Hilbert space coherent states

Once this discussion of the space-inversion operator in the context of optically active isomers is accepted, it follows that a molecular interpretation of the optical activity equation will not be a trivial matter. This is because a molecule is conventionally defined as a dynamical system composed of a particular, finite number of electrons and nuclei it can therefore be associated with a Hamiltonian operator containing a finite number (3 M) of degrees of freedom (variables) (Sect. 2), and for such operators one has a theorem that says the Hamiltonian acts on a single, coherent Hilbert space > = 3 (9t3X)51). In more physical terms this means that all the possible excitations of the molecule can be described in . In principle therefore any superposition of states in the molecular Hilbert space is physically realizable in particular it would be legitimate to write the eigenfunctions of the usual molecular Hamiltonian, Eq. (2.14)1 3 in the form of Eq. (4.14) with suitable coefficients (C , = 0. Moreover any unitary transformation of the eigen-... 

Although this equation looks formally like the semiclassical Schrodinger equation (2), we emphasize that it is still different because it is defined in the enlarged Hilbert space X and the phase 0 does not have a definite value, since it is a dynamical variable on the same footing as x. In order to recover the semiclassical equation from Eq. (39), we have to reduce it to an equation defined in the Hilbert space Ji. From a mathematical point of view, this can be done by fixing a particular value of 0, as we did in Section II.A. Physically, this can be achieved, as we show in the following, by choosing the initial condition of the photon field as a coherent state. 

In the absence of shared entanglement between Alice and Bob the mean fidelity of the output state F = (t/.jpout t/, ) is bounded. For example, for teleporting qubits, F < 2/3, whereas for the teleportation of coherent states in an infinite-dimensional Hilbert space it is bounded by F < 1/2 [Braunstein 2000 (b)]. 

This work is intended as an attempt to present two essentially different constructions of harmonic oscillator states in a FD Hilbert space. We propose some new definitions of the states and find their explicit forms in the Fock representation. For the convenience of the reader, we also bring together several known FD quantum-optical states, thus making our exposition more self-contained. We shall discuss FD coherent states, FD phase coherent states, FD displaced number states, FD Schrodinger cats, and FD squeezed vacuum. We shall show some intriguing properties of the states with the help of the discrete Wigner function. 

Figure 1. Examples of discrete Wigner function on a torus in 19-dimensional Hilbert space i s = 18) (a) vacuum 0) (b) single-photon number state 1) (c) FD preferred phase state ( phase vacuum ) 0o)(5) (d) FD coherent state, a) ss a) (e) FD displaced number state, oc, 1)(j) oc, 1)(j) (0 FD phase coherent state, P, 0o)(j) P,0o)(5), with equal displacement parameters, 7. — 7. — P — P — 0.5 and 0o = 0. The darker is a region, the higher is the value of the Wigner function.

Glauber [8] constructed coherent states in the ID Hilbert space by applying the displacement operator D a, a ) = exp(afit — a a) on vacuum state 0). Analogously, one can define the generalized coherent state [16]... 

Figure 2. Generalized coherent states. The superposition coefficients feW for a) versus displacement parameter amplitude a for (a) n = 0, (b) n = 1, and (c) n = 2 in the Hilbert spaces of different dimensionality s = 1 (dotted), s = 2 (dot-dashed), s = 3 (dashed), and s = oo (solid curves).

Figure 3. Generalized coherent states (black bars) versus truncated coherent states (white bars) photon-number distribution Ps(n) as a function of n in FD Hilbert spaces with s = 5,..., 50 for the same displacement parameters a = a = 4.

Figure 4. Generalized coherent states Wigner function Ws(n,Qm) in FD Hilbert space with s = 18 for a)p8j with different values of displacement parameter a, chosen as fractions of the quasiperiod T i T % 8.8. As in Fig. 1, higher values of Wigner function are depicted darker.

Analogously to the generalized, CS in a FD Hilbert space, analyzed in Section IV. A, other states of the electromagnetic field can be defined by the action of the FD displacement or squeeze operators. In particular, FD displaced phase states and coherent phase states were discussed by Gangopadhyay [28]. Generalized displaced number states and Schrodinger cats were analyzed in Ref. 21 and generalized squeezed vacuum was studied in Ref. 34. A different approach to construction of FD states can be based on truncation of the Fock expansion of the well-known ID harmonic oscillator states. The same construction, as for the... 

Here, we discuss two kinds of FD squeezed vacuum. We will present explicit forms of these states, which reveal the differences and similarities between them and the conventional IF squeezed vacuum [68] or FD coherent state. We will show that our states are properly normalized in of arbitrary dimension and go over into the conventional squeezed vacuum if the dimension is much greater than the square of the squeeze parameter. Squeezing and squeezed states in FD Hilbert spaces were analyzed, in particular, by Wodkiewicz et al. [31], Figurny et al. [32], Wineland et al. [33], Buzek et al. [16], and Opatrny et al. [20]. An FD analog of the conventional squeezed vacuum was proposed by Miranowicz et al. [34],... 

Let us start the discussion of practical possibilities of the FD coherent-state generation from the simplest case, where only superpositions of vacuum and single-photon state are involved (the Hilbert space discussed is reduced to two dimensions). We consider the system governed by the following Hamiltonian defined in the interaction picture (in units of h = 1) to be... 

Again, these solutions are identical to those derived by Miranowicz et al. [3] [compare Eq. (25) in Ref. 1]. Of course, we can write the equations for arbitrary value of the parameter N and hence, get the formulas for the probability amplitudes for the n-photon state expansion of the FD coherent state defined in the iV-dimensional Hilbert space. In general, for any dimension N and arbitrary real periodic function /(f) with the period T, we find that the system evolves at t = kT into the state [33]... 

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