## Abstract

As part of the 20th century scientific revolution, matter in bulk was found to exhibit many remarkable properties including superfluid behavior. Possibly the most general definition of superfluidity involves the ability, in the right circumstances, of a fluid to flow without resistance. It is thus possible to create persistent currents in superfluids which can flow without any measurable dissipation. Shortly after the liquefaction of helium early in the 20th century by H. Kammerlingh Onnes, (1913 Nobel Laureate), a phase transition was discovered in metallic mercury below which all electrical resistance abruptly vanished. Thus superconductivity was discovered, and following this discovery, many other superconducting materials were found. A fascinating feature of superconductors is the Meissner effect which is the strong tendency of superconductors to expel magnetic field.

In the 1930‘s liquid 4He was found to exhibit some remarkable flow properties below 2.17k, which was called the lambda temperature because of the shape of the specific heat curve near this point. Some extraordinarily beautiful flow experiments were performed by Pyotr Kapitza (1978 Nobel Laureate), Jack Allen, John Daunt, Kurt Mendelsohn and others. The results were explained on the basis of the two fluid model which involved two interpenetrating fluids, a zero entropy fluid known as the superfluid component and a normal fluid component which could transport heat and interact strongly with the fluid container walls. Lev Landau (1962 Nobel Laureate) interpreted the normal fluid in terms of a cloud of excitations (phonons and rotons) with the superfluid component constituting a background fluid.

Fritz London played a key role in the theory of superfluids by introducing the concept of quantum mechanics on a macroscopic scale. The idea is that when the thermal de Broglie (1929 Nobel Prize) wave length is on the order of or greater than the mean spacing between atoms in a fluid, the behavior of the fluid will be dominated by quantum mechanics. Thus Bose – Einstein condensation would be expected to play a role in the behavior of liquid helium and especially the onset of superfluidity. The origin of superconductivity remained a mystery for many years, however, since the electrons in metals obey Fermi-Dirac statistics and therefore do not undergo Bose Einstein condensation. It was not until 1957 that Bardeen, Cooper and Schrieffer (1972 Nobel Prize) were able to resolve this problem by introducing a theory (BCS) which involved pairing of electrons to form Bose – Einstein like entities. The BCS theory led to an explosion of research, both experimental and theoretical, which lasted for many years. A major consequence of BCS pairing was suggested by Brian Josephson (1973 Nobel Laureate) who found that the electron pairs of BCS theory (Cooper pairs) could tunnel quantum mechanically through a thin oxide barrier between two superconductors. This discovery has had an enormous impact on science and technology.

Studies of rotating superfluid 4He, quantized vortex lines and persistent currents have been particularly important. My Cornell colleague, John Reppy, invented a superfluid gyroscope as well as other clever devices for studying these phenomena. A closely related phenomenon, the quantization of magnetic flux, was discovered in superconductors by Deaver and Fairbank and by Doll and Nabauer. This phenomenon and the Josephson effect led to the invention of the superconducting quantum interefence detector (SQUID) which has had a myriad of applications including the measurement of very small magnetic fields.

Shortly after the announcement of the BCS theory of superconductivity, theorists began to consider the possibility that the rare isotope of helium, ³He, might also exhibit Cooper pairing and BCS superfluidity. Because the ³He have very strong short range repulsion, it is impossible to form the simple s wave pairs in which the mates move with respect to one another in a collinear fashion. The short wave repulsion prevents this simple pairing from occurring. The ³He Cooper pairs must therefore form in a higher angular momentum state. Atoms of ³He possess a nuclear spin of 1/2, so the nuclear spin state of a pair will be either parallel (S=1) or anti-parallel (S=1) for odd or even angular momentum quantum numbers respectively.

In 1972, three new phases of liquid ³He were found in a series of experiments at Cornell University by Douglas D. Osheroff, Robert C. Richardson, and the author. These new states possessed remarkable nuclear magnetic resonance (NMR) properties, and were identified as Cooper pairing states which ultimately were found to manifest superfluid behavior. Anthony Leggett played a seminal role in showing how a large NMR frequency shift seen in one of the these states could arise as a result of parallel spin pairing and an odd relative angular momentum number characterizing the Cooper pairs. It is now believed that all three phases of superfluid ³He contain triplet pairs (S=1) with angular momentum quantum number (I=1). P-wave superfluid pairing states, corresponding to the observed superfluid phases, were first proposed by Anderson (1977 Nobel Laureate) and Morel, and by Balian and Werthamer. An important feature of higher angular momentum pairing states is that they possess internal degrees of freedom as well as the translational degrees of freedom found in superfluid 4He and conventional superconductors. These internal degrees of freedom must also be described by the macroscopic wave function (order parameter) characterizing the superfluid. Many of the most fascinating properties of superfluid ³He are associated with internal degrees of freedom including the multiplicity of superfluid phases, the bizarre nuclear magnetic properties, the anisotropic flow properties, and the complex vortex states, all of which have been the subject of intense investigation, both experimental and theoretical, for the past 25 years.

In recent years, it has been shown that high temperature superconductors discovered by Bednorz and Müller (1987 Nobel Prize) also exhibit higher order angular momentum pairing. It is thought that the heavy Fermion superconducting states also can be described in terms of higher order angular momentum pairing. At the present time, it appears that at least for the case of high temperature superconductors, I=2 or d wave pairing occurs. Several beautiful experiments have been performed which point to a d wave pairing order parameter. The pairing states for heavy Formion Superconductors have not been well characterized so far.

An exciting development has occurred in recent years which extends the study of superfluids into an entirely new realm. It was shown by Eric Cornell, and Carl Wiemann and their associates at Boulder, Colorado and by Wolfgang Ketterle and his associates at MIT that trapped alkali gases at very low temperatures could undergo Bose-Einstein condensation. Later it was shown by Greytak, Kleppner and their colleagues at MIT that trapped spin polarized hydrogen gas at sufficiently low temperatures also exhibits Bose-Einstein condensation. These highly rarefied weakly interacting gases also have internal degrees of freedom associated with different spin states. Their properties can be probed by powerful optical techniques. Many fascinating properties of these new condensates have already been revealed and future developments are eagerly awaited.

In the 1930‘s liquid 4He was found to exhibit some remarkable flow properties below 2.17k, which was called the lambda temperature because of the shape of the specific heat curve near this point. Some extraordinarily beautiful flow experiments were performed by Pyotr Kapitza (1978 Nobel Laureate), Jack Allen, John Daunt, Kurt Mendelsohn and others. The results were explained on the basis of the two fluid model which involved two interpenetrating fluids, a zero entropy fluid known as the superfluid component and a normal fluid component which could transport heat and interact strongly with the fluid container walls. Lev Landau (1962 Nobel Laureate) interpreted the normal fluid in terms of a cloud of excitations (phonons and rotons) with the superfluid component constituting a background fluid.

Fritz London played a key role in the theory of superfluids by introducing the concept of quantum mechanics on a macroscopic scale. The idea is that when the thermal de Broglie (1929 Nobel Prize) wave length is on the order of or greater than the mean spacing between atoms in a fluid, the behavior of the fluid will be dominated by quantum mechanics. Thus Bose – Einstein condensation would be expected to play a role in the behavior of liquid helium and especially the onset of superfluidity. The origin of superconductivity remained a mystery for many years, however, since the electrons in metals obey Fermi-Dirac statistics and therefore do not undergo Bose Einstein condensation. It was not until 1957 that Bardeen, Cooper and Schrieffer (1972 Nobel Prize) were able to resolve this problem by introducing a theory (BCS) which involved pairing of electrons to form Bose – Einstein like entities. The BCS theory led to an explosion of research, both experimental and theoretical, which lasted for many years. A major consequence of BCS pairing was suggested by Brian Josephson (1973 Nobel Laureate) who found that the electron pairs of BCS theory (Cooper pairs) could tunnel quantum mechanically through a thin oxide barrier between two superconductors. This discovery has had an enormous impact on science and technology.

Studies of rotating superfluid 4He, quantized vortex lines and persistent currents have been particularly important. My Cornell colleague, John Reppy, invented a superfluid gyroscope as well as other clever devices for studying these phenomena. A closely related phenomenon, the quantization of magnetic flux, was discovered in superconductors by Deaver and Fairbank and by Doll and Nabauer. This phenomenon and the Josephson effect led to the invention of the superconducting quantum interefence detector (SQUID) which has had a myriad of applications including the measurement of very small magnetic fields.

Shortly after the announcement of the BCS theory of superconductivity, theorists began to consider the possibility that the rare isotope of helium, ³He, might also exhibit Cooper pairing and BCS superfluidity. Because the ³He have very strong short range repulsion, it is impossible to form the simple s wave pairs in which the mates move with respect to one another in a collinear fashion. The short wave repulsion prevents this simple pairing from occurring. The ³He Cooper pairs must therefore form in a higher angular momentum state. Atoms of ³He possess a nuclear spin of 1/2, so the nuclear spin state of a pair will be either parallel (S=1) or anti-parallel (S=1) for odd or even angular momentum quantum numbers respectively.

In 1972, three new phases of liquid ³He were found in a series of experiments at Cornell University by Douglas D. Osheroff, Robert C. Richardson, and the author. These new states possessed remarkable nuclear magnetic resonance (NMR) properties, and were identified as Cooper pairing states which ultimately were found to manifest superfluid behavior. Anthony Leggett played a seminal role in showing how a large NMR frequency shift seen in one of the these states could arise as a result of parallel spin pairing and an odd relative angular momentum number characterizing the Cooper pairs. It is now believed that all three phases of superfluid ³He contain triplet pairs (S=1) with angular momentum quantum number (I=1). P-wave superfluid pairing states, corresponding to the observed superfluid phases, were first proposed by Anderson (1977 Nobel Laureate) and Morel, and by Balian and Werthamer. An important feature of higher angular momentum pairing states is that they possess internal degrees of freedom as well as the translational degrees of freedom found in superfluid 4He and conventional superconductors. These internal degrees of freedom must also be described by the macroscopic wave function (order parameter) characterizing the superfluid. Many of the most fascinating properties of superfluid ³He are associated with internal degrees of freedom including the multiplicity of superfluid phases, the bizarre nuclear magnetic properties, the anisotropic flow properties, and the complex vortex states, all of which have been the subject of intense investigation, both experimental and theoretical, for the past 25 years.

In recent years, it has been shown that high temperature superconductors discovered by Bednorz and Müller (1987 Nobel Prize) also exhibit higher order angular momentum pairing. It is thought that the heavy Fermion superconducting states also can be described in terms of higher order angular momentum pairing. At the present time, it appears that at least for the case of high temperature superconductors, I=2 or d wave pairing occurs. Several beautiful experiments have been performed which point to a d wave pairing order parameter. The pairing states for heavy Formion Superconductors have not been well characterized so far.

An exciting development has occurred in recent years which extends the study of superfluids into an entirely new realm. It was shown by Eric Cornell, and Carl Wiemann and their associates at Boulder, Colorado and by Wolfgang Ketterle and his associates at MIT that trapped alkali gases at very low temperatures could undergo Bose-Einstein condensation. Later it was shown by Greytak, Kleppner and their colleagues at MIT that trapped spin polarized hydrogen gas at sufficiently low temperatures also exhibits Bose-Einstein condensation. These highly rarefied weakly interacting gases also have internal degrees of freedom associated with different spin states. Their properties can be probed by powerful optical techniques. Many fascinating properties of these new condensates have already been revealed and future developments are eagerly awaited.