Two-photon entangled state

The state (79) is called a two-photon entangled state. In Section VI, we will discuss another method of preparing the system in the two-photon entangled state based on the interaction of two atoms with a squeezed vacuum field. 

The two-photon entangled states cannot be generated by a simple coherent excitation. A coherent field applied to the two-atom system couples to one-photon transitions. The problem is that coherent excitation populates the upper state e) but also populated the intermediate states. v) and u). The two-photon entangled states (92) are superpositions of the collective ground and excited states with no contribution from the intermediate collective states j) and a). 

Note that the states T4), T2) and T3) are the same as for the small sample model, discussed in the preceding section. This means that the presence of the antisymmetric state does not affect the two-photon entangled states, but it can affect the population distribution between the states and the purity of the system. In Fig. 10, we plot the populations P, of the states T,) as a function of the interatomic separation. The figure demonstrates that the atoms are driven into a mixed state composed of two states Tj) and a), and there is a vanishing probability that the system is in the states IT2) and s). 

From the birth-paired photons, so-called path-entangled states can be constructed by overlapping them on a non-polarizing 50/50 beamsplitter (the numbers denote transmission and reflection coefficients of the beamsplitter). Two photons having the same polarization and coupled into different input ports of the beamsplitter will interfere and produce a new state that can be described as ... 

The cavity photon entangled with the material system state is written as Rb 61d5/2 l j)k=x2- Base states Rb 63p3/2) and Rb61d5/2)1 la/) have equal energy they relate to two different states at lower energy a A state. Symbol 4/ 00) reminds the case. 

If a quantum state starting from Rb63p3/2), then we take C4 = 1 and C3 = 0. The process described by the arrival state C4 = 0 and C3 = 1 can be seen in a two-fold manner (1) The > -photon is sent away into the cavity by spontaneous emission (2) The linear superposition becomes periodic in time where the state is an entangled state the energy hco is no longer available. 

Bell s theorem is by now a well-established experimental fact. The most accurate experiments have been based on analogs of the EPR-Bohm experiment measuring photon polarizations rather than spins of massive particles. Instead of spin-up and spin-down states, photons can have right and left circular polarizations. In certain processes, two photons with correlated polarizations—one left, one right— can be emitted in opposite directions. Wheeler had proposed in 1946 that the pair of photons emitted in the annihilation of positronium (see Fig. 7.12) were entangled with opposite polarizations. This was experimentally confirmed by Wu and Shaknov in 1949. 

It is evident from Eq. (110) that in the squeezed vacuum the density matrix of the system is not diagonal, due to the presence of the two-photon coherencies pge and peg. In this case the collective states g), j) and e) are no longer eigenstates of the system. The density matrix can be rediagonalized by including peg and p to give the new (entangled) states... 

There are many different physical realizations of the EPR or entangled states in optics and condensed-matter physics. For example, the creation of two photons with different helicities by a single atom in the process of cascade... 

Second-order perturbational theory then provides the entangled two-photon state /2) (0, to)) for times to sufficiently long so that the nonlinear interaction is... 

The teleportation protocol was proposed by Bennett and coworkers, and applied by Anton Zeilinger s group. The latter used the entangled states (EPR effect) of two photons described above. ... 

Since Alice and Bob have two entangled photons of the same polarization, let us assume that the state of the two photons is the following superposition 00 > + ll >, where the first position in every ket pertains to Alice s photon, and the second one pertains to Bob s. 

It turned out that in the Bohr-Einstein controversy, Bohr was right. The Einstein-Podolsky-Rosen paradox resulted (in agreement with Bohr s view) in the concept of entangled states. These states have been used experimentally teleport a photon state without violating the Heisenberg uncertainty principle. Also, the entangled states stand behind the idea of quantum computing with a superposition (qubit) of two states a 0) + b l) instead of 0) and 1) as information states. 

The superposition principle (also known as quantum entanglement) states that whenever the photon is in one state, it can always be considered to be partly in each of the two states. In other words, the photon actually exists in an indeterminate state that is a linear combination of the two possible translational states, as shown by Equation (3.26). The probability of measuring result T or T2 in a single measurement depends on the relative weights of T and T2 in the superposition. The electrons in the modern-day double-slit experiment act as a superposition of the two different slits. 

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