The topic of quantum noise has become extremely timely due to the rise of quantum information physics and the resulting interchange of ideas between the condensed matter and atomic, molecular, optical--quantum optics communities. This review gives a pedagogical introduction to the physics of quantum noise and its connections to quantum measurement and quantum amplification. After introducing quantum noise spectra and methods for their detection, the basics of weak continuous measurements are described. Particular attention is given to the treatment of the standard quantum limit on linear amplifiers and position detectors within a general linear-response framework. This approach is shown how it relates to the standard Haus-Caves quantum limit for a bosonic amplifier known in quantum optics and its application to the case of electrical circuits is illustrated, including mesoscopic detectors and resonant cavity detectors.

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Introduction to quantum noise, measurement, and amplification

How do closed quantum many-body systems driven out of equilibrium eventually achieve equilibration? And how do these systems thermalize, given that they comprise so many degrees of freedom? Progress in answering these—and related—questions has accelerated in recent years—a trend that can be partially attributed to success with experiments performing quantum simulations using ultracold atoms and trapped ions. Here we provide an overview of this progress, specifically in studies probing dynamical equilibration and thermalization of systems driven out of equilibrium by quenches, ramps and periodic driving. In doing so, we also address topics such as the eigenstate thermalization hypothesis, typicality, transport, many-body localization and universality near phase transitions, as well as future prospects for quantum simulation. Statistical mechanics is adept at describing the equilibria of quantum many-body systems. But drive these systems out of equilibrium, and the physics is far from clear. Recent advances have broken new ground in probing these equilibration processes.

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Quantum many-body systems out of equilibrium

A central goal in condensed matter and modern atomic physics is the exploration of quantum phases of matter--in particular, how the universal characteristics of zero-temperature quantum phase transitions differ from those established for thermal phase transitions at non-zero temperature. Compared to conventional condensed matter systems, atomic gases provide a unique opportunity to explore quantum dynamics far from equilibrium. For example, gaseous spinor Bose-Einstein condensates (whose atoms have non-zero internal angular momentum) are quantum fluids that simultaneously realize superfluidity and magnetism, both of which are associated with symmetry breaking. Here we explore spontaneous symmetry breaking in 87Rb spinor condensates, rapidly quenched across a quantum phase transition to a ferromagnetic state. We observe the formation of spin textures, ferromagnetic domains and domain walls, and demonstrate phase-sensitive in situ detection of spin vortices. The latter are topological defects resulting from the symmetry breaking, containing non-zero spin current but no net mass current.

Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose–Einstein condensate

Cosmological experiments in condensed matter systems

Spinor Bose gases form a family of quantum fluids manifesting both magnetic order and superfluidity. This article reviews experimental and theoretical progress in understanding the static and dynamic properties of these fluids. The connection between system properties and the rotational symmetry properties of the atomic states and their interactions are investigated. Following a review of the experimental techniques used for characterizing spinor gases, their mean-field and many-body ground states, both in isolation and under the application of symmetry-breaking external fields, are discussed. These states serve as the starting point for understanding low-energy dynamics, spin textures and topological defects, effects of magnetic dipole interactions, and various non-equilibrium collective spin-mixing phenomena. The paper aims to form connections and establish coherence among the vast range of works on spinor Bose gases, so as to point to open questions and future research opportunities.

Spinor Bose gases: Symmetries, magnetism, and quantum dynamics

TOPOLOGICAL defects formed during a rapid symmetry-breaking phase transition in the early Universe1,2 could be responsible for seeding large-scale structure, for the anisotropy of the microwave background radiation, and for the predominance of matter over antimatter3,4. The theory describing this cosmological phase transition is formally analogous to that describing the transition to the superfluid state in liquid 3He, so that in principle the process of cosmological defect formation can be modelled in the laboratory. Here we report the results of an experiment in which the 'primordial fireball' is mimicked using a neutron-induced nuclear reaction (n + 3He → p + 3He + 0.76 MeV) to heat small regions of superfluid 3He above the superfluid transition temperature. These bubbles of normal liquid cool extremely rapidly, and we find that their transition back to the superfluid state is accompanied by the formation of a random network of vortices (the superfluid analogue of cosmic strings). We monitor the evolution of this defect state by rotating the superfluid sample, allowing vortices to escape from the network and thus be probed individually. Our results provide clear confirmation of the idea that topological defects form at a rapid second-order phase transition, and give quantitative support to the Kibble–Zurek mechanism5,6 of cosmological defect formation.

Vortex formation in neutron-irradiated superfluid 3He as an analogue of cosmological defect formation

We report the observation of vortex formation upon the absorption of a thermal neutron in a rotating container of superfluid $^3$He-B. The nuclear reaction n + $^3$He = p + $^3$H + 0.76MeV heats a cigar shaped region of the superfluid into the normal phase. The subsequent cooling of this region back through the superfluid transition results in the nucleation of quantized vortices. Depending on the superflow velocity, sufficiently large vortex rings grow under the influence of the Magnus force and escape into the container volume where they are detected individually with nuclear magnetic resonance. The larger the superflow velocity the smaller the rings which can expand. Thus it is possible to obtain information about the morphology of the initial defect network. We suggest that the nucleation of vortices during the rapid cool-down into the superfluid phase is similar to the formation of defects during cosmological phase transitions in the early universe.

https://arxiv.org/pdf/cond-mat/9512117.pdf

Big bang simulation in superfluid 3He-B -- Vortex nucleation in neutron-irradiated superflow

The effective capacitance has been measured in the split Cooper-pair box (CPB) over its phase-gate bias plane. Our low-frequency reactive measurement scheme allows us to probe purely the capacitive susceptibility due to the CPB band structure. The data are quantitatively explained using parameters determined independently by spectroscopic means. In addition, we show in practice that the method offers an efficient way to do nondemolition readout of the CPB quantum state.

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Direct observation of Josephson capacitance

We use neutron absorption in rotating ${}^{3}\mathrm{He}\ensuremath{-}B$ to heat locally a $\ensuremath{\sim}100\ensuremath{-}\ensuremath{\mu}\mathrm{m}\ensuremath{-}\mathrm{size}$ volume into normal phase. When the heated region cools back in microseconds, vortex lines are formed. We record with NMR the number of lines vs the applied superflow velocity and compare to the Kibble-Zurek theory of vortex-loop freeze-out from a random network of defects. The measurements confirm the calculated loop-size distribution and indicate that the superfluid state itself forms as a patchwork of competing $A\ensuremath{-}$ and $B\ensuremath{-}\mathrm{phase}$ blobs. The consequences to the $A\ensuremath{\rightarrow}B$ transition in supercooled ${}^{3}\mathrm{He}\ensuremath{-}A$ are discussed.

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Defect Formation in Quench-Cooled Superfluid Phase Transition

Momentum production during the phase slip process in SNS Josephson junction is discussed. It is caused by the spectral flow of bound states of fermions localized within the junction. This effectively reduces the Magnus force acting on vortices which provides an explanation for the experimental observation of the negligible Magnus force in 2D Josephson junction arrays. The flow of the fermionic levels is similar to that in sphalerons in particle physics, where it gives rise to the baryogenesis.

Yu. G. Makhlin

Papers

Vortex formation in neutron-irradiated superfluid 3He as an analogue of cosmological defect formation

Big bang simulation in superfluid 3He-B -- Vortex nucleation in neutron-irradiated superflow

Direct observation of Josephson capacitance

Defect Formation in Quench-Cooled Superfluid Phase Transition

Spectral flow in Josephson Junctions and effective magnus force