t*r + r*t = O. in .NET

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t*r + r*t = O.
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(4.127)
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From (4.127) it follows that arg(r) - arg(t) = 7r/2. For a 50:50 beam splitter, Irl = It I = 1/.J2. Choosing r as real, we find that r = 1/.J2 and t = i/.J2. In the case of a quantized field, the plane wave complex amplitudes are annihilation operators. So the fields at the output ports of a 50:50 beam splitter can be described by the annihilation operators d and e, given by
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e -+ ie and c -+ ie, we obtain
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Figure 4.7 Diagram of a balanced homodyne detector. The local oscillator field (intense coherent state) is injected in a. The field to be measured, with the same frequency of the local oscillator, is injected in e. The outputs c and d are measured by two different photodetectors whose difference in photocounts is monitored.
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A 1 (A A) c=.J2a+e.
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Suppose now that in the port corresponding to e, we inject squeezed vacuum, and in that corresponding to a, we inject a coherent state la(t)} with the same frequency but with a phase difference e, so that aCt) = B exp(i[e - wt]). The balanced homodyne detection scheme consists of measuring the difference in the photon numbers at ports 1 and 2 (see Figure 4.7). Writing the operators a and e as before, that is,
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(4.131)
and (4.132)
we find that the difference in photon number is given by (4.133) As the coherent state la(t)} is very intense (a laser beam), we can neglect the second term square brackets on the right-hand side of (4.133). Now ~el(e - wt) is the variance of the quadrature el in a reference frame rotated by e - wt in relation to the old reference frame. The rotation by -wt allows it to follow the movement of ein phase space due to its free time evolution, so that ~el(e - wt) does not change in time. The dependence on e allows us to choose the direction in which we want to measure the field fluctuations (i.e., it allows us to choose which quadrature we want to look at).
A BOX OF PHOTONS
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In Section 4.1 we saw how an open system becomes entangled to the immense number of degrees of freedom of the environment. This entanglement leads to a mixed reduced-density matrix for the open system. Controlled entanglement, rather than this ordinary entanglement with the inaccessible degrees of freedom of the environment, has many interesting applications, including teleportation, quantum cryptography, and quantum computing. For more on these applications that are now part of the new field of quantum information, see
[71].
The Glauber-Sudarshan quasiprobability distribution seen in Section 4.1 is not the only one used in quantum optics. Another example of quasiprobability distribution is the Wigner function [644], which is associated with symmetrical ordering rather than normal ordering. It introduces the idea of phase-space in quantum mechanics (for a phase space approach to quantum optics, see Schleich's book [537]). Some of the advantages of the Wigner function are that unlike P, it is not singular for coherent states and for physical quantum states, and that it makes important features such as quantum interference easy to identify visually.
In the 1990s, the Wigner function of a light beam was determined experimentally for the first time using the technique of quantum tomography. For an excellent account of this as well as a list of some of the key references in that period, see [399]. Unfortunately, this technique does not apply to quantum fields in high-Q microwave cavities, as those fields cannot be taken out of the cavity and as there are no photon detectors for those frequencies. For this typical cavity QED scenario, a different technique was proposed in which atoms are used to probe the field: atomic homodyne detection [168, 170, 652].
Outside the former Iron Curtain, interest in moving mirrors and time-dependent boundary conditions was sparked by the possibility of particle generation due to the expansion of the universe [57, 475]. The first known paper on the problem of the quantum radiation field in a cavity with moving walls was published by Gerald T. Moore in 1970 [457]. There are, however, older papers on the classical theory of time-dependent boundary conditions that were published in the former Soviet Union. For a comprehensive review of the moving mirrors problem (also known as the dynamic Casimir effect), see [161]. More recently, Law found an effective Hamiltonian that explicitly describes the physical processes that happen during a change of cavity size [394, 395]. Also of interest are a recent calculation of the decoherence produced by moving boundaries [141] and the proposal by Schwinger [546-550] to explain the phenomenon