Coherent optical processes with an all-optical atomic simulator

We show how novel photonic devices such as broadband quantum memory and efficient quantum frequency transduction can be implemented using three-wave mixing processes in a 1D array of nonlinear waveguides evanescently coupled to nearest neighbors. We do this using an analogy of an atom interacting with an external optical field using both classical and quantum models of the optical fields and adapting well-known coherent processes from atomic optics, such as electromagnetically induced transparency and stimulated Raman adiabatic passage to design. This approach allows the implementation of devices that are very difficult or impossible to implement by conventional techniques.


Introduction
In recent years, optical photonic chips have emerged as one of the most promising platforms for practical quantum computation. This technology reached an important milestone, demonstrating the "quantum supremacy" [1]. The all-optical photonic circuits are scalable [2] including the CMOS-compatibility [3] and potentially reprogrammable, enabling execution of a variety of algorithms on the same device. All-optical control via second order nonlinearity is significantly faster than other reprogramming methods and adds frequency as a synthetic dimension -a substantial improvement over arrays of linear evanescentlycoupled waveguides. Second order nonlinearity enables three-wave mixing (TWM) [4], which includes a second harmonic generation (SHG) and a sum-/difference-frequency generation (SFG/DFG) [5]. Applications of TWM such as entanglement generation [6], quantum frequency conversion [7], and optical parametric oscillators [8,9] have been longknown. Compact and efficient off-the-shelf components for these applications are now readily available [10][11][12][13][14][15][16]. Recent advances in the fabrication of periodically-poled lithium niobate (PPLN) waveguides have allowed very high efficiency of TWM processes with optical propagation losses as low as 0.03 dB/cm [17,18].
Linear and nonlinear waveguide lattices, consisting of an array of evanescently coupled waveguides attracted a lot of attention [19][20][21]. For example, equations describing the propagation of optical fields in an array of identical straight waveguides evanescently coupled to their next neighbors have the same functional form as the Schrödinger equation, with conjugated field amplitudes representing wavefunction coefficients, and with the spatial evolution replacing the temporal one [22]. This model allows one to employ classical optics analogs of both nonrelativistic and relativistic quantum mechanical systems [23]. In particular, this system was used to simulate displaced Fock states [24], W-states [25], and squeezed states [26]. Particularly, by using intensity correlations, it is possible to study the path-entanglement created by propagation of photonic lattices [27]. Waveguide arrays can also provide a classical platform for tight-binding models [28], Majorana physics [29] and the Jaynes-Cummings model [30].
Here we introduce a novel formalism in which classical and quantum optical fields in evanescently coupled waveguides with second-order nonlinearity can be arranged to model atomic systems interacting with external optical fields. This approach reproduces the phenomena from a rich playbook of atomic quantum optics, such as electromagnetically induced transparency (EIT) and stimulated Raman adiabatic passage (STIRAP) [31], in a nonlinear photonic lattice with all-optical quantum states. In the proposed formalism, atomic levels are represented by spatial and spectral optical modes coupled to each other in a controlled manner by TWM nonlinear processes. In contrast to a real atomic system, one can engineer a true 2-or n-level model of the "atomic" system with varying complexity. For example, in the simplest case, all atomic levels can be made non-degenerate and spinless. Such "atomic" systems are not sensitive to external magnetic fields. At the same time, these systems can be tuned via temperature and via TWM nonlinear control.
Here we demonstrate a number of new photonic devices, inspired by the analogy with the atoms. For example, we show how an all-optical analog of EIT in the system of two evanescently coupled waveguides can be used as an ultrafast broadband all-optical switch or on-chip quantum memory. Similarly, a STIRAP optical analogy gives rise to the robust frequency transduction when direct frequency conversion is impractical. Key advantages of nonlinear coupling between modes over linear coupling are: (1) the coupling strength can be changed by the end-user simply by adjusting the pump power, (2) the coupled modes may have same or different frequencies, and (3) a synthetic spectral dimension can be added to the spatial dimension of a waveguide array.

Two-level atom analogy: three-wave mixing in a single nonlinear waveguide
In this section, we show that propagation dynamics of two optical fields and a pump mutually coupled via TWM in a nonlinear waveguide is equivalent to temporal dynamics of a two-level atom interacting with an external optical field. In the following calculations we assume that each waveguide supports a single spatial mode and multiple spectral modes.
We mathematically describe the three-wave mixing (TWM) in such a waveguide using the plane-wave approximation, and note that complete energy conversion from one wavelength to another is theoretically possible [5]. Very high quantum SFG/DFG conversion efficiencies were demonstrated experimentally; for example nearly 100 % internal efficiency for the pulse conversion [32] and more than 80 % efficiency for continuous wave conversion [33,34], were demonstrated. A 100 % conversion has been approached even in a more challenging case of SHG, [35]. In the slowly varying amplitudes approximation and in the presence of a strong pump field A p with frequency ω p , evolution of an input optical field A i with frequency ω i and sum-frequency field A s with frequency ω s , can be written as [5]: where ω s = ω i + ω p , c is speed of light in vacuum, d eff is the effective nonlinear coupling coefficient defined by nonlinear properties of the waveguide, and Δk = k i + k p − k s is the phase mismatch. Here we use the propagation constants k i, s, p = n i, s, p ω i, s, p /c, where n i,s,p are the effective refractive indexes in the waveguides of the input, sum-frequency and pump optical modes, respectively. In the non-depleted pump approximation ∂ z A p ≈ 0. By using the following substitutions: Ω/2 = 2d eff c A p ω i ω s / n i , n s ; ϕ = arg A p ; A i = C i ω i /n i ; A s = C s ω s /n s , (2) one can rewrite the system (1) as: These equations are mathematically identical to those describing Rabi oscillations of an idealized single two-level atom interacting with a single-mode classical field in the absence of dephasing. Here temporal evolution of atomic states is replaced by the spatial evolution of the optical modes during propagation in a nonlinear waveguide (t z), and Ω is equivalent to the atomic Rabi frequency, see Fig. 1. Notice that the average photon flux for input and sum-frequency fields can be written as: Therefore, absolute squares of amplitudes C i,s 2 are proportional to the photon fluxes in the corresponding optical modes {i, s}. At the same time, in this all-optical analogy to the atomic system, C i,s 2 represent populations of the two states of the model two-level atom.
Therefore, TWM process can be used for an all optical simulation of an atomic system, with the population of the "optical atomic states" directly mapping to photon flux. In other words, populations of all-optical atomic states are proportional to optical power in each mode, and thus can be measured using an optical power meter.

Quantum model of three-wave mixing in a nonlinear waveguide
The analogy introduced in the previous section can be extended to the quantum regime.
We consider the extension of the theoretical model to a quantum description of the optical modes in the waveguide, except for the strong undepleted pump field(s) which we treat classically. The interaction Hamiltonian for the nonlinear waveguide describing TWM coupling between two quantized spectral modes for sum (s) and input (i) photons respectively is: [7]: Let us first consider the case of only one photon present in the system: In this case one can look for the solution of the non-stationary Schrödinger equation (NSE) in the following form: By using the substitution (2) and projecting on the Fock basis, one can obtain the following system of equations for complex amplitudes b i,s for single-photon excitation in the nonlinear waveguide: Notice that these equations are identical to the semiclassical approach (3) after substitution b 1 i , 0 s C i and b 0 i , 1 s C s (Fig. 1). Thus, when only a single photon is present, fully quantum optical models of an atom are identical to their semi-classical counterparts.
However, in the quantum regime, the "atomic" state populations now correspond to the populations of the single-photon states in signal and idler modes b 0 i , 1 s 2 and b 1 i , 0 s 2 , rather than to the values of the photon flux in these modes. Therefore, the classical case is analogous to a simulation for mean values of atomic populations and coherences, while the single photon simulates experiments with single atoms, c.f. [21].
In the single-photon case we used the fact that only two states -1 i , 0 s and 0 i , 1 s -exist in the waveguide, allowing their mapping into the Jaynes-Cummings states g, 1 and e, 0 , where g and e are indexes of the ground and excited states of the atom, and "0" and "1" are the number of photons in each mode of the quantized optical field.
With more than one photon in the waveguide, all possible combinations of the two-mode Fock states n i , n s such that n = n i + n s can be present in the waveguide. For n = 2, there are 3 possible combinations: 2 i , 0 s , 0 i , 2 s and 1 i , 1 s . Using the Hamiltonian (4), one can obtain the following system of equations for complex amplitudes of these states: where we simplified the index notations b p i , q s b pq . In the case of two-photon excitation, a spectral NOON state with N = 2 is coupled to a biphoton state 1 i , 1 s , and equivalent to the Hong-Ou-Mandel effect in two coupled linear waveguides for two indistinguishable photons.
For an arbitrary length of the waveguides the Hamiltonian is equivalent to a beamsplitter, but instead of splitting a beam in two spatial directions, the beam is splitted into two different spectral modes, c.f. [20].

Multi-level atoms model: evanescently-coupled waveguides and nonlinearly-coupled multiple optical modes
In a linear waveguide array evanescent coupling of a single spectral mode m in the j th waveguide to its next neighbor is described by the system of ordinary differential equations [36]: Similarly, the poling can be designed to phase-match more than just one set of TWMcoupled modes [37,38]. Thus, multiple coupled modes can co-exist in a single waveguide.
In a full analogy with evanescent coupling, if several TWM processes are phase-matched, coupling between several spectral components can be induced by using multiple classical pumps. The advantages of this configuration are threefold. First, the coupling coefficients Ω j lm /2, phases ϕ j lm and phase mismatch Δk j lm can be controlled by tuning external classical pump fields. Second, coupled fields can have different wavelengths, which increases the range of practical devices that can be built based on this platform. Third, by combining the two types of coupling, we can take advantage of a synthetic dimension due to spectral modes. Particularly, a 1D array of physical waveguides becomes a quasi 2D array of coupled modes. Indeed, in the undepleted pump approximation, and using the substitution (2) for M spectral modes, one writes: since the pump field can oscillate between the neighbor waveguides due to their evanescent coupling. One can take advantage of this effect to engineer Floquet-type interactions in the system [39]. Thus, in contrast with evanescently coupled arrays, taking advantage of the spectral domain allows the flexibility of using the same physical device to simulate different Hamiltonians.
Of particular interest here are novel devices that are inspired by coherent atomic phenomena.
To this end, a three-level atom interacting with two classical fields enables an all-optical analog of the coherent population trapping (CPT), EIT, and STIRAP effects. We will describe practical photonic devices based on EIT and STIRAP using a small number of waveguides and pumps. We will also show that this 1D photonic system can emulate complex, multidimensional hamiltonians that typically occur in photonic lattices filled with cold atomic gases. Here we will present only the semiclassical description, since it can be straightforwardly extended to a single-photon case, as shown in the previous section. For simplicity, we assume that wavelength λ s ≪ λ i . Then Λ i ≫ Λ s ≈ 0 because modes with shorter wavelength are better confined. By using identical pump fields in both waveguides one can eliminate dependence of Ω on z. Further, assuming that TWM is phase-matched in only in the waveguide 1 (Fig. 2), the above equations can then be re-written as:

All-optical EIT-inspired switch
where we dropped the equation for C 0 s since it is decoupled from other modes. Following the analogy of all-optical atomic states, this photonic model can be interpreted as a three level "atomic" system and therefore can be used to simulate EIT-like processes. In the absence of the pump, C 1 p = 0, the idler field oscillates periodically between the two waveguides due to The pump field turns on the coupling between the spectral modes "i" and "s" in waveguide 1. Equations (5) show that a proper choice of the coupling constants suppresses the light transfer between the waveguides due to the EIT-like destructive interference. The results of numerical simulations for this system are shown in Fig. 3, where the nonlinear coupling strength Ω 1 = 0.23 mm −1 corresponds to a modest pump power of ≈ 200 mW and the frequency conversion normalized efficiency η = 2600% W −1 cm −2 , values reported experimentally for nanophotonic PPLN waveguides [18]. The unique property of this effect is that both the input and the output remain at the same frequency (ω i ), while the control field can be significantly red-detuned from both the input and output frequencies to eliminate both Raman and SPDC noise in LiNbO 3 [12,13]. This system can be used in practice as an ultrafast all-optical switch for faint states of light. Its switching time is limited only by the bandwidth of the TWM process in the nonlinear waveguide, and thus can achieve sub-ps switching times [40].
A looped nonlinear waveguide with an integrated switch can store short broadband optical pulses, operating as an ultrafast and broadband quantum memory. For example, a single round trip delay of L loop /(nc) ≈ 1 ns can be realized in a looped ≈ 7 cm-long waveguide (same as the expected nonlinear interaction length of the proposed switch). Due to the high internal losses (of the order of 0.03 dB/cm), the storage time in a lithium niobate waveguide loop device is limited to just a few ns. Longer storage times are achievable by employing low-loss materials such as silicon nitride [41] to fabricate the delay loop waveguide that is evanescently coupled to PPLN waveguide. This is because only one waveguide needs to be nonlinear for our EIT-like switching method to work.

All-optical STIRAP for frequency conversion
Another well-known effect in atomic optics is STIRAP. In STIRAP, the population is transferred between two atomic states via two sequential coherent pulses. Here we propose using the all-optical STIRAP for frequency conversion between the states that are hard to couple in a single-step TWM transduction. For example, the use of a blue detuned pump can be undesirable since it can result in significant background noise due to spontaneous parametric downconversion (SPDC) processes [33]. Alternatively, the two states can be spectrally too close to couple them directly using an accessible optical pump. Indeed, transduction between the two communication bands at 1310 nm and 1550 nm via TWM nonlinear frequency conversion requires a pump at 8.5 μm, but the transparency window in most TWM crystals is limited to ≲ 5 μm in the infrared; the other choice is to use a 710 nm pump, but the SPDC background noise will be significant. If the strong pump is too close to one of the fields spectrally, significant background noise will be generated via Raman scattering. For instance, if transduction between the Rubidium transition at 780 nm ( 87 Rb D 2 (5 2 S 1/2 → 5 2 P 3/2 )) and the telecommunication band at 1550 nm is desired, a strong pump at 1570 nm required, which is detuned by just 20 nm from one of the fields. The STIRAP approach allows coupling the two target spectral modes while avoiding practical limitations of TWM.
Two configurations are possible, depending on the frequencies of the participating optical fields. A Λ-configuration can be used to couple two spectrally close target fields (Fig. 4). A ladder configuration is desirable for spectrally distant target fields. In our first example above, one can couple the two communication bands via an intermediate "dark state" at ≈ 900 nm with the proposed STIRAP-like TWM using two commercially available pumps at ≈2.15 μm and ≈2.87 μm that conveniently occur within the transparency window of lithium niobate. The second example of Rb frequency transduction can be realized using an intermediate "dark state" at 1064 nm and two pumps at ≈ 2.9 μm and ≈ 3.4 μm.
The STIRAP-like evolution of the optical fields is described by the following system of differential equations: i ∂ z C a (z) = − Ω ab (z)e iΔk ab z C b , i ∂ z C b (z) = − Ω ab (z)e −iΛk ab z C a − Ω bc (z)e ±iΔk bc z C c , i ∂ z C c (z) = − Ω bc (z)e ∓iΔk bc z C b where signs of the exponent's arguments are "+" in the second equation and "-" in the third equation for a Λ-configuration. For a ladder-configuration, these signs are flipped. To achieve full population transfer between the target fields, coupling with pump fields should change during propagation. One can implement such controllable change by sending the pumps through two separate waveguides whose transverse position changes with the propagation coordinate. Target fields propagate in a central waveguide. The conceptual waveguide design is shown in Fig. 5 at the bottom. In this model we used strengths are approximately the same as in the EIT model. Alternatively, a propagation-dependent TWM interaction with pump fields can be implemented, for instance, by changing the amplitudes of the Fourier components of the QPM grating along the length of a single waveguide. The desired interaction envelope can be achieved with a Fourier design technique straightforwardly [42].
In addition to enhancing the range of low-noise frequency conversion, the proposed device offers the advantage of unique STIRAP robustness that distinguishes this approach from the "intuitively" ordered sequence of π-pulses [43]. Indeed, the π-pulse sequence is very sensitive to pump parameters and waveguide imperfections.

Complex quantum systems
By where P j lm are the coupling coefficients between m th and l th spectral modes due to TWM processes in the j th nonlinear waveguide with the phase mismatch Δk j lm . C j, j ± 1 m are the coupling constants between m th spectral modes in the neighboring waveguides. Dependence of P j lm coefficients on propagation coordinate z is due to possible evanescent tunneling of pump fields to the neighbor waveguides. The pump field evolution in the undepleted pump approximation is driven by the coupling between the waveguides only: i ∂ z P j lm = − C j, j + 1 lm P j + 1 lm − C j, j − 1 lm P j − 1 lm , where C j, j ± 1 lm are coupling constants for the pump field to neighbor waveguides.
The Hamiltonian (6) describes a 2D photonic lattice, however, the 2 nd dimension is synthetic, i.e. due to different frequency modes within the 1-dimensional nonlinear waveguide array. In contrast to the passive evanescent field coupling between nearestneighbor waveguides, TWM-mediated coupling in the synthetic spectral dimension enables a reconfigurable, tunable system. One can engineer a wide variety of Hamiltonians in the frequency domain: from a 1D Ising model with all-to-all coupling to a Honeycomb lattice, from a stub lattice or a dimer lattice to the Floquet system, in which temporal modulations are introduced via the oscillations of the pump field between neighbor waveguides. Therefore, a physical layer of a 1D array of nonlinear waveguides can be extended to a 2D system with the synthetic spectral dimension. In contrast to conventional photonic waveguide arrays, in which the coupling between the waveguides is mostly limited by the nearest neighbors interactions, the quasi-phase-matching in periodically poled waveguides can be engineered to couple all spectral modes. This coupling is controlled by the pump strength. Thus, such a system can be used to simulate complex band structures and topological effects such as "bosonic" superradiance [44].

Conclusions
In this work, we have introduced a theoretical formalism in which spatial and spectral modes of classical or quantum single-photon fields are mapped into energy levels of a fictitious atomic system interacting with external optical fields. This formal similarity between alloptical systems and well-studied atomic systems is not only of the fundamental interest of the theoretical quantum optics, but it enables new interesting applications and experimental simulations of atoms on integrated platform and with all-optical control. This approach is characterized by multiple potential advantages. First, it eliminates many decoherence mechanisms typical for an atom, thanks to the very weak interaction of photons with the environment. Second, it allows simulating quantum optical phenomena with classical light paired with a simple classical measurement, while it is also valid if applied to quantum single-photon fields. Third, the all-optical model of an atomic system can be engineered for any available wavelength of the optical sources, within the optical transparency window of the waveguides (from 0.3 μm to 5 μm for PPLN waveguides). Finally, the fabrication of PPLN waveguides is CMOS -compatible, which potentially allows integration of the electronics, detectors and light sources on the same chip in a scalable architecture.
To illustrate the potential of our theoretical framework, we also presented two practical photonic applications, based on traditional atomic phenomena. First, we demonstrated how the all-optical analogy of the EIT effect can be used to switch coupling between two adjacent waveguides. Second, we used STIRAP as inspiration for robust and noiseless frequency transduction where direct frequency conversion is not possible. The analogy between TWM of classical fields, the two-level atom interacting with an external field, and TWM of single-photon fields. Effective level configuration for EIT in two coupled waveguides supporting two spectral modes "i" and "s" with TWM in the waveguide #1. A numerical simulation of photon flux in the ultrafast all-optical switch implemented with a coupled linear waveguide (LWG) and nonlinear waveguide (NLWG). In the absence of an optical control pump field, the signal field is transferred from LWG to NLWG (left). When the control field is applied, the signal field remains in LWG due to an EIT-like destructive interference (right). Bottom graphs show a conceptual layout of waveguides in the proposed experimental implementation (disproportionally scaled because waveguide width is of order of 1 μm); color gradients are the artistic representations of the switching process in coupled LWG and NLWG. Simulation parameters: Λ i = 0.1Ω 1 = 0.023 mm −1 , Δk 1 = 0.15 mm −1 (see text). Effective level configuration for STIRAP-like frequency conversion between telecommunication bands.