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
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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. ccsd-00004617, version 2 - 8 Jul 2005 in,wih ntecs fsmcnutrmtras are materials, semiconductor of excita- so-called of case the photonic consist the and in system which, electronic the tions, of of modes superpositions normal linear the regime, microcavities this semiconductor in quan embedded in wells transitions tum excitonic using systems solid-state in opigrgm a enfis bevdi h ae’80s late the cavities in metallic observed in first atoms been using has strong The regime linewidths. coupling transition electronic and mode ity h ocle aumRb rqec Ω frequency Rabi vacuum so-called the u aifeunyΩ frequency Rabi uum rcino h rniinfrequency transition the of fraction gemn iherirsmcascltertclpredic- theoretical Liu by semiclassical tions in earlier transition, intersubband with mid-infrared pho- agreement cavity a a and between mode regime ton coupling strong of stration zdb Dini by ized eieweetenra-oeplrtnsltigis splitting polariton normal-mode exploring the of where using possibility of regime the advantage a important is an transitions detail, intersubband in been detectors show infra-red has will well in- regime we quantum total coupling in of strong observed principle The also the to reflection. thanks ternal microcavity, work a mirrors in embedded whose structure well quantum tiple eie hc sahee hnacvt oei reso- is frequency mode of cavity transition electronic a an when with achieved coupling nant is light-matter which strong so-called regime, the is interaction infrared. far and mid the tunable in providing region, emitting active of sources the thickness the in material via wells chosen semiconductor quantum be the the can deter- rather of but not gap used, system is energy the transitions by intersubband mined of bands, frequency valence cascade and the conduction quantum between transitions the band remarkable as to such lasers leading devices success, opto-electronic considerable a enjoyed transitions tronic eety Dini Recently, n ftems acntn set flight-matter of aspects fascinating most the of One elec- intersubband of study the decade, last the In 2,3,4 ncnrs otemr ovninlinter- conventional more the to contrast In . unu aumpoete fteitrubn aiypolar cavity intersubband the of properties vacuum Quantum 9 nti ido ytm h aumRb rqec Ω frequency Rabi vacuum the system, of kind two a this of in presence in transition intersubband semiconductor ubn rniinfrequency transition subband o ogwvlnt rniin,bcuefragvndoping given Ω a ratio for the wells, because quantum transitions, wavelength long for orltdpoo ar u ftevcu i unu electr quantum via effect. vacuum Casimir quantum the dynamical polariton of the the out of pairs tunability photon the correlated how out point finally hc a etuned groun the be of can properties which quantum the characterize We length. tal. et h ilcrcFbyPrtsrcuereal- structure Fabry-Perot dielectric The . polaritons 1 aoaor ireAgan cl oml Sup´erieure, 24 Normale Ecole Aigrain, Pierre Laboratoire epeetaqatmdsrpino lnrmcoaiypho microcavity planar a of description quantum a present We 2 R E-NMadDpriet iFsc,Uiestad Tre Universit`a di Fisica, di Dipartimento and BEC-INFM CRS 1 tal. et 8 nsmcnutrqatmwlshas wells quantum semiconductor in ossso ouaindpdmul- doped modulation a of consists 8 nbt hs ytm,tevac- the systems, these both In . 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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. 1). Each quantum well wavevector along the growth direction, which depends is assumed to be doped with a two-dimensional density on the boundary conditions imposed by the specific mir- N2DEG of electrons, which, at low temperatures, popu- ror structures. In the simplest case of metallic mirrors, π late the first quantum well subband. Due to the presence kz = Lcav , with Lcav the cavity thickness. of the two-dimensional electron gas, it is possible to have transitions from the first to the second subband of the quantum well. We will callhω ¯ the considered intersub- 12 II. SECOND QUANTIZATION HAMILTONIAN band transition energy. If we denote with z the growth direction of the multiple quantum well structure, then the dipole moment of the transition is aligned along z, In this Section, we introduce the system Hamiltonian i.e., d12 = d12zˆ. This property imposes the well known in a second quantization formalism. In the following, polarization selection rule for intersubband transitions in we will call ak† the creation operator of the fundamental quantum wells, i.e., the electric field must have a com- cavity photon mode with in-plane wave-vector k. Note ponent along the growth direction. We point out that in that, in order to simplify the notation, we will omit the the case of a perfect planar structure, the in-plane wave- polarization index of the photon mode, which is meant vector is a conserved quantity, unlike the wave-vector to be Transverse Magnetic (TM)-polarized (also known component along the z direction. Therefore, all wave- as p-polarization). This photon polarization is neces- vectors k will be meant as in-plane wave-vectors, unless sary to have a finite value of the electric field compo- differently stated. nent along the growth direction z of the multiple quan- 3
tum well structure, direction along which the transition Hanti =¯h iΩR,k akb−k ak† b† k − − dipole of the intersubband transition is aligned. bk† will k n be instead the creation operator of the bright intersub- X +D a a− + a† a† . (8) band excitation mode of the doped multiple quantum k k k k −k well structure. In the simplified case of n identical o QW H in Eq. (6) describes the energy of the bare cavity quantum wells that are identically coupled to the cavity 0 photon and intersubband polarization fields, which de- photon mode, the only bright intersubband excitation is the totally symmetric one, with an oscillator strength pend on the numbers ak† ak, bk† bk of cavity photons and intersubband excitations, respectively. nQW times larger than the one of a single quantum well. Hres in Eq. (7) is the resonant part of the light-matter The nQW 1 orthogonal excitations are instead dark and will be neglected− in the following. The creation operator interaction, depending on the vacuum Rabi energyh ¯ΩR,k corresponding to the bright intersubband transition can and on the related coupling constant Dk. The terms pro- be written as portional to ΩR,k describe the creation (annihilation) of one photon and the annihilation (creation) of an inter- nQW 1 (j) (j) , subband excitation with the same in-plane wavevector. bk† = c2,q†+kc1,q (1) nQW N2DEGS In contrast, the term proportional to Dk contains only j=1 q The specific values of the coupling constants ΩR,k and H = H0 + Hres + Hanti (5) Dk depend on the microscopic parameters of the inter- which consists of three qualitatively different contribu- subband microcavity system. tions, namely The so-called vacuum Rabi energyh ¯ΩR,k is the Rabi energy obtained with the electric field corresponding to 1 5,30 8,9 H = ¯hω a† ak + + ¯hω b† bk , (6) one photon . For the system under consideration , 0 cav,k k 2 12 k k k the polariton coupling frequency for the TM-polarized X X mode31 reads 2 1/2 Hres =¯h iΩR,k a† bk akb† 2πe k − k eff 2 k ΩR,k = eff N2DEG nQW f12 sin θ , (9) n ǫ m0Lcav X ∞ +Dk ak† ak + akak† , (7) eff where ǫ is the dielectric constant of the cavity, Lcav the o ∞ 4 0.45 infra-red, by increasing the number of quantum wells and 0.4 by choosing semiconductors with smaller effective mass. Let be θres the cavity propagation angle corresponding 0.35 to kres. From the relation 0.3 ω12 kres = √ǫ sin θres , (12) 12 ∞ 0.25 c ω / we get that for metallic mirrors 0.2 R,k Ω λ12 0.15 Lcav = , (13) 2√ǫ cos θres ∞ 0.1 where 2π/λ12 = ω12/c. Under these conditions, the light- 0.05 matter coupling ratio at the resonance angle is 0 res 0 1 2 3 ΩR,k 10 10 10 10 = η λ , (14) λ (µ m) 12 12 ω12 with p FIG. 2: Coupling ratio ΩR,k/ω12 as a function of the intersub- 2 2 band emission wavelength λ12 (µm). Parameters: f12 = 12.9 e f12 sin θres cos θres N2DEGnQW η = 2 . (15) (GaAs quantum well), cavity spacer refraction index √ǫ∞ = s πm0c √ǫ eff 11 −2 ◦ ∞ 3, nQW = 50, N2DEG = 5 10 cm and θ = 60 . The Fabry-Perot resonator is a λ/×2-microcavity. Results obtained Note that the prefactor given in Eq.(15) has a weak de- from the analytical expressions in Eqs. (14) and (15). pendence on λ12. In fact, in the limit case of a rectan- gular quantum well with high potential barriers, f12 = ⋆ 0.96 m0/m and does not depend at all on λ12. More refined calculations32 including the non-parabolicity of effective thickness of the cavity photon mode (which de- the semiconductor band and the finite depth of the po- pends non-trivially on the boundary conditions imposed tential well show that f12 has a moderate dependence on eff by the specific mirror structures), and nQW the effective the emission wavelength λ12 (it actually increases with eff number of quantum wells (nQW = nQW in the case of λ12). Hence, the normalized vacuum Rabi frequency quantum wells which are identically coupled to the cav- ΩR,kres /ω12 increases at least as √λ12. The predictions ity photon field and which are located at the antinodes of Eqs. (14) and (15) are reported in Fig. 2 for a sys- of the cavity mode electric field). The oscillator strength tem of 50 GaAs quantum wells and a doping density 11 2 of the considered intersubband transition reads N2DEG =5 10 cm− . For an intersubband emission wavelength of× 100µm, the ratio Ω /ω can be as high as 2 R 12 f12 =2m0ω12d12/¯h , (10) 0.2. The values in Fig. 2 can be significantly increased using semiconductors with smaller effective mass, such as where m is the free electron mass and d is the electric 0 12 InGaAs/AlInAs-on-InP33. dipole moment of the transition. Under the approxima- To complete our description, we need to provide the ex- tion of a parabolic energy dispersion of the quantum well plicit expression for the coefficient D , which quantifies subbands, the oscillator strengths of the different inter- k the effect of the squared electromagnetic vector potential subband transitions satisfy the f-sum rule32 in the light-matter interaction. Generalizing Hopfield’s procedure29 to the case of intersubband transitions, we f = m /m∗, (11) 1j 0 find that all the intersubband transitions give a contri- j X bution to Dk, namely where m∗ is the effective electron mass of the conduction 2 j f1j ΩR,k band. In particular, for our case of a deep rectangular Dk = . (16) well, the sum rule is almost saturated by the first inter- Pf12 ω12 subband transition f m /m∗. Finally, θ is the prop- However, as the oscillator strength of a deep rectangu- 12 ≃ 0 agation angle inside the cavity (which is different from lar well is concentrated in the lowest transition at ω12, the propagation angle in the substrate), and is related to the effect of the higher transitions is a minor correction, the in-plane wavevector k by k/kz = sin θ/ cos θ. namely As we will see in the next section, the relevant pa- 2 2 ΩR,k ΩR,k rameter quantifying the importance of the quantum ef- Dk 1.04 . (17) fects considered in this paper is the dimensionless ratio ≃ ω12 ≈ ω12 ΩR,kres /ω12, where kres is the resonance in-plane wavevec- Note that for a quantum well with a parabolic confine- 2 2 tor such ashω ¯ cav,kres =hω ¯ 12. In the system studied ment potential V (z) = (1/2)m∗ω12z , the expression 8 2 by Dini et al. , this ratio is already significant, namely Dk =ΩR,k/ω12 would be exact, since in this case all the ΩR,k/ω12 =0.05. Here, we show that the ratio ΩR,k/ω12 intersubband oscillator strength is exactly concentrated can be largely increased designing structures in the far in the lowest transition ω12. 5 2.5 1.1 0.18 2 1 |x | (a) LP,k 0.16 (b) 2 0.9 UP 0.14 0.8 2 0.12 |z | 1.5 LP,k 0.7 12 0.1 ω / 0.6 ω 1 0.08 0.5 0.06 0.4 0.5 LP 2 |w | 0.04 0.3 LP,k 0 0.2 0.02 0 0.2 0.4 0.6 0.8 1 2 Ω /ω |y | R,k 12 0.1 0 LP,k 0 0.5 1 0 0.5 1 Ω /ω Ω /ω R,k 12 R,k 12 FIG. 3: Normalized polariton frequencies ωLP,k/ω12 and 2 ωUP,k/ω12 as a function of ΩR,k/ω12 for Dk = ΩR,k/ω12. The calculation has been performed with ωcav,k = ω12. Note that FIG. 4: Mixing fractions for the Lower Polariton (LP) 12 for a given microcavity system, ΩR,k/ω12 can be tuned in-situ mode as a function of ΩR,k/ω (see Eq. (18) in the text). The calculation has been performed for the resonant case by an electrostatic bias, which is able to change the density 2 ωcav,k = ω12 , as in the previous figure. Panel (a): wLP,k of the two-dimensional electron gas. 2 | | (thin solid line), xLP,k (thick solid line). Note that for | |2 2 ΩR,k/ω12 1, wLP,k xLP,k 1/2. Panel (b): 2 ≪ | | ≃ | 2 | ≃ yLP,k (thin dashed line), zLP,k (thick dashed line). For | | 2 | 2 | ΩR,k/ω12 1, yLP,k zLP,k 0. The Upper Polariton IV. INTERSUBBAND POLARITONS ≪ | | ≃ | | ≃ 2 2 (UP) fractions (not shown) are simply wUP,k = xLP,k , 2 2 2 2 | 2| | 2 | xUP,k = wLP,k , yUP,k = zLP,k , zUP,k = yLP,k . As all the terms in the Hamiltonian H = H0 + Hres + | | | | | | | | | | | | Hanti are bilinear in the field operators, H can be exactly diagonalized through a Bogoliubov transformation. Fol- lowing the pioneering work by Hopfield29, we introduce The Hopfield-like matrix for our system reads the Lower Polariton (LP) and Upper Polariton (UP) an- nihilation operators ω +2D iΩ 2D iΩ cav,k k − R,k − k − R,k iΩR,k ω12 iΩR,k 0 Mk = − . pj,k = wj,k ak + xj,k bk + yj,k a† k + zj,k b† k , (18) 2Dk iΩR,k ωcav,k 2Dk iΩR,k − − − − − − iΩR,k 0 iΩR,k ω12 where j LP,UP . The Hamiltonian of the system − − (24) ∈ { } can be cast in the diagonal form The four eigenvalues of Mk are ωLP,k, ωUP,k . Under 2 {± ± } the approximation Dk =ΩR,k/ω12 (i.e., all the oscillator H = EG + ¯hωj,k pj,† kpj,k , (19) strength concentrated on the ω12 transition), det Mk = j LP,UP k (ω ω )2, giving the simple relation ∈{ X } X cav,k 12 where the constant term EG will be given explicitly later. ωLP,k ωUP,k = ω12 ωcav,k , (25) The Hamiltonian form in Eq. (19) is obtained, provided that the vectors i.e., the geometric mean of the energies of the two po- lariton branches is equal to the geometric mean of the T ~vj,k = (wj,k, xj,k,yj,k,zj,k) (20) bare intersubband and cavity mode energies. The de- pendence of the exact polariton eigenvalues as a function satisfy the eigenvalues equation of ΩR,k/ω12 is reported in Fig. 3, for the resonant case ωcav,k = ω12. Mk~vj,k = ωj,k~vj,k (21) with ωj,k > 0. The Bose commutation rule A. Ordinary properties in the limit ΩR,k/ω12 1 ≪ ′ ′ [pj,k,pj†′,k′ ]= δj,j δk,k (22) In the standard case Ω /ω 1, the polariton op- R,k 12 ≪ imposes the normalization condition erator can be approximated as w∗ w ′ + x∗ x ′ y∗ y ′ z∗ z ′ = δ ′ . (23) p k w ak + x bk , (26) j,k j ,k j,k j ,k − j,k j ,k − j,k j ,k j,j j, ≃ j,k j,k 6 2 2 2 2 with wj,k + xj,k 1. This means that the annihi- this affects the ordinary fractions wLP,k , xLP,k as lation| operator| | for a| polariton≃ mode with in-plane wave- well. Owing to the blue-shift of the| cavity| photon| | fre- vector k is given by a linear superposition of the photon quency induced by the light-matter coupling, at the and intersubband excitation annihilation operators with resonance wavevector k = kres the lower polariton be- 2 2 the same in-plane wavevector, while mixing with the cre- comes more matter-like (i.e., xLP,kres > wLP,kres and 2 2 | | | | ation operators (represented by the coefficients y and z res > y res ), while the upper polariton more j,k | LP,k | | LP,k | zj,k) is instead negligible [see Fig. 4]. In this limit, the photon-like. In this resonant case, the UP Hopfield coef- geometric mean can be approximated by the arithmetic ficients (not shown) are simply related to the LP ones 2 2 2 2 mean and Eq. (25) can be written in the more usual by: wUP,kres = xLP,kres , xUP,kres = wLP,kres , | 2 | | 2 | | 2 | 2| | form: yUP,kres = zLP,kres , zUP,kres = yLP,kres . | V.| THE| QUANTUM| | GROUND| | STATE| ω + ω ω + ω . (27) LP,k UP,k ≃ cav,k 12 A. The normal vacuum state 0 for ΩR,k = 0 For the specific resonant wavevector kres such that | i ωcav,kres = ω12, the polariton eigenvalues are In the case ΩR = 0 (negligible light-matter interac- ωLP (UP ),kres ω12 ΩR,kres , (28) tion), the quantum ground state G of the considered ≃ ∓ system is the ordinary vacuum 0 |fori the cavity photon 2 2 | i and the mixing fractions are wLP,kres xLP,kres and intersubband excitation fields. Such ordinary vac- 1/2. | | ≃ | | ≃ uum satisfies the relation ak 0 = bk 0 =0 , (31) B. Ultra-strong coupling regime | i | i which means a vanishing number of photons and inter- When the ratio ΩR,k/ω12 is not negligible compared to subband excitations: 1, then the anomalous features due to the anti-resonant 0 a† ak 0 = 0 b† bk 0 = 0 a† bk 0 = 0 (32) terms of the light-matter coupling becomes truly rele- h | k | i h | k | i h | k | i vant. and no anomalous correlations, i.e., In the resonant ωcav,kres = ω12 case and under the ap- 2 proximation Dk = ΩR,k/ω12, the polariton frequencies 0 akak′ 0 = 0 bkbk′ 0 = 0 akbk′ 0 =0 . (33) are given by h | | i h | | i h | | i 2 2 ωLP (UP ),kres = ω + (ΩR,kres ) ΩR,kres , (29) 12 ∓ B. The squeezed vacuum state q which, as it is apparent in Fig. 3, corresponds to a strongly asymmetric anti-crossing as a function of With a finite ΩR,k, the ground state of the system G | i Ω res /ω . This is due to the combined effect of the is no longer the ordinary vacuum 0 such that: R,k 12 | i blue-shift of the cavity mode frequency due to the terms proportional to D in Eq. (7), and of the anomalous ak 0 = bk 0 =0 , (34) k | i | i coupling terms in Eq.(8). These same effects contribute to the non-trivial evo- but rather the vacuum of polariton excitations: lution of the Hopfield coefficients shown in Fig. 4. The 2 2 anomalous Hopfield fractions y and z signif- pj,k G =0 . (35) LP,k LP,k | i icantly increase because of the| anomalous| | coupling,| and eventually become of the same order as the normal ones As the polariton annihilation operators are linear su- 2 2 xLP,k and wLP,k . Due to the normalization condi- perpositions of annihilation and creation operators for tion| | | | the photon and the intersubband excitation modes, the ground state G is, in quantum optical terms, a squeezed w 2 + x 2 y 2 z 2 =1 , (30) state34,35. By| invertingi Eq.(18), one gets | j,k| | j,k| − | j,k| − | j,k| ak w∗ w∗ y y pLP,k LP,k UP,k − LP,k − UP,k bk x∗ x∗ z z pUP,k = LP,k UP,k − LP,k − UP,k , (36) † a† k yLP,k∗ yUP,k∗ wLP,k wUP,k pLP, k − − − − b† z∗ z∗ xLP,k xUP,k p† k − LP,k − UP,k UP, k − − 7 1 a time-dependent external electrostatic bias. In partic- ular, we shall discuss how this remarkable tunability of 0.8 the system can be used to ”unbind” the virtual photons D /ω by modulating the parameters of the system in a time- 0.6 k 12 dependent way, and generate some radiation which can 0.4 be actually detected outside the cavity. These issues will be the subject of sec. VI. 0.2 ∆ω /ω ZP 12 0 C. The ground state energy −0.2 corr −0.4 ∆ω /ω ZP 12 Also the energy EG of the quantum ground state has a significant dependence on the coupling ΩR,k. Defining E0 −0.6 0 0.2 0.4 0.6 0.8 1 as the ground state energy of the uncoupled (Ω = 0) Ω /ω R,k R,k 12 system, we have that: FIG. 5: Solid line: normalized differential zero-point energy 2 2 EG E0 = ¯hDk ¯hωj,k( yj,k + zj,k ) . (per mode) ∆ωZP (k)/ω12 as a function of ΩR,k/ω12. Dashed − − | | | | k j LP,UP line: Dk/ω12. Dotted line: normalized correlation contribu- X ∈{ X } corr (43) tion ∆ωZP (k)/ω12. The calculation has been performed with ωcav,k = ω12. Note that this energy difference includes only the contri- bution of the zero-point fluctuations of the intersubband polariton field and does not take into account the other contributions coming, e.g. from the change of the elec- from which, using Eq. (35) and the boson commutation trostatic energy of the system (which is imposed by an rules, we obtain that the ground state contains a finite applied bias), as already discussed at the end of Sec. II. number (per mode) of cavity photons and intersubband The (always positive) term Dk in Eq.(43) is the zero- excitations: point energy change due to the mere blue-shift of the bare cavity mode frequency and does not correspond to 2 2 G a† ak G = yLP,k + yUP,k (37) any squeezing effect. The second term is instead due to h | k | i | | | | 2 2 the mixing of creation and annihilation operators into G b† bk G = zLP,k + zUP,k , (38) h | k | i | | | | the polaritonic operators as described in Eq.(18) and is as well as some correlation between the photon and in- proportional to the number of virtual photons and inter- tersubband fields: subband excitations present in the ground state G of the system according to Eqs.(37-38). As it is usual| fori a cor- relation contribution, it tends to lower the ground state G ak† bk G = yLP,k∗ zLP,k + yUP,k∗ zUP,k . (39) h | | i energy. It is interesting to study the differential ”zero- Moreover, significant anomalous correlation exist be- point” energy per modeh ¯∆ωZP (k), whose sum over all tween opposite momentum components of the fields: the k-modes gives the quantum ground state energy dif- ference E E . The differential ”zero-point” frequency G − 0 G aka k G = wLP,k∗ yLP,k wUP,k∗ yUP,k (40) reads h | − | i − − G bkb k G = xLP,k∗ zLP,k xUP,k∗ zUP,k (41) h | − | i − − corr ∆ωZP (k)= Dk + ∆ω (k) , (44) G bka k G = xLP,k∗ yLP,k xUP,k∗ yUP,k . (42) ZP h | − | i − − Note that the finite photonic population which is present where the (negative) correlation contribution reads in the ground state G of our system is composed of ”virtual” photons. In| thei absence of any perturbation or ∆ωcorr(k)= ω ( y 2 + z 2) . (45) modulation of the parameters of the quantum Hamilto- ZP − j,k | j,k| | j,k| j LP,UP nian, these virtual photons can not escape from the cavity ∈{ X } and therefore do not result in any observable emitted ra- diation (indeed, energy would not be conserved in such These quantities (normalized to ω12) are plotted in Fig. a process). 5 as a function of ΩR,k/ω12 for the resonant case ωcav,k = As it has been shown in Fig. 4, the Hopfield coeffi- ω12. Although it is the diagonal blueshift which gives the cients xj,k,yj,k, wj,k,zj,k as well as the ground state G dominant contribution to the ground state energy shift, of the system strongly depend on the vacuum Rabi energy| i the negative contribution due to the correlation effects Ω , which in our case can be dramatically modulated is important, being as large as 0.13¯hω already for R,k − 12 in situ, e.g. by changing the electron density N2DEG via ΩR,k/ω12 =0.5. 8 Vbias stantaneous way. This scheme has the merit of allowing to grasp the essential physics of the problem, providing quantitative estimates without the need of embarking in complicate calculations. A complete and quantitative calculation of the spec- q q res res tral shape and intensity of the emitted radiation for the most relevant case of a periodic modulation of ΩR,k is be- yond the scope of the present paper, as it would require a careful analysis of the coupling of the cavity system to the extra-cavity field as well as of the other non-radiative FIG. 6: Sketch of a possible set-up for the generation of cor- loss mechanisms of the electronic system36. This is work related photon pairs in the intersubband cavity system. The actually in progress and here we shall restrict ourselves vacuum Rabi frequency of the intersubband cavity system can to a very qualitative discussion of its main features. be modulated through an electric gate, which changes the den- sity of the two-dimensional electron gas or alternatively the dipole moment of the intersubband transition. A modulation A. Abrupt switch off of the vacuum Rabi energy of the bias is expected to induce the emission of correlated photon pairs with opposite in-plane wavevectors. This kind of radiation can be optimally guided out of the cavity through Let us suppose that the considered intersubband cavity wedged lateral facets, with inclination equal to the resonance system is in the ground state G . As we have already dis- angle θres. cussed, the squeezed vacuum |Gi contains a finite number of cavity photons and intersubband| i excitations because of the correlations due to the anomalous coupling terms in Eq.(8). VI. TUNING THE QUANTUM VACUUM: If one switches off the vacuum Rabi frequency ΩR,k QUANTUM RADIATION EFFECTS of the system in an abrupt, non-adiabatic way by sud- denly depleting the electron gas, the photon mode does The possibility of tuning in a dramatic way the prop- not have the time to respond to the perturbation and erties (energy and squeezing) of the ground state of the will remain in the same squeezed vacuum state as before. system, as well as the significant number of (virtual) ex- As this state is now an excited state of the Hamiltonian citations already present in the ground state suggests for ΩR,k = 0, the system will relax towards its ground that the present system could be a potential labora- state, which now corresponds to the standard vacuum, tory to study Quantum Electro-Dynamics (QED) phe- by emitting the extra photons as propagating radiation. nomena, which are reminiscent of the dynamic Casimir One possible way to collect this quantum vacuum ra- effects23,24,25. In particular, we shall discuss how a time- diation is through the set-up sketched in Fig. 6, which modulation of the ground-state properties of the sys- allows one to collect the photons which are emitted with tem can parametrically produce real excitations above internal propagation angle θ around the resonance value the ground state of our cavity, which then escape from θres. If one neglects the losses due to the background the cavity as photons and propagate in the external free- absorption by the dielectric material forming the micro- space. cavity, an estimate of the number of emitted photons In the typical arrangement for the observation of the can be obtained as follows. The number of photon states 2 dynamical Casimir effect, one has to modulate in time (per unit area) in the 2D momentum volume d k is sim- 2 2 the properties of an optical cavity and, in particular, its ply d k/(2π) . Hence, the differential density of photons 2 resonance frequencies. Several proposals have appeared (per unit area) in the 2D momentum volume d k is in order to do this: in the simplest ones, one has to peri- d2k † odically move the mirrors so as to modify the boundary dρphot = 2 G akak G , (46) conditions of the field25,26. Other proposals27 deal with (2π) h | | i a time-dependance of the refractive index of a dielectric where the photon number G ak† ak G in the quantum medium placed inside the cavity. A recent work proposes ground state is given by Eq.h (37).| Now,| i all the expecta- to vary the effective length of the cavity by changing the k 28 tion values depend only on and hence we can rewrite reflectivity of a composite mirror . the momentum volume as d|2k| = 2πkdk. Knowing that The main peculiarity of our system as compared to the in-plane wavevector k is given by the relationship previous proposals is due to the possibility of modulat- k = kz tan(θ) and using Eq. (12), we find the final result ing the properties of the ground state in a much stronger way, due to the ultra-strong and tunable light-matter 2 dρphot 1 ω12 coupling. † (θres)= 2 ǫ tan(θres) G akak G . (47) In the next subsection, we shall give a detailed analy- dθ 2π c ∞ h | | i sis of a simple gedanken experiment, where the vacuum To give a numerical application of Eq. (47), let us Rabi frequency is assumed to be switched off in an in- consider an intersubband cavity system withhω ¯ 12 = 140 9 36 meV, resonance angle θres = 65◦ andh ¯ΩR,kres = 7 meV the non-radiative ones . For a complete and quantita- (these are approximately the values in the sample mea- tive treatment of these issues, further investigations are sured by Dini et al8). For these parameters, Eq. (47) in progress. 5 gives the differential photon density dρphot/dθ 1 10 VII. CONCLUSIONS 2 1 ≃ × cm− rad− . k Note that the emission corresponding to the -mode In conclusion, we have shown that in the intersubband is correlated to the emission corresponding to the mode cavity polariton system, a new regime of ultra-strong cou- with opposite in-plane wavevector, as shown in Eq. (40). pling can be achieved, where the vacuum Rabi frequency Indeed, the ”quantum vacuum radiation” here described Ω is a large fraction of the intersubband transition fre- consists in the emission of correlated photon pairs37. R quency ω12. This scenario appears to be easier to achieve in the far infrared, since the ratio ΩR/ω12 scales as the square root of the intersubband transition wave-length. B. Periodic modulation of ΩR,k In the ultra-strong coupling regime, the usually neglected anti-resonant terms of the light-matter coupling start The requirement of a abrupt, non-adiabatic, switch-off playing an important role. In particular, the ground of the Rabi coupling ΩR,k imposes very stringent limits state of system is no longer the ordinary vacuum of pho- on the time-scale τsw over which the electrostatic bias has tons and electronic excitations, but rather a two-mode to be applied. In particular, we expect that in order to squeezed vacuum, whose properties strongly depend on maximize the quantum vacuum radiation generation, τ sw the ratio ΩR/ω12. As this quantity can be dramatically can not be too much longer that the oscillation period of tuned by applying an electrostatic bias, we have pointed the lower polaritonic mode. out the possibility of observing interesting quantum elec- It is then perhaps more accessible from an experimen- trodynamic effects reminiscent of the dynamical Casimir tal point of view to try to detect the vacuum radiation effect, i.e. the generation of correlated photon pairs out by periodically modulating the vacuum Rabi frequency of the initial polariton vacuum state. A quantitative es- at an angular frequency ωmod timate of the number of emitted photons has been given ¯ for the simplest case of an instantaneous switch-off of the ΩR,k(t)= ΩR,k + ∆ΩR,k sin(ωmodt). (48) light-matter coupling, and the results look promising in view of experimental observations. Work is actually in Note that in principle this kind of modulation can be ob- progress in the direction of extending the analysis to the tained not only through a gate-induced depletion of the case of a periodic modulation of Ω , case in which one two-dimensional electron gas11, but also by modulating R,k should be able to enhance the emitted intensity via para- the dipole moment of the intersubband transition or al- metric resonance effects. From the theoretical point of ternatively the reflectivity of the mirrors. As all the rele- view, this study requires a complete treatment of losses vant physical quantities in the present problem (polariton in order to describe the dynamical equilibrium between energies, Hopfield coefficients, ground state energy) de- the parametric process generating the quantum radiation pend in a nonlinear way on the vacuum Rabi frequency and the dissipation. ¯hΩR,k, we expect that for large modulation amplitudes high order harmonics of the fundamental modulation fre- quency ωmod will play a significant role in the paramet- ric process which is responsible for the vacuum radiation Acknowledgments generation25. In particular, emission will be enhanced if ωj,k + ωj′, k = r ωmod , (49) It is our pleasure to thank Raffaele Colombelli and − Carlo Sirtori for many enthusiastic and stimulating meet- with r being a generic positive integer number, and ings about intersubband photonics. CC would like to j, j′ = LP,UP . This is the phase-matching condition thank Alessandro Tredicucci and A. Anappara for show- for the{ parametric} generation of two polaritons with op- ing data concerning the gate-controlled vacuum Rabi posite momentum. As usual, the narrower the polaritonic energy prior to publication. We are grateful to Paolo resonance, the stronger the resonant enhancement. Schwendimann, Antonio Quattropani, Arnaud Verger, As it is generally the case for parametric processes in a Chiara Menotti, Alessio Recati, Mauro Antezza and cavity, the number of photons which are generated in the Maurizio Artoni for discussions and/or for a critical read- cavity and then emitted as radiation is determined by a ing of this manuscript. 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