The work distribution function for a driven quantum system which evolves out of thermal equilibrium encodes some universal features and is a central character in the so-called fluctuation theorems. In this talk I will review recent efforts in developing methods to efficiently sample the work distribution function in modern quantum simulators. Not only I will review some theoretical work but also present the results of a conceptually simple (but technically challenging) experiment which was done to implement such methods and illustrate the validity of the quantum fluctuation theorems.
The work distribution function for a driven quantum system which evolves out of thermal equilibrium encodes some universal features and is a central character in the so-called fluctuation theorems. In this talk I will review recent efforts in developing methods to efficiently sample the work distribution function in modern quantum simulators. Not only I will review some theoretical work but also present the results of a conceptually simple (but technically challenging) experiment which was done to implement such methods and illustrate the validity of the quantum fluctuation theorems.
Work is an essential concept in classical thermodynamics, and in the quantum regime, where the notion of a trajectory is not available, its definition is not trivial. For driven (but otherwise isolated) quantum systems, work can be defined as a random variable, associated with the change in the internal energy. The probability for the different values of work captures essential information describing the behaviour of the system, both in and out of thermal equilibrium. In fact, the work probability distribution is at the core of "fluctuation theorems" in quantum thermodynamics. Here we present the design and implementation of a quantum work meter operating on an ensemble of cold atoms, which are controlled by an atom chip. Our device not only directly measures work but also directly samples its probability distribution. We demonstrate the operation of this new tool and use it to verify the validity of the quantum Jarzynksi identity.
Fluctuation theorems provide a correspondence between properties of quantum systems in thermal equilibrium and a work distribution arising in a non-equilibrium process that connects two quantum systems with Hamiltonians H 0 and H 1 = H 0 + V . Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of H 1 at inverse temperature β ≥ 0 starting from a purification of the thermal state of H 0 . The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is O ~ ( e β ( Δ A − w l ) / 2 ) , where Δ A is the free-energy difference between H 1 and H 0 , and w l is a work cutoff that depends on the properties of the work distribution and the approximation error ϵ > 0 . If the non-equilibrium process is trivial, this complexity is exponential in β ‖ V ‖ , where ‖ V ‖ is the spectral norm of V . This represents a significant improvement of prior quantum algorithms that have complexity exponential in β ‖ H 1 ‖ in the regime where ‖ V ‖ ≪ ‖ H 1 ‖ . The dependence of the complexity in ϵ varies according to the structure of the quantum systems. It can be exponential in 1 / ϵ in general, but we show it to be sublinear in 1 / ϵ if H 0 and H 1 commute, or polynomial in 1 / ϵ if H 0
The thermodynamics of quantum systems driven out of equilibrium has attracted increasing attention in the last decade, in connection with quantum information and statistical physics, and with a focus on non-classical signatures. While a first approach can deal with average thermodynamics quantities over ensembles, in order to establish the impact of quantum and environmental fluctuations during the evolution, a continuous quantum measurement of the open system is required. Here, we provide an introduction to the general theoretical framework to establish and interpret the thermodynamics for quantum systems whose nonequilibrium evolution is continuously monitored. We review the formalism of quantum trajectories and its consistent application to the thermodynamic scenario, where primary quantities such as work, heat, and entropy production can be defined at the stochastic level. The connection to irreversibility and fluctuation theorems is also discussed together with some recent developments, and we provide some simple examples to illustrate the general theoretical framework.
The original motivation to build a quantum computer came from Feynman who envisaged a machine capable of simulating generic quantum mechanical systems, a task that is intractable for classical computers. Such a machine would have tremendous applications in all physical sciences, including condensed matter physics, chemistry, and high energy physics. Part of Feynman's challenge was met by Lloyd who showed how to approximately decompose the time-evolution operator of interacting quantum particles into a short sequence of elementary gates, suitable for operation on a quantum computer. However, this left open the more fundamental problem of how to simulate the equilibrium and static properties of quantum systems. This requires the preparation of ground and Gibb's states on a quantum computer. For classical systems, this problem is solved by the ubiquitous Metropolis algorithm, a method that basically acquired a monopoly for the simulation of interacting particles. In this talk, I will demonstrate that the corresponding quantum problem can be solved by a quantum Metropolis algorithm. This validates the quantum computer as a universal simulator, and proves that the so-called sign ! problem occurring in quantum Monte Carlo methods can be resolved with a quantum computer.
Recent experimental developments in diverse areas-ranging from cold atomic gases to light-driven semiconductors to microcavity arrays-move systems into the focus which are located on the interface of quantum optics, many-body physics and statistical mechanics. They share in common that coherent and driven-dissipative quantum dynamics occur on an equal footing, creating genuine non-equilibrium scenarios without immediate counterpart in equilibrium condensed matter physics. This concerns both their non-thermal stationary states and their many-body time evolution. It is a challenge to theory to identify novel instances of universal emergent macroscopic phenomena, which are tied unambiguously and in an observable way to the microscopic drive conditions. In this review, we discuss some recent results in this direction. Moreover, we provide a systematic introduction to the open system Keldysh functional integral approach, which is the proper technical tool to accomplish a merger of quantum optics and many-body physics, and leverages the power of modern quantum field theory to driven open quantum systems.
At non-zero temperature classical systems exhibit statistical fluctuations of thermodynamic quantities arising from the variation of the system's initial conditions and its interaction with the environment. The fluctuating work, for example, is characterised by the ensemble of system trajectories in phase space and, by including the probabilities for various trajectories to occur, a work distribution can be constructed. However, without phase space trajectories, the task of constructing a work probability distribution in the quantum regime has proven elusive. Here we use quantum trajectories in phase space and define fluctuating work as power integrated along the trajectories, in complete analogy to classical statistical physics. The resulting work probability distribution is valid for any quantum evolution, including cases with coherences in the energy basis. We demonstrate the quantum work probability distribution and its properties with an exactly solvable example of a driven quantum harmonic oscillator. An important feature of the work distribution is its dependence on the initial statistical mixture of pure states, which is reflected in higher moments of the work. The proposed approach introduces a fundamentally different perspective on quantum thermodynamics, allowing full thermodynamic characterisation of the dynamics of quantum systems, including the measurement process.