State Schrodinger cat

T. Kobayashi I would like to make the comment that an interesting application of wavepacket control [1] is phonon squeezing in molecular systems and the creation of the Schrodinger cat state. It was theoretically predicted that there are several mechanisms that lead to squeezing of phonon states. 

We have created and analyzed thermal, Fock, squeezed, coherent, Schrodinger-cat states, and other superpositions of Fock states [21,24,25] here we briefly describe the creation and measurement of coherent and Schrddinger-cat states [21,24], We note that a scheme recently proposed for producing arbitrary states of the electromagnetic field [26] should be directly applicable to the ion case for producing arbitrary states of motion. 

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. 

The above results are applied to a harmonic oscillator coupled to a two-level system, that serves as the repeatedly measured ancilla. Relatively sparse measurements are shown to destroy the coherence of the oscillator whereas, in the Zeno-limit, the coherence is preserved for all times. This is demonstrated by a periodic generation of a Schrodinger cat-like state. The decoherence process is highly nonlinear in the initial state amplitude and the decoherence time decreases rapidly for increasing amplitude. 

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... 

We consider the generation of the Schrodinger cat-like state and the coherence loss as function of both the initial coherent amplitude (3 and the measurement cycle time At. We study the cases with At = 10-4 T and At = 10-9 T,... 

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. 

Interactions between adjacent particles of condensed phases can lead to quantum correlations, quantum interference, entanglement and decoherence, delocalization and "Schrodinger s cat" states. Such effects are theoretically expected to be extremely short-lived, due to environmental disturbances. Therefore, it has been widely believed that they cannot be experimentally detected. However, based on previous theoretical work (cf. [Chatzidimitriou-Dreismann 1995 Chatzidimitriou-Dreismann 1997 (b)]), we proposed to detect QE in condensed systems by means of sufficiently "fast" scattering techniques. Particularly suitable for this purpose is the NCS method. Our NCS investigations (on liquid H2O - D2O mixtures [Chatzidimitriou-Dreismann 1997 (a)]) started 1995 and have provided, for the first time, direct experimental evidence of attosecond QE between a proton and its adjacent particles. 

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. 

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... 

Analogously one can construct FD superpositions of several CS, that is, FD Schrodinger cat-like or kitten states, which in the limit go over into the conventional ID ones [66,67],... 

Here, we rq>ort related trapped-ion research at NIST on (1) the study of the dynamics of a two-level atomic system coupled to harmonic atomic motion, (2) the creation and characterization of nonclassical states of motion such as Schrodinger-cat superposition states, and (3) quantum logic for the generation of highly entangled states and for the investigation of scaling in a quantum computer. 

A Schrbdinger-cat state is taken to be a coherent superposition of classical-like motional states. In Schrodinger s original thought experiment [30], he described how we could, in principle, entangle a superposition state of an atom with a macroscopic-scale superposition of a live and dead cat. In our experiment [24], we construct an analogous state, on a snuiller scale, with a single atom. We create the state... 

The state represented by Eq. (15) is of the same form as that ofEq. (10). Both involve entangled superpositions and both are subject to the destructive effects of decoherence. Creation of SchrOdinger cats like Eq. (10) is particularly relevant to the ion-based quantum computer because the primary source of decoherence will probably be due to decoherence of the n=0,l) motional states during the logic operations. 

Finally, Alex Brown et al. present in Chap. 9 some applications in the context of quantum computing. The possibilities offered through quantum computation have been well known for many years now [257,258]. A quantum computer is a computation device that makes direct use of quantum-mechanical phenomena, mainly the fact that the system can be in a coherent superposition of different eigenstates due to the superposition principle. This has no classical counterpart as illustrated by the famous Schrodinger cat as explained above. In classical computers, the basic unit of information is a bit that can have only two values often denoted 0 and 1. As explained by Brown et al, in quantum computers, the unit of information is a qubit that is a coherent superposition of two quantum sates denoted 0 and 1. More precisely, it is a two-state quantum-mechanical system that can be written as a 0 > >. The advantage of a quantum computer... 

The remarkable conclusion is that the microscopic quantum state, specified by the wave function ip, can be described on a macroscopic level by the probability distribution Pj. A single pure state corresponds to a macroscopic ensemble. The interference terms that are typical for quantum mechanics no longer appear. Incidentally, this resolves the paradox of Schrodinger s cat and, in general, the quantum mechanical measurement problem. )... 

Volumes have been written about the red herring known as Schrodinger s cat. Without science writers looking for sensation, it is difficult to see how such nonsense could ever become a topic for serious scientific discussion. Any linear differential equation has an infinity of solutions and a linear combination of any two of these is another solution. To describe situations of physical interest such an equation is correctly prepared by the specification of appropriate boundary conditions, which eliminate the bulk of all possible solutions as irrelevant. Schrodinger s equation is a linear differential equation of the Sturm-Liouville type. It has solutions, known as eigenfunctions, the sum total of which constitutes a state function or wave function, which carries... 

---