Physics papers: XIII. States of matter, the vacuum and particles

Kapitza 1938 — Viscosity of Liquid Helium below the λ-Point

Nature 141, 74 experiment

Kapitza measured the viscosity of liquid helium below the λ-point and reported that he found it very small; the name ‘superfluidity’ for the phenomenon is associated with this note.

Helium-4 passes, below about 2.17 K (the λ-point), into a separate state called liquid helium II. Below this temperature it was known to behave like a liquid that conducts heat unusually well and seeps from the walls of vessels through thin cracks. Kapitza’s question was simple: what, in fact, is the viscosity of this liquid, that is, its internal resistance to flow?

The short note in Nature describes a set-up that measures the passage of the liquid through a very thin gap between two flat surfaces. The gap is so narrow that an ordinary fluid would hardly pass at all. Helium II, however, flows easily through this gap. The measured viscosity is much smaller than that of liquid helium just above the λ-point; Kapitza presents this as a result pointing to a state very different from any known liquid, and he names this state ‘superfluidity’.

The note is short and proposes no theory; it only reports the measurement and the surprising size of the measurement. The question of what kind of mechanism it is is left open. In the same issue the independent experiment of Allen and Misener was also published.

One thing to note when reading: the note uses the word ‘viscosity’ as the name of a measurement. The question of whether the viscosity is really zero or has merely fallen below a measurable level is not a question this short text tries to answer. For this reason it is fitting to read the note not as ‘frictionless flow was proven’ but as ‘the measured viscosity came out much smaller than expected’. For the details of the experimental set-up and the numerical limits one has to look at the note itself; what is described here is only the framework.

This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.

Afterwards: In the same year Tisza, and then Landau, proposed two-fluid models for helium II; one part of the liquid behaves frictionlessly, another part behaves ordinarily. The phenomenon is related to quantum Bose condensation, but since interactions are strong in liquid helium, the two are not the same thing.

Allen 1938 — Flow of Liquid Helium II

Nature 141, 75 experiment

Allen and Misener measured the flow of helium II through narrow capillary tubes and showed that it differs from the usual viscous flow.

This short paper, which appeared in the same issue as Kapitza’s note, approaches the same phenomenon with an independent experiment. The question is again this: how does liquid helium below the λ-point behave while passing through a narrow tube?

The authors pass the liquid through very fine capillary tubes and measure the flow rate. In an ordinary viscous liquid, as the tube narrows the flow rate falls quickly, and it rises in proportion to the pressure difference. Helium II, however, passes through even very narrow tubes with unexpected ease; the flow rate does not obey the laws expected for ordinary liquids. The authors interpret this behavior as the liquid’s viscosity being very small and report it as a finding in conflict with the familiar laws of flow.

Like Kapitza’s, this paper proposes no mechanism; its main contribution is that the same phenomenon was measured in two separate laboratories with two separate set-ups. That two mutually independent measurements point in the same direction strengthens the impression that the phenomenon is not a measurement artifact.

It is suggested that these two short papers be read together: Kapitza treats the viscosity through a gap between two surfaces, Allen and Misener indirectly through capillary tubes. The common result of the two methods is that helium II displays a behavior that does not obey the known laws of liquid flow. For detailed numerical values and the design of the set-up one should look at the paper itself; what is given here is only the general framework, and it was written without seeing the paper’s text.

This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.

Afterwards: In the following years experiments measuring the flow behavior of the same liquid multiplied, and the two-fluid model was used to organize this behavior quantitatively.

Bardeen 1957 — Theory of Superconductivity

Phys. Rev. 108, 1175 theory

Bardeen, Cooper and Schrieffer gave the first microscopic theory of superconductivity, starting from electrons.

Some metals, below a critical temperature, lose electrical resistance entirely and push magnetic fields out of their interior. Up to 1957 these phenomena had been observed, but how electrons established such an order was not known. The question was: how do electrons that repel one another establish a collective state that can flow without friction?

The starting point of BCS theory is a result Cooper had shown earlier: two electrons above the Fermi sea can be bound to each other by a weak attraction mediated by the vibrations of the lattice (phonons). The paper shows that these ‘Cooper pairs’ form a coherent ground state in the whole electron population: electrons with opposite momenta and opposite spins pair up, and the pairs take part in a single quantum wave function. Between this state and the smallest energy needed to break the pairs there is an ‘energy gap’.

The gap explains why the resistance is zero: small excitations cannot break the pairs, so the electrons cannot lose energy through friction. The theory also predicts the dependence of the critical temperature on isotope mass and the exponential behavior in the specific heat; both have agreed with observations. The paper builds the theory with detailed mathematics and for metallic superconductors.

BCS theory rests on the assumption of a single electron–phonon attraction and describes metallic, weakly coupled superconductors well. The idea at the center of the theory is that the absence of resistance is not a property of individual electrons but a collective state of paired electrons. For this reason there is the same kind of idea of collective coherence between superconductivity and superfluidity; however, helium-4 consists of bosons whereas the electrons in a superconductor are fermions, and this difference explains why pairing is needed. The details of the paper’s long mathematical sections are not reproduced here.

This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.

Afterwards: The temperature and behavior of the copper oxide superconductors discovered in 1986 (high-temperature superconductors) could not be explained by the plain BCS–phonon picture; their mechanism continues to be a subject of debate.

Casimir 1948 — On the attraction between two perfectly conducting plates

Proc. K. Ned. Akad. Wet. B51, 793 — citation verified second-hand theory

Casimir calculated the attractive force between two perfectly conducting parallel plates from the zero-point energy of the vacuum.

Casimir asks how two uncharged, perfectly conducting plates, parallel to each other, behave in empty space. In classical electrodynamics no force is expected between them. Casimir, however, counts the allowed modes of the electromagnetic field between the plates: the field’s boundary conditions at the conducting surfaces allow only certain wavelengths to fit between the plates. Outside there is no such restriction.

Each mode, according to quantum theory, carries a zero-point energy of at least ½ħω. The sum over all modes is infinite; but the difference between the region between the plates and the outside is finite and depends on the plate separation. The result is an attractive force per unit area of about π²ħc/(240 d⁴); here d is the plate separation, ħ is the reduced Planck constant and c is the speed of light. That is, the force is inversely proportional to the fourth power of the separation and depends only on fundamental constants and geometry.

This paper is a short calculation; while it proposes an experiment, it does not carry out the measurement. The assumptions the calculation rests on (perfect conductivity, zero temperature, smooth surfaces) are not met in real materials; these deviations are treated in later measurements as corrections.

An important point is that the finite result in the formula comes out of the difference of infinite sums. This difference gives only the change of the energy with plate separation; the absolute zero-point energy is still infinite and cannot be measured on its own. What is measured is the force arising from the separation-dependent energy difference. The plates used in real experiments are not perfect conductors and the temperature is not zero; for this reason the theory compared with measurements is the corrected form of the 1948 formula. For the paper’s detailed derivation one has to look at the paper itself.

This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.

Afterwards: The interpretation of the effect is disputed: some authors tie it directly to the zero-point energy of the vacuum, others say the same result can also be derived as the effect of the material’s charges on one another. Measurements do not separate the two interpretations, since both predict the same force.

Lamoreaux 1997 — Demonstration of the Casimir force in the 0.6 to 6 μm range

Phys. Rev. Lett. 78, 5 experiment

Lamoreaux measured the Casimir force with a torsion pendulum over a range of 0.6–6 μm.

Casimir’s 1948 prediction could not be tested directly for half a century, because the force is noticeable only at separations of a micrometer and below, and at this scale it is very difficult to keep two surfaces parallel. Lamoreaux’s question was: does this attractive force really exist, and is it of the predicted size?

In the experiment, in place of one of the plates a gold-coated spherical surface is used, so that the problem of parallelism disappears. The force between the sphere and the flat plate is read from the angular deflection of a torsion pendulum. The separation is varied between 0.6 and 6 micrometers. Electrostatic forces are first nulled with a separate measurement; thus the attraction that remains corresponds to the Casimir force between uncharged surfaces.

The result the author reports is that the measured force agrees with the theory’s prediction within an error margin of a few percent. The theoretical comparison is a calculation corrected for the geometry of the sphere and for the imperfect conductivity of gold.

Using the sphere–plate geometry in the experiment is a choice: keeping two flat plates parallel at micrometer separation is much harder than bringing a sphere close to a plate. In the sphere–plate geometry the force formula differs from the plate–plate formula, so the comparison is made with the theory recalculated for the sphere. The result’s detailed error margin and correction terms are given in the paper; here a general framework written without access to the text is presented.

This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.

Afterwards: The experiment is cited as one of the first direct measurements; more precise measurements followed, one of which is Mohideen and Roy’s atomic force microscope measurement.

Mohideen 1998 — Precision Measurement of the Casimir Force from 0.1 to 0.9 μm

Phys. Rev. Lett. 81, 4549 experiment

Mohideen and Roy measured the Casimir force much more precisely, with an atomic force microscope, over a range of 0.1–0.9 μm.

This work, carried out a few years after Lamoreaux’s measurement, aims to measure the same force at much smaller separations and with smaller uncertainty. The question: does the Casimir force agree with theory at the one percent level once the effects of real materials and of temperature are taken into account?

The authors measure the force between a metal-coated sphere 196 micrometers in diameter and a flat plate with an atomic force microscope. The sphere is glued to a cantilever attached to the microscope’s tip; as it approaches the plate the cantilever bends and the bending gives the force. The separation is scanned between 0.1 and 0.9 micrometers.

The results are consistent with theoretical calculations that include corrections for finite conductivity, surface roughness and temperature. The root-mean-square deviation between theory and experiment is 1.6 pN, which corresponds to about one percent at the smallest separation.

A measurement at this precision turns the question of the force’s existence into a question of detail: what is now being discussed is how the corrections are to be calculated.

The importance of the measurement is that the error margin has come down to the level of one percent. At this level, the theory’s assumption of a perfect conductor is seen to fall short, and corrections that take into account the optical properties of real metals become necessary. The abstract says that these corrections (finite conductivity, roughness, temperature) are consistent with the theory taken together; the paper gives the detail and the data tables.

Afterwards: Later studies repeated this measurement with different geometries and different materials; the question of interpretation (whether the force is to be read as zero-point energy or as material interaction) was not resolved by measurement.

Higgs 1964 — Broken Symmetries and the Masses of Gauge Bosons

Phys. Rev. Lett. 13, 508 theory

Higgs showed how gauge bosons can acquire mass through symmetry breaking.

Gauge theories derive forces from a symmetry principle; but in these theories the gauge bosons come out massless. The weak force has a short range, so its carriers must be heavy. Another obstacle is the Goldstone theorem: when a symmetry is spontaneously broken, massless scalar particles were expected, and these were not observed. Higgs’s question is whether this obstacle can be overcome.

The two-page paper shows that in a model in which a gauge symmetry is broken, the Goldstone theorem does not hold. The massless Goldstone mode turns into an extra component of the gauge field; the gauge boson thereby acquires mass. What remains is a massive excitation of a scalar field: the paper points out that the presence of a multiplet of scalar fields in incomplete multiplets requires the existence of a massive scalar boson.

The paper gives the basic idea of the mechanism in a short and general form; it is not a complete particle physics model. Its application to the Standard Model and the detailed solution of the mass question were carried out in other works.

The shortness of the paper also shows that its scope is limited: it puts forward the idea of the spontaneous symmetry breaking of a scalar field as the source of mass, but it does not answer the question ‘what is the mass of the Higgs boson’. The mass is a parameter and has to be determined by experiment. Also, this mechanism explains the mass of the fundamental particles (the W and Z bosons, quarks, charged leptons); most of the proton’s mass comes from the energy of the strong interaction that binds the quarks.

This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.

Afterwards: In the same year Englert and Brout, and Guralnik, Hagen and Kibble, reached similar results independently. The gauge sector of the Standard Model was built with this idea; the search for the scalar particle took decades.

Englert 1964 — Broken Symmetry and the Mass of Gauge Vector Mesons

Phys. Rev. Lett. 13, 321 theory

Englert and Brout solved the same problem independently, with a model in which gauge fields acquire mass through spontaneous symmetry breaking.

This paper was published a few weeks before Higgs’s and asks the same question: when a gauge symmetry is spontaneously broken, can vector mesons acquire mass?

The authors examine a gauge theory in which the vacuum expectation value of a scalar field is non-zero and compute the quantum corrections. The result is that the gauge field acquires mass; the massless Goldstone particle does not appear as a separate physical particle.

The approach the paper takes is presented in a language different from Higgs’s (through the perturbative calculation of field theory), and the emphasis on the ‘remaining massive scalar boson’ that is explicitly stressed in Higgs’s paper does not appear here in the same form. The two papers are two independent presentations of the same mechanism.

This paper and Higgs’s can be read as two writings of the same idea. The difference between the two lies in their emphases and in the tools they use. The question of which first showed which detail is a matter for the writing of history and is not decided here. For the paper’s full calculations one has to look at the paper itself; this summary was written without access to the text.

This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.

Afterwards: For this reason the mechanism is sometimes referred to by more than one name (Brout–Englert–Higgs, Guralnik–Hagen–Kibble). The 2013 Nobel Prize in Physics was given to Englert and Higgs.

Aad 2012 — Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC

Phys. Lett. B 716, 1 (ATLAS Collaboration) experiment

The ATLAS collaboration reported a new boson around 126 GeV at the LHC.

The last untested piece of the Standard Model was the Higgs boson. ATLAS’s question is whether there is an excess in proton–proton collisions at the LHC that fits this particle.

The analysis uses about 4.8 fb⁻¹ of data collected in 2011 at 7 TeV and about 5.8 fb⁻¹ collected in 2012 at 8 TeV. Three decay channels are examined in detail: H→ZZ*→4 leptons, H→γγ and H→WW→eνμν; these are combined with earlier 7 TeV searches.

The reported result is clear evidence for the production of a neutral boson with a measured mass of 126.0 ± 0.4 (statistical) ± 0.4 (systematic) GeV. The significance of the observation is 5.9 standard deviations; that is, the probability of its arising from a background fluctuation is about 1.7 × 10⁻⁹. The authors say that this is compatible with the production and decay of the Standard Model Higgs boson. Detailed measurement of the properties (spin, couplings) lies outside the scope of this paper.

Here the expression ‘new particle’ is deliberately cautious: the title says that a new particle was observed in the search for the Standard Model Higgs boson, but whether it is exactly the Higgs boson predicted by the Standard Model has to be determined by later measurements. The paper says ‘compatible’; it does not say identical. A significance above 5 standard deviations is the threshold commonly used in particle physics to announce a discovery. The systematic uncertainty of the mass and the statistical uncertainty are of the same size.

Afterwards: CMS reported a similar result with an independent analysis in the same period; the particle’s properties were examined with later measurements.

Chatrchyan 2012 — Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC

Phys. Lett. B 716, 30 (CMS Collaboration) experiment

The CMS collaboration, independently and in the same period, reported a new boson around 125 GeV.

CMS, the second experiment asking the same question as ATLAS, works with the Compact Muon Solenoid detector at the LHC. Its data are about 5.1 fb⁻¹ at 7 TeV and about 5.3 fb⁻¹ at 8 TeV.

The search is carried out in five decay modes: γγ, ZZ, WW, ττ and bb̄. Above the expected background an excess is found around 125 GeV. The local significance is 5.0 standard deviations; the significance expected for the Standard Model Higgs boson is 5.8. The excess is seen most clearly in the two modes with the best mass resolution (γγ and ZZ); from these modes the mass is found to be 125.3 ± 0.4 (statistical) ± 0.5 (systematic) GeV. Decay in the two-photon mode shows that the particle’s spin is different from 1.

The authors report the finding as a signal pointing to the existence of a new particle; this paper has no detailed coupling measurements.

ATLAS and CMS are two separate detectors; they work by separate teams, with separate software and analysis chains. That they report an excess in the same region, in the same direction, shows the value of this separation. The mass values (126.0 and 125.3 GeV) are consistent within their uncertainties. The CMS abstract reports the result as ‘a signal pointing to the existence of a new particle’; it leaves its identity with the Standard Model Higgs boson to more detailed measurements.

Afterwards: The ATLAS and CMS results were interpreted together; later experiments measured the particle’s spin and coupling properties and compared them with the Standard Model expectation.

Feynman 1982 — Simulating physics with computers

Int. J. Theor. Phys. 21, 467 theory

Feynman argued that classical computers could not efficiently simulate quantum systems and proposed a computer made of quantum elements.

Feynman’s question is what it means to simulate physics with a computer and what limits it is subject to. He argues that this is possible for classical and probabilistic systems; the difficulty begins with quantum systems.

His reasoning is this: writing down the state of a quantum system made of N two-level particles requires a number of complex amplitudes that grows exponentially with N. An ordinary computer must store these amplitudes one by one, so as N grows the computation ceases to be practical. Feynman also discusses that a classical probabilistic computer could not reproduce quantum results with local hidden variables (a point related to Bell-type arguments).

As a solution he puts forward the idea of a computer that itself consists of quantum-mechanical elements: one quantum system should be able to imitate another directly. The text gives no working design or algorithm; it sets out the subject as a question and a program. It is a paper derived from a talk, and its line of thought remains intuitive.

The language of the paper is less that of a computer science paper than a physicist’s stream of thought carried on with himself. Rather than saying ‘a classical computer cannot simulate a quantum system’, Feynman argues that this cannot be done efficiently and in general; the distinction is important, because small systems can already be computed on a classical computer. Many other subjects also appear in the text; here only the main thesis is conveyed, and it was written without seeing the text.

This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.

Afterwards: The paper is cited as one of the starting texts of quantum computing research; the proposal was given mathematical form by Lloyd in 1996.

Lloyd 1996 — Universal Quantum Simulators

Science 273, 1073 theory

Lloyd showed that quantum computers can efficiently simulate local quantum systems.

In 1982 Feynman had intuitively proposed that a quantum computer could imitate quantum systems. Lloyd’s question is whether this is mathematically correct.

The paper’s abstract reports the result thus: Feynman’s 1982 conjecture, that is, that quantum computers can be programmed to simulate any local quantum system, is correct. The word ‘local’ describes cases in which the system’s interactions concern only a few particles at a time; most physical systems fall into this class.

The main idea of the work is to divide the system’s time evolution into small steps and to carry out each step approximately with a series of simple quantum operations. The approximation error is controllable, and the number of operations grows with the size of the system not exponentially but reasonably. This overcomes the problem of exponential growth on a classical computer.

The paper establishes the concept of a ‘universal quantum simulator’; it does not state how this is to be built as a physically working device.

Lloyd’s result is a ‘possibility’ result: if a quantum computer exists as a sufficiently large and properly working device, it can simulate local quantum systems. The difficulty of building such a device is not the paper’s subject. The result shows why simulation is logically possible; it should also be kept in mind that not all quantum systems (for example those with nonlocal interactions) fall within this framework.

Afterwards: In the following years, in place of a direct universal computer, special-purpose devices simulating particular models (ultracold atoms, trapped ions) were developed; the experiment of Greiner and colleagues is one of them.

Greiner 2002 — Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms

Nature 415, 39 experiment

Greiner and colleagues observed the transition from a superfluid to a Mott insulator in ultracold atoms in an optical lattice.

This experiment is cited as one of the first strong examples of the idea of carrying out quantum simulation not with a computer but with a quantum system itself. The question is whether the behavior of a model known from solid-state physics can be reproduced in a controllable assembly of atoms.

In the work rubidium atoms are cooled to a very low temperature and placed in a standing-wave lattice formed by laser beams, that is, an ‘optical lattice’. The depth of the lattice can be changed with the laser intensity. When the lattice is shallow, the atoms form a coherent superfluid that can flow along the lattice. When the lattice is deepened, the repulsion between atoms becomes dominant and a fixed number of atoms remains at each lattice site; this state is the Mott insulator, and the atoms can no longer flow.

The authors show that as the lattice depth is changed the two states turn into each other as a phase transition; this is called a quantum phase transition because the transition is driven not by temperature but by quantum fluctuations. The result points to an assembly of atoms being able to imitate this model with adjustable parameters.

The importance of this work is that the simulation here is not a ‘program’ but a ‘physical set-up’: the atoms in the lattice take the place of the model’s electrons. Such devices are sometimes called an ‘analog quantum simulator’, and they are a narrow application of Feynman’s proposal; they imitate not every model but a particular model. For the paper’s detailed experimental measurements one has to look at the paper itself.

This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.

Afterwards: This approach was developed in the following years for the simulation of other models and of larger systems.