Simulations and experiments on polarization squeezing in optical fiber

We investigate polarization squeezing of ultrashort pulses in optical fiber, over a wide range of input energies and fiber lengths. Comparisons are made between experimental data and quantum dynamical simulations to find good quantitative agreement. The numerical calculations, performed using both truncated Wigner and exact $+P$ phase-space methods, include nonlinear and stochastic Raman effects, through coupling to phonon variables. The simulations reveal that excess phase noise, such as from depolarizing guided acoustic wave Brillouin scattering, affects squeezing at low input energies, while Raman effects cause a marked deterioration of squeezing at higher energies and longer fiber lengths. We also calculate the optimum fiber length for maximum squeezing.

Figure 1

(a) Representation in phase space of the evolution of a coherent beam (bottom right) under effect of the Kerr nonlinearity, which generates a quadrature (or Kerr) squeezed state (upper left). The arrow indicates the direction of state evolution with propagation. (b) Polarization squeezing generated by overlapping two orthogonally (i.e., $x$- and $y$-) polarized quadrature-squeezed states.

Figure 2

Representation of the variances of a polarization squeezed (upper left) and a coherent state (lower right) on the Poincaré sphere.

Figure 3

Schematic of the single-pass method for the efficient production of polarization squeezed states. The Stokes measurement after the fiber scans the dark ${\widehat{S}}_1-{\widehat{S}}_2$ plane of the circularly $⟨{\widehat{S}}_3⟩\ne 0$ polarized output.

Figure 4

(Color online) Noise power against phase-space rotation angle for the rotation of the measurement half-wave plate for a pulse energy of $83.7\mathrm{pJ}$ using $13.3\mathrm{m}$ of fiber I. Inset: Schematic of the projection principle for angle $\theta$.

Figure 5

(Color online) Linear noise reduction against optical transmission for the polarization squeezing generated by pulses of an energy of $81\mathrm{pJ}$ in $3.9\mathrm{m}$ of fiber I.

Figure 6

(Color online) Plot showing a stable squeezing of $-4.0\pm 0.3\mathrm{dB}$ over $100\min$. A $30\mathrm{m}$ optical fiber with a pulse energy of $80\mathrm{pJ}$ was used.

Figure 7

(Color online) Measurement results and theoretical simulations for $13.2\mathrm{m}$ 3M FS-PM-7811 fiber (run II) as a function of pulse energy: (a) the squeezing angle, (b) the squeezing and (c) antisqueezing noise. Solid and dashed lines show the simulation results with and without additional phase noise, with linear losses taken to be 13%. The shading indicates simulation uncertainty. The simulation result without third-order dispersion is given by the dots in (b). Diamonds represent the experimental results, with experimental uncertainty indicated by the error bars in the squeezing. Both the simulation and the experimental errors were too small to be plotted distinctly for the squeezing angle and antisqueezing. The measured noises are corrected for $-85.1\pm 0.1\mathrm{dBm}$ electronic noise.

Figure 8

Illustration of the effect of excess phase noise on different squeezing ellipses. Dashed line gives the Kerr-squeezed ellipse and the solid line gives the ellipse with added phase noise. The effect on the squeezing and the angle is less for the ellipse with larger Kerr squeezing (lower ellipse).

Figure 9

Simulation pulse-spectrum at pulse energies (a) $1.5E_s$, (b) $E_s$, and (c) $0.5E_s$ at rescaled distances of $\zeta =0$ (dot-dashed line) and $\zeta =25$ with (solid line) and without (dashed line) third-order dispersion.

Figure 10

Simulation pulse shape (a) and spectrum (b) at pulse energies $1.5E_s$ and normalized propagation length $\zeta =25$, with (solid line) and without (dashed line) Raman effects.

Figure 11

Squeezing degradation at high intensity: (a) squeezing and (b) antisqueezing measurements for $L=13.35\mathrm{m}$ of fiber I. Solid and dashed lines shows the simulation results with and without Raman effects (i.e., a purely electronic nonlinearity), respectively. The data points marked by squares are the results of exact $+P$ calculations with error bars indicating estimated sampling error. The points at 109 and $134\mathrm{pJ}$ were calculated with 10 000 trajectories; the other 4 $+P$ points were calculated with 1000 trajectories. Note that these results were obtained in an experimental setup with higher losses than that of Fig. 7, giving a reduced magnitude of raw squeezing. The simulations here assume 19.9% loss.

Figure 12

Experiments (corrected for dark noise) and simulations (with and without fitted phase noise) of the polarization squeezing, antisqueezing, and squeezing angle for 3.9 [(a), (c), and (e)] and $13.3\mathrm{m}$ [(b), (d), and (f)] of fiber I.

Figure 13

Experiments (corrected for dark noise) and simulations (with and without fitted phase noise) of the polarization squeezing, antisqueezing, and squeezing angle for 20 [(a), (c), and (e)] and $30\mathrm{m}$ [(b), (d), and (f)] of fiber II. The lighter data points in (c) are from a corrected experimental run.

Figure 14

Experiments (corrected for dark noise) and simulations (with and without fitted phase noise) of the polarization squeezing, antisqueezing, and squeezing angle for 50 [(a), (c), and (e)] and $166\mathrm{m}$ [(b), (d), and (f)] of fiber II. The amount of dispersion over $166\mathrm{m}$ makes the simulations impractical for this case.

Figure 15

Simulated squeezing as a function of fiber length for various total input energies: $E=74.4\mathrm{pJ}$ (triangles), $E=83.7\mathrm{pJ}$ (diamonds), $E=93\mathrm{pJ}$ (squares), $E=108\mathrm{pJ}$ (circles), and $E=119\mathrm{pJ}$ (crosses). Linear loss and phase noise have not been included (Fiber I).