ultra-strong coupling regime, in which ternal reflection. The strong coupling regime has been the vacuum Rabi frequency ΩR is a large fraction of the 10 also observed in quantum well infra-red detectors . As intersubband transition frequency ω12. To this purpose, we will show in detail, an important advantage of using transitions in the far infrared are most favorable, because intersubband transitions is the possibility of exploring the ratio ΩR/ω12 scales as the square root of the inter- a regime where the normal-mode polariton splitting is subband emission wavelength. Within a second quanti- 2 zation formalism, we characterize the polaritonic normal (a) z modes of the system in the weak excitation limit, in which the density of intersubband excitations is much smaller than the density of the two-dimensional electron gas in each quantum well (in this very dilute limit, the inter- q subband excitations behave as bosons). We point out Lcav the non-classical properties of the ground state, which consists of a two-mode squeezed vacuum. As its proper- ties can be modulated by applying an external electro- static bias, we suggest the possibility of observing quan- tum electrodynamics effects, such as the generation of (b) a(c) correlated photon pairs from the initial vacuum state. E2 Such an effect closely reminds the so-called dynamical Casimir effect23,24,25, whose observation is still an open E1 hw12 challenge and is actually the subject of intense effort. 2DEG Many theoretical works have in fact predicted the gener- ation of photons in an optical cavity when its properties, z q e.g. the length or the dielectric permittivity of the cavity spacer material, are modulated in a rapid, non-adiabatic FIG. 1: (a) Sketch of the considered planar cavity geom- way26,27,28. etry, whose growth direction is called z. The cavity spacer of thickness Lcav embeds a sequence of nQW identical quan- The present paper is organized as follows. In Sec. I we tum wells. The energy of the cavity mode depends on the describe the system under examination and in Sec. II we cavity photon propagation angle θ. (b) Each quantum well introduce its Hamiltonian. The scaling of the coupling in- contains a two-dimensional electron gas in the lowest subband tensity with the material parameters is discussed in Sec. (obtained through doping or electrical injection). The transi- III, while Sec. IV is devoted to the diagonalization of the tion energy between the first two subbands is ¯hω12. Only the Hamiltonian and the discussion of the polaritonic nor- TM-polarized photon mode is coupled to the intersubband mal modes of the system in the different regimes. The transition and a finite angle θ is mandatory to have a finite quantum ground state is characterized in Sec. V and dipole coupling. (c) Sketch of the energy dispersion E1(q) and its quantum properties are pointed out. Two possible E2(q)= E1(q) + ¯hω12 of the first two subbands as a function schemes for the generation of photon pairs from the ini- of the in-plane wavevector q. The dispersion of the inter- tial vacuum by modulating the properties of the ground subband transition is negligible as compared to the one of the cavity mode. For a typical value of the cavity photon in-plane state are suggested in Sec. VI. Conclusions are finally wavevector k, one has in fact E2( k + q ) E1(q) ¯hω12. drawn in Sec. VII. | | − ≃ I. DESCRIPTION OF THE SYSTEM In the following, we will consider the fundamental cav- ity photon mode, whose frequency dispersion is given by c 2 2 ωcav,k = kz + k , where ǫ is the dielectric con- In the following, we will consider a planar Fabry-Perot √ǫ∞ ∞ stant of the cavity spacer and k is the quantized photon resonator embedding a sequence of nQW identical quan- p z tum wells (see the sketch in Fig.
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