User:Hpmadathil/sandbox/two photon interferometry

Two Photon Interferometry
Two Photon Interferometry is a type of interferometric technique that exploit non-classical nature of photons released in a two photon cascade process. The pair of photons released during a Spontaneous Parametric Down Conversion (SPDC) process is a good example for this. This type of a two two-photon interference effect led to the first observation of the photon bunching process or popularly known as the 'HOM' effect. The fourth order interference effects enabled them to demonstrate the "HOM-Dip" even when the second-order interference effects where zero (no fixed phase relationship). This paved the way to precise measurements techniques using two photon and multi-photon interference. The same idea can be applied to study non-locality present in a quantum state like a bell-test for example.

Theoretical Background
We can observe correlations between photons scattered from two multi-level atoms and people have made use of these photons for various tests for Bell Inequality. The same could be said for two photons produced in a cascade emission between energy levels in a single atom. In order to really appreciate the theory behind the second-order correlation between the emitted photons we need to know how a cascade process work.

Two Photon Cascade Process
Suppose we have a three level atomic system in which at time t=0, the system is found in the excited state $$| e\rangle$$. As time progress the atom in the excited state will decay to an intermediate state $$|i\rangle$$ by releasing a photon k of frequency $$\omega_k$$. After a finite amount of time, the atom in the intermediate state will decay into the ground state $$|g\rangle$$ and in the process will emit a photon q having frequency $$\omega_q$$. In order to study these spontaneous processes, it is always beneficial to work in the Heisenberg picture. The interaction picture Hamiltonian for the atom-field interaction is given by the equation,

where $$\sigma^{(1)}_{+}=| e \rangle \langle i|,\sigma^{(2)}_{+}=| i \rangle \langle g|$$ and $$g^{*}_{e,k}, g^{*}_{i,q}$$ are the coupling contants for the two transitions. The atom-field system can be described by the state given below,

Here, at t=0, $$ C_{i,k}(0)=C_{g,k,q}(0)=0 $$. The equations of motion for the probability amplitudes can be calculated as follows, $$\frac{d C_{e}}{dt}=-\frac{i}{\hbar} \langle e,0|H^{(I)}|\Psi (t) \rangle =$$ $$ -\frac{i}{\hbar} \langle e,0|\hbar \sum_{k} \left[g^{*}_{e,k} \sigma^{(1)}_{+} \hat{a}_{k} e^{i(\omega_{ei}-\omega_{k}) t} + HC \right]|\Psi (t) \rangle$$

Similarly, the other two equations of motions are,

From Weisskopf-Wigner approximation, which deals with the theory of atomic decay, we can express RHS of equation ($$) with the rate of spontaneous decay from the excited state to the intermediate state with a decay constant $$\Gamma_e$$.

Similarly, we can also express the second term on the RHS part of equation ($$) with respect to the rate of decay from the intermediate state to the ground state with a decay constant $$\Gamma_{i}$$.

Substituting equations ($$) and ($$) into the equations of motion given above. We will get,

The end purpose of this derivation is to understand the final two photon state after the cascade emission is complete. In order to do that, we will look at a time t which is way longer than the decay time for both the spontaneous emission processes $$(t>>\Gamma^{-1}_{e}\& \Gamma^{-1}_{i})$$. We are interested in estimating $$C_{g,k,q}$$ since the atom will be in the ground state at the time t mentioned before. Integrating equation ($$) which is a pretty straight forward integral and plugging the result into equation ($$), we will get, $$C_{i,k}(t)=-ig_{e,k}\int^{t}_{0}dt' e^{-i(\omega_{e i}-\omega_{k}) t'-\frac{\Gamma_{e}}{2}t'}e^{\frac{\Gamma_{i}}{2}(t-t')}$$. As $$t \rightarrow \infty$$, we can evaluate the integral and we can arrive at a solution given below,

Substituting equation ($$) into equation ($$) and integrating as before, we will arrive at an expression for the coeffiecient for the ground-two photon state.

Therefore the final two photon state emitted due to the cascade process is given by,

The above equation gives you information about the emitted two photon state. First of all it tells you that the photons can be created in any mode that belongs to the $$4\pi$$ solid angle. Another important aspect here is that the two photons are correlated in energy and momentum. So, these photons can exhibit spatial correlations in between them. An example of a non-linear optical process that satisfies these energy and momentum conservation requirements is called Spontaneous-Parametric-Down-Converion(SPDC) process. We can see that most of the experiments that make use of two photon interferometry will have an SPDC source to create a two photon state mentioned above.

Second Order Correlations
If these photon are send to two separate locations such that the distance between them is greater than the coherence length between the two photons. Then we will not expect a first order correlation between them since the relative phase relationship between the two photons is no more. But that does not stop us from looking at the higher order correlations. The second-order correlation or the photon-photon correlation function in this case is given as,

Here, $$|\Psi\rangle$$ is a two photon state mentioned in equation ($$). We are sending the two photons to two separate detectors at points $$\textbf{r}_1$$ and $$\textbf{r}_2$$ respectively. These detectors are switched on to receive a signal at times $$t_1$$ and $$t_2$$ respectively. We look at the coincidence counts from the two detectors to understand more about the correlations. We can look into the two-photon correlation a little bit more, $$\langle \Psi|\textbf{E}^{(-)}(\textbf{r}_1,t_{1})\textbf{E}^{(-)}(\textbf{r}_2,t_{2})\textbf{E}^{(+)}(\textbf{r}_1,t_{1})\textbf{E}^{(+)}(\textbf{r}_2,t_{2})|\Psi\rangle=\sum_{n}\langle \Psi|\textbf{E}^{(-)}(\textbf{r}_1,t_{1})\textbf{E}^{(-)}(\textbf{r}_2,t_{2})|\{n\}\rangle \langle\{n\}|\textbf{E}^{(+)}(\textbf{r}_1,t_{1})\textbf{E}^{(+)}(\textbf{r}_2,t_{2})|\Psi\rangle$$

$$=\langle \Psi|\textbf{E}^{(-)}(\textbf{r}_1,t_{1})\textbf{E}^{(-)}(\textbf{r}_2,t_{2})|0\rangle \langle 0|\textbf{E}^{(+)}(\textbf{r}_1,t_{1})\textbf{E}^{(+)}(\textbf{r}_2,t_{2})|\Psi\rangle=\Psi^{*(2)}(\textbf{r}_1,t_{1}:\textbf{r}_2,t_{2})\Psi^{(2)}(\textbf{r}_1,t_{1}:\textbf{r}_2,t_{2})$$

Here, $$\Psi^{(2)}(\textbf{r}_1,t_{1}:\textbf{r}_2,t_{2})=\langle 0|\textbf{E}^{(+)}(\textbf{r}_1,t_{1})\textbf{E}^{(+)}(\textbf{r}_2,t_{2})|\Psi\rangle$$ is called the two-photon 'wave function'. We can evaluate this two photon by utilizing the field operator given as $$\textbf{E}(\textbf{r}_i,t_i)=\sum_{p}\hat{a}_{p}e^{-i\omega_{p}t_{i}+i\textbf{k.r}_i}$$. $$\Psi^{(2)}(\textbf{r}_1,t_{1},\textbf{r}_2,t_{2})=\sum_{k,q}\sum_{p,s}\langle 0|\hat{a}_{p}\hat{a}_{s}e^{-i\omega_{p}t_{1}+i\textbf{k.r}_1}e^{-i\omega_{s}t_{2}+i\textbf{k.r}_2}\times\dfrac{-g_{e,k}g_{i,q}e^{-i(\textbf{k}+\textbf{q}).\textbf{r}_{0}}}{[i(\omega_{k}+\omega_{q}-\omega_{e g})-\frac{\Gamma_{e}}{2}][i(\omega_{q}-\omega_{i g})-\frac{\Gamma_{i}}{2}]}|1_{k} 1_{q}\rangle$$

We will integrate the above expression by changing the summation to an intergral. Evaluating everything at resonances, and choosing z-axis for $$\textbf{k}$$ and $$\textbf{q}$$ such that $$\textbf{k.}\Delta\textbf{r}=k\Delta r\cos\theta$$ and $$\textbf{q.}\Delta\textbf{r}=qr\Delta r\cos\theta$$. Here, $$\Delta r_1=\textbf{r}_1-\textbf{r}$$ and $$\Delta r_2=\textbf{r}_2-\textbf{r}$$. The final result after carrying out all the calculation which includes estimating the contours to work in, we will arrive at a solution, given by,

$$\implies \Psi^{(2)}(\textbf{r}_1,t_{1},\textbf{r}_2,t_{2})=\frac{-k}{\Delta\textbf{r}_1 \Delta\textbf{r}_2}\exp\left[-\left(i\omega_{ei}+\frac{\Gamma_{e}}{2}\right)\left(t_{1}-\frac{\Delta r_{1}}{c} \right)\right]\Theta\left( t_{1}-\frac{\Delta r_{1}}{c} \right)$$

The step function that we see above is a physical consequence that the signal cannot travel faster than the speed of light. So, the two photon wavefunction can be written in a simpler manner as shown below,

So, the second-order correlation function will have cross terms that can demonstrate interference between the two emitted photons.

Experimental Realization
R.Chosh and L. Mandel in 1987 were the first first group to show that the pair of photons produced from an SPDC process exibhit non-classical correlations that can only be explained through the quantum theory of light. They calculated the joint probability of observing a photon being detected at a point $$\textbf{x}_1$$ at time $$t_{1}$$ and a photon being detected at a point $$\textbf{x}_2$$ at time $$t_{2}$$. Assuming we have a inital two photon state of the form $$|\Psi\rangle=|1_{k},1_{q}\rangle$$. The joint probability is given by the equation, $$P_{12}(x_{1},x_{2})\delta x_{1}\delta x_{2}=2K_{1}K_{2}\langle \Psi|\hat{E}^{(-)}(x_{1})\hat{E}^{(-)}(x_{2})\hat{E}^{(+)}(x_{1})\hat{E}^{(+)}(x_{2})|\Psi\rangle \delta x_{1}\delta x_{2}$$ where $$K_{1} \& K_{2}$$ are characteristic features of the detectors. $$\delta x_{1}$$ and $$\delta x_{2}$$ are spatial ranges where we expect to see the photons. After carrying out the calculation, we will arrive at a final equation given as,

where, $$L\approx L/\theta$$ is the interference spacing between the fringes. The cosine term in the correlation fucntion signifies that there is some interference going on, and according to their results, the interference can be attributed to the non-classical feature of the light.

Let's look at the experimental setup for a better understanding. The two photons are created from an SPDC source which is a nonlinear crystal of $$LiIO_{3}$$ that is pumped by a UV laser and in the process produces two photons called the signal photon and the Idler photon. They emerge from the crystal making an angle of 3 degrees wrt each other. The two mirrors $$M_{1} \& M_{2}$$ make the two photons come together at an angle of 2 degrees just after passing through an interference filter (IF). The two photons produces an interference in a plane that is 1.1 m away from the crystal. The interference occurs because the detectors are unable distinguish the incoming photon whether its k or q. A convex lens (L) is placed after the point of interference, in order to magnify the fringe spacing to a value of 0.34 mm. Two movable glass plates (MGP) at $$x_{1}\&x_{2}$$ collect photons that come their way and they feed these photons into the two photomultipliers (PMT) present in the setup. The signal from the two photo-multiplier outputs are amplified and shaped and finally they end up in a start and stop inputs of a Time-to-Digital converter. Pulses arriving within a 5 ns interval are treated as coincident and the rest of the pulses are rejected. Coincident pulses from a 10 hour long interval for three different values of $$(x_{1}-x_{2})$$ is taken and the joint probability is calculated and plotted. The joint probability function derived theoretically and shown in equation ($$) agrees with experimentally plotted points. They ran a Chi-square analysis test taking the difference between the experimentally plotted point and the theoretical curve and obtained $$\Chi^{2}$$ value of 0.44 which led to the conclusion that the photons released in an SPDC indeed have higher-order correlation between them which can only be explained by the quantum theory of light.

Immediately after this, C.K. Hong, Z.Y. Ou and L. Mandel demonstrated they can measure the time delay in the range of picoseconds between the signal and Idler photons using the second order correlations shared between the two emitted photons. This was the first experiment to shown the 'Photon-Bunching' phenomenon. This effect is popularly known as the HOM effect. There is a nice Wikipedia page on the same that you can take a look at.