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Physics Today
Coherence and precision in classical systems Spencer Klein
Published Jun 1, 2012 • 0 citations
Abstract The Quick Study “Collaboration and precision in quantum measurement” (PHYSICS TODAY, December 2011, page 72) by Rob Sewell and Morgan Mitchell points out that some quantum mechanical coherence, “quantum collaboration” in their language, allows for the magnetization of a gas to be measured with a precision of 1/N, where N is the number of photons. For large N, 1/N is smaller than 1/√N, so improves upon the usual 1/√N measurement limit; the authors comment that for noninteracting particles, the central limit theorem precludes better classical measurements.However, coherence is not merely a quantum mechanical effect; many classical systems exhibit similar behavior. For example, one can search for ultrahigh-energy neutrino interactions in Antarctic ice by observing the coherent radio pulses emitted by the resulting particle showers. The observed electric field strength of the pulse scales as the square of the number of particles in the shower (reference 11. P. W. Gorham et al., Phys. Rev. D 72, 023002 (2005). https://doi.org/10.1103/PhysRevD.72.023002; see also the article I wrote with Francis Halzen, PHYSICS TODAY, May 2008, page 29), so for a given uncertainty in field-strength measurement, the uncertainty in the number of shower particles scales as 1/N. That is purely classical electromagnetism.There are also examples of 1/N scaling without coherence. Consider a system consisting of a noninteracting gas in a reservoir at pressure P, and a small valve that controls access to a gas sensor. The best measurement of the valve’s opening time comes from the first gas molecule observed by the sensor. As one increases the pressure (number of probe molecules N), the time delay between the gate opening and the sensor decreases in a 1/N fashion. For large N, that is more accurate than finding the mean arrival time of the molecules, with an accuracy of 1/√N, and trying to correct for the average delay time.These comments are not to take anything away from the nice study by Sewell and Mitchell. However, measurements that exhibit 1/N scaling are not limited to quantum systems, and are more common than one might imagine.REFERENCESSection:ChooseTop of pageREFERENCES <<1. P. W. Gorham et al., Phys. Rev. D 72, 023002 (2005). https://doi.org/10.1103/PhysRevD.72.023002, Google ScholarCrossref© 2012 American Institute of Physics.
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The Quick Study “Collaboration and precision in quantum measurement” (PHYSICS TODAY, December 2011, page 72) by Rob Sewell and Morgan Mitchell points out that some quantum mechanical coherence, “quantum collaboration” in their language, allows for the magnetization of a gas to be measured with a precision of 1/N, where N is the number of photons. For large N, 1/N is smaller than 1/√N, so improves upon the usual 1/√N measurement limit; the authors comment that for noninteracting particles, the central limit theorem precludes better classical measurements.However, coherence is not merely a quantum mechanical effect; many classical systems exhibit similar behavior. For example, one can search for ultrahigh-energy neutrino interactions in Antarctic ice by observing the coherent radio pulses emitted by the resulting particle showers. The observed electric field strength of the pulse scales as the square of the number of particles in the shower (reference 11. P. W. Gorham et al., Phys. Rev. D 72, 023002 (2005).
https://doi.org/10.1103/PhysRevD.72.023002 ; see also the article I wrote with Francis Halzen, PHYSICS TODAY, May 2008, page 29), so for a given uncertainty in field-strength measurement, the uncertainty in the number of shower particles scales as 1/N. That is purely classical electromagnetism.There are also examples of 1/N scaling without coherence. Consider a system consisting of a noninteracting gas in a reservoir at pressure P, and a small valve that controls access to a gas sensor. The best measurement of the valve’s opening time comes from the first gas molecule observed by the sensor. As one increases the pressure (number of probe molecules N), the time delay between the gate opening and the sensor decreases in a 1/N fashion. For large N, that is more accurate than finding the mean arrival time of the molecules, with an accuracy of 1/√N, and trying to correct for the average delay time.These comments are not to take anything away from the nice study by Sewell and Mitchell. However, measurements that exhibit 1/N scaling are not limited to quantum systems, and are more common than one might imagine.REFERENCESSection:ChooseTop of pageREFERENCES <<1. P. W. Gorham et al., Phys. Rev. D 72, 023002 (2005).
https://doi.org/10.1103/PhysRevD.72.023002 , Google ScholarCrossref© 2012 American Institute of Physics.
In his letter, Christopher Monroe recommends that the term “quantum nondemolition” be abandoned (PHYSICS TODAY, January 2011, page 8). I agree, but for somewhat different reasons. Quantum measurement theory has always used the concept of ideal measurements, or “measurements of the first kind,” in which the state of the measured system does not change—that is, it is not “demolished”—if it is already in an eigenstate of the observable. Although an ideal measurement is often a good approximation when dealing with conserved particles, its realization requires special effort in the case of photons, which are usually absorbed while being measured. The concept of ideal measurements is certainly meaningful, but I do not agree with the suggestion that the repeated preparation of the same photon state should be regarded as being equivalent to its nondemolition during a measurement. The term nondemolition may be quite useful in photon experiments or in the construction of gravitational-wave antennae, where it was first used, as far as I know. But in a general context, the traditional terminology would be more to the point, since nonabsorption of the measured photon does not necessarily define an ideal measurement.Much of the confusion originates in the abridged description of a measurement as a stochastic jump from a superposition a∣0〉 + b∣1〉 of a microscopic system into ∣0〉 or ∣1〉 with probabilities ∣a∣2 or ∣b∣2, respectively. That description is a relic from the times when quantum theory was expected to apply only to isolated atoms, electrons, and so forth. (It does not seem to be a genuine part of the Copenhagen interpretation, which describes measurement outcomes in classical terms and often assumes that the wavefunction loses its meaning thereafter.) In most situations, though, one has to include other interacting systems in the unitary description. In fact, all of the much-discussed weirdness of quantum theory has been derived by using entangled quantum states for all involved systems. Therefore, most of the inflationary confusing vocabulary of quantum weirdness—including spooky action, quantum teleportation, quantum eraser, quantum Darwinism, sudden death, quantum information, and even quantum jumps—could be abandoned if one used instead a consistent description in terms of entangled wavefunctions, at least as far as this entanglement remains relevant.11. See, for example, H. D. Zeh, Foundations of Physics 40 9-10 1476 (2010). https://doi.org/10.1007/s10701-009-9383-9 Usually this is so until irreversible decoherence, which was itself derived from unitary interaction with the environment, has occurred in the chain of interactions leading to observation. However, I think it is definitely wrong to say that decoherence theory allows us to “think of any measurement as a [quantum nondemolition] measurement plus a possible interaction with a reservoir.” Although I personally prefer the assumption that unitary dynamics is universal, I agree that some stochastic dynamics has to be used somewhere along the chain of interactions that ends at the observer. It does not matter whether you describe this observed indeterminism in terms of a collapse or a branching of a universal wavefunction into dynamically autonomous components (unlike the collapse, the branching does not require any new dynamical postulate; it is a consequence of the Schrödinger equation with local interactions). Such a real or apparent collapse is an important part of the effective dynamics of “our quantum world”—our “branch,” if you like—that should be carefully analyzed rather than being used ad hoc in a pragmatic and confusing manner. In neither case would it describe a mere increase of information about an initially incompletely known quantum state of the involved systems; the indeterminism is described by a change of a state, which is known to be different from an ensemble of potential states.REFERENCESSection:ChooseTop of pageREFERENCES <<1. See, for example, H. D. Zeh, Foundations of Physics 40 9-10 1476 (2010). https://doi.org/10.1007/s10701-009-9383-9 , Google ScholarCrossref, ISI© 2011 American Institute of Physics.
Abstract In all quantitative sciences, it is common practice to increase the signal-to-noise ratio of noisy measurements by measuring identically prepared systems N times and averaging the measurement results. This leads to a scaling of the sensitivity as 1/√N, known in quantum measurement theory as the “standard quantum limit” (SQL). It is known that if one puts the N systems into an entangled state, a scaling as 1/N can be achieved, the socalled “Heisenberg limit” (HL), but decoherence problems have so far prevented implementation of such protocols for large N. Here we show that a method of coherent averaging inspired by a recent entanglement-free quantum enhanced measurement protocol is capable of achieving a sensitivity that scales as 1/N in a purely classical setup. This may substantially improve the measurement of very weak interactions in the classical realm, and, in particular, open a novel route to measuring the gravitational constant with enhanced precision.
Permit me to add some information to the article entitled, “Bohr’s molecular model, a century later.” The authors discuss the idea of “dimensional scaling,” approximating the spectra of atoms and molecules by employing perturbation theory to expand around the infinite-dimension/classical limit. That technique of approximating the Schrödinger equation, originally called the 1/N expansion, was developed and applied to numerous problems in the 1980s, including simple atoms, quarkonium, and the hydrogen molecule.11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0,22. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305,33. T. Imbo, U. Sukhatme, Phys. Rev. D 31, 2655 (1985); https://doi.org/10.1103/PhysRevD.31.2655M. Kumar, A. Srivastava, J. K. Bhattacharjee, K. Banerjee, Phys. Lett. A 117, 226 (1986). https://doi.org/10.1016/0375-9601(86)90081-2 It grew out of attempts in the 1970s to formulate quantum chromodynamics in the limit of a large number of colors. The use of the 1/N expansion in atomic physics was discussed by Edward Witten (Physics Today, July 1980, page 38), who pointed out that his discussion was based on work that applied the method to hydrogen and helium atoms.11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0The work in reference 11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0 made use of algebraic methods for the analysis, but the coordinate space method was also developed and was initially used to treat the strong-field Zeeman effect.22. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305 A nice description of the status of the field at the time was given by Laurence Yaffe (Physics Today, August 1983, page 50). Many additional applications of that method in the physics literature may be found in reference 44. A. Chatterjee, Phys. Rep. 186, 249 (1990). https://doi.org/10.1016/0370-1573(90)90048-7.REFERENCESSection:ChooseTop of pageREFERENCES <<1. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 , Google ScholarCrossref, ISIL. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0, , Google ScholarCrossref, ISI2. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305, Google ScholarCrossref, ISI3. T. Imbo, U. Sukhatme, Phys. Rev. D 31, 2655 (1985); https://doi.org/10.1103/PhysRevD.31.2655, Google ScholarCrossref, ISIM. Kumar, A. Srivastava, J. K. Bhattacharjee, K. Banerjee, Phys. Lett. A 117, 226 (1986). https://doi.org/10.1016/0375-9601(86)90081-2, , Google ScholarCrossref, ISI4. A. Chatterjee, Phys. Rep. 186, 249 (1990). https://doi.org/10.1016/0370-1573(90)90048-7, Google ScholarCrossref, ISI© 2014 American Institute of Physics.
The precision of typical thermometers consisting of
N N N particles scales as
1 / N ~1/\sqrt{N} 1/ N . For high-precision thermometry and thermometric standards, this presents an important theoretical noise floor. Here it is demonstrated that thermometry may be mapped onto the problem of phase estimation, and using techniques from optimal phase estimation, it follows that the scaling of the precision of a thermometer may in principle be improved to
1 / N ~1/N 1/ N , representing a Heisenberg limit to thermometry.
PPT|High resolutionIn the Quick Study by Eve Vavagiakis, Thomas Bachlechner, and Matthew Kleban (Physics Today, August 2021, page 62), the authors’ claim about the ontology of the electromagnetic vector fields seems too simple. As indicated in the figure above (adapted from the authors’), an electron taking either of the paths around the solenoid has an electric field E extending into the solenoid, where there is a nonzero magnetic field H. For a short time, the electron creates a Poynting vector S carrying momentum. That momentum has to be taken away from the initial momenta and thus affects the phase difference between the paths. That quantum mechanical phenomenon, called the magnetic Aharonov–Bohm effect, depends on only the magnetic flux. The obtained phase change does not depend on the distance to the solenoid. Its size is easy to calculate for an infinite solenoid.Usually, a properly renormalized electron can be thought of in quantum mechanics without considering the constantly emitted and absorbed photons building up the electric field. But for an electron passing around a solenoid, there is an exception, as noted by Lev Vaidman.11. L. Vaidman, Phys. Rev. A 86, 040101(R) (2012). https://doi.org/10.1103/PhysRevA.86.040101The electrostatic version nicely described by J. J. Sakurai and Jim Napolitano is more straightforward.22. J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics, 2nd ed., Addison-Wesley (2011). At some point, the electrostatic potential has to be switched on, which, independent of the geometric details, has to involve an electric field. That electric field crosses the particle path and takes away or adds momentum, resulting in the observed phase difference between both paths.ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. L. Vaidman, Phys. Rev. A 86, 040101(R) (2012). https://doi.org/10.1103/PhysRevA.86.040101, Google ScholarCrossref, ISI2. J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics, 2nd ed., Addison-Wesley (2011). Google Scholar© 2022 American Institute of Physics.
The Search and Discovery report titled “An electrical insulator turns metallic within a femtosecond” (Physics Today, February 2013, page 13) is a compelling account of the recent breakthrough experiments on dynamic Stark effects performed by Ferenc Krausz and collaborators. It describes how a strong nonresonant 4-fs laser pulse can be used to generate electric currents along a nanojunction on a femtosecond time scale. Phenomenologically, the current arises from the nonlinear interaction of the active material in the silica glass nanojunction with an incident laser pulse of low temporal symmetry. By varying the degree of time asymmetry of the laser, one can change the sign and magnitude of the photoinduced current. Microscopically, the underlying rectification mechanism of that rather spectacular effect is Stark shifts so large that they can dramatically modify the electronic structure of the silica glass and even bridge its large energy gap.Significantly, and in a broader context, the groundbreaking experiment by Krausz and coworkers falls into a class of symmetry-breaking laser-control scenarios known to induce net currents in spatially symmetric systems through laser fields of low temporal symmetry.11. E. Dupont et al., Phys. Rev. Lett. 74, 3596 (1995); https://doi.org/10.1103/PhysRevLett.74.3596A. Haché et al., Phys. Rev. Lett. 78, 306 (1997); https://doi.org/10.1103/PhysRevLett.78.306S. Flach, O. Yevtushenko, Y. Zolotaryuk, Phys. Rev. Lett. 84, 2358 (2000); https://doi.org/10.1103/PhysRevLett.84.2358G. Kurizki, M. Shapiro, P. Brumer, Phys. Rev. B 39, 3435 (1989); https://doi.org/10.1103/PhysRevB.39.3435S. Kohler, P. Hänggi, Nat. Nanotechnol. 2, 675 (2007). https://doi.org/10.1038/nnano.2007.357 The idea of using Stark effects as the main microscopic mechanism for the production of currents arose in an earlier theoretical proposal to use Stark effects to bridge the energy gap of a semiconducting material.22. I. Franco, M. Shapiro, P. Brumer, Phys. Rev. Lett. 99, 126802 (2007). https://doi.org/10.1103/PhysRevLett.99.126802 The experiments demonstrate how such ideas can be applied to induce currents in a material with an energy gap as large as 9 eV.REFERENCESSection:ChooseTop of pageREFERENCES <<1. E. Dupont et al., Phys. Rev. Lett. 74, 3596 (1995); https://doi.org/10.1103/PhysRevLett.74.3596, Google ScholarCrossref, ISIA. Haché et al., Phys. Rev. Lett. 78, 306 (1997); https://doi.org/10.1103/PhysRevLett.78.306, , Google ScholarCrossref, ISIS. Flach, O. Yevtushenko, Y. Zolotaryuk, Phys. Rev. Lett. 84, 2358 (2000); https://doi.org/10.1103/PhysRevLett.84.2358, , Google ScholarCrossref, ISIG. Kurizki, M. Shapiro, P. Brumer, Phys. Rev. B 39, 3435 (1989); https://doi.org/10.1103/PhysRevB.39.3435, , Google ScholarCrossref, ISIS. Kohler, P. Hänggi, Nat. Nanotechnol. 2, 675 (2007). https://doi.org/10.1038/nnano.2007.357, , Google ScholarCrossref2. I. Franco, M. Shapiro, P. Brumer, Phys. Rev. Lett. 99, 126802 (2007). https://doi.org/10.1103/PhysRevLett.99.126802, Google ScholarCrossref© 2013 American Institute of Physics.