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Physical Review A

Quantum-nondemolition measurements and the ‘‘collapse of the wave function’’

Franz X. Kärtner, H. A. Haus

Published Jun 1, 1993• 25 citations
Publisher

Abstract

This paper intends to clarify some issues in the theory of quantum measurement by taking advantage of the self-consistent quantum formulation of nonlinear optics. A quantum-nondemolition measurement of the photon number of an optical pulse can be performed with a nonlinear Mach-Zehnder interferometer followed by a balanced detector. The full quantum-mechanical treatment shows that the shortcut in the description of the quantum-mechanical measurement, the so-called ``collapse of the wave function,'' is not needed for a self-consistent interpretation of the measurement process. Coherence in the density matrix of the signal to be measured is progressively reduced with increasing accuracy of the photon-number determination. The quantum-nondemolition measurement is incorporated in the double-slit experiment and the contrast ratio of the fringes is found to decrease systematically with increasing information on the photon number in one of the two paths. The ``gain'' in the measurement can be made arbitrarily large so that postprocessing of the information can proceed classically.

DOI
10.1103/physreva.47.4585

Related papers

On the theory of quantum measurement

H. A. Haus, Franz X. Kärtner

Many so called paradoxes of quantum mechanics are clarified when the measurement equipment is treated as a quantized system. Every measurement involves nonlinear processes. Self consistent formulations of nonlinear quantum optics are relatively simple. Hence optical measurements, such as the quantum nondemolition (QND) measurement of photon number, are particularly well suited for such a treatment. It shows that the so called 'collapse of the wave function' is not needed for the interpretation of the measurement process. Coherence of the density matrix of the signal is progressively reduced with increasing accuracy of the photon number determination. If the QND measurement is incorporated into the double slit experiment, the contrast ratio of the fringes is found to decrease with increasing information on the photon number in one of the two paths.

•May 1, 1994•1 citation

Optical quantum nondemolition measurements and the Copenhagen interpretation

H. A. Haus, Franz X. Kärtner

We start from the reasonable assumption that a quantum measurement can be properly described only when the measurement apparatus itself is also treated quantum mechanically. Using idealized optical measurements as an example and introducing a small loss into the measurement apparatus, we show that the combined density matrix of the system and apparatus decoheres in a manner that the outcome of the measurement can be described classically. The loss can be arbitrarily small when the classical limit of the measurement apparatus is taken, i.e., the gain is made to approach infinity. Further, we show that the von Neumann postulate of the collapse of the wave function of the measured system into an eigenstate of the measured observable can be deduced from the analysis of two quantum nondemolition experiments in cascade and need not be introduced as a separate postulate. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Jun 1, 1996•13 citations

Decoherence of the field in quantum nondemolition photon number measurement using the Kerr nonlinearity

A. V. Kozlovskiĭ

The role of decoherence of an electromagnetic field in the process of quantum nondemolition measurement of the number of photons using a nonlinear Mach-Zehnder interferometer is investigated. This decoherence is caused by distributed losses in a Kerr medium. A long interaction time between the field and the medium is required to achieve a high accuracy in the photon number measurement. The losses in the Kerr medium accompanying the resonance four-wave interaction between the measured and the probe fields lead to a measurement error close to unity at γτ ≫1 (γ is the rate of losses in the medium, and τ is the time of interaction between the field and the medium); consequently, nondemolition quantum measurement turns out to be impossible in the scheme considered. Under these conditions, an increase in the intensity of the probe field does not result in achievement of the required measurement accuracy.

Optics and Spectroscopy•Nov 1, 2004•0 citations

Noise reduction in quantum nondemolition measurement with a nonlinear Mach–Zehnder interferometer using squeezed vacuum

Masataka Shirasaki, H. A. Haus

A quantum nondemolition measurement of photon number taken with a nonlinear Mach–Zehnder interferometer is analyzed. When this interferometer is excited by a coherent input, it shows a large amount of noise, caused by the self-phase modulation. We demonstrate how this noise can be suppressed by applying a squeezed vacuum to the unexcited port of the interferometer.

Journal of the Optical Society of America B•Mar 1, 1991•9 citations

Demolishing quantum nondemolition

Christopher R. Monroe

Most physicists are taught that quantum physics has two parts: coherent wave mechanics, in which quantum states evolve according to the reversible Schrödinger wave equation, and the measurement process, in which quantum states irreversibly collapse into a particular state given by the details of the measurement. So when a quantum superposition state such as a|0〉 + b|1〉 is measured with a device that discriminates perfectly between |0〉 and |1〉, the system evolves to the definite state |0〉 or |1〉 with respective probabilities |a|2 and |b|2. Of course, when such an ideal measurement is repeated on the surviving state, the same definite state will appear again and again. Many of us, starting with Albert Einstein, do not like the apparent disconnect between those two rules and may not even subscribe to that conventional “Copenhagen interpretation” of quantum mechanics. It is nevertheless useful because it correctly predicts the results of all experiments performed to date, even though it gives the appearance that the observer necessarily affects the system.In the past two decades, the advent of quantum information science and the ability of experiments to control quantum systems such as single atoms have provided the happy opportunity to once again ponder the disconnect between the fundamental quantum concepts of coherent evolution and measurement. Quantum systems can now be measured with nearly perfect efficiency, meaning that the observed distribution of a measurement is given almost exactly by the probabilities of the underlying superposition, as prescribed by the conventional Copenhagen rules that we are taught.However, a new term, the quantum nondemolition, or QND, has crept into the literature these past decades. Also known as “quantum back-action-evading measurement,” it’s a needlessly confusing term that seems to highlight the mundane aspect of a perfect quantum measurement: If one already knows that the system is in a particular eigenstate of the measuring device, then, obviously, a measurement on the system will produce that eigenstate and leave the system intact. Zero information is gained from the repeated measurement. On the other hand, when the system is not in an eigenstate of the measuring device, the quantum state can be thought to collapse to one of its eigenstates. In that case, information is gained from the system, and the QND measurement most certainly demolishes the system. The concept of QND measurement adds nothing to the usual rules of quantum measurement, regardless of interpretation. But worse, the term confounds students and newcomers who may even be comfortable with the prescriptions of quantum physics.Not all measurements are perfect, of course, and some people may wish to use the QND nomenclature to distinguish the efficiency of a measuring device. However, in the spirit of quantum decoherence theory (see, for example, Wojciech Zurek’s article in PHYSICS TODAY, October 1991, page 36), we can still think of any measurement as a QND measurement plus a possible interaction with a reservoir that can either mask the information obtained from the system or even uncontrollably change the quantum state. The interaction with the reservoir is what can make quantum problems difficult, and it is there that we confront the conceptual difficulties with quantum mechanics and the quantum–classical divide.As a common example of an imperfect measurement, consider photodetection. When a single photon strikes a photomultiplier tube and generates a pulse of photocurrent, the photon is lost forever. Or is it? The photocurrent may interact with a macroscopic system of a bulk conductor, and we may measure the resulting voltage across the conductor. Sure, the photon has disappeared, but if our detector indicates that we had one photon, we can always create another and get the same answer again and again, exactly like a QND measurement. Even if we begin with a coherent superposition of zero and one photon, which is not an eigenstate of our measurement device, that approach works. After the first measurement collapses the state into |0〉 or |1〉, we know which one came out and we either prepare zero or one photon forever after. A real photodetector is not 100% efficient, of course, meaning that a single photon striking the detector could register nothing, or that even in the absence of photons the noisy detector fires anyway. Again we can think of this as a QND measurement in which we have lost information to the environment. In practice, we trace over the environment and then must describe the system in terms of a density matrix containing the probabilities of various outcomes.In every case, the concept of QND measurement is confusing and unnecessary. It wasn’t needed in the formative years of quantum mechanics; and now with quantum information theory, entanglement, and decoherence theory, we still do not need it. Marvelous advances in many experimental quantum systems, such as individual atoms and superconducting circuits, are allowing quantum measurements to be performed almost exactly like in textbooks, as most of us have learned. Why not demolish the term “QND”?© 2011 American Institute of Physics.

Physics Today•Dec 30, 2010•12 citations

A microscopic model of wave-function dephasing and decoherence in the double-slit experiment

Satish Ramakrishna

The act of measurement on a quantum state is supposed to "dephase" (dephasing refers to the phenomenon that the states lose phase coherence; then the phases get randomized in interaction with a bath of other oscillators, which is referred to as "decoherence"), then "decohere" and "collapse" (or more precisely "register" and "reduce") the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schrödinger's equation. Progress has, however, been made in studying this problem with two main calculation tools-one uses a time-independent Hamiltonian, while a rather more general approach proving that decoherence occurs under some generic conditions. The two general approaches to the study of wave-function collapse are as follows. The first approach, called the "consistent" or "decoherent"' histories approach, studies microscopic histories that diverge probabilistically and explains collapse as an event in our particular history. The other, referred to as the "environmental decoherence" approach studies the effect of the environment upon the quantum system, to explain wave-function decoherence which is produced by irreversible effects of various sorts. However, as we know, wave-function collapse is not related to thermal connection with the environment, rather, it is inherent to how measurements are performed by macroscopic apparata. In the "environmental decoherence" approach, one studies decoherence using a Markovian-approximated Master equation to study the time-evolution of the reduced density matrix (post dephasing) and obtains the long-time dependence of the off-diagonal elements of this matrix. The calculation in this paper studies the evolution of a quantum system starting with "dephasing" followed by the effects of the environment with some differences from prior analyses. We start from the Schrödinger equation for the state of the system, with a time-dependent Hamiltonian that reflects the actual microscopic interactions that are occurring. Then we systematically solve (exactly) for the time-evolved state, without invoking a Markovian approximation when writing out the effective time-evolution equation, i.e., keeping the evolution unitary until the end. This approach is useful, and it shows that the system wave-function will explicitly "un-collapse" if the measurement apparatus is sufficiently small. However, in the limit of a macroscopic system, this "dephasing" quickly leads to "decoherence"-collapse is a temporary state that will simply take extremely long (of the order of multiple universe lifetimes) to reverse. This has been attempted previously and our calculation is particularly simple and calculable. We make some connections to the work by Linden et al. while doing so. The calculation in this paper has interesting implications for the interpretation of the Wigner's friend experiment, as well as the Mott experiment, which is explored in "Connection to some general theoretical results" and "Recurrence times" (especially the enumerated points in "Recurrence times"). The upshot is that as long as Wigner's friend is macroscopically large (or uses a macroscopically large measuring instrument), no one needs to worry that Wigner would see something different from his friend. Indeed, Wigner's friend does not even need to be conscious during the measurement that she conducts. It also allows one to reasonably interpret some of the more recent thought experiments proposed. In particular, as a result of the mathematical analysis, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential. Instead, it is the usual Fermi-golden rule result. The long-term behavior is, of course, still exponential. This is a second novel feature of the paper-we connect the short-term Fermi-golden rule (quadratic-in-time behavior) transition probability to the exponential long-time behavior of a collapsing wave-function in one continuous mathematical formulation.

arXiv:1911.08586•Scientific Reports•Oct 25, 2021•3 citations
This paper intends to clarify some issues in the theory of quantum measurement by taking advantage of the self-consistent quantum formulation of nonlinear optics. A quantum-nondemolition measurement of the photon number of an optical pulse can be performed with a nonlinear Mach-Zehnder interferometer followed by a balanced detector. The full quantum-mechanical treatment shows that the shortcut in the description of the quantum-mechanical measurement, the so-called collapse of the wave function,'' is not needed for a self-consistent interpretation of the measurement process. Coherence in the density matrix of the signal to be measured is progressively reduced with increasing accuracy of the photon-number determination. The quantum-nondemolition measurement is incorporated in the double-slit experiment and the contrast ratio of the fringes is found to decrease systematically with increasing information on the photon number in one of the two paths. The gain'' in the measurement can be made arbitrarily large so that postprocessing of the information can proceed classically.
Many so called paradoxes of quantum mechanics are clarified when the measurement equipment is treated as a quantized system. Every measurement involves nonlinear processes. Self consistent formulations of nonlinear quantum optics are relatively simple. Hence optical measurements, such as the quantum nondemolition (QND) measurement of photon number, are particularly well suited for such a treatment. It shows that the so called 'collapse of the wave function' is not needed for the interpretation of the measurement process. Coherence of the density matrix of the signal is progressively reduced with increasing accuracy of the photon number determination. If the QND measurement is incorporated into the double slit experiment, the contrast ratio of the fringes is found to decrease with increasing information on the photon number in one of the two paths.
We start from the reasonable assumption that a quantum measurement can be properly described only when the measurement apparatus itself is also treated quantum mechanically. Using idealized optical measurements as an example and introducing a small loss into the measurement apparatus, we show that the combined density matrix of the system and apparatus decoheres in a manner that the outcome of the measurement can be described classically. The loss can be arbitrarily small when the classical limit of the measurement apparatus is taken, i.e., the gain is made to approach infinity. Further, we show that the von Neumann postulate of the collapse of the wave function of the measured system into an eigenstate of the measured observable can be deduced from the analysis of two quantum nondemolition experiments in cascade and need not be introduced as a separate postulate. \textcopyright{} 1996 The American Physical Society.
The role of decoherence of an electromagnetic field in the process of quantum nondemolition measurement of the number of photons using a nonlinear Mach-Zehnder interferometer is investigated. This decoherence is caused by distributed losses in a Kerr medium. A long interaction time between the field and the medium is required to achieve a high accuracy in the photon number measurement. The losses in the Kerr medium accompanying the resonance four-wave interaction between the measured and the probe fields lead to a measurement error close to unity at γτ ≫1 (γ is the rate of losses in the medium, and τ is the time of interaction between the field and the medium); consequently, nondemolition quantum measurement turns out to be impossible in the scheme considered. Under these conditions, an increase in the intensity of the probe field does not result in achievement of the required measurement accuracy.
Most physicists are taught that quantum physics has two parts: coherent wave mechanics, in which quantum states evolve according to the reversible Schrödinger wave equation, and the measurement process, in which quantum states irreversibly collapse into a particular state given by the details of the measurement. So when a quantum superposition state such as a|0〉 + b|1〉 is measured with a device that discriminates perfectly between |0〉 and |1〉, the system evolves to the definite state |0〉 or |1〉 with respective probabilities |a|2 and |b|2. Of course, when such an ideal measurement is repeated on the surviving state, the same definite state will appear again and again. Many of us, starting with Albert Einstein, do not like the apparent disconnect between those two rules and may not even subscribe to that conventional “Copenhagen interpretation” of quantum mechanics. It is nevertheless useful because it correctly predicts the results of all experiments performed to date, even though it gives the appearance that the observer necessarily affects the system.In the past two decades, the advent of quantum information science and the ability of experiments to control quantum systems such as single atoms have provided the happy opportunity to once again ponder the disconnect between the fundamental quantum concepts of coherent evolution and measurement. Quantum systems can now be measured with nearly perfect efficiency, meaning that the observed distribution of a measurement is given almost exactly by the probabilities of the underlying superposition, as prescribed by the conventional Copenhagen rules that we are taught.However, a new term, the quantum nondemolition, or QND, has crept into the literature these past decades. Also known as “quantum back-action-evading measurement,” it’s a needlessly confusing term that seems to highlight the mundane aspect of a perfect quantum measurement: If one already knows that the system is in a particular eigenstate of the measuring device, then, obviously, a measurement on the system will produce that eigenstate and leave the system intact. Zero information is gained from the repeated measurement. On the other hand, when the system is not in an eigenstate of the measuring device, the quantum state can be thought to collapse to one of its eigenstates. In that case, information is gained from the system, and the QND measurement most certainly demolishes the system. The concept of QND measurement adds nothing to the usual rules of quantum measurement, regardless of interpretation. But worse, the term confounds students and newcomers who may even be comfortable with the prescriptions of quantum physics.Not all measurements are perfect, of course, and some people may wish to use the QND nomenclature to distinguish the efficiency of a measuring device. However, in the spirit of quantum decoherence theory (see, for example, Wojciech Zurek’s article in PHYSICS TODAY, October 1991, page 36), we can still think of any measurement as a QND measurement plus a possible interaction with a reservoir that can either mask the information obtained from the system or even uncontrollably change the quantum state. The interaction with the reservoir is what can make quantum problems difficult, and it is there that we confront the conceptual difficulties with quantum mechanics and the quantum–classical divide.As a common example of an imperfect measurement, consider photodetection. When a single photon strikes a photomultiplier tube and generates a pulse of photocurrent, the photon is lost forever. Or is it? The photocurrent may interact with a macroscopic system of a bulk conductor, and we may measure the resulting voltage across the conductor. Sure, the photon has disappeared, but if our detector indicates that we had one photon, we can always create another and get the same answer again and again, exactly like a QND measurement. Even if we begin with a coherent superposition of zero and one photon, which is not an eigenstate of our measurement device, that approach works. After the first measurement collapses the state into |0〉 or |1〉, we know which one came out and we either prepare zero or one photon forever after. A real photodetector is not 100% efficient, of course, meaning that a single photon striking the detector could register nothing, or that even in the absence of photons the noisy detector fires anyway. Again we can think of this as a QND measurement in which we have lost information to the environment. In practice, we trace over the environment and then must describe the system in terms of a density matrix containing the probabilities of various outcomes.In every case, the concept of QND measurement is confusing and unnecessary. It wasn’t needed in the formative years of quantum mechanics; and now with quantum information theory, entanglement, and decoherence theory, we still do not need it. Marvelous advances in many experimental quantum systems, such as individual atoms and superconducting circuits, are allowing quantum measurements to be performed almost exactly like in textbooks, as most of us have learned. Why not demolish the term “QND”?© 2011 American Institute of Physics.
The act of measurement on a quantum state is supposed to "dephase" (dephasing refers to the phenomenon that the states lose phase coherence; then the phases get randomized in interaction with a bath of other oscillators, which is referred to as "decoherence"), then "decohere" and "collapse" (or more precisely "register" and "reduce") the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schrödinger's equation. Progress has, however, been made in studying this problem with two main calculation tools-one uses a time-independent Hamiltonian, while a rather more general approach proving that decoherence occurs under some generic conditions. The two general approaches to the study of wave-function collapse are as follows. The first approach, called the "consistent" or "decoherent"' histories approach, studies microscopic histories that diverge probabilistically and explains collapse as an event in our particular history. The other, referred to as the "environmental decoherence" approach studies the effect of the environment upon the quantum system, to explain wave-function decoherence which is produced by irreversible effects of various sorts. However, as we know, wave-function collapse is not related to thermal connection with the environment, rather, it is inherent to how measurements are performed by macroscopic apparata. In the "environmental decoherence" approach, one studies decoherence using a Markovian-approximated Master equation to study the time-evolution of the reduced density matrix (post dephasing) and obtains the long-time dependence of the off-diagonal elements of this matrix. The calculation in this paper studies the evolution of a quantum system starting with "dephasing" followed by the effects of the environment with some differences from prior analyses. We start from the Schrödinger equation for the state of the system, with a time-dependent Hamiltonian that reflects the actual microscopic interactions that are occurring. Then we systematically solve (exactly) for the time-evolved state, without invoking a Markovian approximation when writing out the effective time-evolution equation, i.e., keeping the evolution unitary until the end. This approach is useful, and it shows that the system wave-function will explicitly "un-collapse" if the measurement apparatus is sufficiently small. However, in the limit of a macroscopic system, this "dephasing" quickly leads to "decoherence"-collapse is a temporary state that will simply take extremely long (of the order of multiple universe lifetimes) to reverse. This has been attempted previously and our calculation is particularly simple and calculable. We make some connections to the work by Linden et al. while doing so. The calculation in this paper has interesting implications for the interpretation of the Wigner's friend experiment, as well as the Mott experiment, which is explored in "Connection to some general theoretical results" and "Recurrence times" (especially the enumerated points in "Recurrence times"). The upshot is that as long as Wigner's friend is macroscopically large (or uses a macroscopically large measuring instrument), no one needs to worry that Wigner would see something different from his friend. Indeed, Wigner's friend does not even need to be conscious during the measurement that she conducts. It also allows one to reasonably interpret some of the more recent thought experiments proposed. In particular, as a result of the mathematical analysis, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential. Instead, it is the usual Fermi-golden rule result. The long-term behavior is, of course, still exponential. This is a second novel feature of the paper-we connect the short-term Fermi-golden rule (quadratic-in-time behavior) transition probability to the exponential long-time behavior of a collapsing wave-function in one continuous mathematical formulation.