1. Introduction

Non-reciprocal asymmetric transport is a highly desirable feature in integrated quantum photonics networks, and could be used in designing isolators, circulators, or quantum/optical diodes. This feature has been achieved for different wave types such as optical waves [1, 2], spin waves [3, 4], and acoustic waves [5, 6]. Isolators, diodes, and circulators have been fashioned through different spatial dimensions i.e. 2D and 3D planer arrays of waveguides [1, 3, 7, 8], 1D linear systems [9, 10], and a combination of 1D and 2D networks [8]. Uni-directional transport of optical waves could be achieved by applying magnetic fields [11–15]. While the manipulation of magnetic field is not handy for building small networks, non-magnetic non-reciprocal transport has been investigated [8, 10, 16] for integrated systems. Regarding non-reciprocal photon transport, there are many theoretical proposals [12, 17–20], and experimental demonstrations of nonreciprocal phase shifts [18]. In [17] Yuan et al used photonic Aharonov Bohm effect to obtain non-reciprocal uni-directional single-photon transport to demonstrate quantum information processing tasks [9, 10, 18]. A different route towards realizing asymmetric transport in discrete optical systems is proposed in [7] through a dissipative Aharonov Bohm diode. They considered an effective magnetic field introduced via periodic modulation of optical parameters [12, 19, 20] that is a technique using for linear lattices with no nonlinear or magnetic media.

In this work we propose different dynamic or static networks to be used for non-magnetic unidirectional transport in small integrated circuits. The networks are expanded in various spatial dimensions and designed for non-reciprocal transport of quantum-carriers and Gaussian beams. They could be implemented by different site-based dynamic networks like ion traps [21, 22] and waveguide arrays [20] to be used in quantum or optical integrated networks. The two transportation mechanisms in such networks are energy oscillation within sites whose positions could be dynamic, and optical tunneling within spatially modulated optical waveguides. For dynamic site designs, we consider a scheme where the energy excitation is uni-directionally transported within the network via the dynamical close proximity of pairs of sites. We initially consider a linear array of moving sites. The proposed dynamic networks could be implemented either as curved optical waveguides or cold atoms trapped in dynamic optical lattices [23, 24]. We then wrap up the dynamic linear array to form a close spatial ring and we are able to show that simple oscillatory motions of the individual sites on the ring can lead to uni-chiral (clockwise or counter clockwise), transport of energy excitations on the ring via energy oscillation between the sites. The corresponding ring configurations, which could be considered as optical circulators, could be implemented by static curved waveguides on cylindrical substrates or cold atoms trapped in dynamic optical lattices. We also introduce two methods to achieve non-reciprocal transport within parallel waveguide arrays along planar and cylindrical substrates. For linear planar arrays we consider a sawtooth spatial modulation of the propagation constants via a proposed width pattern of individual waveguides in which the gradient of the width profile periodically flips along the waveguides axes. In this structure we find that an initial Gaussian excitation, which would normally execute a Bloch oscillation, now experiences a homogenous spatial force which periodically reverses sign and this combination leads the Bloch wave to move in one direction. We again are able to wrap this method of uni-directional transport into a ring and deduce the spatial modulations of the propagation constants and waveguides widths required to yield uni-chiral transport on a cylindrical shell of parallel waveguides. While all proposed designs could be implemented by curved or width-pattened static waveguide arrays that could be built by direct laser writing of waveguides [25, 26], our non-reciprocal directed devices have advantage over time-dependent couplings/refractive indices designs.

2. Discussion

To achieve directed energy propagation we consider an N-site nearest-site interacting system. The Hamiltonian of this system is $H={\displaystyle {\sum }_{i=1}^{N_0}(J_{j,i+1}{\widehat{\sigma }}_i^{†}{\widehat{\sigma }}_{i+1}+CC)+}{\displaystyle {\sum }_{i=1}^N{\omega }_i{\widehat{\sigma }}_i^{†}{\widehat{\sigma }}_i}$, where J_i,_i+1 is the interaction energy of two nearest sites, ω_i is the the self-energy of site i, ${\widehat{\sigma }}_i^{†}({\widehat{\sigma }}_i)$ is the creation (annihilation) operator of an excitation at site i, N₀ = N for a ring network where N + 1^st site is the first site, and N₀ = N − 1 for a linear network. The evolution of this system is determined by the equation: dϕ(q)/dq = −iαHϕ(q). Considering q = t, α = −1/ħ, the Schrodinger equation appears with ϕ(t) as the total quantum state of a system of N sites. On the other hand, by considering q = z, α = 1, and ω_i = β_i where β_i represents the propagation constant of waveguide i along z axis, the evolution equation turns to the coupled mode equations of a propagating wave through an N−waveguide parallel array and ϕ(z) = (a₁(z); …; a_n(z)) is a column vector of electric field amplitudes on each waveguide.

We now seek suitable Hamiltonians for directed transport in both dynamic site networks and waveguide arrays. In the first case we consider a linear array i.e. planar waveguide arrays or particle networks on a line, and find the appropriate conditions to achieve uni-directional transport. In the second case, we couple the first and last sites of the linear network to obtain the corresponding ring/cylindrical systems.

2.1. Linear arrays

In this section we introduce two methods to achieve directed energy propagation in linear arrays. The first method is represented by a linear array of eight moving sites where the position of each site along x axis as a function of time is shown in Fig 1(a). The first and last sites are fixed on x axis while the middle sites start oscillating in due times. The initial separation of sites 1 and 2 is 0.007, the positions of pair of middle sites i = n, n + 1 at time t are x_i = (n/2)a_i+b_isin(0.08t+π−0.3125π(n/2−1))+d_Block(n/2−1); where (a_n, b_n) = (0.5, 0.495), (a_n+1, b_n+1) = (0.507, 0.49), and d_Block = 1.009, and the last site is permanently located at x_N = d_Block(N/2−1)+0.005. We initially launch an excitation on the first site, while considering zero self-energies for all sites (ω_i = 0), and dipolar interaction energies between the next nearest sites $(J_i{ }_{,i+1}=1/r_{i,i+1}^3)$. We choose these parametrization for the time modulation of the positions of sites to yield at type of trap for the light made from pairs of sites that lead to directed motion. We numerically calculated the probabilities of finding energy excitation on each site in time as shown in Fig. 1(b). While assumed ħ = 1, the Planck units could be considered for the physical quantities. It can be seen that the directed energy transport could be achieved via the proposed pattern of dynamic particle network. This is due to the fact that the initial energy excitation on first site which is fixed at the origin would oscillate between sites 1 and 2 in terms of a superposition state. It then transfer along the x axis within the superposition states of different pair of sites in subsequent intervals. In other words, the initial energy follows closely the trajectory of increased interaction energy along the nearest sites which are close together. Such networks could be straightforwardly scaled up by adding pair of sites along the networks, however, one should adjust the pair distances in accordance with the total network size and the number of including sites.

 

Fig. 1 Graph (a) shows an eight-site one-dimensional network which achieves directed energy transport. x_i is the position of site i along x axis in time. Graph (b) is an enlargement of graph (a) and shows the pair of sites that are close together. Graph (c) is the dynamic evolution of the excitation initially located at site 1 at t = 0. |ϕ_i|² is the probability of finding the energy excitation on site i in time where the variation of the position of all sites in time are depicted in black color. We see that due to the pattern of pairing of the sites leads the excitation to move in a directed fashion to the right.

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These dynamically controlled particle networks are non-reciprocal so that by initializing the system on Nth site, the input energy spreads within the network and the efficiency of transport towards the reversed direction tends to zero. It has been checked that such isolators are completely non-reciprocal, if the initial energy is launched to the last site after the duration of one network cycle i.e. the time by which all middle sites have spanned their geometrical domains for once. A complete reciprocal version of such arrays can be obtained by flipping the dynamic pattern of site positions in time after each network cycle. These particle networks could be implemented by cold atoms in optical lattices [23, 24] and couple to optical beams in larger photonic circuits [27]. Moreover, as discussed in the beginning it would be also possible to achieve directed transport via a counterpart network of such dynamic site arrays in terms of 2D curved optical waveguide arrays with the waveguides curvature pattern introduced in Fig. 1.

Now we present another method that yields directed transport in parallel waveguide arrays. The uni-directional transport of a Gaussian beam has been previously achieved in a tight binding model by the periodic flipping in time of a static force applied across a next nearest sites interacting 1D lattice [28]. We consider a more detailed implementation of this concept using a linear optical waveguide array whose widths vary spatially to effect equivalent to the periodic flipping of a homogenous static force. An alternative to waveguides width pattering would be to fashion the waveguides from electro-optic materials and apply a different z-dependent electric field along each waveguide to yield a modulated homogenous force on the Bloch oscillating Gaussian beam resulting in non-reciprocal transport. A lateral increase in waveguide width in a parallel array changes the propagation constants of individual waveguides (a schematic shown on a line on top of Fig. 1(a) in [29]). To uni-direct an incident Gaussian beam across a waveguide array we apply a specific width pattern shown schematically in Fig. 2(a) where the lateral increase of widths is reversed periodically along the waveguides extension on z direction. For such array with N = 65 waveguides the resulting profile of waveguides propagation constants has been shown in Fig. 2(b) with |β_i − β_i+1| = δβ = 520 m⁻¹ and the pattern change at intervals Z_B = 0.0063 m along z direction that is related to the corresponding Bloch oscillation. While the difference of propagation constants of adjacent waveguides (δβ) are considered rather small, we neglect the generation of backward traveling modes on each waveguide due to the mismatch of β’s, and therefore suppose equal couplings for two nearest waveguides i.e. J_i,_i+1 = J_i+1,i [29,30]. The constant coupling for all adjacent waveguides is considered J_i,_i+1 = 1240 m⁻¹ which requires the corresponding variation of waveguides’ separation distances. We shine a Gaussian beam of ϕ(z = 0) = exp(−(n − N/2)²/10) on waveguide number n = N/2 and numerically solve the coupled mode equations to find the intensity on each waveguide at different cross sections along z direction. Here we also add a nonlinear term −6.5|ϕ(z)|²ϕ(z) to the right side of the evolution equation which well describes the experimental situation in low power excitation for chosen parameters according to [29], and could be equivalent to the environmental noise effects in counterpart Schrodinger equation. Figure 2(c) shows the numerical values of intensities on all waveguides at different cross sections along z direction, where a_n is the amplitude of the electric field on waveguide n. It can be confirmed that the introduced width pattern of waveguide and the corresponding profile of propagation constants yield uni-directional propagation of Gaussian beams.

 

Fig. 2 Directed wave transport achieved through a width-patterned array of N = 65 waveguides exposed to a Gaussian beam of ϕ(z = 0) = exp(−(n − N/2)²/10). Graph (a) shows the schematic of the proposing width pattern array with waveguides indexed n extended along z axis where z_B = 0.0063. The spatial modulation of the propagation constants of waveguides are shown in graph (b) where |β_i − β_i+1| = δβ = 520 m⁻¹. Same couplings of adjacent waveguides of J_i,_i+1 = 1240m⁻¹ are considered which could be achived by adjusting the nearest guides separations. Graph (c) represents the numerical simulation of intensity |a_n|² on waveguide n along z direction which confirms the uni-directed transport of a Gaussian beam through such width patterned arrays.

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Morandoty et al [29] experimentally demonstrated some similar width-varied waveguide arrays, and assumed the ideal lossless case in which only one mode propagates on each waveguide. Our simulated network parameters (β and J_i,_i+1) are chosen according to that of their experimental setup in which the widths of 25 waveguides are varied from 2 to 3.4 μm. Supposing a linear increase of the guides’ width with the number of guides, the maximum width of our chosen number of 65-guide array do not exceed 10.5 μm which is the typical maximum width of a single-mode fiber. So all waveguides of the simulated 65-guide array could be considered single mode. In addition, one could decrease the minimum guides’ width below 2 μm to avoid any multimode propagation on all waveguides.

To check the non-reciprocity of these networks, we first note that these unidirectional waveguide arrays should be always initiated from a middle site. This is due to the boundary limitation that spreads and reshapes the initial Gaussian beam which is expanded over several waveguides. To use such array as an isolator, one may consider a linear bench array and design input/output ports which are well-beyond the boundary waveguides. In such isolators, a beam launched on the output port would propagate away from the input port, which represents a perfect non-reciprocity. Such waveguide arrays could also be implemented by quantum site networks. The quantum nodes that are the counterpart elements of waveguides should maintain a time dependent self-energy profile as of the propagation constants in Fig. 2(b) to yield non-reciprocal transport of energy excitation.

2.2. Ring networks

In previous subsection we focused on uni-directional transport in linear arrays. We now seek to use these methods in spatially periodic arrays to achieve chiral transport e.g. transport whose handedness is directed. In particular we use time modulated systems to achieve chiral energy transport in ring networks which is analogues to non-magnetic optical circulators. As the first example, we consider a time modulated dynamic network of nine sites each moving along a curve on a circular path. Figure 3(a) shows a schematic of such network where the first site is fixed at (x₁ = 1, y₁ = 0) and the other sites are oscillating on a ring centered at the origin of coordinates. The time variation of the angles subtended by each site is shown in Fig. 3(b). Next nearest sites are dipolar coupled to each other i.e. $J_i{ }_{,i+1}=1/r_{i,i+1}^3$ where r_i,_i+1 is the distance between the sites i and i + 1, and the site energies are considered to vanish (ω_i = 0). Figure 3(c) shows the evolution of an energy excitation initially injected at the first site. It can be seen that it continually travels along the neighboring pair of sites through the ring and returns to the first site. This behaviour is due to the designed dynamics of the pair of sites so that the initial energy excitation evolves to a superposition state of two close sites and so is being handed over the ring to reach the first site. Figure 4(d) shows the evolution of the initial excitation in a longer duration. To explain the azimuthal non-reciprocity of such dynamic networks, we note that by launching the initial excitation on any intermediate site, the pair transportation mechanism would lead the energy excitation around the ring in a clockwise or counterclockwise way which is determined by the position pattern of sites. Such energy circulator networks could be implemented by cold atoms in dynamically controlled optical lattices [23, 24].

 

Fig. 3 A non-magnetic optical circulator via a dynamically controlled ring network of nine sites. Graph (a) shows the trajectories of nine sites (R_{i; i=1 − 9}) in time that are initially located equidistantly on a circle on xy plane centered at the origin. The first site is fixed in time at (x₁ = 1, y₁ = 0). Graph (b) shows the modulation of angles trajectories (in radian) for each site in time. Graphs (c) and (d) represent the dynamics of the probability of energy excitation initially launched at site 1, through the network in shorter and longer time intervals, respectively. The color scale is the total probability of energy excitaiton (|ψ|²) which varies in time at the location of each site.

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Fig. 4 A non-magnetic optical circulator/retarder made of a cylindrical width-patterned waveguide array. Graph (a) shows the schematic of a width varying waveguide array on a cylindrical shell that yields chiral transport of an incident Gaussian beam. Graph (b) shows the profile of propagation constants as a function of waveguide number n along z direction where n ∈ [1, N = 200], $\delta \beta ={(-1)}^{\lfloor \frac{z}{zB/4}\rfloor }d.F$, waveguides separations of d = 2π, effective lateral force of F = 0.005, and the nearest guides’ couplings of J_i,_i+1 = J_i+1,i = 0.2485. The numerical solutions of wave amplitudes or the corresponding intensities are shown in graph (c) for a cylindrical waveguide array that is initially launched by a Gaussian beam of ϕ(z = 0) = exp(−(n − N/2)²/100). It could be confirmed that the incident Gaussian beam would non-reciprocally propagate across the cylinderical shell. The chiral direction could be inversed via mirroring the β(n, z) pattern with respect to line n = 1. By adjusting network parameters like cylinder length or β(n, z) profile, such networks could deviate a Gassian beam for an arbitrary angle to be used as a non-magnetic optical circulator or an optical retarder.

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As the second example of uni-directed chiral transport, we consider a waveguide array with a specific profile for propagation constants. In section 2.1, directed transport was achieved in a linear waveguide array by patterning the waveguides’ widths as schemed in Fig. 2(a), which is equivalent to pattern the propagation constants profile as shown in Fig. 2(b). As discussed in [28] one way to achieve such profile is to periodically flipping the direction of an applied electric field across a waveguide array of photorefractive materials. The electric field creates a gradual increase of refractive indices (and consequently the propagation constants) along the azimuthal direction of waveguide array leading to Bloch oscillations. By flipping the electric force (or reversing the upward trend of β pattern) an initially launched Gaussian beam could be directed non-reciprocally across the array. If the parallel waveguides are separated by distance d then the increase of propagation constants for nearest waveguides is |β_i − β_i+1| = δβ = d.F where F is the electric force [28]. supposing By ħ = 1 the distance and force could be considered in Planck units. In this section, to achieve a chiral non-reciprocal transport via waveguides array on a cylindrical substrate, we propose a new width pattern to obtain a linear increase of propagation constants from the last waveguide to the first one. The schematic of the new width profile and the corresponding cylindrical array are shown in Fig. 4(a). The left graph of Fig. 4(a) indicates that at each increment along the z axis the waveguides widths vary linearly and follow a sawtooth function to maintain a continuous increase of propagation constant from the last guide towards the first one. The rate of width variation along the z and n axes are related to the velocity of corresponding Bloch waves (V_B). The right graph of Fig. 4(a) shows a schematic of a cylindrical array where an input Gaussian beam reaches the end of network after about two round trips. To simulate such system, we consider an array of N = 200 waveguides on a cylindrical substrate and shine a Gaussian beam of ϕ(z = 0) = exp(−(n − N/2)²/100) centered at waveguide number n = N/2. The chosen network parameters are d =2π which is the equivalent nearest guides separations, F = 0.005 is the equivalent lateral force due to the increase of propagation constants (|δβ| = d.F), and J_i,_i+1 = J_i+1,i = ∆/4; (∆ = 0.994) is the nearest guides couplings. The velocity of the corresponding Bloch wave is obtained by V_B = 2γ/(z_B/2) = 0.165 where γ = δ/F is the amplitude of the Bloch wave along the waveguide numbers and z_B = 2πħ/(d.F) is the length of one Bloch oscillation. While assumed ħ = 1, all quantities could be considered in Planck units. Figure 4(b) shows the corresponding β profile of such width patterned arrays where β ∈ [δβ, Nδβ] and $\delta \beta ={(-1)}^{\lfloor \frac{z}{zB/4}\rfloor }d.F$ that flips sign at length separations z_B/4. The variation of propagation constants along z direction is considered δβ/δz = V_B to minimize the power loss after each round trip. Figure 4(c) shows the simulated intensity on each waveguide (|a_n|²) along z direction. It can be seen that chiral transport of an initial Gaussian beam is achieved via the proposed modulations of propagation constants shown in Fig. 4(b). By reversing β variation at z = 0 one would achieve the reverse propagation of chiral transport of Gaussian beam along the cylinder axis.

We verify the azimuthal non-reciprocity of such networks by supposing the initial launch of Gaussian beam from the other side of the cylindrical isolator. For a clockwise circulator transferring beams from the left to the right, the beam launched from the right side would propagate counterclockwise towards the left, which is clockwise from the reversed point of view. So by inversing the direction of input beam, the intrinsic azimuthal direction of the isolator does not change which indicates the non-reciprocity of such networks. Such proposed chiral beam transporters could be built by direct laser-writing and 3D printing [25,26] and be used as non-magnetic optical circulators in photonic circuits. While magnetic Faraday optical circulators have only a few output ports, these cylindrical arrays could change the direction of an incident Gaussian beam for an arbitrary angle via the manipulation of network parameters. In addition, the proposed width-patterned networks could be used as optical retarders, and the torus version of such cylindrical networks could be studied as beam trappers for optical computing applications. The formerly proposed dynamic ring networks could be also used as quantum memories or energy excitation trappers as well as non-reciprocal energy transporters towards arbitrary angles.

3. Conclusion

We introduced four different linear and ring networks implemented by dynamic sites, curved, and width-patterned parallel waveguides for non-reciprocal and uni-directional/uni-chiral transport of energy excitations or Gaussian beams. Since each introduced system has a counterpart implementation, i.e. dynamic sites or waveguides array, eight non-reciprocal and uni-directional networks have been potentially presented. In linear and ring dynamic sites (curved waveguides) the initial energy excitation would non-reciprocally transfer through the network via oscillating between closely positioned pair of sites. One-way transport of Gaussian beams is via tunning across specific width pattered waveguide array that are equivalent to time dependent site energies of dynamic site networks. Such non-magnetic networks could be used in linear, ring, planer, and cylindrical geometries as quantum/optical diodes and circulators in quantum/photonic circuits.