Macroscopic quantum effects for classical light

Optical analogies of macroscopic quantum effects (Schrödinger cat states, squeezing, collapse, and revival) for light beams propagating in an inhomogeneous linear medium are demonstrated theoretically using exact analytical solutions of the wave equation. It is shown that the coherent superposition of macroscopically distinguishable states is generated via mode interference from an initial off-axis single wave packet. Squeezed cat states with a fidelity >99% arise periodically and disappear rapidly within limited intervals of a propagation distance. Collapse and revival of wave packets at long-term nonparaxial evolution due to mode interference is demonstrated. Oscillations of the beam trajectory occur with extremely small amplitude, of the order of $10{}^{-19}$ m, which is typical of the estimated displacement caused by cosmic gravitational waves in gravity-wave detectors.

Figure 1

Trajectory (solid line) and linear momentum (dashed line) of the beam of wavelength $\lambda =0.63$ μm as a function of distance for $p$₀ = 0 and $x$₀ = 15 μm (a) and $x$₀ = 30 μm (c), and their higher-resolution plots, indicating oscillations with period of $L_{cl}$ (b, d).

Figure 2

Phase-space plots for different values of $x$₀: $x$₀ = 1 μm (a, b), $x$₀ = 20 μm (c, d), and $x$₀ = 30 μm (e, f). Solid lines correspond to the exact solution (16), and dashed lines to the solution (15).

Figure 3

Representation in phase space of the paraxial (a) and nonparaxial (b) evolution of a coherent beam in an optical waveguide: $\left|\alpha \left(0\right)\right|=\left|\alpha \left(z\right)\right|$ (a), and $\left|\alpha \left(0\right)\right|\ne \left|\alpha \left(z\right)\right|$ (b).

Figure 4

Evolution of the second-order moments ${\sigma }_x$ (a) and ${\sigma }_p$ (c) and their higher-resolution plots (b, d), and the uncertainty products of Heisenberg (e) and Schrödinger-Robertson (f) for $x$₀ = 20 μm.

Figure 5

Position-momentum covariance (a), correlation radius (b), and relative squeezing (c) variations with distance for $x$₀ = 20 μm, and their higher-resolution plots (d, e, f).

Figure 6

The autocorrelation function $P\left(z\right)={\left|\langle \psi (x,0)|\psi (x,z)\rangle \right|}^2\mathrm{for}$ a value of $\left|\alpha \right|=3.74$ (a) and its high-resolution plot (b).

Figure 7

Wave-packet intensity distribution evolution with distance for $x$₀ = 20 μm. Generation of a cat state at $L$ = $L_{\mathrm{rev}}/2$ (d, j) and revival of initial wave packet at $L$ = $L_{\mathrm{rev}}$ (h, k). Wave-packet shapes (c, e, g) at fractional revivals.

Figure 8

Variation of the fidelity of wave packet with Yurke-Stoler state with distance for $x$₀ = 20 μm (a). Higher-resolution plot of the fidelity, indicating regular spikes of the fidelity with distance $\delta z\ll L_{cl}$(b).

Figure 9

Scheme for coupling a mechanical oscillator's position to optical butt-jointed waveguides $W1$ and $W2$: $L$–laser, $P$–photodetector.