The von Neumann interaction between a particle and an apparatus, both of arbitrary mass, has been considered in the measurement of the position of a simple harmonic oscillator acted on by an external force. When the measurement has finite duration, both the motion of the pointer and the oscillator influence the result of the measurement. Provided that the oscillator is in an eigenstate of its position at the start of the measurement, the pointer will indicate the arithmetic average between the initial and final position of the particle with an added term which depends on the duration of the measurement and the frequency of the oscillator. This additional term is determined by the external force which also causes the appearance of a phase factor in the wave function at the end of the measurement. This phase factor depends on the average of the initial and final positions of the particle. Furthermore, the probability that the pointer will indicate a given average value is equal to the transition probability for the undisturbed free oscillator to experience the change in position. If the initial state of the pointer is a narrow wavepacket, then for any initial state of the oscillator, the measurement yields, approximately, the undisturbed probability distribution for the position of the free oscillator at the end of the measurement. The transition probability for the pointer to experience a change in its position has also been evaluated.
The von Neumann interaction between a particle and an apparatus has been considered in the measurement of the position of a particle when the interaction lasts for a finite amount of time. When the measurement has finite duration, both the motion of the pointer and the particle influences the result of the measurement. Provided that the particle is in an eigenstate of its position at the start of the measurement, the pointer will indicate the arithmetic average between the initial and final position of the particle. Furthermore, the probability that the pointer will indicate a given average value is equal to the transition probability for the undisturbed particle to experience the change in position. If the initial state of the pointer is a narrow wave packet, then for any initial state of the particle, the measurement yields, approximately, the undisturbed probability distribution for the position of the particle at the end of the measurement.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
The monitoring of a quantum-mechanical harmonic oscillator on which a classical force acts is important in a variety of high-precision experiments, such as the attempt to detect gravitational radiation. This paper reviews the standard techniques for monitoring the oscillator, and introduces a new technique which, in principle, can determine the details of the force with arbitrary accuracy, despite the quantum properties of the oscillator. The standard method for monitoring the oscillator is the "amplitude-and-phase" method (position or momentum transducer with output fed through a narrow-band amplifier). The accuracy obtainable by this method is limited by the uncertainty principle ("standard quantum limit"). To do better requires a measurement of the type which Braginsky has called "quantum nondemolition." A well known quantum nondemolition technique is "quantum counting," which can detect an arbitrarily weak classical force, but which cannot provide good accuracy in determining its precise time dependence. This paper considers extensively a new type of quantum nondemolition measurement---a "back-action-evading" measurement of the real part
X1 (or the imaginary part
X2) of the oscillator's complex amplitude. In principle
X1 can be measured "arbitrarily quickly and arbitrarily accurately," and a sequence of such measurements can lead to an arbitrarily accurate monitoring of the classical force. The authors describe explicit Gedanken experiments which demonstrate that
X1 can be measured arbitrarily quickly and arbitrarily accurately. In these experiments the measuring apparatus must be coupled to both the position (position transducer) and the momentum (momentum transducer) of the oscillator, and both couplings must be modulated sinusoidally. For a given measurement time the strength of the coupling determines the accuracy of the measurement; for arbitrarily strong coupling the measurement can be arbitrarily accurate. The "momentum transducer" is constructed by combining a "velocity transducer" with a "negative capacitor" or "negative spring." The modulated couplings are provided by an external, classical generator, which can be realized as a harmonic oscillator excited in an arbitrarily energetic, coherent state. One can avoid the use of two transducers by making "stroboscopic measurements" of
X1, in which one measures position (or momentum) at half-cycle intervals. Alternatively, one can make "continuous single-transducer" measurements of
X1 by modulating appropriately the output of a single transducer (position or momentum), and then filtering the output to pick out the information about
X1 and reject information about
X2. Continuous single-transducer measurements are useful in the case of weak coupling. In this case long measurement times are required to achieve good accuracy, and continuous single-transducer measurements are almost as good as perfectly coupled two-transducer measurements. Finally, the authors develop a theory of quantum nondemolition measurement for arbitrary systems. This paper (Paper I) concentrates on issues of principle; a sequel (Paper II) will consider issues of practice.
We analyze the time-behaviour of the Gaussian wave packet of a quantum particle moving in the field of a linear force, undergoing a continuous nondemolition observation of its position. In particular, the time-dependence of position and momentum dispersions, as well as position-momentum uncertainty is investigated. The discussion includes the case of a free particle and initially coherent state of a harmonic oscillator. It is found that in the beginning of observation the momentum dispersion and position-momentum uncertainty always increase in contrast to the position dispersion. It is also proved that all considered functions asymptotically take the form of harmonic oscillations, decaying to finite values. For initially coherent states, the asymptotic bounds for all considered functions are given.
A unified treatment by the methods of the operator mechanics is given of the quantum theory of the harmonic oscillator subjected to an applied force which is a given, but arbitrary, function of time. Few of the results are new, and most have appeared before scattered among papers devoted to a variety of topics and using a variety of mathematical techniques. Emphasis is placed on numerical methods for the evaluation of the relevant transition probabilities, and on the equality, under certain conditions, of the classical and quantum values of the average energy transfer to an oscillator by an impressed force. Generating functions for the transition probabilities are derived. Some numerical results are presented for illustration.