Coherent superposition state

Figure 5. Truncated coherent states. Superposition coefficients ftW of a)versus displacement parameter amplitude a for the same cases as in Fig. 2.

Figure 3. Field-matter interactions for a pair of electronic states. The zero and first excited vibrational levels are shown for each state (A). The fields are resonant with the electronic transitions. A horizontal bar represents an eigenstate, and a solid (dashed) vertical arrow represents a single field-matter interaction on a ket (bra) state. (See Refs. 1 and 54 for more details.) A single field-matter interaction creates an electronic superposition (coherence) state (B) that decays by electronic dephasing. Two interactions with positive and negative frequencies create electronic populations (C) or vibrational coherences either in the excited (D) or in the ground ( ) electronic states. In the latter cases (D and E) the evolution of coherence is decoupled from electronic dephasing, and the coherences decay by the vibrational dephasing process.

It is possible to create a coherent linear superposition of energy eigenstates via laser excitation. This is a state of the form J2n cn n(%,t), where c = c exp(i ), with a particular phase relation between the coefficients c . The value of these coefficients is determined by the pulsed laser excitation. 

The interaction between light and matter can be viewed as the creation of a coherent quantum superposition of initial and final electron states that has an associated polarization [3], as shown in Figure 1. The coherence between states with different wave vector requires an intermediate virtual state and the presence of a coherent phonon. A transition between the initial and final states may occur when the coherence of the system is broken either due to the finite width of an optical wave packet or by scattering from the environment. The transition results in the absorption of a photon and the creation of a hot electron-hole pair. Otherwise, the photon is re-radiated with a different phase and, perhaps, polarisation. 

To understand any coherence other than SQC, we need a new and more general definition of coherence. Coherence arises from the quantum mechanical mixing or overlap of spin states ( superposition ). In the two spin system (I, S = ll, 13C) we have four spin states (aa, up, pa, and PP), which are all stable states of defined energy. Let s talk about a single - C pair (one molecule). It is possible for this pair to be in any one of the four energy states, but it is also possible for the pair to be in a mixture or overlap or superposition of two states. This is one of the fundamental tenets of quantum mechanics Sometimes you cannot be sure which energy state a particle is in. Let s say that this particular pair is in a mixture of states aa and pp ... 

In the preceding sections we have taken the limit h —> oo in which the coherent states become a 8i/2(0 — 9o) function. For the discussion of the exchanged photons we consider a large but finite h, such that the coherent state is represented by a sharply peaked function that can be written as a superposition (64). Under this condition, the relations (62) and (63) imply that the initial condition... 

If one considers an initial coherent state for the photon field instead of a photon-number state, the superpositions of states have the additional optical phase, giving for (286)... 

We study the various superpositions of states that can be created by adiabatic passage in a robust way with respect to variations of the field amplitude, using the topological analysis with resonances of Section V.D (see also Section IV.B.3 for the case of one laser). We assume that one starts (at time t = ti) with a coherent state for the photon field and in the atomic state 1). We study here the A-system. Our results are easily extended to the other system (ladder and V), using the appropriate signs accompanying the field frequencies. We study the creation of a superposition of states at the final time t = tf. 

A Schrodinger cat-like state is a superposition of two macroscopically distinguishable classical states, [Schrodinger 1935 (a)], which for the harmonic oscillator are represented by strongly excited and sufficiently well separated (thus orthogonal) coherent states. To evolve a coherent state into a superposition, we may apply a unitary operator... 

Figure 3. Attempt to approach a unitary evolution U = e ln (fc+3) 2,r jn the Zeno-limit by performing 104 measurements per period, At = 10 4 T. At the first period, k = 0, the coherent state has evolved into a superposition in the case (a) (3 = 2 but decoherence has been faster than the build-up in the case of the doubled amplitude (b) fi 4. At the second period, k I, the superposition reappears slightly decohered for the smaller amplitude (c) (3 = 2 whereas no change has occurred for (d) fi 4 since in this case the asymptotic state had already been obtained during the previous period.

Figure 4. In order to obtain a superposition even for the doubled amplitude, we have to increase the number of measurements per period by five orders of magnitude, so that we now realize At = 10-9T. At the first period, k = 0, the coherent state has evolved into a superposition for both (a) (3 = 2 and (b) j3 = 4. After 104 periods, k = 104, the superposition is still present for (c) f3 = 2 but is decohered for the (d) f3 = 4 case.

We have applied the above approach to a harmonic oscillator coupled to a spin by means of a photon number - nondemolition Hamiltonian. The spin is being measured periodically, whereas the measurement outcome is ignored. For a sufficiently high measurement frequency, the state of the harmonic oscillator evolves in a unitary manner which can be influenced by a choice of the meter basis. In practice however, the time interval At between two subsequent measurements always remains finite and, therefore, the system evolution is subject to decoherence. As an example of application, we have simulated the evolution of an initially coherent state of the harmonic oscillator into a Schrodinger cat-like superposition state. The state departs from the superposition as time increases. The simulations confirm that the decoherence rate increases dramatically with the amplitude of the initial coherent state, thus destroying very rapidly all macroscopic superposition states. 

Abstract We review two experiments in microwave cavity QED where we have demonstrated entanglement between a two-level atom and a mesoscopic field and probed the coherence of the field state superpositions produced in the process. These studies constitute an exploration of the quantum-classical boundary and open the way to studies of non-local mesoscopic state superpositions. 

Einstein statistics and collective dynamics are represented with phonons totally decoupled from the lattice. The ground states can be represented as 2Ag + Bg + 2Au + Bu symmetry species [Lucazeau 1973], The g and u species correspond to collective oscillations of the singlet and triplet states, respectively. The ground state must be regarded as a superposition of coherent states of protons fully entangled with respect to spins and positions. This long-range quantum coherence can be probed with neutron diffraction. 

Quantum coherence is extremely sensitive to environmental interactions. This is a main stumbling block in the attempts to build quantum computers, and in spite of the fact that such devices are planned to be based on very weakly interacting systems (entanglement of photons or atoms well isolated in cavities) it is extremely difficult to preserve coherence over a sufficiently large number of basic operations steps. Coherent states in molecules are still more perturbed, as displayed for instance by the difference between the spectra of NHs and AsHs gases [Omnes 1994], Here, the H-atom in NH3 is delocalized in a quantum superposition, being on both sides of the //.rplane, while the spatial coherence of the heavier As-atom disappears during the time of observation which results in quite different optical properties. 

Another nonclassical effect is referred to as sub-Poissonian photon statistics (see, e.g., Refs. 7 and 8 and papers cited therein). It is well known that in a coherent state dehned as an inhnite superposition of the number states... 

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).

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... 

Gerry [29] has proposed a similar method based on a dispersive interaction of the atoms with a cavity mode prepared in a coherent state a). The atoms enter the cavity in superposition states... 

If the initial modes are in a superposition of coherent states (with amplitudes Oyo, where j = a, b) and chaotic fields (with intensities (n j)), then the evolution of the frequency converter is described by the following Glauber-Sudarshan P function... 

Figure 11. Clasical-field evolution of the parameters 7y(r) for initial superpositions of coherent state with thermal field (a) p, (0) and (b) p2(0) given by Eqs. (74) and (76), respectively. Labels are the same as in Fig. 9.

Whilst the above is perfectly adequate for the description of processes observed with continuous-wave (cw) input, proper representation of the optical response to pulsed laser radiation requires one further modification to the theory. It is commonly thought difficult to represent pulses of light using quantum field theory indeed, it is impossible if a number state basis is employed. However by expressing the radiation as a product of coherent states with a definite phase relationship, it is relatively simple to construct a wavepacket to model pulsed laser radiation [39]. The physical basis for this approach is that pulses necessarily have a finite linewidth and therefore in fact entail a large number of radiation modes, so that for the pump radiation, it is appropriate to construct a coherent superposition... 

The Gaussian wave packet is a coherent state and can be expressed as a superposition of oscillator states. This means that... 

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