Cascaded cold atomic ensembles in a diamond configuration as a spectrally entangled multiphoton source
H. H. Jen




sappyjen@gmail.com
Institute of Physics, Academia Sinica, Taipei 11529, Taiwan





    We theoretically investigate the spectral entanglement of a multiphoton source generated from the cascade emissions in the cascaded cold atomic ensembles. This photon source is highly directional, guaranteed under the four-wave mixing condition, and is also highly frequency-correlated due to finite driving pulse durations and superradiant decay constants. We utilize Schmidt decomposition to study the bipartite entanglement of the biphoton states projected from the multiphoton ones. This entropy of entanglement can be manipulated by controlling the driving parameters and superradiant decay rates. Moreover the projected biphoton states in the cascaded scheme can have larger entanglement than the one produced from only one atomic ensemble, which results from larger capacity in multipartite entanglement. This cascaded scheme enables a multiphoton source useful in quantum information processing. It also allows for potential applications in multimode quantum communication and spectral shaping of high-dimensional continuous
    frequency entanglement. 
    Cascaded cold atomic ensembles in a diamond configuration as a spectrally entangled multiphoton source
        H. H. Jen
        December 30, 2023
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    § INTRODUCTION
    
    
    Quantum computation and quantum information processing <cit.> promise to outperform classical implementations for efficient algorithm, secure communication <cit.>, and genuine teleportation of quantum states <cit.>. This superiority of quantumness even envisions a quantum network or quantum internet <cit.> which links various quantum systems to process tasks that are intractable in classical regimes. These quantum systems include, to name a few, photonic qubits, trapped ions, atomic ensembles, and solid state systems <cit.>, in which unfortunately both strengths and weaknesses coexist <cit.>. The fact that there is no prefect quantum system (high efficiency, long coherence time, strong coupling, and scalability, for example) demands an integrability of the well-controlled interfaces between these quantum systems. 
    
    A good quantum interface <cit.> involves an efficient generation, distribution, and storage of quantum information. One manifestation of these functionalities is long-distance quantum communication <cit.> using cold atomic ensembles for a quantum repeater <cit.>. The building blocks for this quantum repeater protocol therefore rely on a generation of light-matter entanglement <cit.> and storage of it <cit.> using Λ-type atomic configurations, making the atomic ensemble (mostly alkali metals) a good candidate for quantum network. However one drawback for the fiber-based quantum information transmission is the attenuation loss for D line transitions in alkali metals. To reach a minimal-loss optical fiber transmission, telecommunication (telecom) bandwidth from diamond-type atomic configurations <cit.> serves the purpose and allows for an optimal operation in long-distance quantum communication. 
    
    In addition to the advantage of the telecom bandwidth in fiber transmission, continuous frequency entanglement of cascade emissions from diamond-type atomic transitions can be generated <cit.> and spectrally shaped <cit.> to create a highly entangled biphoton source. This high communication capacity in continuous variables <cit.> is also present in various degrees of freedom, for example, the transverse momentum <cit.>, space <cit.>, time <cit.>, and orbital angular momenta of light <cit.>. In the perspective of spectral shaping, in Ref. <cit.>, we have investigated the frequency-entangled two-photon state in the scheme of multiplexed cold atomic ensembles. This two-photon state is generated from the cascade emissions in the diamond-type atomic configuration. Its upper and lower transitions (also denoted as signal and idler) are respectively in the telecom and infrared bandwidths. In the multiplexed scheme, a multiple common excitation pulses are applied simultaneously to the respective atomic ensembles along with individually controlled frequency shifters for the cascade emissions. Under the condition of weak excitations as in the quantum repeater protocol <cit.>, we effectively create a biphoton state with additive spectral functions where individual frequency shifts for the signal and idler photons can be manipulated. Therefore the spectral property of this biphoton state can be shaped in either modifying the central frequencies or controlling the phases of the photons <cit.>. We further analyze the entropy of entanglement of the multiplexed two-photon source by Schmidt decomposition, which grows as the number of multiplexed ensembles increases. 
    
    Here in contrast we propose to generate a multiphoton source out of the cascade emissions from the cascaded cold atomic ensembles. We take a two-photon source of quantum-correlated signal and idler photons emitted from a diamond-type atomic configuration, as an initial seed for multiphoton generation in the cascaded atomic ensembles. This is motivated by the multiphoton generation from spontaneous parametric down conversion (SPDC) where multiple cascaded nonlinear crystals are pumped sequentially <cit.>. Similar to the sequential pumping scheme in SPDC, we can take the infrared (telecom) photon of the biphoton source along with a telecom (infrared) driving field to generate another spontaneously emitted cascade emissions satisfying the four-wave mixing condition. In this way we enable a k-photon source with m infrared and n telecom photons fulfilling k=m+n. Since the last cascaded atomic ensemble always produce two photons in the infrared and telecom bandwidths respectively, at least one infrared or telecom photon is generated. However if the infrared and telecom photons can be converted back and forth with each other <cit.>, we can have arbitrary m and n. Furthermore, we investigate its entanglement property in continuous frequency spaces. Our studies allow for an alternative setting for multiphoton generation from cold atomic ensembles, and demonstrate spectral shaping to control the entanglement of this multiphoton source <cit.>, advancing the development of multimode quantum communication. 
    
    In this paper, we first discuss the Hamiltonian and four-wave mixing condition in Sec. II, and propose to generate a multiphoton source in the scheme of cascaded atomic ensembles in Sec. III. Then in Sec. IV we study the bipartite entanglement properties of the multiphoton source, and we conclude in Sec. V.
    
    
    
    
    § CASCADE EMISSIONS AND FOUR-WAVE MIXING CONDITION
    
    
    
    We consider a Rb atomic ensemble (AE) with a diamond-type configuration shown in the inset of Fig. <ref>(a). The correlated cascade emissions of signal and idler photons (â_s(i)^† as shorthands for â_k_m,λ_m^† below) are created by pumping the atomic system with two classical fields. With dipole approximation of light-matter interactions and rotating-wave approximation (RWA) <cit.>, we express the Hamiltonian in the interaction picture using the same notation of <cit.>,
    
        V_ I   =   -∑_m=1,2Δ_m∑_μ=1^N|m⟩_μ⟨ m|-∑_m=a,b(Ω_m/2P̂_m^†+ h.c.)
              -i∑_m=s,i{∑__̨m,λ_mg_mâ__̨m,λ_mQ̂_m^† e^-iΔω_m t- h.c.},
     
    where we set ħ=1 for simplicity, and denote λ_m as the polarizations of photons. The collective dipole operators are defined as P̂_a^†≡∑_μ|1⟩_μ⟨ 0|e^i_̨a·_̊μ, P̂_b^†≡∑_μ|2⟩_μ⟨ 1|e^i_̨b·_̊μ, Q̂_s^†≡∑_μ|2⟩_μ⟨ 3|e^i_̨s·_̊μ, and Q̂_i^†≡∑_μ|3⟩_μ⟨ 0|e^i_̨i·_̊μ, with Rabi frequencies Ω_a(b) for two pump fields. Central frequencies and wavevectors of these four fields are ω_a(b),s(i) and _̨a(b),s(i) respectively. Signal and idler photon coupling constants are g_s(i) where we have absorbed (ϵ__̨m,λ_m·d̂_m^*) into g_s(i) for concise expressions. ϵ__̨m,λ_m is the polarization direction of the quantized bosonic fields â__̨m,λ_m, and d̂_m is the unit direction of the dipole operators. The detunings are Δ_1=ω_a-ω_1 and Δ_2=ω_a+ω_b-ω_2, and for later convenience we define Δω_s≡ω_s-ω_2+ω_3-Δ_2 and Δω_i≡ω_i-ω_3 with the atomic level energies ω_1,2,3. The upper level |2⟩ allows for a telecom wavelength within 1.3-1.5 μm <cit.> if 6S_1/2, 7S_1/2, or 4D_3/2(5/2) levels are considered.
    
    With the Hamiltonian of Eq. (<ref>), we can construct self-consistent Schrödinger equations assuming there is only one atomic excitation <cit.>. This is valid when weak and large detuned excitation pulses are considered, satisfying |Δ_1,2|≫|Ω_a,b|. After adiabatically eliminating the atomic levels |1⟩ and |2⟩ in the coupled Schrödinger equations, we derive the probability amplitude of the biphoton state |1__̨s,1__̨i⟩<cit.>,
    
        D_s,i(t)   =    g_i^∗g_s^∗∑_μ=1^Ne^iΔ·̨_̊μ∫_-∞^t∫_-∞^t^'dt^''dt^'
        e^iΔω_it^'e^iΔω_st^''
              × b(t^'')e^(-Γ_3^ N/2+iδω_i)(t^'-t^''),
    
    where b(t)=Ω_a(t)Ω_b(t)/(4Δ_1Δ_2), resulting from the required adiabatic conditions for driving pulses. Four-wave mixing (FWM) condition ∑_μ=1^N e^iΔ·̨_̊μ represents the phase-matching condition when Δ$̨=_̨a+_̨b-_̨s-_̨i→0in the limit of large number of atomsN, and guarantees to create a highly correlated photon pair. Furthermore the idler photon is superradiant <cit.> thatΓ_3^ N=(Nμ̅+1)Γ_3quantifies a superradiant decay rate in the level of|3⟩with an intrinsic decay rateΓ_3. The geometrical constantμ̅<cit.> relates to the shape of the AE, and the associated collective frequency shift <cit.> is denoted asδω_i. This reflects the nature of collective radiation <cit.> due to induced dipole-dipole interactions in the dissipation <cit.>. Note that a complete description of the collective frequency shift requires non-RWA terms in the Hamiltonian <cit.>.
    
    This biphoton state|1__̨s,1__̨i⟩has two main features. One is the strong directionality of the emitted photons, which is determined by FWM. In thermodynamic limit of AE, we haveΔ$̨=0. If counter-propagating excitations <cit.> are used as in Fig. <ref>(a) with excitation and photon emission angles denoted as θ_a,b,s,i, we can derive the condition for the emitted angles as 
          λ_b/λ_a(cosθ_a-cosθ_i)=cosθ_b-cosθ_s,
          λ_b/λ_a(sinθ_a+sinθ_i)=sinθ_b+sinθ_s,
    where λ_a,b are the wavelengths of the excitation fields, and we have assumed |_̨i|=|_̨a| and |_̨s|=|_̨b|. Since θ_a,b are given when the excitations are applied, θ_s,i can be decided from Eq. (<ref>). To have some estimate for the angles, we have (θ_i,θ_s) = (4.9^∘,9.9^∘) when we set (θ_a,θ_b) =(5^∘,10^∘) and λ_b/λ_a=2. For the same ratio of the wavelengths, we have (θ_i,θ_s) = (7.9^∘,13.9^∘) when we set (θ_a,θ_b) =(4^∘,10^∘). If in radians that θ_a,b,s,i≪1 and setting θ_b=2θ_a, we derive θ_s (≈θ_b) ≈2θ_i, which indicates that the signal and idler photons are emitted with corresponding excitation angles of θ_b and θ_a respectively, and they follow the directions almost tangent to the long axis of AE. This direction is preferential in experiments, which allows for strong light-matter couplings. 
    
    The other feature is that the biphoton state is probabilistic. According to Eq. (<ref>), we can express the complete and normalized state as
    
    |Ψ⟩=1/√(1+|D_s,i|^2)|0^⊗ N⟩[| vac⟩ + D_s,i|1__̨s,1__̨i⟩],
    
    which involves N atomic ground with vacuum and biphoton states respectively. Since |D_s,i|≪1, most of the time AE does not generate any photons. Therefore experimentally it requires repeated excitations until the biphoton state is created, which can be confirmed via photon detections. The degree of correlation of the photons (second-order correlation function for example) can be measured as well by a conditional detection <cit.> of the idler photon after the signal one is detected. Below and throughout the paper, we focus on the effective biphoton and multiphoton states, therefore the normalization for these effective states is neglected. Due to their probabilistic feature, these effective states can be confirmed only via conditional detections or post selections.
    
    Specifically we use normalized Gaussian pulses where Ω_a(b)(t)=[√(π)τ_a(b)]^-1Ω̃_a(b)e^-t^2/τ_a(b)^2 with the pulse areas Ω̃_a,b and pulse widths τ_a(b). Considering the long time limit that D_s,i(t→∞), we derive the probability amplitude D_si after inserting the Gaussian forms into Eq. (<ref>) and redefining Δω_s(i) as Δω_s-δω_i and Δω_i+δω_i respectively, which reads
    
        D_si(Δω_s,Δω_i)=Ω̃_aΩ̃_b g_s^*g_i^*∑_μ=1^N e^iΔ·̨_̊μ/4Δ_1Δ_2√(2π)√(τ_a^2+τ_b^2)f(ω_s,ω_i).
    
    The spectral function of the biphoton state is
    
        f(ω_s,ω_i)=e^-(Δω_s+Δω_i)^2τ_eff^2/8/Γ_3^ N/2-iΔω_i,
    
    where τ_eff≡√(2)τ_aτ_b/√(τ_a^2+τ_b^2). This spectrally correlated biphoton state involves a Gaussian weighting modulated by a Lorentzian. The spectral function is most significant when Δω_s≈-Δω_i within the spectral range of 1/τ_eff. More entangled biphoton state can be made with an increasing optical density of the AE or longer pulses <cit.>, making f(ω_s,ω_i) less factorizable aligning on the axis of Δω_s=-Δω_i<cit.>. In the next section we propose to use this two-photon source as a seed to create multiphoton states in the scheme of cascaded AEs.
    
    
    
    
    § MULTIPHOTON STATES FROM THE CASCADE EMISSIONS IN THE CASCADED SCHEME
    
    
    Before we investigate the multiphoton states generated from the cascaded atomic ensembles, we note that throughout the paper, we focus only on the effective pure states generation. In general for an open quantum system, the system becomes mixed states inevitably <cit.> due to the interactions with the reservoir or the imperfections in experiments. The mixed states can be interpreted as statistical mixtures of pure states, which can be expressed as a density operator, ρ̂=∑_k p_k|ψ_k⟩⟨ψ_k| with the constraints of ∑_kp_k=1 and 0<p_k≤1. ρ̂ becomes a pure state when one of p_k's is unity, therefore this density operator shows a general representation of any quantum system. For example of the spontaneous emission process in a two-level atom (|0⟩ and |1⟩), it acts like an amplitude damping <cit.> to the atomic states. In a description of generalized amplitude damping, the atom evolves to the stationary mixed state ρ_∞=p|0⟩⟨ 0|+(1-p)|1⟩⟨ 1| with some probability p. Similarly, ρ_∞ can describe a loss of photon due to attenuation or decoherence from environment if |0⟩ and |1⟩ are denoted as vacuum and one-photon states respectively. 
    
    In general, a spontaneous emission is a random process <cit.> in time and space, where the emitted direction has a uniformly dΩ̂/(4π) distribution in a solid angle of dΩ̂=sinθ dθ dϕ in spherical coordinates. In contrast to this random process in space, the biphoton state from the cascade emissions in a diamond configuration under FWM condition is highly directional and correlated, which thus makes the effective state |1__̨s,1__̨i⟩ valid if photon loss or other noisy channels that deteriorate its fidelity can be neglected. Although this limits our investigation to the pure states, we note that an entanglement distillation or purification procedure <cit.> can be applied to the mixed entangled states by local operations and classical communication. Therefore the pure state picture we focus here can still give insights to quantum information processing with the effective multiphoton state presented in this work. Other discussions of inseparability criterion, bound entanglement, or multipartite entanglement for mixed states can be referred to Refs. <cit.>. For measuring entanglement, it does not imply Bell nonlocality except for pure states, and quantifying entanglement for bipartite mixed state involves various measures not agreeable to the partial von Neumann entropy <cit.>. That being the case, our investigations using Schmidt decomposition to quantify the entanglement of bipartite pure states in Sec. IV provide an upper bound to the entanglement measure.
    
    To generate a multiphoton state from the cascade emissions, we propose to couple one of the two photons with other AE along with a corresponding pump field, such that (k+1)-photon source can be created from a k-photon state. As demonstrated in Fig. <ref>, three and four-photon states are created in the cascaded scheme. AE' and AE” are two other atomic ensembles used to generate correlated photons â_s',i'^† and â_s”,i”^† respectively under the FWM condition. Since the initial seed of two-photon source is highly entangled in frequency space, the generated multiphoton source is expected to be also entangled in continuous frequency spaces. 
    
    The generated multiphoton spectral functions can be derived by products of the two-photon spectral ones of Eq. (<ref>) and invoke the condition for central frequencies of the annihilated photon, that is ω_i=ω_s'+ω_i'-ω_b for example in the case of AE' in Fig. <ref>(a). This condition also satisfies the energy conservation as if four fields are plane waves. Below we demonstrate three and four-photon state spectral functions, and their entanglement properties will be discussed in the next section.
    
    
    
     §.§ Three-photon state
    
    
    As in Fig. <ref>(a) which we denote a route B1 for the three-photon generation, the effective three-photon state involving two signal and one idler photons can be expressed as
    
    |Ψ⟩_3,B1=f_3,B1â^†_sâ_s'^†â_i'^†|0⟩.
    
    We derive the dimensionless and effective spectral function by multiplying a typical biphoton state spectral function of Eq. (<ref>) with the one generated by a plane-wave idler photon â_i^† and a driving field Ω_b in AE',
    
    f_3,B1/Γ_3^2=e^-(Δω_s+Δω_i)^2τ_eff^2/8/Γ_3^ N/2-iΔω_ie^-(Δω_s'+Δω_i')^2τ_b^2/4/Γ_3^ N/2-iΔω_i',
    
    where τ_b appears when we let τ_a→∞ in τ_eff. The above becomes, after setting ω_i=ω_s'+ω_i'-ω_b and using Δ_2=ω_a+ω_b-ω_2,
    
    f_3,B1/Γ_3^2   =   e^-(Δω_s+Δω_s'+Δω_i'+Δ_a3+δω_i)^2τ_eff^2/8/Γ_3^ N/2-i(Δω_s'+Δω_i'+Δ_a3+δω_i)
       ×   e^-(Δω_s'+Δω_i')^2τ_b^2/4/Γ_3^ N/2-iΔω_i'.
    The extra frequency shift Δ_a3≡ω_a-ω_3 in the above can be removed along with δω_i by applying an external Zeeman field, such that we can simplify Eq. (<ref>) as
    f_3,B1/Γ_3^2=e^-(Δω_s+Δω_s'+Δω_i')^2τ_eff^2/8/Γ_3^ N/2-i(Δω_s'+Δω_i')e^-(Δω_s'+Δω_i')^2τ_b^2/4/Γ_3^ N/2-iΔω_i'.
    
    
    Similarly, if we annihilate signal photon of the initial two-photon seed, we have alternatively the effective three-photon state as (B2 to denote the second route for three-photon state generation, involving two idler and one signal photons)
    
    |Ψ⟩_3,B2=f_3,B2â^†_iâ_s'^†â_i'^†|0⟩,
    
    where its dimensionless spectral function is [after removing the extra frequency shift as in deriving Eq. (<ref>)]
    f_3,B2/Γ_3^2   =   e^-(Δω_i+Δω_s'+Δω_i')^2τ_eff^2/8e^-(Δω_s'+Δω_i')^2τ_a^2/4/(Γ_3^ N/2-iΔω_i)(Γ_3^ N/2-iΔω_i').
    
    Note that the overall constants in spectral functions do not make an effect on spectral distributions, thus we regularize them in dimensionless forms. The overall constants however determine the generation rates which are small in general since weak and large detuned excitations are used.
    Eqs. (<ref>) and (<ref>) are two of the main results in this subsection. Obviously these three-photon states are entangled in frequency spaces, meaning there is no possible ways to factorize these spectral functions. Meanwhile |Ψ⟩_3,B1 differs from |Ψ⟩_3,B2 specifically in a modulated Lorentzian function on signal distribution Δω_s'. This results from the annihilation of the idler photon in the route B1, which replaces the Lorentzian with the correlated signal and idler photon pair. Also different timescales τ_b(a) in routes B1(2) respectively for a joint Gaussian profile of the photon pair â_s'^†â_i'^† suggest an independent control over their spectral functions by varying pulse durations.
    
    Below we derive the spectral function for the four-photon state of Fig. <ref>(b) using the three-photon states in the cascaded scheme of Fig. <ref>(a), and also the other possible spectral functions in alternative routes.
    
    
    
    
     §.§ Four-photon state
    
    
    Here using the same fashion to generate three-photon states, the four-photon ones can be also created in the cascaded scheme. As in Fig. <ref>(b), the idler photon â_i'^† emitted from AE' is annihilated with an extra coupling field Ω_b in AE” to generate a newly correlated pair of photons â_s”^†â_i”^†. The effective four-photon state with three signal and one idler photons becomes
    
    |Ψ⟩_4,C1=f_4,C1â^†_sâ_s'^†â_s”^†â_i”^†|0⟩,
    
    where again the dimensionless spectral function can be derived as 
    f_4,C1/Γ_3^3=e^-(Δω_s+Δω_s'+Δω_s”+Δω_i”)^2τ_eff^2/8e^-(Δω_s'+Δω_s”+Δω_i”)^2τ_b^2/4e^-(Δω_s”+Δω_i”)^2τ_b^2/4/[Γ_3^ N/2-i(Δω_s'+Δω_s”+Δω_i”)][Γ_3^ N/2-i(Δω_s”+Δω_i”)](Γ_3^ N/2-iΔω_i”).
    
    Other four possible routes to generate four-photon states are demonstrated in Appendix A. In principle there should be six different spectral functions where three of them are from routes B1 and B2 respectively. Since there is one spectral function which is symmetric to each other in respective routes, making a total of five possible spectral functions in our cascaded scheme. This symmetric four-photon state is generated by annihilating â_s(i)^† from the routes B1(2) respectively along with the coupling fields Ω_a(b). The symmetry is satisfied when â_s”↔â_s' and â_i”↔â_i', and this effective four-photon state with two signal and two idler photons is
    
    |Ψ⟩_4,C3=f_4,C3â^†_s'â_i'^†â_s”^†â_i”^†|0⟩,
    
    with the spectral function
    f_4,C3/Γ_3^3=e^-(Δω_s'+Δω_i'+Δω_s”+Δω_i”)^2τ_eff^2/8e^-(Δω_s'+Δω_i')^2τ_a^2/4e^-(Δω_s”+Δω_i”)^2τ_b^2/4/[Γ_3^ N/2-i(Δω_s”+Δω_i”)](Γ_3^ N/2-iΔω_i')(Γ_3^ N/2-iΔω_i”).
    
    These spectral functions have a common weighting of Gaussian envelope involving four photon frequencies, indicating to possess a genuine k-party entanglement <cit.> for k-photon source in our proposed cascaded scheme. This genuine multiphoton entanglement means to exclude any possible bipartite splittings or groupings <cit.>. For example of Eq. (<ref>) from the route C3, two groups of photons â_s',i'^† and â_s”,i”^† respectively would be able to be factorized if this common weighting of e^-(Δω_s'+Δω_i'+Δω_s”+Δω_i”)^2τ_eff^2/8 is absent. Multipartite entanglement is still an ongoing research even for pure states we consider here, therefore to give an intuitive study of entanglement property of our proposed multiphoton sources, in the next section we introduce Schmidt decomposition to investigate their bipartite entanglements in continuous frequency spaces.
    
    
    
    
    § ENTROPY OF ENTANGLEMENT
    
    
    To gain insights of the spectral entanglement in the multiphoton sources from the cascaded scheme of AEs, we study the bipartite entropy of entanglement in these sources. The multiphoton states can be projected or collapsed to the effective biphoton state by conditional measurements, such that we can quantify and analyze their spectral entanglement by Schmidt decomposition <cit.>. The bipartite entropy of entanglement is S=-∑_n=1^∞λ_n log_2λ_n<cit.>, intended for pure states. Schmidt eigenvalues λ_n determine the probabilities of nth mode functions, with which the state can be expressed as ∑_n√(λ_n)b̂_n^†ĉ_n^†|0⟩, for two effective photon operators b̂_n and ĉ_n associated with mode functions ψ_n and ϕ_n respectively. 
    
    Schmidt decomposition is done numerically as solving the eigenvalue problems of the one-photon spectral kernels <cit.>, 
    ∫ K_1(ω,ω^')ψ_n(ω^')dω^'   =   λ_nψ_n(ω),
    ∫ K_2(ω,ω^')ϕ_n(ω^')dω^'   =   λ_nϕ_n(ω),
    
    where 
    
    K_1(ω,ω^')    ≡   ∫ f(ω,ω_1)f^∗(ω^',ω_1)dω_1, 
    
    K_2(ω,ω^')    ≡   ∫ f(ω_2,ω)f^∗(ω_2,ω^')dω_2. 
    
    The spectral function f comes from the effective biphoton state (a pair of â_s^† and â_i^† for example)
    
    |Ψ⟩   =   ∫ f(ω_s,ω_i)â_s^†(ω_s)â_i^†(ω_i)|0⟩ dω_s dω_i,
       =   ∑_n√(λ_n)b̂_n^†ĉ_n^†|0⟩,
    
    with two mode functions in the Schmidt bases, determining the effective photon operators,
    b̂_n^†   ≡   ∫ψ_n(ω_s)â_s^†(ω_s)dω_s,
    ĉ_n^†   ≡   ∫ϕ_n(ω_i)â_i^†(ω_i)dω_i.
    
    The above decomposition has been used to quantify the spectral entanglement of two-photon source from parametric down conversion <cit.> and cascade emissions <cit.>, and in spectral shaping of biphoton state in a multiplexed scheme <cit.>.
    
    
    
    
     §.§ S in three-photon states
    
    
    For bipartite entanglement property of a three-photon state with the spectral function f_3,B1, three possible projected biphoton states can be derived by either annihilating photons â_s, â_s', or â_i'. In the insets of (a) and (b) of Fig. <ref>, we show the spectral distributions projecting out â_s and â_s' respectively by setting Δω_s(s')=0. The entropy of entanglement S is 2.37 and 0.15 respectively, indicating a more entangled source in the former case. The huge difference in these two projections results from a removal of an entangling function e^-(Δω_s'+Δω_i')^2τ_b^2/4 of Eq. (<ref>) when projecting out â_s'. This Gaussian function supposes to entangle photons â_s' and â_i' in infinite signal and idler frequency spaces. 
    
    
    
    A modulating Lorentzian function of idler photon however confines its spectral distribution into a finite bandwidth of ∼Γ_3^N. Projecting out a signal photon â_s' thus collapses the entangling function into e^-(Δω_i')^2τ_b^2/4, rendering another finite bandwidth of the idler photon with FWHM (full-width at half-maximum) ∼4√(ln 2)/τ_b. In essence these factorizable idler functions, Gaussian and Lorentzian, rotate the spectral distribution toward the axis Δω_i'=0, as we show in Fig. <ref>(b). The third possible projected biphoton state is to project out â_i', which has a similar spectral property to the one annihilating â_s'.
    Likewise in Fig. <ref>, we show the results for the spectral function f_3,B2. The insets of (a) and (b) demonstrate the spectral functions of the projected biphoton states with annihilated photons â_i and â_s' respectively. The entropy of entanglement S is 1.79 and 0.09 respectively, again indicating a more entangled source in the former case. Fig. <ref>(b) is less entangled due to a removal of the signal photon dependence Δω_s' in the entangling function e^-(Δω_s'+Δω_i')^2τ_a^2/4, similar to the cases in Fig. <ref>. The reason why Fig. <ref>(a) is less entangled than Fig. <ref>(a) is that an extra entangling Lorentzian function [Γ_3^ N/2-i(Δω_s'+Δω_i')]^-1 present in Eq. (<ref>). Fig. <ref>(b) shows a relatively smaller entangled biphoton source, resulting from a spectral function with more aligned distributions on Δω_i(i')=0 compared to Fig. <ref>(b), in a somewhat distorted fashion of Fig. <ref>(b). This is due to two factorizable Lorentzian functions of idler photons â_i and â_i' in frequencies Δω_i(i') of Eq. (<ref>) when projecting out the signal photon â_s'.
    
    
    Since the decay constants of two atomic ensembles, AE and AE', are not necessarily the same, we further investigate the spectral property with different Γ_3^N and Γ_3^N'. As an example from Eq. (<ref>) and projecting out â_s, we have
    f_3→ 2,B1/Γ_3^2=e^-(Δω_s'+Δω_i')^2τ_eff^2/8/Γ_3^ N/2-i(Δω_s'+Δω_i')e^-(Δω_s'+Δω_i')^2τ_b^2/4/Γ_3^N'/2-iΔω_i'.
    
    In Fig. <ref> we show the results for different decay constants compared to Fig <ref>(a). The entropy of entanglement S for the insets (a) and (b) are 3.9 and 0.89 respectively. The small Γ_3^N in (a) provides a sharp distribution along the highly entangled axis Δω_s'=-Δω_i', therefore making this projected spectral function more entangled. In contrast the small Γ_3^N' in (b) limits the factorizable Lorentzian idler distribution â_i', allowing for a squeezed distribution in Δω_i' and a less entangled biphoton source. 
    
    In this subsection we have demonstrated a rich spectral property of the projected biphoton state from the three-photon source in the cascaded scheme. The entanglement property can be very different depending on how the counterpart of the source is collapsed. In the perspective of generating a more entangled photon source, the spectral function from route B1 serves better than B2, meanwhile a less(more) superradiant decay constant Γ_3^N(Γ_3^N') of AE(AE') in the route B1 is favorable for this purpose.
    
    
    
    
     §.§ S in four-photon states
    
    
    For four-photon states, there are in general six possible projections to biphoton ones. Though of plenty of possible projected biphoton states for a total of five spectral functions demonstrated in Sec. III.B and Appendix A, there are only a few of qualitatively different spectral functions. As an example, we choose the spectral function f_4,C1 of Eq. (<ref>). Based on the observation from projected three-photon states in the previous subsection, we expect that a projection of â_i” or â_s” in f_4,C1 provides less entangled multiphoton states. For comparisons, in Fig. <ref> we show two projections of f_4,C1 in three-dimensional isosurface plots which provide a qualitative distinction in the projected spectral functions. In Fig. <ref> (a) and (b) we set Δω_s(i”)=0 respectively, which show a tilted and an axial distribution. The tilted distribution potentially allows for a more entangled multiphoton state while we note that further projecting out photons â_i” and â_s respectively in (a) and (b) gives the same spectral function of the projected biphoton states.
    
    
    
    
    Less entangled biphoton states from f_4,C1 can be derived by projecting out a pair of photons (â_s”(i”), â_s') or (â_s”, â_i”). These projections basically collapse f_4,C1 into a single Gaussian function of e^-[Δω_s”(i”)]^2τ_b^2/4 or e^-(Δω_s')^2τ_b^2/4, which again confines its spectral distribution without entangling with other photons. As another example of disentangling photons, we note that the spectral function f_4,C3 of Eq. (<ref>) allows for the least entangled biphoton source. By annihilating a pair of photons of either â_s'(i') or â_s”(i”), we collapse f_4,C3 into two single Gaussian functions. Choosing Δω_s',s”=0 in f_4,C3, in Fig. <ref> we show the collapsed spectral distribution of two idler photons, which has extremely low entropy of entanglement S=0.028. The eigenvalues λ_n are plotted in a logarithmic scale, showing an abrupt decrease of Schmidt numbers. The first two eigenvalues are 0.997 and 0.0028, occupying most of the modes (up to 99.98%) in this biphoton source. In Fig. <ref>(b), we show their mode probability densities. As expected the FWHM of the mode probability density follows the spectral distribution at the cut of Δω_i'(i”)=0. The first mode of the idler â_i” has a shortened linewidth than the one of â_i' due to a squared Lorentzian function of [Γ_3^ N/2-i(Δω_i”)]^-2 in f_4,C3 with Δω_s”=0.
    
    We can also manipulate S by modifying the driving pulse durations τ_a and τ_b. When we increase them to τ_a=τ_b=1Γ_3^-1 in the projected  f_4,C3 as in Fig. <ref>, we find S becomes 0.13, which allows for a more entangled source since the entangling Gaussian function e^-(Δω_i'+Δω_i”)^2τ_eff^2/8 has a tighter photon correlation on the distribution axis Δω_i'=-Δω_i”. On the other hand when we set τ_a=0.25Γ_3^-1 and τ_b=1Γ_3^-1 in the asymmetric setting, we find S becomes 0.023, which is even smaller than the symmetric case in Fig. <ref>. This reflects the competition of this entangling and two other disentangling Gaussian functions. Though τ_eff still increases in the asymmetric setting, a more confined distribution from τ_b limits the overall projected spectral function distribution, thus decreasing S.
    
    
    
    
    
    § DISCUSSION AND CONCLUSION
    
    
    The multiphoton states generated from the cascade emissions in the scheme of cascaded AE are advantageous in the perspective of well-controlled AE preparations. Large-scale implementation of such cascaded scheme is feasible when preparing the atoms either in free-space, multiplexed setting <cit.>, or optical lattices <cit.>, but will suffer from the low generation rate. Low generation rate results from the weak excitations and few-photon level of emissions. Also low light is more subject to propagation attenuation, making the detection of photons difficult. However this can be overcome by raising the repetition rates in excitations. Finite spectral windows for collapsing the multiphoton spectral functions are not considered here. But we expect of no significant modifications in their spectral properties qualitatively, where essentially the finite spectral window of projection averages out the spectral distribution. A gate time of several hundreds of nanoseconds in the photon counting device would be enough to neglect the effect from finite detection windows.
    
    The other merit of the multiphoton source in the cascaded scheme is flexibility to entangle the photons either in signal or idler frequencies, making a repertoire of many possible entangled multiphoton sources. The signal frequency is best for optical fiber transmission while the idler one is preferential in quantum storage. In our scheme a highly entangled photon source can be generated by increasing the excitation pulse durations in the symmetric setting or tighten the spectral distribution on the axis Δω_s+Δω_i=0 that conserves the biphoton energy. By appropriately projecting out the photon  counterparts in the multiphoton spectral function, the entropy of entanglement and the spectral mode functions can be modified and manipulated to serve different purposes requiring either pure or entangled states.
    
    For more entangled biphoton source, we have demonstrated a larger entropy of entanglement S=2.37 or 1.79 from the projected three-photon states compared to 1.33<cit.> from just one AE under the same driving conditions. This shows that a more entangled biphoton source can be generated from a multiphoton source with even larger capacity in the genuine multipartite entanglement. Our cascaded scheme here can also combine with the multiplexed one <cit.>. The multiplexed scheme manipulates the spectral property of the biphoton state by modifying their central frequencies or phases <cit.>. Its maximal entropy of entanglement S_M can be described by S+S_d. S is the entropy of entanglement from one AE while S_d≡log_2 (N_ MP) with N_ MP, the number of multiplexed AEs. Therefore the multiplexed scheme increases s_d as N_ MP increases. Meanwhile the cascaded scheme enables a multiphoton source from sequentially-coupled AEs using diamond configurations. Its bipartite entanglement can be extracted from the biphoton states collapsed from the multiphoton ones. This way the cascaded scheme modifies and manipulates S effectively, making the combination of these two schemes a full control of S_M. In addition to the spectral shaping of the cascade emissions by modifying driving conditions <cit.> or multiplexing AEs <cit.>, the cascaded scheme here provides an alternative route to spectrally shape an even more entangled biphoton source, thus overcomes the limitation of S in the multiplexed scheme.
    
    In conclusion, we propose a cascaded scheme to generate a multiphoton source from the cascade emissions of the atomic ensembles. Highly spectrally entangled (k+1)-photon source can be created using k-photon state as the seed along with an appropriate driving field either in the lower or upper transition of the diamond configuration. Under the FWM condition, this highly directional and frequency-correlated photon source are useful for quantum information processing and applicable to multimode quantum communication. Furthermore such entangled multiphoton source can be spectrally shaped with controllable driving conditions and ensemble properties (for example atomic density and geometry), which could potentially be implemented in quantum spectroscopy <cit.>.
    
    
    
    § ACKNOWLEDGMENTS
    
    This work is supported by the Ministry of Science and Technology (MOST), Taiwan, under Grant No. MOST-103-2112-M-001-011.
    
    
    
    
    § OTHER ROUTES FOR FOUR-PHOTON STATE GENERATION
    
    In Sec. III.B, we have investigated four-photon state generation in the cascaded scheme, and demonstrated the first route C1. Other four possible routes to generate four-photon states can use three-photon states |Ψ⟩_3,B1 and |Ψ⟩_3,B2 as seeds. The route C2 below is to annihilate â_s'^† of |Ψ⟩_3,B1 with an extra coupling field Ω_a in the third AE” to generate a newly correlated pair of photons â_s”^†â_i”^†. The effective state is |Ψ⟩_4,C2=f_4,C2â^†_sâ_i'^†â_s”^†â_i”^†|0⟩ involving two signal and two idler photons with the spectral function
    f_4,C2/Γ_3^3=e^-(Δω_s+Δω_i'+Δω_s”+Δω_i”)^2τ_eff^2/8e^-(Δω_i'+Δω_s”+Δω_i”)^2τ_b^2/4e^-(Δω_s”+Δω_i”)^2τ_a^2/4/[Γ_3^ N/2-i(Δω_i'+Δω_s”+Δω_i”)](Γ_3^ N/2-iΔω_i')(Γ_3^ N/2-iΔω_i”).
    
    The route C3 involves two signal and two idler photons, generated from a symmetric coupling between â_s(i)^† and pump fields Ω_a(b) in the third AE” using three-photon states from routes B1(2) respectively. Its spectral function has been shown in the main paper. For route C4, which annihilates â_i'^† of |Ψ⟩_3,B2 with an extra coupling field Ω_b in the third AE”. The effective state is |Ψ⟩_4,C4=f_4,C4â^†_iâ_s'^†â_s”^†â_i”^†|0⟩ involving two signal and two idler photons with the spectral function
    f_4,C4/Γ_3^3=e^-(Δω_s'+Δω_i+Δω_s”+Δω_i”)^2τ_eff^2/8e^-(Δω_s'+Δω_s”+Δω_i”)^2τ_a^2/4e^-(Δω_s”+Δω_i”)^2τ_b^2/4/(Γ_3^ N/2-iΔω_i)[Γ_3^ N/2-i(Δω_s”+Δω_i”)](Γ_3^ N/2-iΔω_i”).
    
    
    The last route C5 annihilates â_s'^† of |Ψ⟩_3,B2 with an extra coupling field Ω_a in the third AE”. The effective state is |Ψ⟩_4,C5=f_4,C5â^†_iâ_i'^†â_s”^†â_i”^†|0⟩ involving one signal and three idler photons with the spectral function
    f_4,C5/Γ_3^3=e^-(Δω_i+Δω_i'+Δω_s”+Δω_i”)^2τ_eff^2/8e^-(Δω_i'+Δω_s”+Δω_i”)^2τ_a^2/4e^-(Δω_s”+Δω_i”)^2τ_a^2/4/(Γ_3^ N/2-iΔω_i)(Γ_3^ N/2-iΔω_i')(Γ_3^ N/2-iΔω_i”).
    
    99
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