The Two-Photon State Vector

If the two-photon state vector is represented by tff) then we can calculate the expectation value for the product A B according to 

The correlation coefficient E(a, b) can thus be identified as the expectation value of the direct product operator A B.  

In order to conserve angular momentum when the two photons are emitted in opposite directions from a source whose constituents are isotropic before and after emission, it is necessary for the photons to have equal helicity. In terms of right-handed ( / )) and left-handed ( L helicity states we thus expect photon pairs to be represented by the vectors / ), / 2, L), L 2 or a superposition of these, where R)i denotes a photon of right-handed helicity propagating to the left, / )2 a photon of right-handed 

Assuming the photons propagate in the -z and +z directions, it is also possible to describe the two-photon state vector in terms of the linear polarization states x) and y) using the relations 

The choice of orientation of the orthogonal axes x and y in the plane perpendicular to the z axis is, of course, completely arbitrary because of the cylindrical symmetry which is assumed to exist about the z axis. 

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Examination of the two-photon state vector in Eq. (10) or (12) shows that it implies nonlocality and lack of realism. It implies nonlocality since a measurement causes a collapse of, say, )+, Eq. (12a), to either x)i x)2 or ll )il> )2, each possibility occurring with probability one half. Thus, detection of photon 1 to the left with polarization in the x direction ensures that photon 2 to the right behaves as a photon polarized in the x direction also. But, as we have already seen, the choice of x direction is quite arbitrary, so the polarization state measured for photon 2 is, in fact, determined by the measurement we choose to make on photon 1 at a position that may be spatially separated, in the relativistic sense, from the position at which the measurement on photon 2 is carried out. Lack of realism also follows from this argument, since it then is impossible to think of the individual photons possessing properties, in this case polarization, which exist independently of any measurements which may be made on them. 

The physical meaning of is a two-photon detection amplitude, through which one can express the second-order correlation function G j = y l J vIh7 [Scully 1997]. The knowledge of the two-photon wavefunction allows one to calculate the amplitudes of state vector (24) via the two dimensional Fourier transform of 4, at t = t ... 

The transition between the two energy states must cause a change in the charge distribution in the molecule, i.e. a change in the dipole moment. In terms of quantum mechanics absorption of a photon is possible (allowed) if the transition moment M has a non-zero value. Since M is a vector composed of three components parallel to the three coordinates [Eq. (1-2)], at least one component must have a non-zero value. 

In the situation illustrated in Figure 1 where the two photons are emitted in diametrically opposite directions, the polarization part of the associated state vector can be derived from simple consideration of conservation of angular momentum and parity. It should be noted that, in practice, for each photon pair, frequencies Vi and V2 emitted, respectively, in the +2 and -2 directions there is a complementary photon pair, frequencies and P2, emitted, respectively, in the -2 and +2 directions. Because of this symmetry, in what follows it is only necessary to consider the polarization properties of the photon pair. 

While two-photon absorption spectroscopy has been widely applied for precision measurements of atomic structure, the polarization correlation of the simultaneous two-photon emission from the metastable Is state of atomic hydrogen has only been measured very recently. The emission of the coincident two photons can be described by a single state vector which determines the circular and linear two-photon polarization. Compared to the two-photon cascade experiments the polarization correlation of the simultaneous two-photon decay of metastable hydrogen is conceptually closer to the original proposals by Bell and Bohm for tests of the foundation of quantum mechanics. More than SO years have elapsed since the famous Einstein-Bohr debate on microphysical reality and quantum formalism. The present and future outcome of the hydrogen two-photon correlation experiment is considered to be a most crucial test with regard to the rivalry between quantum mechanics and local realistic theories. 

Here, at is the Dirac a-matrix, A(r, t) is the vector potential of the field (divA = 0), and Ho is the Hamiltonian of an isolated atom whose energy in the initial state of the reaction is denoted by Ef. H0 i) = E i). Equation (5) defines the effective two-photon operator Q(2 ui,u> ) which has the dimension of L3 and is a straightforward relativistic generalization of its nonrelativistic counterpart that has been first introduced in [28] to describe the process of two-photon absorption. Generally, the matrix elements of u>,u> ) can be expressed explicitly as... 

The excitation stabilizations D and A have been included in haj0 and h i0. In the expression (2.62) we take as energy origin hd>0 + h 0 and we restrict our investigation to states with total wave vector q equal to that of the incident photon, the resulting hamiltonian H remaining translationally invariant. Then it is appropriate to use two new basis sets ... 

The state of the radiation field is determined by a set of photon numbers nx-The vacuum state, designated 0), contains no photons. The state U) contains one photon of energy hcox, propagation vector kx and polarization ex- The state 12x) contains two such photons, while I lx, Ix ) contains two different photons, A and A. The most general state ofthe radiation fieldwould be designated nx, xs x" ) If the enclosure also contains an atom in quantum state lr , the composite state is designated n x, x, ) ... 

In the last step the erroneous state vector is projected onto the subspace C to recover the initial information. Projection is a non-unitary process which cannot be achieved through a Hamiltonian process, but requires the introduction of irreversibility. To this end, we make use of a path which is symmetric with the pumping step, and consists in two stimulated and one spontaneous emissions. To be more explicit, we apply two left circularly polarized lasers slightly detuned from the transitions (60/ <—> 5d, j = 3/2) and (5d, j = 3/2 <—> 5p, j = 3/2). Due to these laser fields, the atom is likely to fall towards the ground state and emit two stimulated and one spontaneous photons. 

As an example, consider the simple case of two single-mode fields of equal frequencies and polarizations. Assume that there are initially n photons in field 1 and m photons in the field 2, and that the state vectors of the fields are the Fock states /j) = n) and v /2) = m). The initial state of the two fields is the direct product of the single-field states, v /) = n) m). Inserting Eq. (27) into Eq. (28) and taking the expectation value with respect to the initial state of the fields, we find... 

In these equations the position of the molecule is described by the vector R the wavevectors of the two beams of modes r2 and are k2 and k3 respectively, with ( 2) and (q3) the corresponding mean photon numbers (mode occupancies) and is a unit vector describing the polarization state of mode rn. In deriving Eqs. (120) and (121), the state vectors describing the radiation fields have been assumed to be coherent laser states, and so, for example, (<72) = (oc n a(2 ), where a ) is the coherent state representing mode 2 and h is the number operator a similar expression may be written for (<73). Also, the molecular parameters apparent in Eqs. (120) and (121) are the components of the transition dipole, p °, and the index-symmetric second-order molecular transition tensor,... 

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