Creating locally interacting Hamiltonians in the synthetic frequency dimension for photons
Luqi Yuan Avik Dutt Mingpu Qin Shanhui Fan Xianfeng Chen


















ALPGEN
EVTGEN
PYTHIA












    ⟨⟩[2]
        
        ^1State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 
        ^2Department of Electrical Engineering, and Ginzton Laboratory,
        Stanford University, Stanford, CA 94305, USA
        ^3School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 
        ^4Collaborative Innovation Center of Light Manipulations and
        Applications, Shandong Normal University, Jinan 250358China
        ^*Corresponding author: yuanluqi@sjtu.edu.cn
        ^†Corresponding author: shanhui@stanford.edu
        ^Corresponding author: xfchen@sjtu.edu.cn
        The recent emerging field of synthetic dimension in photonics
        offers a variety of opportunities for manipulating different
        internal degrees of freedom of photons such as the spectrum of
        light. While nonlinear optical effects can be incorporated into
        these photonic systems with synthetic dimensions, these nonlinear
        effects typically result in long-range interactions along the
        frequency axis. Thus it has been difficult to use of the synthetic
        dimension concept to study a large class of Hamiltonians that
        involves local interactions. Here we show that a Hamiltonian that
        is locally interacting along the synthetic dimension can be
        achieved in a dynamically-modulated ring resonator incorporating
        χ^(3) nonlinearity, provided that the group velocity
        dispersion of the waveguide forming the ring is specifically
        designed. As a demonstration we numerically implement a
        Bose-Hubbard model, and explore photon blockade effect, in the
        synthetic frequency space. Our work
         opens new possiblities
        for studying fundamental many-body physics in the synthetic space
        in photonics, with potential applications in optical quantum
        communication and quantum computation.
        Creating locally interacting Hamiltonians in the synthetic frequency dimension for photons
            Luqi Yuan^1,*, Avik Dutt^2, Mingpu Qin^3, Shanhui Fan^2,†, and Xianfeng Chen^1,4,
            December 30, 2023
        ==========================================================================================
        
        
        
        
        The concept of synthetic dimension enables one to explore higher
        dimensional physics in lower dimensional systems
        <cit.>.
        In photonics <cit.>, synthetic dimension
        can be achieved by exploiting various degrees of freedom of light
        such as its frequency <cit.>, spatial mode
        <cit.>, or orbital angular momentum <cit.>, and by
        specifically controlling the coupling between these degrees of
        freedom <cit.>.  While
        initially explored mostly theoretically
        <cit.>, very
        recently there have been a number of important experimental
        developments, including the first experimental demonstration of
        topological insulators with synthetic dimensions <cit.>,
        the measurement of band structure along the frequency axis of
        light <cit.>, and the control of light spectrum in
        synthetic space <cit.>.
        
        
        Most of the existing theoretical and experimental works on
        synthetic dimension in photonics concerns linear processes without
        photon-photon interactions. Certainly, it would be of interest to
        consider nonlinear systems where there is photon-photon
        interaction. Moreover, since one of the major objectives by going
        into synthetic dimension is to create a platform to study specific
        interacting Hamiltonians that are of physical significance, it
        would be important to develop a strategy to synthesize these
        interacting Hamiltonians. A very large class of interesting
        interacting Hamiltonians have local interactions
        <cit.>. Attempts to achieve such
        locally interacting Hamiltonians in synthetic space and have been
        made in photonics in the geometric angular coordinate
        <cit.> and also in ultracold atoms with spin being the
        internal degree of freedom <cit.>. On the other hand,
        as is known and we will briefly reiterate below, the standard form
        of nonlinear optics typically leads to a form of interaction that
        is nonlocal in the synthetic space
        <cit.>. How to achieve a Hamiltonian with
        local interactions in the synthetic space for photons, thus
        therefore represents an important open theoretical question.
        
        
        In this Letter, we consider a  synthetic frequency dimension of
        light, formed in a ring resonator incorporating a modulator. To
        create photon-photon interaction we consider χ^(3)
        processes in the waveguide forming the ring.  We show that a local
        photon-photon interaction in the frequency dimension can be
        achieved with a careful design of the group velocity dispersion of
        the waveguide [Fig. <ref>(a)]. As a demonstration we show
        that this system can be used to demonstrate a Bose-Hubbard model,
        and achieve photon-blockade effects along the synthetic axis. This
        work here significantly broadens the range of physics phenomena
        that can be studied in photonic synthetic space, and may lead to
        new opportunities in quantum simulations and in the manipulation
        of light.
        
        
        
        
        
        We start with a brief review of the ring resonator under the
        dynamic modulation as shown in Fig. <ref>(b), which
        naturally leads to a synthetic dimension of light along the
        frequency axis. We consider a ring composed by a single-mode
        waveguide with an effective refractive index n_eff,
        and a length l. For simplicity, assuming  zero group velocity
        dispersion in the waveguide, the ring resonator supports resonant
        modes at the resonant frequencies ω_m = ω_0 + m Ω
        where Ω≡ 2π c/ n_effl ≪ω_0 is
        the free-spectral-range (FSR) for the ring. We place a phase
        modulator inside the ring and choose the modulation frequency to
        be equal to the free spectral range. The transmission of light
        passing through the modulator can be described by a time-dependent
        transmission coefficient
        
            T = e^i 2 αcos(Ω t),
        
        where α is the modulation amplitude. We assume that the
        field experiences small changes for each round-trip. Then after
        one round-trip, the change of the field amplitude E_m for the
        m-th resonant mode can be written as
        
            2π/Ω∂ E_m/∂ T≡Δ
            E_m = i α (E_m+1 + E_m-1),
        
        where T is the slow time variable <cit.>. The dynamics of
        Eq. (<ref>) is described by an effective Hamiltonian:
        
            H_0 = - ħ J ∑_m ( a_m^† a_m+1 + h.c.),
        
        where J = αΩ/2π is the modulation strength and
        a_m (a_m^†) is the annihilation (creation) operator for
        the m-th resonant mode. The Hamiltonian (<ref>)
        describes a one-dimensional tight-binding model along the
        synthetic frequency axis [see Fig. <ref>(b)]. With this
        approach, a wide variety of non-interacting Hamiltonian with
        different connectivity and topological properties can be created
        <cit.>. However, there have been far less works on
        creating important interacting Hamiltonians.
        
        
        As an  illustration of the general difficulty as well as our
        approach of creating locally interacting Hamiltonian in the
        frequency synthetic dimension, we aim to synthesize the effective
        Bose-Hubbard Hamiltonian in the synthetic frequency dimension
        
            H_1 = - ħ J ∑_m ( a_m^† a_m+1 + h.c.)
             -  ħ g/2∑_m a_m^† a_m^† a_m a_m ,
        
        where g is the local nonlinear strength. Although nonlinear
        optical phenomena in many photonic materials have been extensively
        studied, creating the Hamiltonian of Eq. (<ref>) in
        the frequency synthetic space is in fact non-trivial
        <cit.>. The introduction of the nonlinearity typically
        leads to long-range interactions over all synthetic lattice sites.
        For example, the dynamically-modulated ring as shown in Fig.
        <ref>(b), with a third-order nonlinear susceptibility
        χ^(3) is described by an  interacting Hamiltonian
        <cit.>:
        
            H_2 = - ħ J ∑_m ( a_m^† a_m+1 + h.c.) -
            ħ g/2∑_m,n,p,q a_m^† a_n^† a_p a_q -
            ħ g/3∑_m,n( a_m^† a_n^3 + h.c.
            ),
        
        as can be derived using the rotating wave approximation. In Eq.
        (<ref>), the second and third terms describe two
        four-wave-mixing effects. The second term describes the
        hyper-parametric oscillation process involving four modes with
        frequencies with ω_m+ω_n = ω_p+ω_q. The
        third term describes the third harmonic generation (THG) process
        with ω_m = 3ω_n. Comparing Eq. (<ref>)
        with Eq. (<ref>), we see that the interacting term
        in Eq. (<ref>) corresponds to the self-phase
        modulation (SPM) process. In Eq. (<ref>), however,
        in addition to the SPM process,  other terms which describe  the
        cross phase modulation (XPM) process, other hyper-parametric
        processes, and the THG process result in   long-range interactions
        between different resonant modes in the ring (see Fig.
        <ref>). While here for illustration purposes we consider a
        specific type of nonlinearity, the observation is in fact rather
        general: standard nonlinearity does not lead to local interactions
        in the synthetic frequency dimension.
        
        
        
        
        
        In order to synthesize the Bose-Hubbard model where the
        interaction is local  along the frequency dimension, we propose a
        ring consisting of two sections of waveguides. The two sections
        have the same χ^(3) nonlinear susceptibility. However, they
        have opposite group velocity dispersion (GVD), as illustrated in
        Fig. <ref>(a). In the linear regime of light propagation,
        for each complete round-trip as light goes around the ring, the
        dispersion effect in the two waveguide sections cancels. Hence
        this ring also features equally spaced resonances along the
        frequency axis, same as a ring with zero GVD, and a
        one-dimensional synthetic frequency dimension can be created in
        this ring by applying the dynamic modulation. On the other hand,
        in each waveguide section, due to the the group velocity
        dispersion, the only phase-matched processes are the SPM and XPM
        processes. Moreover, as we will show below, in any Hilbert space
        with a fixed photon number the Hamiltonian for the XPM process in
        fact reduces to that for the SPM process. We therefore show that
        our design supports an effective Bose-Hubbard Hamiltonian
        (<ref>) along the frequency axis.
        
        
        In the following, we explain our design in details. We first
        consider light propagating inside a single-mode waveguide. The
        evolution of the field amplitude A(z,ω) with the frequency
        ω at the position z in the waveguide can be described as
        <cit.>:
        
            ∂ A(z,ω)/∂ z = -i β(ω)
            A(z,ω),
        
        where β(ω) denotes the propagation wavevector.
        β(ω) can be expanded around the reference frequency
        ω_0, i.e.,
        
            β (ω) - β (ω_0) =
            .dβ/dω|_ω_0(ω-ω_0) +
            1/2.d^2β/dω^2|_ω_0(ω-ω_0)^2 + … .
        
        Here we set β_0 = β (ω_0), v_g =
        (dβ/dω)^-1 is the group velocity, and
        β_2 = d^2β/dω^2 is the GVD around ω_0. By
        neglecting the higher-order terms in Eq. (<ref>),
        one can write the field amplitude of light after it propagates
        through a waveguide with the length L
            A(z=L,ω) = A(z=0,ω) e^-i β_0 L -i
            (ω-ω_0) L/v_g - i β_2
            (ω-ω_0)^2 L/2 .
        
        Note that the GVD induces a frequency-dependent phase delay.
        
        
        
        
        
        
        Next, we explore the light propagates through the waveguide
        section having the positive GVD (β_2 > 0) and  χ^(3)
        as shown in Fig. <ref>(a). For simplicity, we consider
        two incident waves with the frequencies ω_1 and ω_2
        respectively, in the vicinity of the reference frequency
        ω_0. We look for the solution for a general
        four-wave-mixing process where output fields are generated  at
        frequencies ω_3 and ω_4, satisfying the energy
        conservation relation ω_3 + ω_4 = ω_1 + ω_2
        [see Fig. <ref>(a)]. We also assume the narrowband
        limit, i.e., |ω_0 - ω_m|≪ω_0, m=1,2,3,4. The
        resulting coupled-mode equations under the slowly varying
        amplitude approximation are <cit.>
            ∂Ã_3/∂ z =
            iω_0^2/2β_0 c^2χ^(3)Ã_1 Ã_2
            Ã_4^* e^-iΔ k z,
        
            ∂Ã_4/∂ z =
            iω_0^2/2β_0 c^2χ^(3)Ã_1 Ã_2
            Ã_3^* e^-iΔ k z,
        
        where Ã_m = A_m e^i β(ω_m) z and the momentum
        mismatch is
        
            Δ k = β_2/2[
            (ω_3-ω_0)^2 +
            (ω_4-ω_0)^2 -
            (ω_1-ω_0)^2 -
            (ω_2-ω_0)^2 ] .
        
        At frequencies around  ω_0, the phase matching condition
        (Δ k =0) is satisfied if ω_1 = ω_2 = ω_3 =
        ω_4, which corresponds to  the SPM  process, or ω_1 =
        ω_3(4) and ω_2 = ω_4(3)≠ω_1, which
        corresponds to  the XPM  process. For other FWM processes with the
        phase mismatching condition Δ k ≠ 0, the efficiency for
        frequency conversion  is proportional to the phase mismatch
        factor, i.e., sinc^2(Δ k L), as plotted in  Fig.
        <ref>(b). For |Δ k L| ≥π, the conversion
        efficiency is significantly suppressed. In particular, for
        |Δ k L|=nπ with n being the positive integer,
        sinc^2(Δ k L)=0, meaning that FWM processes other
        than SPM and XPM vanish completely. Similar argument is also valid
        for the THG process, where the conversion efficiency of THG is
        also strongly reduced due to the phase mismatching.
        
        
        Having treated a light propagation effect in a single waveguide
        section, we now discuss the ring in Fig. <ref>(a). The
        waveguide A with the length L=l/2 has a positive GVD
        (β_2,A >0). The waveguide B with the length L
        has a negative GVD (β_2,B<0), we assume that
        β_2,B = - β_2,A. The quadratic
        term of the phase delay in Eq. (<ref>) due to the
        dispersion effect is therefore compensated once the light
        propagates through the entire ring. Hence this ring supports
        equally-spaced resonant modes at the resonant frequencies
        ω_m with the FSR Ω. As light goes through each round
        trip, its amplitude experiences a small change due to both the
        nonlinearity and the dynamic modulation. After a round-trip, the
        change of the field amplitude Ã_m for the m-th mode at
        frequency ω_m can be written from Eqs. (<ref>),
        (<ref>), and (<ref>)
        
            2π/Ω∂Ã_m/∂ T  = i
            α (Ã_m+1 + Ã_m-1) + i
            ω_0^2/2β_0 c^2χ^(3) |Ã_m|^2 Ã_m + i ∑_n≠ m2ω_0^2/β_0 c^2χ^(3)
            |Ã_n|^2 Ã_m,
        
        The corresponding effective Hamiltonian from Eq.
        (<ref>) reads:
        
            H_3 = - ħ J ∑_m ( a_m^† a_m+1 + h.c.)
             -  ħ g/2∑_m a_m^† a_m^† a_m a_m - ħ g ∑_m,n≠ m a_n^† a_m^† a_m a_n.
        
        Here g = 3ħω_0^2 χ^(3)/V n_0^4 ϵ_0
        describes the strength of the nonlinearity
        <cit.>, where V is the effective mode volume,
        n_0 = n(ω_0), and ϵ_0 is the permittivity of free
        space.
        
        
        
        In a Hilbert space where the total number of photons N is
        conserved in the the system,  Eq. (<ref>) can be
        further simplified by noting that
        
            ∑_m,n≠ m a_n^† a_m^† a_m a_n = ∑_m
            a_m^† a_m (∑_n≠ m a_n^† a_n) =
            ∑_m a_m^† a_m (N - a_m^† a_m ) = N^2 - N
            - ∑_m a_m^† a_m^† a_m a_m.
        
        The resulting Hamiltonian reads
        
            H_4 = - ħ J ∑_m ( a_m^† a_m+1 + h.c.)
             +  ħ g/2∑_m a_m^† a_m^† a_m a_m - ħ g (N^2-N).
        
        One notices that the third term ħ g (N^2-N) is just a
        constant shift in energy. The change of sign in the second term
        does not affect the physical observables for bosonic systems.
        Hence, the Hamiltonian (<ref>) [or the
        Hamiltonian (<ref>)] describes the same physics as
        our desired Hamiltonian in Eq. (<ref>). We have
        therefore shown that a Bose-Hubbard model with local interaction
        in the synthetic frequency dimension can be achieved by the ring
        shown in  Fig. <ref>(a).
        
        
        To demonstrate some of the physics effects along the synthetic
        frequency dimension in the Hamiltonian (<ref>), we
        perform numerical simulations with photon number N=2. External
        waveguides are coupled with the ring, where the photons input into
        the ring from the through-port and the detection can be made at
        the drop-port waveguide [see Fig. <ref>(c)]. Hence, our
        system becomes open, and the photon transport property is
        described by the input-output formalism in the Heisenberg picture
        <cit.>:
        
            d a_m(t)/dt = i/ħ[H_3, a_m] - γ a_m (t) +
            i √(γ) c_in,m (t),
        
            c_out,m (t) = c_in,m (t) - i√(γ)
            a_m(t),
        
            d_out,m (t) =  - i√(γ) a_m(t),
        
        where γ is the waveguide-cavity coupling strength.
        c_in,m (c_out,m) and
        d_out,m are the input (out) annihilation operators
        for photons at the frequency ω_m in the through-port and
        drop-port waveguides, respectively.
        
        
        As the input state, we consider a strongly-correlated photon pair
        <cit.>:
        
            | ϕ (p,q) ⟩ =  ∬ dt_1 dt_2
            f(t_1+t_2/2) h(t_1-t_2)
            c_in,q^† (t_2) c_in,p^† (t_1)
            |0⟩,
        
        which satisfies the normalization condition ⟨ϕ | ϕ⟩=1. We further assume  that h(t) has an extremely short
        temporal width. Hence the input state (<ref>)
        describes the scenario where two photons at the frequencies
        ω_p + Δω and ω_q + Δω are
        simultaneously injected into the through-port waveguide. We choose
        f(t) = e^ - (t - t_D)^2/ Δ t^2  e^-i 2Δω t,
        where t_D describes the timing of the incident photons, Δ
        t is the pulse temporal width, and Δω is the small
        frequency detuning. To excite the individual sites in the
        synthetic lattice along the frequency dimension, the condition
        1/Δ t ≪Ω is required.
        
        
        To measure the quantum statistics of the photons inside the ring,
        we define the two-photon correlation function G^(2)_m,n
        (t,t') =⟨ϕ | d_out,m^† (t)
        d_out,n^† (t') d_out,n (t')
        d_out,m (t) | ϕ⟩<cit.>. We then
        compute the two-photon correlation probability that two output
        photons coincide in time
        
            P_m,n =  ∫ dt  G^(2)_m,n (t,t).
        
        
        
        
        The simulation procedure follows the standard formalism
        <cit.> but in the synthetic frequency dimension. We
        consider the synthetic lattice in Fig. <ref>(b) involving
        31 resonant modes (m=-15,…,15), and set γ = 0.2J,
        t_D = 20 J^-1, and Δ t = 6.4 J^-1. We also choose
        Δω =  g to compensate for the overall  frequency
        shift in the effective Bose-Hubbard model. A photon pair | ϕ
        (-4,4) ⟩ is injected into the waveguide. We consider the
        cases with  g=0, g=2J, and g=10J in the simulations.
        Normalized distributions of the two-photon correlation probability
        P_m,n for each choice of g are plotted in Figs.
        <ref>(a)-(c). There is no photon interaction in the g =
        0 case. Notice the existence of non-zero probability P_m,m
        where the output photons have the same frequency. As the
        interaction increases the output photon statistics changes.  For a
        large g=10J, the correlation probability P_m,m=0 for all
        m. High nonlinearity thus introduces a large blockade effect at
        each resonant mode along the synthetic frequency dimension.
        
        
        
        
        
        We next use the photon pair | ϕ (-15,-15) ⟩ as the
        input and assume that there is a sharp boundary at ω_-16
        in the synthetic frequency dimension. Such an artificial sharp
        boundary can be created by adding another ring to strongly couple
        only to the -16th resonant mode  but no other modes
        <cit.>. In this input state the two photons have the
        same carrier frequencies. Figs. <ref>(d)-(f) show the
        simulated normalized distributions of the two-photon correlation
        probability P_m,n for different g. At g = 0, there is a
        strong probability for the two photons to have the same
        frequencies. In contrast, with a large nonlinearity g = 10J,
        away from the input frequency at ω_-15, there is very
        little probability that the output photons exhibit the same
        frequencies. Again, we see strong photon-blockade effect in the
        synthetic frequency dimension.
        
        
        Simulations in Fig. <ref> show the photon-photon
        interaction along the synthetic frequency dimension, which is an
        analog to the on-site nonlinear interaction in the spatial
        dimension <cit.>. Moreover, we note that
        simulation results in Fig. <ref> are the same as results
        performed by simulating Eqs.
        (<ref>)-(<ref>) with H_3
        replaced by  H_1 in Eq. (<ref>), indicating that
        the Hamiltonian H_3 in Eq. (<ref>) indeed
        provides an implementation of the Bose-Hubbard model  in the
        synthetic frequency dimension.
        
        
        Our proposed system requires waveguides with strong nonlinearity
        and large group velocity dispersion, which is experimentally
        challenging but may be achievable with the current
        start-of-the-art technology. For a ring with the length ∼ 1
        mm, Ω/2π is ∼ 10 GHz. The modulation strength
        J/2π can be tuned at the order of 0.1-1 GHz to make the
        system operate in the weak modulation regime. Photonic crystal
        fiber filled with a high-density atomic gas has the potential to
        create a nonlinearity with g/2π up to ∼ 1 GHz in the
        few-photon level <cit.>. Strong group velocity
        dispersion is desired to bring the phase-mismatching condition
        |Δ k L_A| = nπ, which can be simplified to |β_2 L_A
        Ω^2| ∼ 2π. The necessary high dispersion β_2 ∼
        10^-19 s^2/m is possible with  dispersion engineering in the
        photonic crystal fiber <cit.> or by exploiting mode crossing
        in coupled waveguides <cit.>. The theoretical proposal here
        may also be implementable in the microwave frequency range using
        superconducting quantum circuits <cit.>.
        
        
        
        In summary, we show that a ring resonator undergoing dynamic
        modulation can be used to achieve an interacting Hamiltonian where
        the interaction is local along the synthetic frequency dimension,
        provided that the ring incorporates χ^(3) nonlinearity, and
        with a specific design of the group velocity dispersion of the
        ring. Our work complements other emerging efforts in creating
        locally interacting Hamiltonian in synthetic space, by showing
        that such locally interacting Hamiltonian can be achieved in a
        system that has been widely used in photonics. The possibility of
        creating locally interacting Hamiltonian in the synthetic
        frequency dimension is fundamentally important for future studies
        of photon-photon interactions and many-body physics with synthetic
        dimensions in photonics, with promising potentials for a variety
        of quantum optical applications such as quantum information
        science and quantum communication technology
        <cit.>.
        
        
        We notice Ref. <cit.> when we were in the final stage
        of preparing this paper.
        
        
        
        This paper is supported by National Natural Science Foundation of
        China (11974245), National Key R&D Program of China
        (2018YFA0306301 and 2017YFA0303701), Natural Science Foundation of
        Shanghai (19ZR1475700). This work is also partially supported by
        the U.S. Air Force Office of Scientific Research Grant No.
        FA9550-18-1-0379, a Vannevar Bush Faculty Fellowship from the U.
        S. Department of Defense (Grant No. N00014-17-1-3030), as well as
        the U.S. National Science Foundation Grant No. CBET-1641069.
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