Two-photon self-Kerr nonlinearities for quantum computing and quantum optics

The self-Kerr interaction is an optical nonlinearity that produces a phase shift proportional to the square of the number of photons in the field. At present, many proposals use nonlinearities to generate photon-photon interactions. For propagating fields these interactions result in undesirable features such as spectral correlation between the photons. Here we engineer a discrete network composed of cross-Kerr interaction regions to simulate a self-Kerr medium. The medium has effective long-range interactions implemented in a physically local way. We compute the one- and two-photon $S$ matrices for fields propagating in this medium. From these scattering matrices we show that our proposal leads to a high-fidelity photon-photon gate. In the limit where the number of nodes in the network tends to infinity, the medium approximates a perfect self-Kerr interaction in the one- and two-photon regime.

Figure 1

(a) An interaction between two input fields is mediated by a cavity containing a cross-Kerr medium, described by the Hamiltonian $H_{\mathrm{int}}=\chi a^{†}a{b}^{†}b$. The cavity decay rate is $\gamma$ and the cavity Hamiltonian is $\mathrm{\Delta }{a}^{†}a$. (b) By feeding one output mode of the cavity into the other input mode, we expect to obtain an effective self-Kerr medium. (c) Cavity with a self-Kerr medium such that $H_{\mathrm{int}}=\chi a^{†}a{a}^{†}a$, which is equivalent to $H_{\mathrm{int}}=\chi a^{†}{a}^{†}aa$ up to a term proportional to the number operator.

Figure 2

(a) Chain of $N$ interaction sites. The unit cell in Fig. 1 is represented differently here. Closed circles represent cavity modes, while the purple dashed lines represent the cross-Kerr interaction. We enforce chirality of the fields by using circulators, which we represent by open circles with arrows. By cascading $N$ interaction sites and connecting the output of the upper chain into the input of the lower chain, we simulate counterpropagating conditions, the basis of the proposal in [34, 35]. (b) The cascading in (a) can also be seen effectively as a local implementation of a series of nonlocal interactions, similar to a spin-ladder-type interaction [36, 37, 38, 39].

Figure 3

(a) Single cross-Kerr interaction site which we supplement with a mirror to mimic a self-Kerr site. (b) Generalization of (a) to a chain with two sites.

Figure 4

(a) Realization of our interaction site using two three-level atoms. The transitions of each atom ($a$ and $b$) are addressed by different field modes. The level structure is constructed so as to induce a nonlinear effect only if both photons are absorbed, one by each atom. The component proportional to $4\chi$ is never accessed by two photons and so can be dropped from the level structure. (b) Six-level structure equivalent to (a) if there are fewer than two excitations in the field. Each transition is addressed by a different field mode, colored as in Fig. 1.

Figure 5

(a) We interpret the square of the wave packet as the probability to find the photon in a given position. Consider the semiclassical picture where we sample photons from the wave packet at discrete sites. Suppose the first photon is sampled at some position and the second at different distances from it (I–III). If interactions are local, as in (b), only when the second photon is at position II do they interact. With interactions of varying distances, as in (c), the photons are more likely to interact no matter how far apart they were sampled. The nonlocality structure of (c) is effectively mimicked by our mirror arrangement (cf. Fig. 2).

Figure 6

Three cphase gates using Kerr nonlinearities and dual-rail encoding [7]. For simplicity, we chose the encoding such that the ${|1\rangle }_{Q1}$ and ${|1\rangle }_{Q2}$ logical states are given by the ${|01\rangle }_{12}$ and ${|10\rangle }_{34}$ physical states, respectively. (a) Cross-Kerr interaction applied to one mode of each qubit (proposed in [52, 53]). (b) Two modes pass through a balanced beam splitter. Whenever both are occupied with a single photon the photons exit together and are routed to two parallel nonlinear sign-shift gates [7, 8], which can be implemented using our self-Kerr construction. (c) By using half waveplates (HWP) and polarizing beam splitters (PBS), we can store the dual-rail encoding temporarily in the photon polarization, enabling the gate with a single self-Kerr interaction, of course, under the caveat that the interaction must be fully polarization independent.