Abstract

Engineering multiphoton states is an outstanding challenge with applications in multiple fields such as quantum metrology, quantum lithography, or even biological sensing. State-of-the-art methods to obtain them rely on post-selection, multi-level systems, or Rydberg atomic ensembles. Recently, it was shown that a strongly driven two-level system interacting with a detuned cavity mode can be engineered to continuously emit n-photon states. In the present work, we show that spectral filtering of its emission relaxes considerably the requirements on the system parameters even to the more accessible bad-cavity situation, opening up the possibility of implementing this protocol in a much wider landscape of different platforms. This improvement is based on a key observation: in the imperfect case where only a certain fraction of emission is composed of n-photon states, these have a well-defined energy separated from the rest of the signal, which allows one to reveal and purify multiphoton emission just by frequency filtering. We demonstrate these results by obtaining analytical expressions for the relevant figures of merit of multiphoton emission, such as the n-photon coupling rate between cavity and emitter, the fraction of light emitted as n-photon states, and n-photon emission rates. This allows us to make a systematic study of such figures of merit as a function of the system parameters and demonstrate the viability of the protocol in several relevant types of cavity quantum electrodynamics setups, where we take into account the impact of their respective experimental limitations.

© 2018 Optical Society of America

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References

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2017 (7)

A. González-Tudela, V. Paulisch, H. J. Kimble, and J. I. Cirac, “Efficient multiphoton generation in waveguide quantum electrodynamics,” Phys. Rev. Lett. 118, 213601 (2017).
[Crossref]

M. Peiris, K. Konthasinghe, and A. Muller, “Franson interference generated by a two-level system,” Phys. Rev. Lett. 118, 030501 (2017).
[Crossref]

L. de Santis, C. Antón, B. Reznychenko, N. Somaschi, G. Coppola, J. Senellart, C. Gómez, A. Lemaître, I. Sagnes, A. G. White, L. Lanco, A. Auffeves, and P. Senellart, “A solid-state single-photon filter,” Nat. Nanotechnol. 12, 663 (2017).
[Crossref]

C. Hamsen, K. N. Tolazzi, T. Wilk, and G. Rempe, “Two-photon blockade in an atom-driven cavity QED system,” Phys. Rev. Lett. 118, 133604 (2017).
[Crossref]

X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718-719, 1–102 (2017).

K. A. Fischer, Y. A. Kelaita, N. V. Sapra, C. Dory, K. G. Lagoudakis, K. Müller, and J. Vučković, “On-chip architecture for self-homodyned nonclassical light,” Phys. Rev. Appl. 7, 044002 (2017).
[Crossref]

J. C. L. Carreño, E. del Valle, and F. P. Laussy, “Photon correlations from the Mollow triplet,” Laser Photon. Rev. 11, 1700090 (2017).
[Crossref]

2016 (9)

V. Giesz, N. Somaschi, G. Hornecker, T. Grange, B. Reznychenko, L. D. Santis, J. Demory, C. Gomez, I. Sagnes, A. Lemaitre, O. Krebs, N. D. Lanzillotti-Kimura, L. Lanco, A. Auffeves, and P. Senellart, “Coherent manipulation of a solid-state artificial atom with few photons,” Nat. Commun. 7, 11986 (2016).

K. A. Fischer, K. Müller, A. Rundquist, T. Sarmiento, A. Y. Piggott, Y. Kelaita, C. Dory, and K. G. L. J. Vuckovic, “Self-homodyne measurement of a dynamic Mollow triplet in the solid state,” Nat. Photonics 10, 163–166 (2016).
[Crossref]

J. S. Douglas, T. Caneva, and D. E. Chang, “Photon molecules in atomic gases trapped near photonic crystal waveguides,” Phys. Rev. X 6, 031017 (2016).
[Crossref]

O. Firstenberg, C. S. Adams, and S. Hofferberth, “Nonlinear quantum optics mediated by Rydberg interactions,” J. Phys. B 49, 152003 (2016).
[Crossref]

Y. Chang, A. González-Tudela, C. S. Muñoz, C. Navarrete-Benlloch, and T. Shi, “Deterministic down-converter and continuous photon-pair source within the bad-cavity limit,” Phys. Rev. Lett. 117, 203602 (2016).
[Crossref]

E. Sánchez-Burillo, L. Martín-Moreno, J. García-Ripoll, and D. Zueco, “Full two-photon down-conversion of a single photon,” Phys. Rev. A 94, 053814 (2016).
[Crossref]

F. Hargart, M. Müller, K. Roy-Choudhury, S. L. Portalupi, C. Schneider, S. Höfling, M. Kamp, S. Hughes, and P. Michler, “Cavity-enhanced simultaneous dressing of quantum dot exciton and biexciton states,” Phys. Rev. B 93, 115308 (2016).
[Crossref]

K. E. Dorfman, F. Schlawin, and S. Mukamel, “Nonlinear optical signals and spectroscopy with quantum light,” Rev. Mod. Phys. 88, 045008 (2016).
[Crossref]

N. Somaschi, V. Giesz, L. D. Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Anton, J. Demory, C. Gomez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaitre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state,” Nat. Photonics 10, 340–345 (2016).
[Crossref]

2015 (10)

J. López Carreño, C. S. Muñoz, D. Sanvitto, E. del Valle, and F. Laussy, “Exciting polaritons with quantum light,” Phys. Rev. Lett. 115, 196402 (2015).
[Crossref]

P. Lodahl, S. Mahmoodian, and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures,” Rev. Mod. Phys. 87, 347–400 (2015).
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2014 (9)

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Other (4)

Note that the definition of πn introduced in Ref. [32] differs slightly with the one used here. In that work, it was defined as πn=λnλn+λ1, where λn are fitted from photon counting distributions and correspond to λn=γana(n)/n. The comparison was made adopting the definition used in this text and defining πn=λn/nλn/n+λ1.

E. del Valle, Microcavity Quantum Electrodynamics (VDM Verlag, 2010).

Value not provided in the reference. We assumed the typical value of the QD lifetime in a photonic bandgap of ∼5 ns.

We note that the predicted π2f has been obtained by assuming one can perfectly filter the two-photon contribution with respect to the single-photon one. To give a more accurate quantitative estimation, one needs to take into account a realistic filter in the calculation.

Supplementary Material (1)

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Figures (8)

Fig. 1.
Fig. 1. (a) Scheme of the proposed setup: a two-level system is coupled to a cavity and strongly driven by a classical field. By selecting the proper cavity frequency, the system emits a continuous stream of photon pairs. (b) Evidence of two-photon emission in the cavity emission spectrum, plotted as a function of the driving field amplitude. The driving laser is in resonance with the 2LS, and the cavity is detuned by Δa=5g. Mollow sidebands appear in the spectrum at ω=ωL±2Ω. When the cavity frequency lies between a sideband and the central peak, Δa=±Ω, two-photon emission is seen as a clear feature in the spectrum. At higher pumping one can also observe a small feature related to the three-photon resonance when Ω is such that Δa=Δa(3)=2Ω/2, though it is weak because the experimental parameters are not good enough. (c) Spectrum in the regime of two-photon emission at Ω=Δa. (d) Transitions in the ladder of dressed states giving rise to the four peaks featured in panel (c). Simulation parameters: γa=1.3g, γσ=0.01g.
Fig. 2.
Fig. 2. (a) Joint plot of two-photon [na(2), red] and one-photon [na(1), blue] emission rates as a function of cavity (γa) and 2LS (γσ) decay rates. Contour lines correspond to the total population na computed numerically (solid) and analytically (dashed), based on the assumption nana(2)+na(1), and using the analytical approximations of Eqs. (8) and of Eq. (10) to analytically calculate the populations na(n) and na(1), respectively. (b) Two-photon emission rate [na(2), red] and one-photon emission rate [na,f(1), blue] at the cavity frequency. Contour lines correspond to the total filtered emission na,f, computed numerically (solid) and analytically, based on the assumption na,fna(2)+na,f(1) and Eqs. (8) and (23). Ω=20g in panels (a) and (b). (c) Components of the cavity population as a function of Ω when the cavity is fixed at the two-photon resonance (Δa=Ω). Black: total cavity population. Blue: single-photon component of cavity population as given in Eq. (10). Red: two-photon component of cavity population na(2) as given by Eq. (8). Dashed, horizontal lines mark the weak driving limit of na(n) given by Eq. (13). (d) Same as in (c), for the three-photon resonance. Simulation parameters: γa=0.1g and γσ=0.01g.
Fig. 3.
Fig. 3. Cavity emission rates at the two-photon resonance for three different spectral windows: central peak of the Mollow triplet (red), cavity peak (light blue), and total emission (dark blue). These rates have been numerically computed using Eq. (22). The bundle emission γana(2) (dashed, orange) computed from the n-photon master equation given in Eq. (7) closely matches the emission at the cavity peak. This confirms that the light emitted at that frequency is composed of photon bundles, and that spurious emission can be eliminated by frequency filtering. Parameters: Ω/g=20 and γσ/g=0.025.
Fig. 4.
Fig. 4. (a) Numerical calculations for the purity of two-, three-, and four-photon emission for unfiltered (πn, solid) and filtered (πnf, dashed) cases, given by Eq. (11) and Eq. (26), with na(1) and na,f(1) computed numerically from a model truncated at one photon, and with na(n) for n2 computed numerically from the master equation Eq. (7), where the effective n-photon coupling rates are given by the analytical expression of Eq. (5). Parameters: Ω/g=20, γσ/g=0.025,0.005, and 0.001 for n=2,3,4, respectively. (b) Numerically computed unfiltered and filtered emission rates, γana (solid, black) and γana,f (dashed, black), compared to the underlying n-photon components of the cavity emission rate, γana(n), used in the computation of πn and πnf of panel (a).
Fig. 5.
Fig. 5. Purity of two-photon emission in the unfiltered (a) and filtered (b) case, as a function of cavity decay rate γa and 2LS decay rate γσ. Colored points correspond to experimental state-of-the-art samples, with values summarized in Table 1. Circles correspond to semiconductor QDs, and squares correspond to atoms. In panel (a), the values of the observable T defined in Eq. (20) are shown in dashed-dotted red lines for comparison. Parameters: Ω=20g.
Fig. 6.
Fig. 6. (a) Calculated rates for the phonon-induced transitions as a function of temperature. (b) Cavity population emitted at the cavity frequency as a function of Ω for a cavity at the two-photon resonance Δa=Ω and different temperatures. Dashed, black: two-photon population. (c) Spectrum of cavity emission as a function of the amplitude of the driving field for T=30  K. Δa=5g. (d) Spectrum at the cavity frequency as a function of driving field amplitude and different temperatures. Δa=5g.
Fig. 7.
Fig. 7. Ratio between the coherent and incoherent signals at the cavity frequency when the cavity is coherently driven, as a function of the effective driving of the 2LS. In order to describe the situation where the full cavity peak is filtered, the filter linewidth has been taken equal to the cavity linewidth, Γ=γa. For each value of the driving, the cavity frequency is tuned to the two-photon resonance, Δa=Ωeff.
Fig. 8.
Fig. 8. (a) Cavity spectrum as a function of the driving amplitude Ω for set of parameters typical of atomic cavity QED systems, γa=γσ=0.1g. The detuning between cavity and 2LS is Δa=5g. (b) Value of the spectrum at the cavity frequency ωa as a function of the driving amplitude Ω, for four fixed sets of cavity frequencies. A feature indicating two-photon emission appears whenever Ω=Δa, and it appears for driving amplitudes as low as 3g.

Tables (1)

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Table 1. Table of State-of-the-Art Parameters in Cavity QED, with Special Emphasis on Semiconductor Samplesa

Equations (31)

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H=Δaaa+g(aσ+σa)+Ω(σ+σ),
dρdt=i[H,ρ]+γσ2Lσ[ρ]+γa2La[ρ],
H=Ωσ˜z+Δaaa+g2{a(σ˜σ˜+σ˜z)+h.c},
Heff(n)=Ωσ˜z+Δaaa+g(n)(σ˜an+σ˜an),
g(n)=gn2(n1)!2(n24Ω)n1.
γσ2Lσ[ρ](γσ8Lσ˜+γσ8Lσ˜+γσ2Lσ˜σ˜)[ρ],
dρdt=i[Heff(n),ρ]+γa2La[ρ]+γσ˜2Lσ˜[ρ]+Pσ˜2Lσ˜[ρ]+γϕ˜2Lσ˜σ˜[ρ],
na(n)n2κ(n)Γσ˜(nγa+Γσ˜+γϕ˜)+κ(n)nγaPσ˜,
na(n)nκ(n)γaPσ˜κ(n)+Γσ˜,
na(1)2  g2(γa2+28Ω2)(γa2+4Ω2)(γa2+36Ω2)+32  g2Ω2(γa4+432Ω4)γσ˜γa(γa2+4Ω2)2(γa2+36Ω2)2.
πn=na(n)nana(n)na(1)+na(n).
γσ(n,opt)=(gn24Ω)n8Ω3(n!)3.
na(n)n4γσγa,(Ω0),
na(n)g2nΩ2(n1)An1Ω2(n1),
na(1)(g2Ω2)n3(2+n2)γσ+n(n4n2+2)γa16(n21)2γa1Ω2,
An=[16n1γa(2nγa+3γσ)n2(12n)(n1)!3]1.
πn{1+[4g(γa2+28Ω2)(γa2+4Ω2)(γa2+36Ω2)+32  g2Ω2(γa4+432Ω4)γσγa(γa2+4Ω2)2(γa2+36Ω2)2]×[4γanγσ˜+Ω2(n1)g2nAn]}1.
π2(1+718γa2g2+83C+2118C+8  g2γa2C2)1,
Ωopt(n)(4γag2nAnnγσ)12(n1).
T2a2a2aa,
S(ω)=1πβ[(γβ/2)Lβ(ωωβ)2+(γβ/2)2(ωωβ)Kβ(ωωβ)2+(γβ/2)2],
na,f=ωβωaLβ.
na,f(1)R{32  g2(γa2Ω2+4iγaΔaΩ24Δa2Ω28Ω4)γσγa(γa+2iΔa)2(γa+2iΔa4iΩ)2(γa+2iΔa+4iΩ)2}.
πnf=na(n)na,f,
π2f(1+83C+8  g2γa2C2)1.
πnfna(n)n1,f(1)+m=2nna(m).
γσa2Lσa[ρ]+γσa2Lσa[ρ],
γσa/σa=2B2g2R[0dτe±iΔaτ(eϕ(τ)1)],
B=exp[120dωJ(ω)ω2coth(βω/2)],
ϕ(t)=0dωJ(ω)ω2[coth(βω/2)cos(ωt)isin(ωt)],
SC=Γ22|a|2Γ2/4+ω2.

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