Generate tensor network state by sequential single-photon scattering in waveguide QED systems

We propose a scheme to generate photonic tensor network states by sequential scattering of photons in waveguide QED systems. We show that sequential scatterings can convert a series of unentangled photons into any type of matrix product states. We also demonstrate the possibility of generating projected entangled pair states with arbitrary network representation by photon re-scattering.

a real atom, or a solid state multi-level system such as a quantum dot, a color center or a superconducting qubit. We prove that we can generate arbitrary one-dimensional MPS states with their bond dimension equal to the dimension of the Hilbert space of the atom ( Fig. 1(a)). Moreover, unlike the cavity QED approach, in this approach there is an additional flexibility of photon re-scattering, i.e. the scattering of the same photons multiple times against the atom.
We show that such a flexibility can enable one to generate PEPS with arbitrary graph representation ( Fig. 1(b)). As a start, we consider a waveguide QED system where photons propagate along the waveguide and scatter against an atom supporting multiple ground states. The scattering process in general can be described by the single-photon scattering matrix (S matrix) with its matrix element denoted by In this letter we use Greek letters such as α, β to denote the atom's ground states and Latin letters such as i, j to denote the internal degrees of freedom of the photon, which can be either polarizations or discrete frequency bins of photons. Such a waveguide QED system has been extensively discussed in the literature . We show a concrete example of one such waveguide QED system including its Hamiltonian and single-photon S matrix in the later part.
For most of the paper we proceed generally by only assuming that such an S matrix exists. For a system described by the single-photon S matrix (1), if we scatter a photon in an arbitrary input state i d i |i against the atom in the state α I α |α , the output state in general is an entangled state between the photon and the atom having the form: or in shortŜ In (2) and also for the rest of the paper, we use the Einstein summation convention to simplify the expression.
For the system as described by (1) and (2), we first present a protocol to generate MPS by sequentially scattering photons against the atom. Suppose the atom is initially prepared to be in the state I α0 |α 0 and against this atom we sequentially scatter n photons in the states d [1]j1 |j 1 , · · · , d [n]jn |j n , respectively. As shown in (2), after the first scattering, the system is in the state S i1j1 β1α0 d [1]j1 I α0 |i 1 , β 1 . We then perform a unitary operation α1β1 |α 1 β 1 | on the atom to arrive at the state R [1] α1β1 S i1j1 β1α0 d [1]j1 I α0 |i 1 , α 1 . Repeating this process by scattering a second photon against the atom followed by a unitary operationR [2] on the atom, we obtain the state R [2] α2β2 S i2j2 β2α1 d [2]j2 R [1] α1β1 S i1j1 β1α0 d [1]j1 I α0 |i 2 i 1 , α 2 . This process can be repeated for all the other photons. Finally, as the very last step, we disentangle the photons from the atom by projecting the atomic state to some final state F αn |α n .
The resulting n-photon state is of the form We define a series of matrices and represent the initial and final states of the atom as I and F with entries I α0 and F αn , respectively, the state (3) can be written more transparently as which is exactly in the MPS form with the open boundary condition. The bond dimension of such MPS is equal to the number of the ground states of the atom. Furthermore, the unitarity constraint of S matrix S li * γα S lj γβ = δ ij δ αβ gives the normalizations of the A matrices: Eq.(4) provides an explicit connection between MPS representation and the single-photon S matrix of a waveguide QED system. One one hand, given the S matrix and rotation operations, we know immediately the type of MPS that we can generate. To remove the non-uniqueness of MPS representation, one can always convert the MPS in Eq. (4) to its canonical form [51]. On the other hand, for a given A matrix, there are enough degrees of freedom to construct a S matrix that satisfies (4). Therefore, we can generate any type of translation-invariant MPS by designing a waveguide QED system with the proper S matrix.
For illustration, here we provide a simple waveguide QED system and its S matrix that can be used to generate either a W-type state or a GHZ state by sequential scattering of photons. For three photons, these are the only two inequivalent ways of entanglement [52]. Therefore, this construction provides an illustration that arbitrary MPS states can be indeed generated. The system consists of a four-level atom as shown in Fig. 2 and the Hamiltonian is described as where c k (c † k ) is the annihilation (creation) operator of the photon state in the waveguide. These operators satisfy the standard commutation relation [c k , c † k ] = δ(k − k ). Here for simplicity we consider a waveguide consisting of only a single mode in the sense of Ref. [26]. The argument here, however, can be straightforwardly generalized to waveguides supporting multiple modes. ∆ 0 , ∆ 1 and Ω are the respective energy of the ground states |g 0 , |g 1 and the excited state |e of the atom satisfying ∆ 0 < ∆ 1 < Ω. We define ∆ α ≡ Ω − ∆ α for α = 0, 1. The waveguide photons couple to both |g 0 − |e and |g 1 − |e transitions of the atom with respective coupling constants γ 0 /2π and γ 1 /2π. There is an additional energy level |g 2 that does not couple to any other atom levels by waveguide photons and thus does not appear in the waveguide-atom coupling term in the Hamiltonian (6). This can be accomplished, for example, by choosing the frequency bins of the photons such that the photon frequency is off resonance from the transitions that involves |g 2 . The single-photon S matrix for this system is where α, β take values of 0, 1 and is the transmission amplitude of the waveguide photon |k when the initial and final states of the atom are |g α and |g β , respectively [55][56][57][58]. Furthermore, we only consider waveguide photons with two resonate frequencies ∆ 0 and ∆ 1 . That is, we encode the photonic qubit in the frequency degree of freedom as |i ≡ |∆ i for i = 0, 1. Finally, let γ0+γ1 and |∆ 0 − ∆ 1 | γ 0 , γ 1 , the computed single-photon S matrix (7) reduces to the form of (1) with where the row and column index of the matrices are related to the atom's degrees of freedom α, β. For an input state |i, g α , the output state remains the same when i = α but becomes entangled when i = α. To generate a W-state, we initialize the atom to be in the state |g 0 and all three photons to be in the state |0 . We then sequentially scatter three photons against the atom. Finally, we decouple the atom by projecting it to the state |g 1 . The result is a W-type state η |001 + ξη |010 + ξ 2 η |100 . To generate a GHZ state, we set the coupling constants γ 0 = γ 1 such that ξ = 0, η = −1. We first initialize the atom to be in the state |g 0 + |g 2 and sequentially scatter three photons with input state |0 . After each scattering, we rotate the atom state as |g 0 → |g 1 and |g 1 → −|g 0 . At the final step, we decouple the atom by projecting it to the state |g 0 + |g 2 . The result is a GHZ state |000 + |111 .
So far we show that one can generate a MPS by sequential scattering of single photons against an atom. Unlike previous works that use cavity QED systems for sequential generation of multi-photon MPS [18,19], where the different photon states are restricted to either the absence (|0 ) or presence (|1 ) of a photon in a given time bin, here the different photon states can be either different polarizations or frequencies. As result, the physical dimension of the MPS can be higher than two. More importantly, there is a flexibility to rescatter an outgoing photon against the same atom. This flexibility enables one to generate PEPS that are more general than MPS.
For illustration, we first present the procedure for generating a four-photon PEPS ( Fig. 1 (b)). Given four photons in the respective states d [1]j1 |j 1 , · · · , d [4]j4 |j 4 , we sequentially scatter them against an atom with an initial state I α0 |α 0 and perform unitary operations in the ground-state manifold of the atom between two sequential scatterings. As discussed before, after the fourth photon scattering, we have a state A with matrices A defined in (4). Now instead of decoupling the atom, we rescatter the first photon against the atom, followed by an unitary operationÛ [1] on the atom. As a result, the atom-photon state becomes α1α0 I α0 |i 4 i 3 i 2 i 1 , β 1 . We then rescatter the third photon against the atom and do an unitary operation on the atom. Finally, we decouple the atom by projecting it to the state F to end up with a four-photon entangled state: By defining tensors the state (10) can be simplified to the form of which is a PEPS with the graph representation of two triangles as shown in Fig. 1(b).
The above procedure can be easily generalized to generate an arbitrary PEPS through sequential scatterings. Note that any PEPS state can be represented as an undirected graph. For any graph, the number of odd-degree nodes must be even. If the graph contains zero or two odd-degree nodes, we can traverse the whole graph through a single Eulerian path. In this case, we initialize the atom, do sequential scattering of photons following the visiting order of nodes in the Eulerian path, and finally decouple the atom. If the graph contains more than one pair of odd-degree nodes, we can traverse the whole graph through multiple disjoint Eulerian paths. For each Eulerian path, we initialize the atom at the begin of the path, do sequential scattering following the visiting order along the path and decouple the atom at the end of the path. In this process, we visit each edge only once. But we may visit a single node for multiple times, which corresponds to the operations where we re-scatter the same photon multiple times against the atom. Such a capability is unique to our scattering approach and is not available in previous approaches for using cavity QED systems for entangled state generation.
A related application is the generation of two-dimensional cluster states that are crucial in the quantum computing [14]. Recently, two-dimensional cluster states with square lattice structure have been generated using a cavity QED system [22]. In our sequential scattering approach, to generate the cluster state with arbitrary graph representation, we first design a waveguide QED system where two-photon sequential scattering process is identical to a controlled phase-flip gate [53,54]. We then initialize each photon to be in the state |0 + |1 and perform sequential scatterings following the graph representation of the state. In this way, we can generate cluster states with arbitrary graph representation.
To summarize, we propose a sequential scattering approach to generate photonic MPS and PEPS using waveguide QED systems. We show that by performing sequential scattering of initially unentangled photons against an atom, we can generate arbitrary MPS with the bond dimension equal to the number of the ground states of the atom in the waveguide QED system. Moreover, by photon re-scatterings, we can generate PEPS with arbitrary graph representation. This approach provides significant more flexibility in generating MPS and PEPS as compared to previous approach, and points to the importance of waveguide QED systems for quantum state generations.