User:Andrewkforbes/sandbox

Large systems of entangled atoms with a known entangled state have many uses in quantum metrology as well as in the field of quantum computing. By measuring a single photon after it passes through a cloud of atoms, it is possible to prepare the majority of those atoms in an entangled non-Gaussian "cat state". These states are a hallmark of non-classicality, as they represent entanglement between a large number of particles in a given system. This non-classicality can be detected by analysis of the Wigner function, an example of a quasiprobability distribution. Such distributions permit negative valued probabilities, a phenomena heavily associated with quantum mechanics that does not appear in classical mechanics. This makes the Wigner function a well-suited detector of non-classical behavior. By measuring the polarization of a so-called "heralding photon" (it heralds the creation of the desired state, also called the heralded state) after its interaction with a cloud of atoms one can generate an entangled state of nearly 3,000 atoms with a minimum value of the Wigner function of approximately $$W=0.36\pm0.08$$.

Overview of Experiment
In 2015 Ph.D. Vladan Vuletic's research group at the Massachusetts Institute of Technology (MIT) published a paper entitled "Entanglement with negative Wigner function of almost 3,000 atoms heralded by one photon". The experiment uses an optical cavity with ~3,000 Rubidium atoms to enhance the entanglement between the polarization of incoming photons and the spin state of the atomic ensemble. The $$m=1$$ and $$m=-1$$ hyperfine manifold of the $$5S_{1/2},F=1$$ energy level of Rubidium is used as the spin up and spin down state respectively. Incoming photons are prepared in a vertical polarization state $$|v\rangle$$, and after interacting with the atoms they encounter a polarizing beam splitter, which separates out vertically polarized photons from horizontally polarized photons $$|h\rangle$$. The atoms are prepared in a Spin Coherent State (SCS) along the $$\hat{x}$$ axis, which is an eigenstate of the collective spin operator $$\hat{S}_x=\sum_i^Nf_x^{(i)}$$. This state has a small projection noise along the $$\hat{\boldsymbol z}$$ axis, producing a slight Faraday rotation on the polarization of the incoming photons. Upon measuring the polarization of the outgoing photons one will measure $$|h\rangle$$ (called the heralding photon) with small probability, but upon such a measurement the atomic state is thrown into a highly entangled cat state through measurement back action. This method of entangling atomic spins has an entanglement depth of ~90%, meaning it entangles roughly 90% of the atoms present (no less than $$2910\pm190$$ entangled atoms out of ~$$3,100$$ total atoms).

After the state is prepared the researchers attempted to reconstruct the density matrix by sending a second, stronger probe pulse of light. This light undergoes a much stronger Faraday rotation and thus serves as a measurement of the $$\hat S_z^2$$ component of the atomic ensemble. From this data, one is able to construct the critical elements of the density matrix that contribute to the heralded state.

Experimental Setup
The Rb atoms are initially quantized along the $$+\hat{z}$$ axis (fig. 1), which is followed by a $$\frac{\pi}{2}$$ radio wave pulse which orients the atomic spins along the $$\hat{x}$$ axis (perpendicular to the direction of propagation). The $$|v\rangle$$ photons can be written as a superposition of their positive and negative helicity components ,$$|v\rangle=2^{-1/2}\left(|\sigma^+\rangle+|\sigma^-\rangle\right)$$, which drive transitions between $$F=1, m=-1$$ to $$F'=0$$ and $$F=1, m=+1$$ to $$F'=0$$ respectively. For sufficiently large detuning $$\Delta$$ relative to the $$F'=0$$ energy level, we can adiabatically eliminate the $$F'=0$$ transition, leaving us with long-lived coupling between the $$F=1, m=-1$$ and $$F=1, m=+1$$ hyperfine manifolds. The photons exiting the cavity encounter a polarized beam splitter that reflects vertically polarized photons, and transmits horizontally polarized photons. The light then encounters a single-photon detector, allowing one to make a measurement of the outgoing polarization. The cavity serves to increase the coupling strength between atoms and the light, by allowing the light to reflect off of the cavity mirrors multiple times before being sent to the detectors. This helps to increase the probability of measuring a heralding photon, without changing the projection of the SCS along the direction of propagation.

Hamiltonian and Initial State
The SCS after the initial $$\frac{\pi}{2}$$ pulse can be represented as a product state of $$\sigma_x$$ eigenstates (where $$\sigma_x$$ is the Pauli Matrix along the $$\hat{\boldsymbol{x}}$$ axis). This can equivalently be represented in the $$\hat z$$ Dicke basis as $$ $$ where $$|m\rangle$$ are the Dicke state, which are eigenstates of the collective spin operator $$\hat{S}_z=\sum_{i=0}^N\hat f_z^{(i)}$$ and $$ c_m=\frac1{2^S}\sqrt{\frac{(2S)!}{(S+m)!(S-m)!}}. $$ This coefficient is similar to the binomial coefficient, and does the job of containing information about how many states (in the Pauli z basis) have collective spin $$m$$, as well as normalizing the state. The joint state of the atoms and the light can therefore be written as $$ $$ The Hamiltonian corresponding to the Faraday rotation is $$ \frac{\hat H}{\hbar}=\left(\frac{g^2}{\Delta}\right)\hat J_z\hat S_z $$ where $$\hat J_z=\frac12\left(\hat a_+^\dagger \hat a_++\hat a_-^\dagger \hat a_-\right)$$ is the Stokes vector along the $$\hat{z}$$ axis (on the Poincaré sphere), $$\hat a_k$$ and $$\hat a_k^\dagger$$ are the annihilation and creation operators in mode $$k$$ (in this case it is the positive and negative helicity modes), $$\Delta$$ is the detuning from the $$F'=0$$ energy level, and $$2g$$ is the single photon Rabi frequency.
 * S_x\rangle=\sum_mc_m|m\rangle
 * \psi_{\text{AL}}\rangle=\frac1{\sqrt2}\left(\sum_mc_m|m\rangle\right)\left(|\sigma^+\rangle+|\sigma^-\rangle\right).

Evolution and Post-Measurement State
The incoming photons are on resonance with the cavity, and therefore the average time spent in the cavity is $$2/\kappa$$ before being measured, where $$\kappa$$ is the cavity linewidth. The total state after evolution by $$\hat H/\hbar$$ is given by $$\begin{align} e^{-it\hat H}|\psi_{\text{AL}}\rangle&=\exp\left[-i(2g^2/\Delta\kappa)\hat J_z\hat S_z\right]\frac1{\sqrt2}\left(\sum_mc_m|m\rangle\right)\left(|\sigma^+\rangle+|\sigma^-\rangle\right)\\ &=\frac1{\sqrt2}\exp\left[-i(2g^2/\Delta\kappa)\left(+\frac12\right)\hat S_z\right]\sum_mc_m|m\rangle|\sigma^+\rangle+\frac1{\sqrt2}\exp\left[-i(2g^2/\Delta\kappa)\left(-\frac12\right)\hat S_z\right]\sum_mc_m|m\rangle|\sigma^-\rangle\end{align}$$

In the equation above it becomes clear that the time evolution results in a superposition of rotations by equal and opposite angles of the SCS on the Bloch sphere. One can alternatively write the time evolution such that it highlights the effect on the light

$$ e^{-it\hat H}|\psi_{\text{AL}}\rangle=\frac1{\sqrt2}\sum_m\exp\left[-i(2g^2/\Delta\kappa)\hat J_z (m)\right]|m\rangle\left(|\sigma^+\rangle+|\sigma^-\rangle\right) $$

where now it becomes clear that the rotation on the Bloch sphere will depend on the value of $$m$$ that is measured. The rotation of the atomic ensemble on the Bloch sphere about the $$\hat{\boldsymbol z}$$ axis is by an angle $$\pm\phi$$ where $$\phi=\frac{g^2}{\Delta\kappa}=\frac{\eta\Gamma}{4\Delta}$$ and $$\eta=\frac{4g^2}{\kappa\Gamma}$$ is the cavity cooperativity parameter, while the rotation of the polarization of the light is by $$\theta=m\phi=S_z\phi$$ about the $$\hat{\boldsymbol z}$$ axis. Upon measuring a single photon in the $$|h\rangle$$ state the post-measurement state of the ensemble is thrown into $$ $$ $$ =\frac1{\sqrt2}\left(|+\phi\rangle-|-\phi\rangle\right) $$ where $$|+\phi\rangle$$ and $$|-\phi\rangle$$ are each identical to the original SCS, but have been rotated by $$+\phi$$ and $$-\phi$$ respectively. By contrast a photon measured in the $$|v\rangle$$ state will leave the atomic state as $$ $$ $$ =\sum_m\cos(\phi m)c_m|m\rangle $$ $$ \approx\sum_mc_m|m\rangle $$ which is very nearly the initial SCS state, albeit very slightly spin-squeezed. The approximation that $$\phi\ll1$$ is valid because the Faraday rotation is only due fluctuation noise in the projection of the SCS onto the direction of light propagation. In this limit the event of measuring a photon in the vertical polarization does not significantly change the SCS, and therefore multiple photons can be sent through until a heralding photon gets detected.
 * \psi_{\text{A}}\rangle\propto\frac1{\sqrt2}\exp\left[-i\phi\hat S_z\right]\sum_mc_m|m\rangle-\frac1{\sqrt2}\exp\left[+i\phi\hat S_z\right]\sum_mc_m|m\rangle
 * \psi_{\text{A}}\rangle=\frac1{\sqrt2}\exp\left[-i\phi\hat S_z\right]\sum_mc_m|m\rangle+\frac1{\sqrt2}\exp\left[+i\phi\hat S_z\right]\sum_mc_m|m\rangle

The probability of measuring a horizontally polarized photon is given by $$P_h=\sin^2(\theta)$$ where $$\theta$$ is the angle of rotation on the light relative to the $$\hat y$$ axis. Further, measurements corresponding to $$m\approx0$$ will be suppressed, since they produce a very small Faraday rotation. We will observe measurements of $$m$$ biased towards the standard deviation of $$S_z$$ since they produce a larger rotation. Therefore on average $$P_h\approx\theta^2\approx\frac{S\phi^2}{2}$$. However, because $$P_h\approx\theta^2=S_z^2\phi^2$$ any measurement of the polarization angle gives only information about $$S_z^2$$, and not $$S_z$$.

Quantum Tomography
After one or more heralding photons are measured the atomic ensemble is rotated about the $$\hat x$$ axis by an angle $$\beta=0,\pi/4,\pi/2,3\pi/4$$. A pulse of light with vertical polarization (much stronger than the pulse used to create the heralding event, containing approximately $$1.7\times10^4$$ photons instead of one) is sent through the ensemble, after which the polarization along $$|h\rangle$$ is again measured. This stronger pulse undergoes a larger Faraday rotation than the single photon, and is used to probe the total spin along $$\hat S_z$$ of the atomic ensemble. Depending on the angle $$\beta$$, this measurement contains information about the collective spin in the $$\hat z$$, $$\hat z+\hat y$$, $$\hat y$$, or $$\hat y-\hat z$$ direction. Measurements of the horizontal component of the outgoing light are used to construct probability distributions $$g_\beta(n)$$ for $$n$$ horizontally measured photons.

Method of Density Matrix Reconstruction
In order to obtain the state $$\rho$$ from $$g_{\beta}(n)$$ a relationship is needed between $$g_{\beta}(n)$$ and the spin distribution $$f(S_\beta)$$ along a particular direction determined by the rotation angle $$\beta$$. The probability to measure a particular number of photons $$n$$ in $$|h\rangle$$ given a collective spin $$S_\beta$$ is a Poisson distribution, and is given by $$ P(n,S_\beta)=\exp\left[-qn_{\text{in}}(\phi S_\beta)^2\right]\frac{\left(qn_{\text{in}}(\phi S_\beta)^2\right)^n}{n!} $$ where $$n_{\text{in}}$$ is the number of photons in the pulse and $$q$$ is the over all efficiency of the measurement. One can then construct the function $$g_\beta(n)$$ by summing over $$P(n,S_\beta)$$ weighted by the function $$f(S_\beta)$$ $$ g_\beta(n)=\sum_{S_\beta}f(S_\beta)P(n,S_\beta) $$ The probability distribution $$f(S_\beta)$$ can also be obtained from the density matrix $$ f(S_\beta,\rho)=\langle S_\beta|\rho|S_\beta\rangle $$ The density matrix can therefore be approximated by minimizing the least squares deviation $$L$$ between the measured $$g_\beta(n)$$, and the distribution that would theoretically result from a density matrix $$\rho$$ $$ L=\sum_\beta\sum_{n\geq0}\left(\frac{g_{\text{theory}}(\rho,n)-g_{\text{experiment}}(n)}{\sigma_g}\right)^2 $$ where $$\sigma_g$$ is the error in the measurement of $$g_\beta(n)$$.

Implementation
The method for reconstructing the density matrix outlined above involves finding how much the density matrix elements (in the $$\hat x$$ Dicke basis $$|m\rangle_x$$) lie along an eigenvector of collective spin $$|S_\beta\rangle$$. This overlap is $$ G(m,S_\beta)=\langle S_\beta|m\rangle_x=\frac1{\sqrt{2^mm!}}\left(\frac1{\pi N}\right)^{1/4}\exp\left[im\beta-\frac{S_\beta^2}{2N}\right]H_m\left(\frac{S_\beta}{\sqrt N}\right) $$ where $$H_m$$ are the $$m^{\text{th}}$$ order Hermite polynomials. This results in $$\langle S_\beta|m\rangle_x$$ having a component that is a polynomial of order $$m$$ in $$S_\beta$$. The distribution $$f(S_\beta)$$ can therefore be written as $$ f(S_\beta)=\sum_{m,n}G(m,S_\beta)\rho_{mn}G^*(n,S_\beta) $$ This expression of $$f(S_\beta)$$ highlights the fact that measurements of $$S_\beta^2$$ only tell us about how the even terms of $$\rho_{mn}$$ ( when $$m+n$$ mod $$2=0$$) contribute to $$f(S_\beta)$$, since $$f(S_\beta)$$ will contain an even polynomial in $$S_\beta$$ if $$m+n$$ is even. Measurements of $$S_\beta$$ are therefore needed to tell us about odd terms. However, the probability to measure a horizontally polarized photon is proportional to $$S_{\beta}^2$$, and therefore measuring $$g_\beta(n)$$ only contains information about the even terms of $$\rho_{mn}$$ in the Dicke basis. In order to find the odd terms of $$\rho_{mn}$$ the incoming photons are instead initially polarized at $$45^{\circ}$$ from the $$\hat y$$ axis, and detectors are arranged after the beam splitter to measure both $$|v\rangle$$ and $$|h\rangle$$. The difference of these two measurements is proportional to $$S_z$$ since $$ P_v-P_h=\cos^2(\pi/4-\theta)-\sin^2(\pi/4-\theta)=\sin(2\theta)\approx2\theta\propto S_z $$ The SCS is initially centered about the $$\hat x$$ axis, and so $$\langle S_z\rangle=0$$ before the photons pass through the cavity. The result of performing the $$S_z$$ measurement on the state is that it keeps $$\langle S_z\rangle=0$$ the same (the center of the state does not change). Since the odd terms of the density matrix correspond to the $$S_\beta$$ terms, and the heralded state is not displaced by any significant amount compared to the initial SCS, the researches conclude that these odd matrix elements must be commensurate with 0.

Negative-Valued Wigner Function
The Wigner function is an example of a quasiprobability distribution, and is analogous to a probability distribution over phase space in classical mechanics. Notably, quasiprobability distributions permit negative values as outputs, and therefore are not meant to be taken explicitly as probability distributions. The negativity of a quasiprobability distribution in quantum mechanics is the result of interference of quantum states, and is thus seen as a measure of non-classicality. It was used in this research as a measure of the achieved non-classicality of the state. The Wigner function for an angular-momentum state is with $$N$$ spins, and normalization constant $$(\pi/(2S))^{1/2}$$, is given by $$ W(\theta,\phi)=\sqrt{\frac{\pi}{2S}}\sum_{k=0}^{N}\sum_{q=-k}^k\rho_{kq}Y_k^q(\theta,\phi) $$ where $$\rho_{kq}$$ is the density matrix of the state in the spherical harmonic basis and $$Y_k^q(\theta,\phi)$$ are the spherical harmonics. The state $$\rho_{kq}$$ is related to the density matrix in the Dicke state basis along $$\hat z$$, $$\rho_{mm'}$$, by $$ \rho_{kq}=\sum_{m=-S}^S\sum_{m'=-S}^S\rho_{mm'}t_{kq}^{Smm'} $$ where $$t_{kq}^{Smm'}=(-1)^{S-m-q}\langle S,m;S,-m|k,q\rangle$$ are the Clebsch-Gordan coefficients. The measurement of a heralding photon generates a Wigner function with global minimum $$-0.36\pm0.08$$, encompassing roughly $$90\%$$ of the atoms in the ensemble. Previous attempts to generate entangled states with comparably negative Wigner functions had contained only a few atoms. In the limit that the number of atoms $$N$$ is large, the Wigner function in the center of the SCS (where $$\theta=\frac\pi2$$ and $$\phi=0$$) is well approximated by considering only the population terms of the density matrix in the $$\hat x$$ Dicke basis $$ W\left(\frac\pi2,0\right)=\sum_{n=0}^{N}(-1)^n\rho_{nn} $$ where $$n$$ runs over the $$\hat x$$ Dicke states labeled by $$|n=0\rangle$$, $$|n=1\rangle$$, etc. This point is of particular interest since it is where the Wigner function is expected to be most negative (since the center of the heralded state is the same as the SCS). Higher orders of $$\rho_{nn}$$ beyond $$n=1$$ contribute very little, and thus a good approximation of the Wigner function can be obtain by considering only $$\rho_{00}$$ and $$\rho_{11}$$. These values are found through tomography to be $$\rho_{00}\approx0.32\pm0.03$$ and $$\rho_{11}\approx0.66\pm0.04$$, resulting in a Wigner function with minimum value $$W\left(\frac\pi2,0\right)=-0.36\pm0.08$$.