Schrodingers Cat

Atoms and the particles that compose them are unimaginably small. Electrons have a mass of less than a trilUonth of a trilhonth of a gram, and a size so small that it is immeasurable. A single speck of dust contains more electrons than the number of people that have existed on Earth over all the centuries of time. Electrons are small in the absolute sense of the word—they are among the smallest particles that make up matter. And yet, as we have seen, an atom s electrons determine many of its chemical and physical properties. If we are to understand these properties, we must try to understand electrons. 

The thought experiment known as Schrodinger s cat is intended to show that the strangeness of the quantum world does not transfer to the macroscopic world 

Tlie absurdity resolves itself, however, upon observation. When we set ont to measure the emitted particle, the act of measuronent actually forces the atom into one state or other. 

Now here comes the absnrdity if the steel chamber is closed, the whole systan ranains unobserved, and the radioactive atom is in a state in which it has anitted the particle and not emitted the particle (with eqnal probabihty). Therefore, the cat is both dead and undead. Schrodinger put it this way [the steel chamba would have] in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. When the chamber is opened, the act of observation forces the entire systan into one state or the other the cat is either dead or alive, not both. However, while unobserved, the cat is both dead and aUve. The absurdity of the both dead and undead cat in Schrodinger s thonght experiment was meant to demonstrate how quantum strangeness does not transfer to the macroscopic world. 

In this chapter, we examine the quantum-mechanical model of the atom, a model that explains the strange behavior of electrons. In particnlar, we focus on how the model describes electrons as they exist within atoms, and how those electrons determine the chemical and physical properties of elanents. Yon have already learned mnch about those properties. You know, for example, that some elanents are metals and that others are nomnetals. You know that the noble gases are chanically inert and that the alkali metals are chemically reactive. You know that sodinm tends to form 1+ ions and that fluorine tends to form 1 - ions. But we have not explored why. The quantum-mechanical model explains why. In doing so, it explains the modem periodic table and provides the basis for our understanding of chemical bonding. 

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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. 

Figure 16.2 SchrOdinger Cat experiment. A spray of catnip is released when a decay from a small radioactive sample is detected. Version suggested by Sarah Jane Blinder and Amy Rebecca Blinder, A/ . J. Phys. 69, 633 (2001).

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. 

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... 

Superpositions of two CS have attracted much attention [13,66] as simple examples of Schrodinger cats. In this Section, we will discuss two kinds of FD... 

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],... 

Figure 9. Generalized Schrodinger cats. Wigner function for ao)(18) (a-c) and oti)(]8j (d-f) with a given by fractions of the quasiperiod T = T t = 8.8.

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. 

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. 

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