Spatiotemporal few-photon optical nonlinearities through linear optics and measurement

It is well known that optical nonlinearities are extremely weak at the quantum, or single-photon, level. This has been one of the major difficulties for optical implementations of universal, scalable quantum computation. Knill, Laflamme, and Milburn [Nature (London) 409, 46 (2001)] showed, among other things, that one could perform the elusive two-qubit logic gates with only linear-optical elements if one also uses extra single photons and measurement. In this work, we apply linear-optics techniques to produce effects in few-photon beams that are more familiar to strong-field nonlinear optics. Specifically, we show that these methods are sufficient to change the spatial (or temporal) properties of a light beam with strong dependence on its constituent number of photons; such phenomena cannot occur via linear optics alone.

Figure 1

Method for producing spatial nonlinear optical effects at the few-photon level via linear optics and projective measurement. An $N$-photon state and a one-photon state are incident on a beam splitter of reflectivity $R$. These incident modes do not, in general, have the same beam shapes. A spatial filter is used to separate output mode $B$ photons into a single spatial mode $F$ and all other spatial modes that are orthogonal to $F$, labeled $\overline{F}$. The state of interest in mode 2 is produced conditional on having exactly one photon in mode $F$ and no photons in $\overline{F}$.

Figure 2

Two specific examples of conditional beam intensities for one-, two-, three-, and four-photon states at the beam-splitter output. Normalized intensity predictions as a function of position are shown and were produced using the correlation function method described in the text. An input $N$-photon state with intensity rms width $100\mu \mathrm{m}$ and an ancilla photon with rms intensity width $75\mu \mathrm{m}$ were mixed at a beam splitter with reflection probability (a) $R=83\%$ and (b) $R=76\%$. In (a), the rms width of the outgoing beam increases with increasing photon number—higher photon-number states expand more than lower. In (b) we move beyond this perturbative limit and the widths obey no such simple trend. Beam distortion is very apparent for the three-photon state due to strong destructive interference.

Figure 3

Intensity width vs beam-splitter reflection probability. For incident one-, two-, three-, and four-photon states with initial rms widths of $100\mu \mathrm{m}$, the rms width of the outgoing conditional intensity is shown as a function of the beam-splitter reflectivity for ancilla widths (a) 20, (b) 75, (c) 125, and (d) $500\mu \mathrm{m}$. The behavior in the limits as $R=1$ and $R\to 0$, the high reflectivity perturbative behavior, and the rapid, photon-number-dependent changes in the width are discussed in the text. The most important feature, however, is that the conditional intensity profiles are very different for different photon-number states. Such number-dependent beam profiles cannot occur with linear optics alone.