Quantum states of matter and metrology

Two characteristics of states of matter are associated with them being referred to as quantum. One characteristic is the importance of quantum statistics of particles, i.e., that the system is composed of particles that obey Fermi-Dirac or Bose-Einstein statistics. The second characteristic is that a macroscopic property is quantized with values determined by Planck’s constant. I now discuss each of these with respect to emergence.

Quantum statistics. 

For a system of non-interacting  fermions and bosons at high temperatures the properties of the system are those of a classical ideal gas. As the temperature decreases there is a smooth crossover to low-temperature properties that are qualitatively different for fermions, bosons, and classical particles. This crossover occurs around a temperature, known as the degeneracy temperature, that is dependent on the particle density and Planck’s constant. 

Many of the properties resulting from quantum statistics also occur in systems of strongly interacting particles and this is central to the concept of Landau’s Fermi liquid and viewing liquid 4He as a boson liquid. If liquid 3He and the electron liquid in elemental metals are viewed as a gas of non-interacting fermions, the degeneracy temperature is about 1 K and 1000 K, respectively. Thermodynamic properties are qualitatively different above and below the degeneracy temperature. Low-temperature properties can have values that differ by orders of magnitude from classical values and have a different temperature dependence. In contrast to a classical ideal gas, a fermion gas has a non-zero pressure at zero temperature and its magnitude is determined by Planck’s constant. This degeneracy pressure is responsible for the gravitational stability of white dwarf and neutron stars.  

These properties of systems of particles can be viewed as emergent properties, in the sense of novelty, as they are qualitatively different from high-temperature properties. However, they involve a crossover as a function of temperature and so are not associated with discontinuity. They also are not associated with unpredictability as they are straightforward to calculate from a knowledge of microscopic properties.

Quantised macroscopic properties.

These provide a more dramatic illustration of emergence. Here I consider four specific systems: superconducting cylinders, rotating superfluids, Josephson junctions, and the integer Quantum Hall effect. All of these systems have a macroscopic property that is observed to have the following features.

i. As an external parameter is varied the quantity varies in a step-like manner with discrete values on the steps. This is contrast to the smooth linear variation seen when the material is not condensed into the quantum state of matter.

ii. The value on the steps is an integer multiple of some specific parameter.

iii. This parameter (unit of quantisation) only depends on Planck’s constant h and other fundamental constants. 

iv. The unit of quantisation does not depend on details of the material, such as chemical composition, or details of the device, such as its geometrical dimensions.

v. The quantisation has been observed in diverse materials and devices.

vi. Explanation of the quantisation involves topology.

Superconducting cylinders. A hollow cylinder of a metal is placed in a magnetic field parallel to the axis of the cylinder. In the metallic state the magnetic flux enclosed by the cylinder increases linearly with the magnitude of the external magnetic field. In the superconducting state, the flux is quantized in units of the magnetic flux quantum, Φ0 = h/2e where e is the charge on an electron. It is also found that in a type II superconductor the vortices that occur in the presence of an external magnetic field enclose a magnetic flux equal to Φ0.  

Rotating superfluids. When a cylinder containing a normal fluid is rotated about an axis passing down the centre of the cylinder the fluid rotates with a circulation proportional to the speed of rotation and the diameter of the cylinder. In contrast, in a superfluid, as the speed of rotation is varied the circulation is quantised in units of h/M where M is the mass of one atom in the fluid. This quantity is also the circulation around a single vortex in the superfluid. 

Josephson junctions. In the metallic state the current passing through a junction increases linearly with the voltage applied across the junction. In the superconducting state the AC Josephson effect occurs. If a beam of microwaves of constant frequency is incident on the junction, jumps occur in the current when the voltage is an integer multiple of h/2e. The quantisation is observed to better than one part in a million (ppm).

Integer Quantum Hall effect. In a normal conductor the Hall resistance increases linearly with the external magnetic field for small magnetic fields. In contrast, in a two-dimensional conductor at high magnetic fields the Hall resistance is quantized in units of h/2e^2. The quantisation is observed to better than one part in ten million. Reflecting universality, the observed value of the Hall resistance for each of the plateaus is independent of many details, including the temperature, the amount of disorder in the material, the chemical composition of system (silicon versus gallium arsenide), or whether the charge carriers are electrons or holes.

Other examples of macroscopic quantum effects are seen in SQUIDs (Superconducting Quantum Interference Devices). They exhibit quantum interference phenomena analogous to the double-slit experiment. The electrical current passing through the SQUID has a periodicity defined by the ratio of the magnetic flux inside the current loop of the SQUID and the quantum of magnetic flux.

The precision of the quantisation provides a means to accurately determine fundamental constants. Indeed, the title of the paper announcing the discovery of the integer quantum Hall effect was, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance.” It is astonishing that a macroscopic measurement of a property of a macroscopic system, such as the electrical resistance, can determine fundamental constants that are normally associated with the microscale and properties of atomic systems. 

Laughlin and Pines claimed that the quantisation phenomena described above reflect organizing principles associated with emergent phenomena, and their universality supports their claim of the unpredictability of emergent properties. 

Quantum states of matter and metrology

The universality of these macroscopic quantum effects has practical applications in metrology, the study of measurement and the associated units and standards. In 1990 new international standards were defined for the units of voltage and electrical resistance, based on the quantum Hall effect and the AC Josephson effect, respectively.

Prior to 1990 the standard used to define one volt was based on a particular type of electrical battery, known as a Weston cell. The new standard using the AC Josephson effect allowed voltages to be defined with a precision of better than one part per billion. This change was motivated not only by improved precision, but also improved portability, reproducibility, and flexibility. The old voltage standard involved a specific material and device and required making duplicate copies of the standard Weston cell. In contrast, the Josephson voltage standard is independent of the specific materials used and the details of the device. 

Prior to 1990 the international standard for the ohm was defined by the electrical resistance of a column of liquid mercury with constant cross-sectional area, 106.3 cm long, a mass of 14.4521 grams and a temperature 0 °C. Like the Josephson voltage standard, the quantum Hall resistance standard has the advantage of precision, portability, reliability, reproducibility, and independence of platform. The independence of the new voltage and resistance standards from the platform used reflects the fact that the Josephson and quantum Hall effects have the universality characteristic of emergent phenomena.

This post is an adaptation of material in Condensed Matter Physics: A Very Short Introduction

Quantum matters for the public

I have now finished my first draft of  Chapter 7 of Condensed Matter Physics: A Very Short Introduction. The main purpose of the chapter is to introduce quantum states of matter. It is arguably the most challenging of the chapters to write and to understand. But, it is potentially the most fascinating.

I welcome comments and suggestions. However, bear in mind that my target audience is not the typical reader of this blog, but rather your non-physicist friends and family.

I think it still needs a lot of work, particularly to be less technical. For example, I should probably drop Aharonov Bohm ...

The goal is for the chapter to be interesting, accessible, and bring out the excitement and importance of condensed matter physics.

The most accurate equations in all of physics

A beautiful and profound article, Superconductivity and the Quantization of Energy, by D.G. McDonald was published in Science in 1990.

Ideas about quantized energy levels originated in atomic physics, but research in superconductivity has led to unparalleled precision in the measurement of energy levels. A comparison of levels produced by two Josephson junctions shows that they differ by no more than 3 parts in 10^19 at an energy of 0.0003 electron volt. The fact that the myriad of interactions of 10^12 particles in a macroscopic body, a Josephson junction, can produce sharply defined energy levels suggests a dynamical state effectively divorced from the complexities of its environment. The existence of this state, the macroscopic quantum state of superconductors, is well established, but its isolation from intrinsic perturbations has recently been shown to be extraordinary. These new results, with an improved precision of about ten orders of magnitude, are discussed in the context of highly accurate results from quantum electrodynamics, atomic spectroscopy, and the standards of metrology.  

McDonald suggests that the equations associated with the AC Josephson effect are the ``most accurate equations in all of physics". The central equation is the relationship between the voltage V across a junction and the frequency, nu, of the radiation shone on the junction.

The proportionality constant is the Josephson constant,

where e is the electronic charge and h is Planck's constant. Thus, the Josephson effect provides a means to make a macroscopic measurement of fundamental constants that are normally associated with measurements of atomic systems.

The relative value of the Voltage V in two different Josephson junctions can be measured with great precision using a SQUID. One obtains the same value for a range of systems of different chemical composition and Josephson junctions of different design.

On a philosophical level, these results are a beautiful illustration that superconductivity is an emergent state of matter and that the Josephson effect is an emergent effect. An important characteristic of emergent phenomena are that they are independent of most of the chemical and structure details of the materials involved.

These observations had important implications for metrology (the scientific study of measurement and the associated units and standards). In 1990 an international agreement was made that the voltage standard based on Weston cells (a particular type of electrical battery) would be replaced by the Josephson voltage standard. This change was not just one of precision, but also portability, reproducibility, and flexibility. The old voltage standard involved a specific material and device and required duplication in order to have an accurate standard. In contrast, the Josephson standard is independent of the materials used and the details of the device.

All of the above illustrates how much Josephson matters.