PhyslibAlpha.Mathematics.LadderSystem.OccupationBasis

For natural numbers $d$ and $n$, a count function of degree $n$ on $d$ colors is a function $\alpha : \{0, 1, \dots, d-1\} \to \mathbb{N}$ such that the sum of its values over all colors is equal to $n$, i.e., $\sum_{c=0}^{d-1} \alpha(c) = n$. In the context of quantum mechanics, this represents the occupation numbers of $n$ identical bosons distributed among $d$ states.

For natural numbers $d$ and $n$, there is an equivalence (bijection) between the set of count functions of degree $n$ on $d$ colors, denoted $\text{CountFun}(d, n)$, and the symmetric power $\text{Sym}^n(\{0, 1, \dots, d-1\})$. A count function $\alpha$ is a mapping from the set of indices $\{0, 1, \dots, d-1\}$ to the natural numbers such that the sum of its values is $n$, i.e., $\sum_{i=0}^{d-1} \alpha(i) = n$. The symmetric power $\text{Sym}^n(\text{Fin } d)$ represents the collection of multisets of size $n$ whose elements are drawn from the set $\{0, 1, \dots, d-1\}$. In physical terms, this identifies the occupation-number representations of $n$ identical bosons in $d$ states with the possible configurations of those bosons.

For any natural numbers $d$ and $n$, the set of count functions of degree $n$ on $d$ colors, denoted $\text{CountFun}(d, n)$, is finite. Here, a count function $\alpha$ is defined as a function $\alpha : \{0, 1, \dots, d-1\} \to \mathbb{N}$ such that the sum of its values is $n$, i.e., $\sum_{i=0}^{d-1} \alpha(i) = n$.

For any natural numbers $d$ and $n$, the number of degree-$n$ count functions on $d$ colors is given by the binomial coefficient $\binom{d + n - 1}{n}$. A degree-$n$ count function $\alpha$ is a mapping from the set of indices $\{0, 1, \dots, d-1\}$ to the natural numbers such that the sum of its values is $n$, i.e., $\sum_{i=0}^{d-1} \alpha(i) = n$. In physics, this corresponds to the number of ways to distribute $n$ identical bosons into $d$ energy levels.

Let $L$ be a ladder system on a vector space $V$ with $d$ colors. For any two distinct indices $i, c \in \{0, 1, \dots, d-1\}$ (i.e., $i \neq c$), any natural number $k$, and any vector $y \in V$, the $k$-th power of the annihilation operator $a_i$ commutes with the creation operator $a^\dagger_c$. That is: $$(a_i)^k (a^\dagger_c y) = a^\dagger_c ((a_i)^k y)$$ where $a_i$ denotes the annihilation operator for color $i$ and $a^\dagger_c$ denotes the creation operator for color $c$.

Let $L$ be a ladder system on a vector space $V$ over a field $K$ with $d$ modes. Let $a_i$ and $a^\dagger_i$ denote the annihilation and creation operators for mode $i \in \{0, \dots, d-1\}$, respectively. Given a vector $x \in V$ that is annihilated by $a_i$ (i.e., $a_i x = 0$) and a list of indices $v$, let $k = \text{count}(v, i)$ be the number of occurrences of $i$ in $v$. Then, applying the annihilation operator $a_i$ exactly $k$ times to the creation-operator word $a^\dagger_{v_1} \dots a^\dagger_{v_n} x$ removes all occurrences of $a^\dagger_i$ and scales the result by $k!$: $$(a_i)^k (a^\dagger_{v_1} \dots a^\dagger_{v_n} x) = k! \cdot (a^\dagger_{w_1} \dots a^\dagger_{w_m} x)$$ where $[w_1, \dots, w_m]$ is the list $v$ after filtering out all indices equal to $i$.

Given a ladder system $L$ on a vector space $V$ over a field $K$ with $d$ modes (or "colors"), a list of indices $cs$, and a function $\alpha: \{0, \dots, d-1\} \to \mathbb{N}$ providing exponents, the joint annihilation monomial is the endomorphism of $V$ defined by the product: $$\prod_{c \in cs} (a_c)^{\alpha(c)}$$ where $a_c$ denotes the annihilation operator corresponding to the mode $c$.

For a ladder system $L$ and any exponent function $\alpha: \{0, \dots, d-1\} \to \mathbb{N}$, the joint annihilation monomial associated with an empty list of indices, denoted as $L.\text{annMono} \ [] \ \alpha$, is equal to the identity operator $1$.

For a ladder system $L$ with $d$ modes, let $a_c$ denote the annihilation operator corresponding to the mode $c \in \{0, \dots, d-1\}$. For any index $c$, list of indices $cs$, and exponent function $\alpha: \{0, \dots, d-1\} \to \mathbb{N}$, the joint annihilation monomial $\text{annMono}$ for the list starting with $c$ followed by $cs$ is given by: $$\text{annMono}(c :: cs, \alpha) = a_c^{\alpha(c)} \circ \text{annMono}(cs, \alpha)$$ where the multiplication on the right-hand side represents the composition of endomorphisms.

Let $L$ be a ladder system on a vector space $V$ over a field $K$ with $d$ modes. Let $x \in V$ be a vector annihilated by all annihilation operators, such that $a_i x = 0$ for all $i \in \{0, \dots, d-1\}$. Suppose $cs$ is a list of distinct color indices, $\alpha : \{0, \dots, d-1\} \to \mathbb{N}$ is a function providing exponents, and $v$ is a list of indices representing a creation-operator word. If for every color $c$ in the list $cs$, the number of occurrences of $c$ in $v$ is exactly $\alpha(c)$, then applying the joint annihilation monomial to the word applied to $x$ yields: $$\left( \prod_{c \in cs} a_c^{\alpha(c)} \right) (a^\dagger_{v_1} a^\dagger_{v_2} \dots a^\dagger_{v_n} x) = \left( \prod_{c \in cs} \alpha(c)! \right) \cdot (a^\dagger_{w_1} a^\dagger_{w_2} \dots a^\dagger_{w_m} x)$$ where $a_c$ and $a^\dagger_i$ are the annihilation and creation operators of the ladder system, respectively, and the sequence of indices $[w_1, \dots, w_m]$ is the sublist of $v$ containing only the elements not present in $cs$.

For any natural number $d$ and any count function $\alpha : \{0, \dots, d-1\} \to \mathbb{N}$, the word $\text{countWord}(d, \alpha)$ (which repeats each color $c$ exactly $\alpha(c)$ times) contains only elements that are valid color indices from the set $\{0, \dots, d-1\}$. Consequently, filtering the list $\text{countWord}(d, \alpha)$ to retain only those elements not contained in the range of all possible colors (`List.finRange d`) results in an empty list: $$(\text{countWord}(d, \alpha)).\text{filter}(\lambda y. y \notin [0, \dots, d-1]) = [].$$

For any natural numbers $d$ and $b$ where $b > 0$, and for any functions $\gamma, \gamma' : \text{Fin } d \to \mathbb{N}$ representing sequences of digits, if all digits satisfy $\gamma(i) < b$ and $\gamma'(i) < b$ for all $i \in \text{Fin } d$, then the equality of their base-$b$ positional encodings $$\sum_{i \in \text{Fin } d} \gamma(i) b^i = \sum_{i \in \text{Fin } d} \gamma'(i) b^i$$ implies that the sequences are identical, i.e., $\gamma = \gamma'$.

For a natural number $n$ and a count function $\gamma : \{0, \dots, d-1\} \to \mathbb{N}$, the function calculates the base-$(n+1)$ positional encoding of $\gamma$, defined by the formula: $$\text{countEncode}(n, \gamma) = \sum_{i=0}^{d-1} \gamma(i) (n + 1)^i$$ where $i$ ranges over the indices of the modes (or "colors") and $\gamma(i)$ represents the occupation number of the $i$-th mode. Because the total number of particles is $n$ (i.e., $\sum \gamma(i) = n$), each $\gamma(i)$ is strictly less than the base $n+1$, ensuring the encoding is injective.

For any natural numbers $d$ and $n$, and any two count functions $\gamma, \gamma' : \{0, \dots, d-1\} \to \mathbb{N}$, if the sum of the values of each function is equal to $n$ (i.e., $\sum_{i=0}^{d-1} \gamma(i) = n$ and $\sum_{i=0}^{d-1} \gamma'(i) = n$), then the equality of their base-$(n+1)$ positional encodings $$\sum_{i=0}^{d-1} \gamma(i) (n + 1)^i = \sum_{i=0}^{d-1} \gamma'(i) (n + 1)^i$$ implies that the functions themselves are identical, $\gamma = \gamma'$.

Let $L$ be a ladder system with vacuum state $\Omega$. For any count function $\alpha : \{0, \dots, d-1\} \to \mathbb{N}$ that assigns an occupation number to each of the $d$ colors, the corresponding occupation-number state $$a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} \Omega$$ is non-zero, where the sequence of indices $i_1, i_2, \dots, i_n$ is the canonical word associated with $\alpha$ (containing each color $c$ exactly $\alpha(c)$ times).

Let $L$ be a ladder system on a $K$-vector space $V$ with $d$ modes, and let $\Omega \in V$ be a vacuum state. For any total particle number $n \in \mathbb{N}$ and any count function $\alpha$ of degree $n$ (representing occupation numbers $\alpha(i)$ such that $\sum_{i=0}^{d-1} \alpha(i) = n$), the occupation-number state $$\psi_\alpha = a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} \Omega$$ (constructed by applying the creation operators according to the canonical word of $\alpha$) is an eigenvector of the operator $$M = \sum_{i=0}^{d-1} (n+1)^i N_i$$ where $N_i$ is the number operator for the $i$-th mode. The corresponding eigenvalue is the base-$(n+1)$ positional encoding of $\alpha$, given by $$\sum_{i=0}^{d-1} \alpha(i) (n+1)^i$$ evaluated in the field $K$.

Let $L$ be a ladder system over a field $K$ and a vector space $V$. Suppose $\Omega \in V$ is a vacuum state. For any total particle number $n \in \mathbb{N}$, the set of occupation-number states $$\left\{ (a^\dagger_0)^{\alpha(0)} (a^\dagger_1)^{\alpha(1)} \dots (a^\dagger_{d-1})^{\alpha(d-1)} \Omega \right\}_{\alpha \in \text{CountFun}(d, n)}$$ indexed by count functions $\alpha : \{0, \dots, d-1\} \to \mathbb{N}$ such that $\sum_{i=0}^{d-1} \alpha(i) = n$, is linearly independent over $K$. Here, $a^\dagger_i$ represents the creation operator associated with the $i$-th mode of the ladder system.

Let $L$ be a ladder system over a field $K$ with $d$ modes and a vacuum state $\Omega \in V$. For any natural number $n$, the $n$-th excitation-number sector $\text{vacuumSpan}(L, \Omega, n)$ is equal to the $K$-linear span of the occupation-number states indexed by count functions of degree $n$: $$\text{vacuumSpan}(L, \Omega, n) = \text{span}_K \left\{ (a^\dagger_0)^{\alpha(0)} (a^\dagger_1)^{\alpha(1)} \dots (a^\dagger_{d-1})^{\alpha(d-1)} \Omega \mid \alpha \in \text{CountFun}(d, n) \right\}$$ where $\alpha: \{0, \dots, d-1\} \to \mathbb{N}$ satisfies $\sum_{c=0}^{d-1} \alpha(c) = n$, and $a^\dagger_i$ denotes the creation operator for the $i$-th mode.

For a $d$-mode ladder system $L$ over a field $K$ and a vacuum state $\Omega$ in a vector space $V$, the occupation-number basis of the $n$-th excitation-number sector $\text{vacuumSpan}(L, \Omega, n)$ is the basis indexed by count functions $\alpha \in \text{CountFun}(d, n)$. Each basis vector is formed by the application of creation operators to the vacuum state: $$\left( a^\dagger_0 \right)^{\alpha(0)} \left( a^\dagger_1 \right)^{\alpha(1)} \dots \left( a^\dagger_{d-1} \right)^{\alpha(d-1)} \Omega$$ where $\alpha(i)$ is the number of excitations in the $i$-th mode and the total number of excitations is $\sum_{i=0}^{d-1} \alpha(i) = n$.

Let $L$ be a $d$-mode ladder system over a field $K$ acting on a vector space $V$, and let $\Omega \in V$ be a vacuum state. For any $n \in \mathbb{N}$ and any count function $\alpha \in \text{CountFun}(d, n)$, the basis vector in the occupation-number basis $\text{vacuumBasis}(L, \Omega, n)$ indexed by $\alpha$ is equal to the vector obtained by applying the corresponding creation operators to the vacuum state: $$\text{vacuumBasis}(L, \Omega, n)_\alpha = (a^\dagger_0)^{\alpha(0)} (a^\dagger_1)^{\alpha(1)} \dots (a^\dagger_{d-1})^{\alpha(d-1)} \Omega,$$ where $a^\dagger_i$ denotes the creation operator for the $i$-th mode and $\alpha(i)$ is the number of excitations in that mode.

Let $L$ be a $d$-mode ladder system over a field $K$ acting on a vector space $V$, and let $\Omega \in V$ be a vacuum state for $L$. For any $n \in \mathbb{N}$, the dimension of the $n$-th excitation-number sector $\text{vacuumSpan}(L, \Omega, n)$—defined as the subspace spanned by applying $n$ creation operators to $\Omega$—is given by the binomial coefficient $$\dim_K(\text{vacuumSpan}(L, \Omega, n)) = \binom{d + n - 1}{n}.$$ This value represents the degeneracy of the $n$-th energy level of a $d$-dimensional bosonic oscillator.

Let $L$ be a ladder system with $d$ modes acting on a $K$-vector space $V$. For any vacuum state $\Omega \in V$ and any excitation number $n \in \mathbb{N}$, the $n$-th excitation-number sector $\text{vacuumSpan}(L, \Omega, n)$ is a finite-dimensional subspace of $V$ over $K$. Here, $\text{vacuumSpan}(L, \Omega, n)$ is defined as the $K$-linear submodule spanned by all vectors of the form $$a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} \Omega$$ where each $i_j \in \{0, 1, \dots, d-1\}$.

Let $L$ be a ladder system with $d$ modes on a vector space $V$ over a field $K$, and let $\Omega \in V$ be a vacuum state (satisfying the property `HasVacuum`). If the number of modes $d$ is strictly positive ($d > 0$), then for any natural number $n$, the $n$-th excitation-number sector $\text{vacuumSpan}(L, \Omega, n)$ is not the trivial subspace $\{0\}$.