• Photonics Research
  • Vol. 11, Issue 7, 1203 (2023)
Simon Messelot, Solen Coeymans, Jérôme Tignon, Sukhdeep Dhillon, and Juliette Mangeney*
Author Affiliations
  • Laboratoire de Physique de l’Ecole Normale Supérieure, Ecole normale supérieure, PSL University, Sorbonne Université, Université Paris Diderot, Sorbonne Paris Cité, CNRS, 24 rue Lhomond, 75005 Paris, France
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    DOI: 10.1364/PRJ.482195 Cite this Article Set citation alerts
    Simon Messelot, Solen Coeymans, Jérôme Tignon, Sukhdeep Dhillon, Juliette Mangeney, "High Q and sub-wavelength THz electric field confinement in ultrastrongly coupled THz resonators," Photonics Res. 11, 1203 (2023) Copy Citation Text show less

    Abstract

    The control of light–matter coupling at the single electron level is currently a subject of growing interest for the development of novel quantum devices and for studies and applications of quantum electrodynamics. In the terahertz (THz) spectral range, this raises the particular and difficult challenge of building electromagnetic resonators that can conciliate low mode volume and high quality factor. Here, we report on hybrid THz cavities based on ultrastrong coupling between a Tamm cavity and an LC circuit metamaterial and show that they can combine high quality factors of up to Q=37 with a deep-subwavelength mode volume of V=3.2×10-4λ3. Our theoretical and experimental analysis of the coupled mode properties reveals that, in general, the ultrastrong coupling between a metamaterial and a Fabry–Perot cavity is an effective tool to almost completely suppress radiative losses and, thus, ultimately limit the total losses to the losses in the metallic layer. These Tamm cavity-LC metamaterial coupled resonators open a route toward the development of single photon THz emitters and detectors and to the exploration of ultrastrong THz light–matter coupling with a high degree of coherence in the few to single electron limit.
    H^=ωA(a^a^+12)+ωB(b^b^+12)+G(a^b^+a^b^a^b^a^b^),

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    |+1=cosθ|1A,0B+sinθ|0A,1B,|1=sinθ|1A,0Bcosθ|0A,1B,

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    ω±=ωA+ωB2±12(ωAωB)2+4G2.

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    {dAdt=(iωAΓA2)A+iCBCAGB+Γrad1,ACAs+,1dBdt=(iωBΓB2)B+iCACBGA,

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    Γ±=Γrad1,Tamm+Γrad2,LC+Γloss,Tamm+Γloss,LC2.

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    V=(ϵE2μH2)d3r2ϵ(r0)E2(r0),

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    EA(r)=iEA(afA(r)afA*(r)),EB(r)=iEB(bfB(r)bfB*(r)).

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    V+=ϵ(r)|+1|E(r)2|+1|d3rϵ(r0)|+1|E(r0)2|+1|.

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    +1|E(r0)2|+1=32cos2θEA2+32sin2θEB2+2cosθsinθEAEB,

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    V+=VBsin2θω+ωB11+23ωAωBVBVAcotθ+ωAωBVBVAcot2θ,

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    V=VBcos2θωωB1123ωAωBVBVAtanθ+ωAωBVBVAtan2θ.

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    κi*=tiCi.(A1)

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    {dAdt=(iωA1τA)A+igA*B+tACAs+,1,1τA=|tA|22CA+1τloss,A,dBdt=(iωB1τB)B+igB*A,1τB=|tB|22CB+1τloss,B,(A2)

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    s,1=rAs+,1+tAAands,2=rBs+,2+tBB.(A3)

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    {0=(i(ωAω)1τA)A+igA*B0=(i(ωBω)1τB)B+igB*A.(A4)

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    0=gA*gB*+(i(ωAω)1τA)(i(ωBω)1τB)(A5)

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    0=gA*gB*ω2+ω((ωA+ωB)+i(1τA+1τB))i(ωAτB+ωBτA)+1τAτBωAωB.(A6)

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    0=gA*gB*ω2+ω(ωA+ωB)ωAωB,(A7)

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    ω±=ωA+ωB2±(ωAωB)24+gA*gB*.(A8)

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    {dαdt(t)=i(ωAω)α(t)+igA*β(t)dβdt(t)=i(ωBω)β(t)+igB*α(t),(A9)

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    (α(t)β(t))=(cos(Gt)iGgA*sin(Gt)),withG2=gA*gB*.(A10)

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    W(t)=CA|A(t)|2+CB|B(t)|2=CAcos2(Gt)+CB(GgA*)2sin2(Gt).(A11)

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    CAgA*=CBgB*.(A12)

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    gA*=g*CAandgB*=g*CB,(A13)

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    G=g*CACB.(A14)

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    {tACAs+,1=(i(ωAω)1τA)A+ig*CAB0=(i(ωBω)1τB)B+ig*CBA,(A15)

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    A(ω)=(i(ωωB)+1τB)+tACAs+,1(i(ωωA)+1τA)(i(ωωB)+1τB)+G2,B(ω)=ig*CBtACAs+,1(i(ωωA)+1τA)(i(ωωB)+1τB)+G2.(A16)

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    r=s,1s+,1=rA+As+,1tAandt=s,2s+,1=Bs+,1tB(A17)

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    r(ω)=rA(i(ωωB)+1τB)(i(ωωA)+1τATACA)+G2(i(ωωA)+1τA)(i(ωωB)+1τB)+G2,(A18)

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    t(ω)=ig*tAtBCACB(i(ωωA)+1τA)(i(ωωB)+1τB)+G2.(A19)

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    R=(G2(ωωA)(ωωB)+1τB(1τATACA))2+((ωωB)(1τATACA)+1τB(ωωA))2(G2(ωωA)(ωωB)+1τA1τB)2+(1τA(ωωB)+1τB(ωωA))2(A20)

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    T=G2TACATBCB(G2(ωωA)(ωωB)+1τA1τB)2+(1τA(ωωB)+1τB(ωωA))2.(A21)

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    Γrad1/2,i=2τrad,i=TiCi,Γloss,i=2τloss,i,with1τi=1τrad,i+1τloss,i=Γi2.(A22)

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    R=(G2(ωωA)(ωωB)+ΓB2(ΓA2Γrad1,A))2+((ωωB)(ΓA2Γrad1,A)+ΓB2(ωωA))2(G2(ωωA)(ωωB)+ΓAΓB4)2+(ΓA2(ωωB)+ΓB2(ωωA))2(A23)

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    T=G2Γrad1,AΓrad2,B(G2(ωωA)(ωωB)+ΓAΓB4)2+(ΓA2(ωωB)+ΓB2(ωωA))2.(A24)

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    R=ΓB2(Γloss,AΓrad1,A2)2+G2(Γloss,A+Γloss,B+Γrad2,BΓrad1,A)2(ΓAΓB2)2+G2(ΓA+ΓB)2,T=4G2Γrad1,AΓrad2,B(ΓAΓB2)2+G2(ΓA+ΓB)2.(A25)

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    Γrad1,LC=|t1,LC|2CLCandG=g*CTammCLC.(A28)

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    Γrad1,LCnLCandG(nLC)12,(A29)

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    G2CTammΓrad1,LC=(g*)2|t1,LC|2,(A30)

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    g*=nSub|t1,LC|.(A31)

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    V=(ϵE2μH2)d3r2ϵ(r0)E2(r0),(D1)

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    E(r,t)=modejαj(t)fj(r).(E1)

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    V=ϵ(r)|E(r)|2d3rϵ(r0)|E(r0)|2=ϵ(r)ϵ0|f(r)|2d3r.(E2)

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    |+1=cosθ|1A,0B+sinθ|0A,1B,|1=sinθ|1A,0Bcosθ|0A,1B,(E3)

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    EA(r)=iEA(afA(r)afA*(r)),EB(r)=iEB(bfB(r)bfB*(r)),(E4)

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    +1|E(r)2|+1=3cos2θEA2|fA(r)|2+3sin2θEB2|fB(r)|2+2sinθcosθEAEBRe(fAfB*).(E5)

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    ϵ(r)+1|E(r)2|+1d3r=32cos2θωA+32sin2θωB+2sinθcosθωAωBVAVBϵr(r)Re(fAfB*)d3r,(E6)

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    ϵ(r0)+1|E(r0)2|+1=32cos2θωAVA+32sin2θωBVB+sinθcosθ2ωAωBVAVB.(E7)

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    V+=ω+cos2θωAVA+sin2θωBVB+23sinθcosθωAωBVAVB,(E8)

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    V=ωsin2θωAVA+cos2θωBVB23sinθcosθωAωBVAVB,(E9)

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    Simon Messelot, Solen Coeymans, Jérôme Tignon, Sukhdeep Dhillon, Juliette Mangeney, "High Q and sub-wavelength THz electric field confinement in ultrastrongly coupled THz resonators," Photonics Res. 11, 1203 (2023)
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