Down Converted Light

Spontaneous Parametric Down-Conversion Conceptual Overview

SPDC is an important process in quantum optics. Spontaneous parametric down conversion is the nonlinear process whereby two photons (called idler and signal) are created from a parent photon (called the pump photon). This is done by sending the pump photon through a nonlinear crystal which splits incoming photons into pairs of photons of lower energy whose combined energy and momentum are equal to the energy and momentum of the original photon. The state of the crystal is left unchanged in the process, which is why energy and momentum must be conserved. This dictates that the photon pair be entangled in the frequency domain. SPDC is stimulated by random vacuum fluctuations, as such the photon pairs are created at random times. However, if one of the pair (the "signal") is detected at any time then we know its partner (the "idler") is present.
SPDC allows for the creation of a single photon. This is the predominant mechanism used to create single photons for use in experiments.

The two down-converted photons exist in an entangled state and SPDC is the most efficient and useful method of obtaining two such entangled photons. Prior to the discovery SPDC, the calcium cascade effect was used in experiments as a photon source and actually was the first source used to verify the existence of single photons. The idea of using such a process as SPDC to produce entangled photon pairs was introduced in 1970 by David Burnham and Donald Weinberg who referred to the process as “parametric fluorescence.” The process was perfected by Leonard Mandel of the University of Rochester in 1985 who demonstrated the non-local effects of entangled states that could not be explained classically.

Implied in its title, SPDC is a spontaneous process in which there is only a chance that a photon incident on a non-linear crystal will split into two, meaning not all of the light incident on the crystal is down-converted. This brings up the issue of measuring the efficiency of the crystal used in order to gauge our experimental photon counts with how many we should expect. Now the two photons are produced simultaneously within the crystal, so energy and momentum must both be conserved. These two photons are also entangled, meaning they exist together in a superposition state of their individual states. However, entanglement means their individual states are inseparable under the entangled state and that the entangled state cannot simply be expressed as a product of the two states. In fact, we should not even be allowed to speak of the states composing the entangled superposition as “individual” due to their inseparable nature. The superposition state thus corresponds to a continuous range of possible energy and momentum values. We typically choose the degenerate case in which the frequency of the two down-converted photons are equal and half that of the pump. We also cut the optical axis of the crystal so that the two down-converted photons (known as the signal and idler) emerge from the crystal at a defined angle (three degrees in our case) relative to the propagation direction of the pump. However, the signal and idler can exit the crystal in different directions and as a result when multiple photon pairs are produced the signals and idlers form their own respective light cones. In our case, we are using Type-1 SPDC in which both the signal and idler are plane-polarized perpendicular to the polarization of the pump, meaning the signal and idler light cones overlap each other.

In order to select a single pair out of the cones we must place a pinhole in the light cone of the signal effectively “choosing” one photon in that light cone. This means its partner can be found by maneuvering a detector in the opposite light cone until a coincidence is counted. Having chosen a pair of photons in this manner, we can exploit the entangled nature of the pair in order to produce our single photon source. Since the photons are entangled, any measurement of the state of one of the photons will collapse its partner into the same state. Thus, by detecting the signal photon of our pair, we are ensured that there is a single photon on the chosen path in the opposite light cone. This is our single photon source [1].

The down converted (DC) photons are produced simultaneously and are correlated in energy and momentum [2]:

\begin{equation} E_p=E_{DC-signal} + E_{DC-idler}k_p = k_{DC-signal}cos(q_s) + k_{DC-idler}cos(q_i) \end{equation}

where $E = {hc \over l}$ and $k = {2p \over l}$ are respectively the energy and the wave number of the photons of wavelength l. For the case $l_{DC} = l_{DC-signal} = l_{DC-idler}$ these laws impose the following condition on the indices of refraction of pump and DC light [2]:

\begin{equation} n(l_{pump}) = n(l_{DC})cos(q_i) \end{equation}

where q is the angle that the DC beams form with the direction of the pump inside the crystal.
For Type-I phase matching, DC light is polarized perpendicular to the optic axis (OA) of the crystal and $n(l_{DC}) = n_O$(ordinary index of refraction). The polarization of the pump beam is in the same plane as OA:

\begin{equation} n(l_p) = n_e(f_{pm}) \end{equation}


\begin{align} n_e(f_{pm}) = {cos(2f_{pm}) \over n_o^2} + {sin^2(f_{pm}) \over n_e^2}^{-(1/2)} \end{align}

with fpm being the phase matching angle. We use BBO crystals, which have indices of refraction given by [2]:

\begin{align} n(l) = {A + B \over l^2 + C + D l^{2}}^{1/2} \end{align}
1. Greenstein and Zajonc. The Quantum Challenge. Jones and Bartlett Publishers: Massachusetts. (2006).
2. E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer, "Interference with correlated photons: Five quantum mechanics experiments for undergraduates," Am. J. of Phys. 73, 127-140 (2005)
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