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Structure of the harmonic oscillator in the space of $n$-particle Glauber correlators. E. Zubizarreta Casalengua, J. C. López Carreño, E. del Valle and F. P. Laussy in J. Math. Phys. 58:062109 (2017).  What the paper says!?

Here, we "chart" the Hilbert space of the harmonic oscillator by providing some intuition of where its quantum states live. The basic intuition is: if we take a random state of the harmonic oscillator, where is it likely to be found?

This needs to clarify two questions:

1. What is meant by "taking a random state of the harmonic oscillator"?
2. What is mean by "where is it found?"

The former is understood as a uniform probability over the space itself. If the space is truncated, to $N$ particles at most, this has a clear answer. For instance for $N=2$, a random state is $$\ket{\psi}=\alpha_0\ket{0}+\alpha_1\ket{1}+\alpha_2\ket{2}$$ and a random pick is taken by [doing what with the $\alpha$s?].

Mathematically, this corresponds to taking the Haar measure, i.e., [what?]

The second question is answered by locating points of the Hilbert space in a map of useful or familiar or well-known attributes (which in reality are never entirely any of this). To characterise quantum states, Glauber's $n$th-order correlators $g^{(n)}$ are typical, so we can ask,

• The unnormalised correlators $G^{(n)}=\sum_k k(k-1)\cdots(k-n+1)P_k$ are in bijection with the $P_k$, inverted by $P_i=\sum_{j\ge i}(-1)^{i+j}G^{(j)}/[i!(j-i)!]$ (Eq. 10).
• A uniform sampling of $\mathcal H_N$ gives the joint density $\mathcal P_g(n_0,g^{(2)},\dots,g^{(N)})=n_0^{(N^2+N-2)/2}/\mathrm{sf}(N-1)$ on its support (Eq. 15): it depends on the population only; the structure of the chart is all in the shape of the support.
• $\mathcal H_2$ (Fig. 2): support $\lfloor n_0\rfloor(2n_0-\lfloor n_0\rfloor-1)/n_0^2\le g^{(2)}\le1/n_0$, density $n_0^2$, marginals in closed form; the most likely antibunching is $g^{(2)}=1/2$, that of $|2\rangle$.
• $\mathcal H_3$ (Fig. 3): bounds on each of $n_0,g^{(2)},g^{(3)}$ in terms of the two others (Eqs. 26–29), marginals as piecewise polynomials (Appendix A), $\mathcal P_g(g^{(2)},g^{(3)})=U^6/12$ with $U$ the population upper bound (Eq. 30); no closed form found for $\mathcal P_g(g^{(3)})$.
• For every $N$: the population marginal is the Irwin–Hall law (Eq. 32), and in each $(n_0,g^{(k)})$ plane $\frac{(N-1)!}{(N-k)!}\frac1{n_0^{k-1}}\ge g^{(k)}\ge\frac{\lfloor n_0\rfloor!}{(\lfloor n_0\rfloor-k)!\,n_0^k}\big(1+\frac{k(n_0-\lfloor n_0\rfloor)}{\lfloor n_0\rfloor+1-k}\big)$ (Eq. 33), both reached by two-state "coin" superpositions; the upper bound recedes with $N$, so $\mathcal H_\infty$ keeps only the lower one (Fig. 4).
• Physics: states with $g^{(2)}<1/2$ and $n_0>1$ exist (e.g. $\sqrt p|1\rangle+\sqrt{1-p}|2\rangle$), so $g^{(2)}<1/2$ certifies fewer than two particles on average, not single-photon emission, for which only $g^{(2)}=0$ will do; correlators of different orders are unconstrained with respect to each other, only with respect to $n_0$, so arbitrary bunching at one order coexists with antibunching at another; a diverging correlator forces the state onto vacuum in $\mathcal H_N$ but not in $\mathcal H_\infty$.
• Monte Carlo charts for $N=2$ to $5$ (Fig. 5) show the coin-state "scars" fading with $N$.

This paper was the TFG (Trabajo de Fin de Grado) of Eduardo Zubizarreta Casalengua, and is a testimony of his exceptional abilities that it led to a publication even before his TFM (Trabajo de Fin de Master).