Entanglement-seeded, dual, optical parametric amplification: Applications to quantum imaging and metrology

The study of optical parametric amplifiers (OPAs) has been successful in describing and creating nonclassical light for use in fields such as quantum metrology and quantum lithography [Agarwal et al., J. Opt. Soc. Am. B 24, 2 (2007)]. In this paper we present the theory of an OPA scheme utilizing an entangled state input. The scheme involves two identical OPAs seeded with the maximally path-entangled $\mid N00N⟩$ state $(\mid 2,0⟩+\mid 0,2⟩)∕\sqrt{2}$. The stimulated amplification results in output state probability amplitudes that have a dependence on the number of photons in each mode, which differs greatly from two-mode squeezed vacuum. A large family of entangled output states are found. Specific output states allow for the heralded creation of $N=4$ $N00N$ states, which may be used for quantum lithography, to write sub-Rayleigh fringe patterns, and for quantum interferometry, to achieve Heisenberg-limited phase measurement sensitivity.

Figure 1

(Color online) Schematic of optical parametric amplification and spontaneous parametric down-conversion. The bottom diagram shows the crystal being pumped by a laser, along with the signal and idler output modes. The top left diagram depicts momentum conservation between the pump, signal, and idler modes, otherwise known as the phase-matching condition. The top right picture diagrammatically shows energy conservation of the system, such that the frequency of the pump is equal to the signal and idler frequencies added together. These restrictions lead to the path entanglement we desire in our scheme.

Figure 2

(Color online) Diagram of our entanglement-seeded, dual, optical amplification scheme. On the far left is the relatively weak pump beam pumping a nonlinear crystal in the low-gain regime in order to produce spontaneous parametric down-conversion. This output state, taken initially to be ∣1,1⟩, is then incident on a beam splitter, which leads to the maximally spatially entangled state $(\mid 2,0⟩+\mid 0,2⟩)∕\sqrt{2}$. This state is then incident into one mode of each of the two OPAs. The other two modes are left as vacuum inputs. We then assume that OPA I and OPA II are pumped by the same high gain laser in order to achieve parametric amplification. Note that the pumps for all three of the nonlinear crystals are to be phase locked. All four modes are amplified, resulting in entanglement between modes $B$ and $C$. We have drawn photodetectors at modes $A$ and $D$, which are to be used in heralded production of specific states.

Figure 3

(Color online) Probability of obtaining output states with $n=m$ for a fixed gain of $r=1.08$. A joint detection of equal photon numbers at modes $A$ and $D$ results in the inner two modes being in the state $\mid n+2,n⟩+\mid n,n+2⟩$. We see that for this experimentally feasible gain that the vacuum term $n=0$ is no longer the most probable outcome, whereas the desirable $n=1$ output is.

Figure 4

(Color online) Probabilities of obtaining $n$ and $m$ photon states for fixed $r=1.08$. The most likely joint photodetection at detectors $D_A$ and $D_D$ is when each mode contains only one photon. The vacuum $n=m=0$ term is the top diagonal term, and is not visible because the $n=m=1$ term is more probable. A joint photodetection of $n=m=1$ leads to the entangled state $\mid 3,1⟩+\mid 1,3⟩$ between modes $B$ and $C$, which when incident on a beam splitter leads to the $N=4$ $N00N$ state $\mid 4,0⟩+\mid 0,4⟩$.