Classical-Quantum Boundary

The location of the quantum/classical boundary across a covalent bond also has implications for the energy terms evaluated in the Emm term. Classical energy terms that involve only quantum atoms are not evaluated. These are accounted for by the quantum Hamiltonian. Classical energy terms that include at least one classical atom are evaluated. Referring to Figure 2, the Ca—Cp bond term the N — Ca—Cp, C — Ca—Cp, Ha— Ca—Cp, Ca—Cp — Hpi, and Ca—Cp — Hp2 angle tenns and the proper dihedral terms involving a classical atom are all included. 

The use of QM-MD as opposed to QM-MM minimization techniques is computationally intensive and thus precluded the use of an ab initio or density functional method for the quantum region. This study was performed with an AMi Hamiltonian, and the first step of the dephosphorylation reaction was studied (see Fig. 4). Because of the important role that phosphorus has in biological systems [62], phosphatase reactions have been studied extensively [63]. From experimental data it is believed that Cys-i2 and Asp-i29 residues are involved in the first step of the dephosphorylation reaction of BPTP [64,65]. Alaliambra et al. [30] included the side chains of the phosphorylated tyrosine, Cys-i2, and Asp-i 29 in the quantum region, with link atoms used at the quantum/classical boundaries. In this study the protein was not truncated and was surrounded with a 24 A radius sphere of water molecules. Stochastic boundary methods were applied [66]. 

TOWARD COHERENT CONTROL AROUND THE QUANTUM-CLASSICAL BOUNDARY... 

We have developed ultrahigh-precision coherent control based on this WPI, in which we have succeeded in visualizing and controlling the ultrafast evolution of a WP interference in a molecule with precisions on the picometer spatial and attosec-ond (as) temporal scales [37-39], This is the cutting edge of coherent control. We have utilized this ultrahigh-precision coherent control to develop a molecular computer that executes ultrafast Fourier transform with molecular wave functions in 145 fs [40,41], More recently, we have extended the target of our coherent control to wave functions delocalized in a bulk solid [42,43], In this account, we will describe these developments of our experimental toolbox and the outlook toward the coherent control around the quantum-classical boundary. 

The concept of coherent control, which we have developed with isolated molecules in the gas phase, is universal and should apply to condensed matter as well. We anticipate that the coherent control of wave functions delocalized over many particles in solids or liquids will be a useful tool to track the temporal evolution of the delocalized wave function modulated by many-body interactions with other particles surrounding itself. We may find a clue to better understand the quantum-classical boundary by observing such dynamical evolution of wave functions of condensed matter. In the condensed phase, however, the coherence lifetime is in principle much shorter than in the gas phase, and the coherent control is more difficult accordingly. In this section, we show our recent efforts to develop the coherent control of condensed matter. 

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. 

Analysis of this state is interesting from the point of view of the quantum measurement problem, an issue that has been debated since the inception of quantum theory by Einstein, Bohr, and others, and continues today [31]. One practical approach toward resolving this controversy is the introduction of quantum decoherence, or the environmentally induced reduction of quantum superpoations into clasacal statistical mbrtures [32], Decoherence provides a way to quantify the elusive boundary between classical and quantum worlds, and almost always precludes the existence of macroscopic Schrodinger-cat states, except for extremely short times. On the othm hand, the creation of mesoscopic Schrddinger-cat states like that of q. (10) may allow controlled studies of quantum decoherence and the quantum-classical boundary. This problem is directly relevant to quantum computation, as we discuss below. 

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