Quantum theory of measurement

 

We are mainly interested in continuous (prolonged in time) quantum measurements, especially in continuous quantum measurements of energy of a discrete-level system.

In quantum mechanics the transition between different energy levels is mathematically described by gradual changing coefficients in the superposition of two states, each one on a definite level. Thus, the transition is a process prolonged in time. However, usually the transition is understood as an instantaneous ``quantum jump". This is justified by the circumstance that the precise measurement of energy can discover the system only in one of the energy eigenstates. Measuring energy repeatedly one will each time discover the system on the same level, until once it will turn out to be on another one, and then the system will be observed each time on the new energy level. This is a picture of a quantum jump. Gradual change of the coefficients of the superposition leads to change of the probabilities to discover the system in the corresponding state at the given time.

This argument seemingly excludes the possibility to observe experimentally the gradual transition of the system from one level to the other. Moreover, it was proved that measuring energy too often an experimentalist will prevent the transition which otherwise would occure (so-called Zeno effect).

The impression of inevitability to observe a quantum transition as a jump is however only illusive. It is caused by a specific definition of the energy measurement. Namely, the measurement is conventionally understood as an infinitely precise (sharp). Then the measurement results in the projection (reduction) of the measured system state on one of the energy eigenstates.

In a series of works (Phys. Rev. A56, 44-54 (1997), Physics Letters A 237, 1-9 (1997), Intern. J. Theor. Phys. 37, 215-217 (1998)) we showed that a series of unsharp (soft) measurements of energy or, in a limiting case, a continuous fuzzy measurement of energy is a good instrument for observation time evolution of the system in the process of the level transition. The investigation was performed with exploiting both concrete quantum-mechanical models and the phenomenological restricted-path-integral (RPI) approach developed by Michael Mensky.

Our results include the following issues:

• In the framework of the restricted-path-integral and complex-Hamiltonian methods it was proved (Phys. Rev. A56, 44-54 (1997)) that a fuzzy continuous quantum measurement of energy of a driven two-level system enables one to monitor its Rabi transition between energy levels. The probability of the transition is diminished as a result of the back action of the measuring system.
• An experimental setup is suggested (Physics Letters A 237, 1-9 (1997)) to realize the continuous quantum measurement of energy of a two-level atom by scattering of a long series of electrons on it. A much more wide set of realizations may be suggested, including quantum-optical setups.
• It is shown (quant-ph/9808062) that a long series of essentially arbitrary but short and weak interactions of the two-level atom with a subsidiary system (meter) leads to the same universal behavior of the measured system described by the complex-Hamiltonian Schroedinger equation (a quantum analogue of the Central Limiting Theorem).