Immanants

Immanants generalize both permanents and determinants, and they appear naturally in multiphoton interference when exchange symmetry is not purely bosonic or purely fermionic. In GroupFunctions.jl, these quantities are recovered from sums of matrix elements computed by group_function.

Setup and notation

For a photon in mode $k$ centered at time $\tau_j$, use

\[\hat{A}^\dagger_k(\tau_j)= \int d\omega\,e^{i\omega\tau_j}\varphi(\omega)\hat{a}^\dagger_k(\omega),\]

with bosonic commutator

\[[\hat{a}_i(\omega),\hat{a}^\dagger_j(\omega')] = \delta_{ij}\delta(\omega-\omega').\]

With a Gaussian spectrum,

\[|\varphi(\omega)|^2=\frac{e^{-(\omega-\omega_0)^2/(2\sigma^2)}}{\sqrt{2\pi}\sigma},\]

the temporal overlap factor is

\[\zeta_{ij}=e^{-\sigma^2\tau_{ij}^2}, \qquad \tau_{ij}=\tau_i-\tau_j.\]

For a matrix $M$, $M_{v \to u}$ denotes the submatrix with rows $u$ and columns $v$; repeated row or column labels are allowed (for example, an output pattern like pp4).

Definition and special cases

For a partition $\lambda \vdash n$, the immanant of an $n \times n$ matrix $M$ is

\[\mathrm{Imm}^{\lambda}(M)=\sum_{\pi \in S_n}\chi^\lambda(\pi)\prod_{i=1}^n M_{i,\pi(i)},\]

where $\chi^\lambda$ is the character of the irrep $\lambda$ of S_n.

Two important limits are

\[\mathrm{Per}(M)=\mathrm{Imm}^{(n)}(M), \qquad \mathrm{Det}(M)=\mathrm{Imm}^{(1,\ldots,1)}(M).\]

For $3 \times 3$ matrices and mixed symmetry $(2,1)$:

\[\mathrm{Imm}^{(2,1)}(M)=2M_{11}M_{22}M_{33}-M_{12}M_{23}M_{31}-M_{13}M_{21}M_{32}.\]

Example: permanent from a symmetric group function

This reproduces the relation tested in the package:

using GroupFunctions

permanent3(A) = A[1,1]*A[2,2]*A[3,3] +
                A[1,1]*A[2,3]*A[3,2] +
                A[1,2]*A[2,1]*A[3,3] +
                A[1,2]*A[2,3]*A[3,1] +
                A[1,3]*A[2,1]*A[3,2] +
                A[1,3]*A[2,2]*A[3,1]

alpha1, beta1, gamma1 = rand(Float64, 3)
block12_a = su2_block(4, 1, (alpha1, beta1, gamma1))
alpha2, beta2 = rand(Float64, 2)
block23_a = su2_block(4, 2, (alpha2, beta2, alpha2))
alpha3, beta3, gamma3 = rand(Float64, 3)
block12_b = su2_block(4, 1, (alpha3, beta3, gamma3))
alpha4, beta4 = rand(Float64, 2)
block34_a = su2_block(4, 3, (alpha4, beta4, alpha4))
alpha5, beta5 = rand(Float64, 2)
block23_b = su2_block(4, 2, (alpha5, beta5, alpha5))
alpha6, beta6, gamma6 = rand(Float64, 3)
block12_c = su2_block(4, 1, (alpha6, beta6, gamma6))

U = block12_a * block23_a * block12_b * block34_a * block23_b * block12_c

basis = basis_states([3,0,0,0])
state_x = filter(s -> pweight(s) == [0,1,1,1], basis)[1]
state_y = filter(s -> pweight(s) == [1,0,2,0], basis)[1]

M = U[[1,2,3], [2,2,4]]
group_function([3,0,0,0], state_x, state_y, U) ≈ permanent3(M) / sqrt(2)

#todo write a funcion that return the occupation numbers

Application: Two-photon coincidence rate with delay

#todo add another folder with applications. and add my paper. ==so we have application, tutorial, and background. == For two photons entering input modes $k,l$ and detected at outputs $m,n$, with relative delay $\tau_{ab}$:

\[R(kl \to mn;\tau_{ab}) =\frac{1}{2}(1+\zeta_{ab})\left|\mathrm{Per}(U_{kl\to mn})\right|^2 +\frac{1}{2}(1-\zeta_{ab})\left|\mathrm{Det}(U_{kl\to mn})\right|^2.\]

Limits:

\[\tau_{ab}=0 \Rightarrow R=\left|\mathrm{Per}(U_{kl\to mn})\right|^2,\]

\[\tau_{ab}\to\infty \Rightarrow R=\frac{1}{2}\left|\mathrm{Per}(U_{kl\to mn})\right|^2 +\frac{1}{2}\left|\mathrm{Det}(U_{kl\to mn})\right|^2.\]

So the delay continuously interpolates between fully indistinguishable interference (permanent) and the incoherent permanent/determinant mixture.

Reading the rate notation

The shorthand

\[R(\text{inputs} \to \text{outputs}; \text{delays})\]

means: coincidence probability for a fixed input occupation pattern, output occupation pattern, and set of delays.

Examples:

• \[R(23 \to 13; \tau_{ab})\]

  : one photon enters input 2 and one enters input 3; detect one at output 1 and one at output 3.

• \[R(234 \to \alpha\alpha4; \tau_{13})\]

  : one photon enters each of inputs 2,3,4; detect two photons at output $\alpha$ and one at output 4.

Repeated output labels encode multiplicity.

Three photons with one delayed input

For inputs $2,3,4$ in a four-mode interferometer, with $\tau_1=\tau_2 \neq \tau_3$, and output pattern pp4:

\[R(234\to pp4;\tau_{13}) =\frac{1}{3}(1+2\zeta_{13})\left|\mathrm{Per}(U_{234\to pp4})\right|^2 +\frac{2}{3}(1-\zeta_{13})\left|\mathrm{Imm}^{(2,1)}(U_{234\to pp4})\right|^2.\]

For distinct outputs $\alpha \neq \beta$:

\[\begin{aligned} R(234\to \alpha\beta4;\tau_{13}) &=|A|^2+|B|^2+|C|^2 \\ &\quad+\zeta_{13}\left[(A+B)^*C+(B+C)^*A+(C+A)^*B\right], \end{aligned}\]

with

\[U_{xyz}^{(\alpha\beta4)} \equiv U_{xyz\to\alpha\beta4},\quad P=\mathrm{Per}(U_{234}^{(\alpha\beta4)}),\quad I_{xyz}=\mathrm{Imm}^{(2,1)}(U_{xyz}^{(\alpha\beta4)}),\]

and

\[\begin{aligned} A&=\frac{1}{3}\left(P-I_{243}-I_{324}+I_{342}\right),\\ B&=\frac{1}{3}\left(P-I_{234}+I_{243}-I_{324}-I_{342}\right),\\ C&=\frac{1}{3}\left(P+I_{234}+I_{324}\right). \end{aligned}\]

Here $\tau_{13}$ is the only independent delay because $\tau_1=\tau_2$. At $\tau_{13}=0$, photons are fully indistinguishable and only the permanent term survives in the pp4 rate.

References

• B. Kostant, "On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents"
• Amaro-Alcala, D et al., "Coincidence landscapes for three-channel linear optical networks"
• Immanant (Wikipedia)

#todo there's a typo here.