Switchable particle statistics with an embedding quantum simulator

We propose the implementation of a switch of particle statistics with an embedding quantum simulator. By encoding both Bose-Einstein and Fermi-Dirac statistics into an enlarged Hilbert space, the statistics of the simulated quantum particles may be changed in situ during the time evolution, from bosons to fermions and from fermions to bosons, as many times as desired before a measurement is performed. We illustrate our proposal with few-qubit examples, although the protocol is straightforwardly extendable to larger numbers of particles. This proposal can be implemented on different quantum platforms such as trapped ions, quantum photonics, and superconducting circuits, among others. The possibility to implement permutation symmetrization and antisymmetrization of quantum particles enhances the toolbox of quantum simulations for unphysical operations as well as for symmetry transformations.

Figure 1

Scheme of the proposed embedding quantum simulator for switchable two-particle statistics.

Figure 2

Scheme of the proposed protocol for $N$-particle statistics when ${log}_{2}N$ is a natural number. Rotation gate $R=exp(-i{\sigma }_y\pi /4)$. The $S$ gate is used to introduce the minus sign. And white and black control dots stand for spin up and down, respectively.

Figure 3

Probability distributions $|\psi (x_1,x_2){|}^2$ at time (a) $t=0$ for initial Gaussian wave packets and $t=\pi /2g$ for (b) Bose-Einstein and (c) Fermi-Dirac statistics, where $x_1$ and $x_2$ are in units of variances of the distributions ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, respectively. (d) Time-dependent expectation value for Bose-Einstein (solid red line) and Fermi-Dirac (dashed blue line) statistics of the nontrivial correlation $M=x_1^2{x}_2^2{\sigma }_1^x{\sigma }_2^x$.

Figure 4

Probability distributions $|\psi (x_1,x_2){|}^2$ with higher initial Fock states for (a) Bose-Einstein and (b) Fermi-Dirac statistics for $t=\pi /2g$ versus $x_1/{\mathrm{\Delta }}_1$ and $x_2/{\mathrm{\Delta }}_2$. (c) Time-dependent measurements for Bose-Einstein (solid red line) and Fermi-Dirac (dashed blue line) statistics versus $gt$ of the nontrivial correlation $M=x_1^2{x}_2^2{\sigma }_1^x{\sigma }_2^x$.

Figure 5

Probability distributions $|\psi (x_1,x_2){|}^2$ of all spin-up subspace at time $t=1.4\pi /g$ for (a) Bose-Einstein and (b) Fermi-Dirac statistics, where $x_1$ and $x_2$ are in units of the position variances ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, respectively.

Figure 6

Probability distributions $|\psi (x_1,x_2){|}^2$ for (a) Bose-Einstein and (b) Fermi-Dirac statistics in the subspace of all spins up at $t=1.4\pi /g$ with $x_3=0$ versus $x_1/{\mathrm{\Delta }}_1$ and $x_2/{\mathrm{\Delta }}_2$. Probability distributions $|\psi (x_1,x_3){|}^2$ for (c) Bose-Einstein and (d) Fermi-Dirac statistics in the subspace of all spins up at $t=1.4\pi /g$ with $x_2=0.5{\mathrm{\Delta }}_2$ versus $x_1/{\mathrm{\Delta }}_1$ and $x_3/{\mathrm{\Delta }}_3$.