Multiphoton, multimode state classification for nonlinear optical circuits

Abstract

We introduce a new classification of multimode states with a fixed number of photons. This classification is based on the factorizability of homogeneous multivariate polynomials and is invariant under unitary transformations. The classes physically correspond to field excitations in terms of single and multiple photons, each of which being in an arbitrary irreducible superposition of quantized modes. We further show how the transitions between classes are rendered possible by photon addition, photon subtraction, and photon-projection nonlinearities. We explicitly put forward a design for a multilayer interferometer in which the states for different classes can be generated with state-of-the-art experimental techniques. Limitations of the proposed designs are analyzed using the introduced classification, providing a benchmark for the robustness of certain states and classes.

Figures (6)

• Figure 1: Scheme of all possible classes of $5$-photon states for (a) $M=2$ and (b) $M\geq 3$ bosonic modes.
• Figure 2: (a) Scheme of the multilayer interferometer with photon additions $\hat{a}^\dagger$ in the first channel and linear transformations $U^{(1)},\ldots,U^{(N)}$. (b) Scheme of the transitions between states enabled by the multilayer system. Allowed transitions are indicated by arrows. Blue regions correspond to the classes $[1^n]_M$, while the white region indicates other classes.
• Figure 3: Effect of unitary transformation and photon addition. The bold dots represent the states that are connected via unitary transformation. After photon addition, the states $\ket{\phi}$ and $\ket{\tilde{\phi}}= U\ket{\phi}$ are transformed into $\ket{\psi} \propto \hat{a}^\dagger\ket{\phi}$ and $\ket{\tilde{\psi}} \propto \hat{a}^\dagger\ket{\tilde{\phi}}$, respectively. The resulting states $\ket{\psi}$ and $\ket{\tilde{\psi}}$ are not necessarily connected via a unitary transformation anymore.
• Figure 4: Scheme of the multilinear interferometer with $N$-additions and rotations together with (a) a single subtraction and (b) a single projection. Gray insets represent the transitions between classes under subtraction and projection. Different colors represent different classes, and allowed transitions are indicated by arrows.
• Figure 5: The structure of multivariate polynomials for the states obtained via (a) the single-photon subtraction and (b) the single-photon projection from a state belonging to the class $[1^4]_M$ with $\gamma^4_1=0$. Different terms are highlighted in different colors. The coefficients $\gamma^i_1$ are indicated by the colored squares. In (b), the dotted squares represent the absence of these coefficients. See the main text for details. Gray insets represent the corresponding transitions between classes under subtraction and projection, different colors represent different classes, and allowed transitions are indicated by arrows.

Theorems & Definitions (10)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Corollary 1
• Remark 1