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I review how, driven originally by foundational concerns and more recently by prospects of quantum computing, the idea of applying quantum mechanics to the motion of macroscopic variables such as the flux in a Josephson qubit went over the last 50 years from a theorists’ pipe-dream to an everyday engineering subject.

I draw an analogy between the “measurement paradox” of quantum mechanics in 2019 and the “Gibbs paradox” of statistical physics in 1875, and use it to argue that we have good reason to believe that the answer to the question in my title is “no”.

We discuss a very simple and essentially exactly solvable model problem which illustrates some nice features of Andreev bound states, namely, the trapping of a single Bogoliubov quasiparticle in a neutral s-wave BCS superfluid by a wide and shallow Zeeman trap. In the quasiclassical limit, the ground state is a doublet with a splitting which is proportional to the exponentially small amplitude for “normal” reflection by the edges of the trap. We comment briefly on a prima facie paradox concerning the continuity equation and conjecture a resolution to it.

A history is given of liquid 3He research from the time when 3He first became available following World War II through 1972 when the discovery of the superfluid phases was made. The Fermi liquid nature was established early on, and the Landau Fermi liquid theory provided a framework for understanding the interactions between the Fermions (quasiparticles). The theory’s main triumph was to predict zero sound, which was soon discovered experimentally. Experimental techniques are treated, including adiabatic demagnetization, dilution refrigerator technology, and Pomeranchuk cooling. A description of the superfluid 3He discovery experiments using the latter two of these techniques is given. While existing theories provided a basis for understanding the newly discovered superfluid phases in terms of l>0 Cooper pairs, the unexpected stability of the A phase in the high-P, high-T region of the phase diagram needed for its explanation a creative leap beyond the BCS paradigm. The use of sum rules to interpret some of the unusual magnetic resonance in liquid 3He is discussed. Eventually a complete theory of the spin dynamics of superfluid 3He was developed, which predicted many of the exciting phenomena subsequently discovered.

After a brief review of the phenomenology of superconductivity and of its generic explanation in terms of the concept of off-diagonal long-range order, I first survey the original Bardeen-Cooper-Schrieffer (BCS) weak-coupling model and some extensions of it. I then turn to systems such as the heavy fermions and the cuprates where the superconductivity is generally believed to be due to an all-electronic mechanism, and ask how much information we get about this mechanism from quite general energetic and other considerations, without committing ourselves to any particular microscopic model.

Following the success of the original BCS theory as applied to superconductivity in metals, it was suggested that the phenomenon of Cooper pairing might also occur in liquid 3- He , though unlike the metallic case the pairs would most likely form in an anisotropic state, and would then lead in this neutral system to superfluidity. However, what had not been anticipated was the richness of the phenomena which would be revealed by the experiments of 1972. In the first place, even in a zero magnetic field there is not one but two superfluid phases, and the explanation of this involves ideas concerning "spin fluctuation feedback" which have no obvious analog in metals. Secondly, the anisotropic nature of the pair wave function, which in the case of the B phase is quite subtle, and the fact that the orientation must be the same for all the pairs, leads to a number of qualitatively new effects, in particular to a spectacular amplification of ultra-weak interactions seen most dramatically in the NMR behavior. In this chapter I review the application of BCS theory to superfluid 3- He with emphasis on these novel features.

I present the motivation for experiments which attempt to generate, and verify the existence of, quantum superpositions of two or more states which are by some reasonable criterion “macroscopically” distinct, and show that various a priori objections to this program made in the literature are flawed. I review the extent to which such experiments currently exist in the areas of free-space molecular diffraction, magnetic biomolecules, quantum optics and Josephson devices, and sketch possible future lines of development of the program.

The element helium comes in two (stable) forms. 4 He and 5 He; at low temperatures and pressures both form liquids rather than solids. The liquid phase of the common isotope, ' 1 He, was realized' nearly a century ago, and since 1938 has been known to show, at temperatures below about 2 K, the property of superfluidity-the ability to flow through the narrowest capillaries without apparent friction. The light isotope, 2 He, is believed to be of quite a different nature; however,because of its similarity to the electrons in metals, which at low temperatures sometimes form Cooper pairs and thereby become superconducting, theorists in the 1960s and early 1970s had speculated that something similar might happen in liquid 3 He, which would then also show superfluidity though for reasons rather different than 4 He. In 1972 nuclear magnetic resonance (NMR) experiments at Cornell University revealed the existence, below 3 millidegrees, if two new phases, one of which displayed extraordinary NMR properties. Anthony Leggett is one of the theorists who succeeded in fitting the experimental properties into the Cooper-pairing scenario; in particular, he explained the NMR behavior and predicted further novel NMR phenomena which were subsequently found.

By using small inclusions of liquid 3 He trapped in a solid 4 He matrix, it may be possible to study a sample of the superfluid A phase in the regime where quantum fluctuations of the relative phase of the “up” and “down” Cooper pairs are no longer negligible. Under these conditions there is little doubt that a standard quantum-mechanical (QM) calculation of the NMR dynamics should correctly describe the behavior of the total magnetization of a large ensemble of independent inclusions. However, it is less obvious that it should describe the behavior of a single inclusion on a single run of the experiment, and indeed it is tempting to argue that a “spontaneous breaking of the symmetry” corresponding to the relative phase should occur, analogously to what (allegedly) happens in the interference of two independent Bose condensates. Here I argue that the prima facie analogy is misleading, and that the QM calculation indeed describes the behavior of a single inclusion.

For a harmonically trapped dilute Bose gas with uniformly repulsive interactions which is assumed to satisfy a certain condition on the extensivity of fluctuations, I find on upper bound on the condensate fraction f. If BEC is defined by the condition that f>const.N−α, α

I present the motivation for experiments which attempt to generate, and verify the existence of, quantum superpositions of two or more states which are by some reasonable criterion `macroscopically' distinct, and show that various a priori objections to this programme made in the literature are flawed. I review the extent to which such experiments currently exist in the areas of free-space molecular diffraction, magnetic biomolecules, quantum optics and Josephson devices, and sketch possible future lines of development of the programme.

The author presents a tutorial review of some ideas that are basic to our current understanding of the phenomenon of Bose-Einstein condensation (BEC) in the dilute atomic alkali gases, with special emphasis on the case of two or more coexisting hyperfine species. Topics covered include the definition of and conditions for BEC in an interacting system, the replacement of the true interatomic potential by a zero-range pseudopotential, the time-independent and time-dependent Gross-Pitaevskii equations, superfluidity and rotational properties, the Josephson effect and related phenomena, and the Bogoliubov approximation.

These lectures address Bose and Fermi ultracold dilute alkali gases. For Bose gases I discuss general BEC concepts, superfluidity, the Josephson effect, and issues related to the establishment of a "phase standard". For Fermi gases I discuss prospects for Cooper pairing and its detection. The four lectures on alkali Bose gases which I actually gave at the School were based on a review paper in preparation for Reviews of Modern Physics. In order to avoid extensive duplication of the contents of that review, what I present here, condensed into three lectures, is a melange of informal commentary on the results presented there and some material which for space and other reasons had to be omitted. This material is ideally read in conjunction with the review. The fourth lecture, on the Fermi alkali gases, is presented here in (nearly) the form in which it was actually given.

The ground state of a rotating Bose-Einstein condensate with attractive interaction in a quasi-one-dimensional torus is studied in terms of the ratio $\gamma$ of the mean-field interaction energy per particle to the single-particle energy-level spacing. The plateaus of quantized circulation are found to appear if and only if $\gamma, Comment: 4 pages, 2 figures, Accepted for publication in Physical Reveiw Letters

I first raise the question: “In what regions of wave vector and frequency does the saving (?) of Coulomb energy associated with the superconducting transition in the cuprates predominantly take place?”, and point out that in principle, and probably in practice, it can be answered quite directly by differential EELS measurements. I then conjecture the answer “In the region of long wavelengths and mid-infrared frequencies”, and show that, if true, this hypothesis explains, inter alia, the systematics of Tc in the cuprates as a function of the layering structure.

A new method for calculating the free energy of an inhomogeneous superconductor is presented. This method is based on the quasiclassical limit (or Andreev approximation) of the Bogoliubov-de Gennes (or wave function) formulation of the theory of weakly coupled superconductors. The method is applicable to any pure bulk superconductor described by a pair potential with arbitrary spatial dependence, in the presence of supercurrents and external magnetic field. We find that both the local density of states and the free energy density of an inhomogeneous superconductor can be expressed in terms of the diagonal resolvent of the corresponding Andreev Hamiltonian, resolvent which obeys the so-called Gelfand-Dikii equation. Also, the connection between the well known Eilenberger equation for the quasiclassical Green's function and the less known Gelfand-Dikii equation for the diagonal resolvent of the Andreev Hamiltonian is established. These results are used to construct a general algorithm for calculating the (gauge invariant) gradient expansion of the free energy density of an inhomogeneous superconductor at arbitrary temperatures., Comment: REVTeX, 28 pages

We discuss some of the unusual characteristics of a Fermi superfluid formed by Cooper pairing in a system, such as spin-polarized6Li, which possesses triple nuclear spin degeneracy. In particular we show (a) that the “naive” BCS expression for the transition temperature is (unlike in the usual spin-1/2 case) exact in the low-density limit, and (b) that there must always be a gapless branch of the excitation spectrum even for s-wave pairing, so that the normal density is nonzero atT=0.

(1996). Josephson experiments on the high-temperature superconductors. Philosophical Magazine B: Vol. 74, David Pines-Forty-five Years of Many-Body Theory, pp. 509-522.

We consider the behavior of an ultracold Fermi gas across a narrow Feshbach resonance, where the occupation of the closed channel may not be negligible. While the corrections to the single-channel formulas associated with the nonzero chemical potential and with particle conservation have been considered in the existing literature, there is a further effect, namely, the ``interchannel Pauli exclusion principle,'' associated with the fact that a single hyperfine species may be common to the two channels. We focus on this effect and show that, as intuitively expected, the resulting corrections are of the order of ${E}_{F}/\ensuremath{\eta}$, where ${E}_{F}$ is the Fermi energy of the gas in the absence of interactions and $\ensuremath{\eta}$ is the Zeeman energy difference between the two channels. We also consider the related corrections to the fermionic excitation spectrum, and briefly discuss the collective modes of the system.

We present some general considerations on the properties of a two-component ultracold Fermi gas along the BEC-BCS crossover. It is shown that the interaction energy and the free energy can be written in terms of a single dimensionless function $h(\ensuremath{\xi},\ensuremath{\tau})$, where $\ensuremath{\xi}=\ensuremath{-}{({k}_{F}{a}_{s})}^{\ensuremath{-}1}$ and $\ensuremath{\tau}=T∕{T}_{F}$. The function $h(\ensuremath{\xi},\ensuremath{\tau})$ incorporates all the many-body physics and naturally occurs in other physical quantities as well. In particular, we show that the average rf-spectroscopy shift $\overline{\ensuremath{\delta}\ensuremath{\omega}}(\ensuremath{\xi},\ensuremath{\tau})$ and the molecular fraction ${f}_{c}(\ensuremath{\xi},\ensuremath{\tau})$ in the closed channel can be expressed in terms of $h(\ensuremath{\xi},\ensuremath{\tau})$ and thus have identical temperature dependence. The conclusions should have testable consequences in future experiments.

Quantum mechanics has been enormously successful in describing nature at the atomic level and most physicists believe it is, in principle, the “whole truth” about the world even at the everyday level. However, such a view, at first glance, leads to a severe problem. In certain circumstances, the most natural interpretation of the theory implies that no definite outcome of an experiment occurs until the act of observation. For many decades this problem was regarded as merely philosophical–it was thought it had no consequences that could be tested in experiment. However, in the last dozen years or so, the situation has changed dramatically in this respect. The problem, some popular resolutions of it, the current experimental situation and prospects for the future are discussed.

Although it is only in the very simplest cases that the order parameter of a superfluid or superconductor can be regarded literally as a Schrodinger wave function, it is quite generally “very like” one, and thinking of it in this way can often be quite helpful to one’s intuition, particularly in cases involving internal degrees of freedom. In this chapter I shall briefly sketch the basis for this point of view, starting with the very simplest case, that of a noninteracting Bose gas, and proceeding through progressively more complicated examples, to end with a Fermi superfluid with “exotic” pairing.

I present a tutorial review of the behavior of a simple 2-state quantum system (“bit”) in interaction with a dissipative environment, with particular attention to the questions of when we can regardthe system as effectively isolatedandof “true” versus “false” decoherence. The last lecture discusses the quantum measurement paradox in the light of recent experiments on Josephson systems.

I draw an analogy between the “measurement paradox” of quantum mechanics in 2019 and the “Gibbs paradox” of statistical physics in 1875, and use it to argue that we have good reason to believe that the answer to the question in my title is “no”.

In this talk I address the question: Under what conditions can we legitimately describe a quantum-mechanical system by a Schrodinger equation in its own right, and how are these conditions related to the degree of “entanglement” with its environment? As examples of systems that are often claimed to be strongly entangled with their environments but nevertheless seem to be well described by one-particle-like Schrodinger equations, I consider (a) Cooper pairs tunnelling between two different “boxes” and (b) quantum-optical systems confined to a cavity. In both cases I argue that the most “obvious” arguments grossly overestimate the true degree of entanglement.

Recently it has been predicted that if a gas of spin-1/2 bosonic atoms is cooled down while conserving its total spin, it can form a Bose-Einstein condensate in a new type of fragmented state. In that fragmented state two orbital wave functions are macroscopically occupied. We derive a set of Gross-Pitaevskii equations that describe the two occupied wave functions, taking into account the effects of interatomic interactions. We then analyze the solutions of those equations in some special cases. In particular, we find that if the fragmented state is realized in a toroidal trap, under certain conditions the system will spontaneously acquire macroscopic orbital angular momentum. Finally, we discuss the nature of the phase transition from an uncondensed gas to a fragmented-state condensate.

I briefly examine the motivation for experiments designed to test quantum mechanics against an alternative, common-sense view of the everyday world which I denote macrorealism, and review how far existing experiments go towards settling the issue.

The Eighth Eugene Feenberg Medal is awarded to Philippe Nozieres in recognition of his many pathbreaking contributions to many-body theory, including• His definitive work on the properties of the free electron gas, in particular in the region of realistic metallic densities,• his rigorous development of the theory of a normal Fermi liquid, which provided a firm microscopic foundation for the Landau theory,• his analysis of the nonequilibrium thermodynamics of 3-He solid-liquid mixtures,• his exact solution to the X-ray edge problem,• his elegant formulation of the low-temperature solution to the single-channel Kondo problem in the language of Fermi-liquid theory,• his introduction of the many-channel problem as a new class of quantum impurity systems, and• his innovative work on the static and dynamic behavior of the liquid-solid interface.

We study two aspects of measurement theory in spinor Bose-Einstein condensates of $F=1$ atoms: the probability of obtaining a certain outcome of the measurement and the evolution of the state of the condensate due to the measurement. We also study the interference patterns arising from the spatial overlap of two spinor condensates. We show that neither a measurement on a small number of escaping atoms nor an interference experiment can distinguish between an antiferromagnetic coherent state condensate, i.e., a condensate in which all the atoms have ${S}_{z}=0$ along an a priori unknown direction, and a spin-singlet condensate, i.e., a condensate with ${S}_{\mathrm{total}}=0.$ We also show that a singlet-state condensate evolves into a coherent state as a result of the measurement.

The third moment frequency sum rule for the density-density correlation function is rederived in the presence of Umklapp processes. Upper and lower bounds on the electron-electron Coulomb energy are derived in two-dimensional and three-dimensional media, and the Umklapp processes are shown to be crucial in determining the spectrum of the density fluctuations (especially for the two-dimensional systems). This and other standard sum rules can be used in conjunction with experimental spectroscopies (electron-energy loss spectroscopy, optical ellipsometry, etc.) to analyse changes of the electron-electron Coulomb energy at the superconducting transition in cuprates.

I consider the general problem of decoherence in Josephson systems, with particular reference to the question of how far we can reliably infer the degree of such decoherence, in an experiment of the MQC type, either by a priori order-of-magnitude arguments or from other types of experiment which may be easier at the initial stages of the program, such as measurements of dissipation in the quasiclassical regime or of tunnelling rates out of a metastablestate. Potential sources of decoherence which will be considered include normal electrons, phonons, nuclear spins, the radiation field and the “passing truck.” It is concluded that the effects of all but the last can be reaosnably estimated, and that there are good prospects (confirmed by recent experiments) of reducing them to manageable levels. The implications for the use of suchsystems for practical quantum computation are briefly considered.

We study the Josephson effect in alkali atomic gases within the two-mode approximation and show that there is a correspondence between the Bogoliubov description and the harmonic limit of the phase representation. We demonstrate that the quanta of the Josephson plasmon can be identified with the Bogoliubov excitations of the two-site Bose fluid. We thus establish a mapping between the Bogoliubov approximation for the many-body theory and the linearized pendulum Hamiltonian., 9 pages, LaTeX, submitted to J. Phys. B

A Bose-Einstein condensate with attractive interaction can be metastable if it is spatially confined and if the number of condensate bosons $N_0$ is below a certain critical value $N_{\rm c}$. By applying a variational method and the instanton techinique to the Gross-Pitaevskii energy functional, we find analytically the frequency of the collective excitation and the rate of macroscopic quantum tunneling (MQT). We show that near the critical point the tunneling exponent vanishes according to $(1-N_0/N_c)^\frac{5}{4}$ and that MQT can be a dominant decay mechanism of the condensate for $N_0$ very close to $N_{\rm c}$., Comment: RevTex 4 pages with 1 postscript figure. Accepted for publication in Physical Review Letters

At the macroscopic level, we are used to the idea that a system (ensemble) which has available to it two or more different states may, under certain conditions, be represented as occupying not one or other of these states but rather a linear superposition: for example, an atom driven by an electromagnetic field may be in a linear combination of its ground (s) state and an excited (p) state, and its properties are then different from what they would be in either state separately (e.g., the expectation value of the atomic polarization is zero in either an s-state or a p-state, but nonzero for the linear combination). Thus we say that in such atomic-level situations no “actualization” of one state or the other has taken place: formally, the linear-superposition state $$\matrix{ {{\psi _{at}} = \alpha |s\rangle + \beta |p\rangle ,} & {|\alpha {|^2} + |\beta {|^2}} \cr } = 1$$ (1) cannot be interpreted as indicating the occurrence of either an s or a p state for each particular member of the ensemble with probability |α|2 (|β|2).

Decoherence is thought to be a major obstacle in using macroscopic quantum states for quantum computing applications. In his Perspective, [Leggett][1] highlights the two independent experiments by [ Vion et al ][2]. and [ Yu et al ][3]. showing that long coherence times can be achieved. The reports indicate that whatever the difficulties that may be encountered in the attempt to build a quantum computer with Josephson circuits, decoherence need not be among them. [1]: http://www.sciencemag.org/cgi/content/full/296/5569/861 [2]: http://www.sciencemag.org/cgi/content/short/296/5569/886 [3]: http://www.sciencemag.org/cgi/content/short/296/5569/889

We consider the coupling between two {ital T}-violating systems by a non-{ital T}-violating interaction. We show that the configuration in which the signs of the {ital T} breaking in the two systems are opposite generally has lower energy. In the context of those theories of the cuprate superconductors which postulate breaking of {ital T} invariance in the CuO{sub 2} planes, a crude estimate of the Coulomb-induced antiferromagnetic'' coupling for reasonable domain sizes indicates that it is an edge'' effect but at least competitive with the ferromagnetic'' one due to Josephson and magnetic effects.

When I was originally asked to give two lectures at this workshop, one on this subject and one on “charge quantization” effects, I felt that there was at best a tenuous link between the two topics. Now I feel that there is a fundamental connection, and that it is absolutely essential to understand what we mean by “dephasing” before we can assess many of the results obtained in the last few years on charge quantization and related effects.

Very often in physics and chemistry we are interested in the motion of some variable which is described by quantum mechanics and is strongly coupled to an “environment,” that is, one or (more likely) many other variables whose behavior is of no particular interest to us in its own right, but only because of the effect it may exert on the motion of the “system,” that is, the variable of primary interest. Some familiar examples of such “systems,” with the “environment” indicated in parentheses, are an electron participating in a chemical reaction (the vibronic modes), a paraelectric defect in a solid, such as OH- in KCl, (phonons), a μ-meson in a metal (conduction electrons) and the \(\rm{K_O-\bar K_O}\) system (surrounding matter). All these are old problems. A class of problem which is of rather more recent interest is when the “system” variable is in some sense macroscopic: examples include charge density waves (normal electrons), the magnetization of small ferromagnetic particles (magnons, phonons), the early Universe (preexisting mesons) and, par excellence, the phase of the Cooper pairs in devices incorporating one or more Josephson junctions (normal electrons, phonons, radiation field, nuclear spins …). The last example has been studied intensively both experimentally and theoretically over the last few years and provides probably the best test-bed for ideas in this area. One further example which is of great current interest is the quantum tunnelling motion of a pair of deuterium nuclei in a matrix of a metal such as Pd or Ti; in this case the “environment” is the conduction electrons, the nuclei of the metal and any “third-party” deuterons which may be present.

I formulate a 'pseudo-paradox' in the theory of a dilute Bose gas with repulsive interactions: the standard expression for the ground state energy within the Gross–Pitaevskii (GP) approximation is lower than that in the Bogoliubov approximation, and hence, by the standard variational argument, the former should prima facie be a better approximation than the latter to the true ground state—a conclusion which is of course opposite to the established wisdom concerning this problem. It is shown that the pseudo-paradox is (unsurprisingly) resolved by a correct transcription of the two-body scattering theory to the many-body case; however, contrary to what appears to be a widespread belief, the resolution has nothing to do with any spurious ultraviolet divergences which result from the replacement of the true interatomic potential by a delta-function pseudopotential. Rather, it relates to an infrared divergence which has the consequence that (a) the most obvious form of the GP 'approximation' actually does not correspond to any well-defined ansatz for the many-body wavefunction, and (b) that the 'best shot' at such a wavefunction always produces an energy which exceeds, or at best equals, that calculated in the Bogoliubov approximation. In fact, the necessity of the latter may be seen as a consequence of the need to reduce the Fock term in the energy, which is absent in the two-particle problem but dominant in the many-body case; it does this by increasing the density correlations, at distances less than or approximately equal to the correlation length , above the value extrapolated from the two-body case. As a by-product I devise an alternative formulation of the Bogoliubov approximation which does not require the explicit replacement of the true interatomic potential by a delta-function pseudopotential.

Recently Javanainen and Wilkens [Phys. Rev. Lett. 78, 4675 (1997)] have analysed an experiment in which an interacting Bose condensate, after being allowed to form in a single potential well, is "cut" by splitting the well adiabatically with a very high potential barrier, and estimate the rate at which, following the cut, the two halves of the condensate lose the "memory" of their relative phase. We argue that, by neglecting the effect of interactions in the initial state before the separation, they have overestimated the rate of phase randomization by a numerical factor which grows with the interaction strength and with the slowness of the separation process., Comment: 2 pages, no figures, to appear in Phys. Rev. Lett

I consider various ways in which the recently stabilized systems of alkali gases in the Bose-condensed state can be used to test rather basic ideas of quantum mechanics, particularly with respect to questions involving the meaning of the "identity" of elementary particles. An essential role, here, is played by the ability to switch tunneling "contact" on and off over timescales small compared to any characteristic dynamical timescale of the system. Apart from their intrinsic interest, these questions may also be relevant to speculations about the "quenching" behavior of the early Universe.