Optimizing the multiphoton absorption properties of maximally path-entangled number states

In this paper we examine the $N$-photon absorption properties of maximally path-entangled number states ($N00N$ states). We consider two cases. The first involves the $N$-photon absorption properties of the ideal $N00N$ state, one that does not include spectral information. We study how the $N$-photon absorption probability of this state scales with $N$, confirming results presented by others in a previous paper by a different method. We compare this to the absorption probability of various other states. The second case is that of two-photon absorption for an $N=2$ $N00N$ state generated from a type-II spontaneous down-conversion event. In this situation we find that the absorption probability is both better than analogous coherent light (due to frequency entanglement) and highly dependent on the optical setup. We show that the poor production rates of quantum states of light may be partially mitigated by adjusting the spectral parameters to improve their two-photon absorption rates. This work has application to quantum imaging, particularly quantum lithography, where the $N$-photon absorbing process in the lithographic resist must be optimized for practical applications.

Figure 1

Multiphoton absorption rate of various sources as a function of the average number of photons in the field ($N\text{th}$ order absorption vs $\overline{N}$ average photons). All rates normalized relative to coherent states (solid line). The dashed line is a thermal state. The dotted line is a Fock (number) state and the dot-dashed line is a $N00N$ state.

Figure 2

(Color online) The basic setup. A nonlinear crystal (BBO in this case) creates a degenerate pair of photons. Each photon is subjected to a filter. A polarization rotator ensures that the two photons are indistinguishable. A beam splitter creates a $|2::0⟩$ state that results in an interference pattern.

Figure 3

(Color online) Contour plot of the absolute value of the biphoton amplitude for our setup $\left|A\right|$. ${\sigma }_e={\sigma }_o={\sigma }_p={10}^{13}\text{Hz}$ and $L=1.5\text{cm}$. $t_1^d$ and $t_2^d$ are in units of ${10}^{-13}\text{sec}$.

Figure 4

(Color online) A plot of the scaled two-photon absorption probability for the realistic $|2::0⟩$ state as a function of the logs (base 10) of the bandwidths of the filters in the arms of the interferometer (in hertz). $L=2.3\text{mm}$, ${\sigma }_p={10}^{12}\text{Hz}$, and ${\kappa }_f={10}^{14}\text{Hz}$.

Figure 5

(Color online) A plot of the scaled two-photon absorption probability for the realistic $|2::0⟩$ state as a function of the log (base 10) of the bandwidth of the pump (in hertz) for three separate settings of the width of the final state of the absorber (in hertz). $L=2.3\text{mm}$ and ${\sigma }_e={\sigma }_o={10}^{13}\text{Hz}$.

Figure 6

(Color online) A plot of the scaled two-photon absorption probability for the realistic $|2::0⟩$ state as a function of the length of the crystal for two different settings of the filters (in hertz). ${\sigma }_p={10}^{12}\text{Hz}$ and ${\kappa }_f={10}^{14}\text{Hz}$.

Figure 7

A plot of the scaled two-photon absorption probability for the $|2::0⟩$ state (in the limit of no filtering) as a function of the bandwidth of the pump. We indicate three separate settings of the width of the absorber. We also set $L=1\text{mm}$. The scaling is the same as on the other pulse-pumped plots.

Figure 8

(Color online) Plot of the two-photon absorption rate from a cw-pumped crystal, as a function of crystal length, for four separate settings of the filters (in hertz).

Figure 9

(Color online) Plot of the two-photon absorption rate from a cw-pumped crystal as a function of the bandwidths of the filters in the ordinary and extraordinary arms. Here $L=6\text{mm}$.