Quantum gate description for induced coherence without induced emission and its applications

We introduce unitary quantum gates for photon pair creation in spontaneous parametric down-conversion nonlinear crystals (NLs) and for photon path alignment. These are the two key ingredients for the method of induced coherence without induced emission and many ensuing variations thereof. The difficulty in doing so stems from an apparent mixing of the mode picture (such as the polarization of photons) and the Fock picture (such as the existence of the photons). We illustrate utility of these gates by obtaining quantum circuits for the experimental setups of the frustrated generation of photon pairs, identification of a pointlike object with undetected photons, and creation of a Bell state. We also introduce an effective nonunitary description for the action of NLs in experiments where all the NLs are pumped coherently. As an example, by using this simplifying picture, we show how NLs can be used to create superposition of given quantum states in a modular fashion.

Figure 1

Unitary gate for an NL. Here a filled circle implies that the corresponding $not$ gate (which transforms $|H\rangle$ and $|0\rangle$ to each other) applies on the target photon when the state of the control photon is $\left|H\right.〉$; otherwise, it acts as an identity.

Figure 2

Setups for (left) frustrated generation of photon pairs and (right) generation of a maximally entangled pair.

Figure 3

Quantum circuit for frustrated down-conversion.

Figure 4

Quantum circuit for identification of a pointlike object with undetected photons. ${\mathcal{E}}_O$ is the object, $U_H$ is the Hadamard gate, and BS denotes the beam splitter.

Figure 5

Quantum circuit for creation of uniform superposition.

Figure 6

Unitary gate for an NL when pump is a single photon. Filled circle denotes the corresponding $not$ gate applied on the target photon when the control photon is in $\left|H\right.〉$, and unfilled circle denotes that the $not$ gate is applied on the target when the control photon is in $\left|0\right.〉$; otherwise, it acts as an identity gate.

Figure 7

Experimental setup (left) and quantum circuit (right) for the frustrated generation of photon pairs with a single-photon pump.

Figure 8

A system with $N_s$ signal and $N_i$ idler paths, where an NL is applied on paths $n_s{n}_i^{\prime }$