Quantum-circuit refrigeration of a superconducting microwave resonator well below a single quantum

In a series of papers, we intend to take the perspective of open quantum systems and examine from their nonequilibrium dynamics the conditions when the physical quantities, their relations, and the laws of thermodynamics become well defined and viable for quantum many-body systems. We first describe how an open-system nonequilibrium dynamics (ONEq) approach is different from the closed combined system+ environment in a global thermal state (CGTs) setup. Only after the open system equilibrates will it be amenable to conventional thermodynamics descriptions, thus quantum thermodynamics (QTD) comes at the end rather than assumed in the beginning. The linkage between the two comes from the reduced density matrix of ONEq in that stage having the same form as that of the system in the CGTs. We see the open-system approach having the advantage of dealing with nonequilibrium processes as many experiments in the near future will call for. Because it spells out the conditions of QTD's existence, it can also aid us in addressing the basic issues in quantum thermodynamics from first principles in a systematic way. We then study one broad class of open quantum systems where the full nonequilibrium dynamics can be solved exactly, that of the quantum Brownian motion of N strongly coupled harmonic oscillators, interacting strongly with a scalar-field environment. In this paper, we focus on the internal energy, heat capacity, and the third law. We show for this class of physical models, amongst other findings, the extensive property of the internal energy, the positivity of the heat capacity, and the validity of the third law from the perspective of the behavior of the heat capacity toward zero temperature. These conclusions obtained from exact solutions and quantitative analysis clearly disprove claims of negative specific heat in such systems and dispel allegations that in such systems the validity of the third law of thermodynamics relies on quantum entanglement. They are conceptually and factually unrelated issues. Entropy and entanglement will be the main theme of our second paper on this subject matter.

Classicality and anticlassicality measures of pure and mixed quantum states

We present explicit evaluations of quantum speed limit times pertinent to the Markovian dynamics of an open continuous-variable system. Specifically, we consider the standard setting of a cavity mode of the quantum radiation field weakly coupled to a thermal bosonic reservoir. The evolution of the field state is ruled by the quantum optical master equation, which is known to have an exact analytic solution. Starting from a pure input state, we employ two indicators of how different the initial and evolved states are, namely, the fidelity of evolution and the Hilbert-Schmidt distance of evolution. The former was introduced by del Campo {\em et al.} who derived a time-independent speed limit for the evolution of a Markovian open system. We evaluate it for this field-reservoir setting, with an arbitrary input pure state of the field mode. The resultant formula is then specialized to the coherent and Fock states. On the other hand, we exploit an alternative approach that employs both indicators of evolution mentioned above. Their rates of change have the same upper bound, and consequently provide a unique time-dependent quantum speed limit. It turns out that the associate quantum speed limit time built with the Hilbert-Schmidt metric is tighter than the fidelity-based one. As apposite applications, we investigate the damping of the coherent and Fock states by using the characteristic functions of the corresponding evolved states. General expressions of both the fidelity and the Hilbert-Schmidt distance of evolution are obtained and analyzed for these two classes of input states. In the case of a coherent state, we derive accurate formulas for their common speed limit and the pair of associate limit times.

<sec>High-order photon correlations of light fields are important for characterizing the quantum nature. Since Hanbury Brown and Twiss conducted the pioneering experiments in the 1950s, the HBT effect has inspired extensive research on high-order photon correlation in quantum optics, quantum information, and quantum imaging. The Single-photon counting module is one of the most widely used single-photon detectors. Due to its high detection efficiency and low dark counts in the visible and near-infrared region, it is reasonably chosen for basic research on quantum mechanics. Many researches have demonstrated that the maximum value of second-order photon correlation <i>g</i><sup>(2)</sup>(<i>τ</i>) at zero delay (<i>τ</i> = 0) can be used to distinguish different light fields. Therefore, the HBT scheme containing two single photon detectors have been widely used in many advanced studies, such as space interference, ghost imaging, single photon detection with high efficiency, etc. However, higher-order photon correlations <i>g</i><sup>(<i>n</i>)</sup> (<i>n</i> > 2) can reveal more measurable characteristics of light fields, such as information about the non-Gaussian scattering process, the skewness and kurtosis of photon number distribution, etc. When the extended HBT scheme is used to measure higher-order photon correlations, the experimental conditions including quantum efficiency and background noise greatly affect the photon correlation measurement. The influences of the counting rate and resolution time of the detection system on the measurements are also very important and cannot be ignored. Therefore, the comprehensive considering of various influence factors is necessary for accurately measuring the high-order photon correlations and also a challenge.</sec><sec>In this paper, we present a method based on double Hanbury Brown-Twiss scheme for the accurate measuring of high-order photon correlations <i>g</i><sup>(<i>n</i>)</sup> (<i>n</i> > 2). The system consists of four single photon counting modules and is used to detect and analyze the joint distribution probability of temporal photon correlation. Considering the effects of the background noise and overall efficiency, theoretically, we analyze the correlations of the third- and fourth-order photon with the incident light intensity, squeezing parameter and photon number respectively for thermal state, coherent state, squeezed vacuum state, and Fock state. Meanwhile, experimentally we study the influences of resolution time and counting rate on correlations of the coherent state and thermal state with third- and fourth-order photon. On condition that the resolution time is 2<sup>10</sup> ns and the counting rate is 80 kc/s, the correlations of third and fourth-order photon with the thermal state at zero time delay are accurately measured, and the relative statistical deviations of the measured vales from the theoretical values are 0.3% and 0.8%, respectively. In addition, the third- and fourth-order photon correlations of the thermal state at different delay times are also observed. It is demonstrated that the high-order photon correlations of light fields are measured accurately by comprehensively analyzing various influencing factors. This technique provides a promising and useful tool to investigate quantum correlated imaging and quantum coherence of light fields.</sec>

It is known that a uniformly accelerating detector gets excited in the inertial vacuum of a field, as if it were immersed in a thermal bath. We study the effect that the choice of the quantum state has on the response of the detector. The ``correlation function'' 〈\ensuremath{\Psi}\ensuremath{\Vert}\ensuremath{\varphi}(x)\ensuremath{\varphi}(x')\ensuremath{\Vert}\ensuremath{\Psi}〉 is computed in a Fock state, in the coherent state, and in the thermal state. It is then shown that the response function is a sum of two terms---the first being the same as the response in the vacuum state and a second term which is state dependent. The limit of the response function as \ensuremath{\Elzxh}\ensuremath{\rightarrow}0 is discussed and the response of the inertial and accelerated detectors compared in this limit. The response in the large-${n}_{\mathrm{k}}$ limit of the Fock state is compared with the response in the coherent state. We conclude that the thermal response of an accelerated detector in the inertial vacuum has interesting similarities to its response in other states.

Conventional Glauber coherent states (CS) can be defined in several equivalent ways, e.g., by displacing the vacuum or, explicitly, by their infinite Poissonian expansion in Fock states. It is well known that these definitions become inequivalent if applied to finite $d$-level systems (qudits). We present a comparative Wigner-function description of the qudit CS defined (i) by the action of the truncated displacement operator on the vacuum and (ii) by the Poissonian expansion in Fock states of the Glauber CS truncated at $(d\ensuremath{-}1)$-photon Fock state. These states can be generated from a classical light by its optical truncation using nonlinear and linear quantum scissors devices, respectively. We show a surprising effect that a macroscopically distinguishable superposition of two qudit CS (according to both definitions) can be generated with high fidelity by displacing the vacuum in the qudit Hilbert space. If the qudit dimension $d$ is even (odd), then the superposition state contains Fock states with only odd (even) photon numbers, which can be referred to as the odd (even) qudit CS or Schr\"odinger's cat state. This phenomenon can be interpreted as an interference of a single CS with its reflection from the highest-energy Fock state of the Hilbert space, as clearly seen via phase-space interference of the Wigner function. We also analyze nonclassical properties of the qudit CS including their photon-number statistics and nonclassical volume of the Wigner function, which is a quantitative parameter of nonclassicality (quantumness) of states. Finally, we study optical tomograms, which can be directly measured in the homodyne detection of the analyzed qudit cat states and enable the complete reconstructions of their Wigner functions.

Boson sampling is a specific quantum computation, which is likely hard to implement efficiently on a classical computer. The task is to sample the output photon-number distribution of a linear-optical interferometric network, which is fed with single-photon Fock-state inputs. A question that has been asked is if the sampling problems associated with any other input quantum states of light (other than the Fock states) to a linear-optical network and suitable output detection strategies are also of similar computational complexity as boson sampling. We consider the states that differ from the Fock states by a displacement operation, namely the displaced Fock states and the photon-added coherent states. It is easy to show that the sampling problem associated with displaced single-photon Fock states and a displaced photon-number detection scheme is in the same complexity class as boson sampling for all values of displacement. On the other hand, we show that the sampling problem associated with single-photon-added coherent states and the same displaced photon-number detection scheme demonstrates a computational-complexity transition. It transitions from being just as hard as boson sampling when the input coherent amplitudes are sufficiently small to a classically simulatable problem in the limit of large coherent amplitudes.

A laser field may be idealized as a coherent state, but it is often more convenient to use Fock states fQr quantum electrodynamical calculations. A Fock state implies fluctuations in the field intensity, aptl a constant field intensity (coherent state) implies fluctuations in the photon number. Both these effects are discussed rigorously. When the average photon number tends to infinity, these different states become asymptotically indistinguishable.

Quantum non-Gaussian (QNG) mechanical states from inherently nonlinear quantum processes are already required in quantum sensing, become essential in quantum thermodynamics, and also open directions in quantum computing with continuous variables. The discrete building blocks of such states are the energy eigenstates - Fock states. Despite the progress in their experimental preparation, the remaining energy uncertainty can still invisibly cause loss of the critical quantum non-Gaussian aspects of the phonon distribution relevant for the applications.

Quantum simulations are becoming an essential tool for studying complex phenomena, e.g. quantum topology, quantum information transfer and relativistic wave equations, beyond the limitations of analytical computations and experimental observations. To date, the primary resources used in proof-of-principle experiments are collections of qubits, coherent states or multiple single-particle Fock states. Here we show a quantum simulation performed using genuine higher-order Fock states, with two or more indistinguishable particles occupying the same bosonic mode. This was implemented by interfering pairs of Fock states with up to five photons on an interferometer, and measuring the output states with photon-number-resolving detectors. Already this resource-efficient demonstration reveals topological matter, simulates non-linear systems and elucidates a perfect quantum transfer mechanism which can be used to transport Majorana fermions.

We derive a general expression of the expectation value of the parity operator in a Mach–Zehnder interferometer with a Fock state in one input port and an arbitrary state in the other input. We obtain the quantum Crámer–Rao bound via the error propagation method from parity detection for the phase shift around zero. In addition, for an arbitrary state with the same mean photon number, our results show that the same phase resolution and the same phase uncertainty can be obtained for the phase shift around zero. That is to say, when a Mach–Zehnder interferometer is powered by a Fock state into one input port, all quantum states with same mean photon number at the other input may have the same performance in the phase estimation via parity detection. As some specific applications, we consider the arbitrary state being a Fock state, a coherent state, and a squeezed-vacuum state.

The minimum number of photons necessary for activating the sense of vision has been a topic of research for over a century. The ability of rod cells to sense a few photons has implications for understanding the fundamental capabilities of the human visual and nervous system and creating new vision technologies based on photonics. We investigate the fundamental metrological capabilities of different quantum states of light to probe the retina, which is modeled using a simple neural network. Stimulating the rod cells by Fock, coherent, and thermal states of light, and calculating the Cramer-Rao lower bound and Fisher information matrix for the signal produced by the ganglion cells in various conditions, we determine the volume of minimum error ellipsoid. Comparing the resulting ellipsoid volumes, we determine the metrological performance of different states of light for probing the retinal network. The results indicate that the thermal state yields the largest error ellipsoid volume and hence the worst metrological performance, and the Fock state yields the best performance for all parameters. This advantage persists even if another layer is added to the network or optical losses are considered in the calculations.

Quantum thermodynamics aims at investigating both the emergence and the limits of the laws of thermodynamics from a quantum mechanical microscopic approach. In this scenario, thermodynamic processes with no heat exchange, namely, adiabatic transformations, can be implemented through quantum evolutions in closed systems, even though the notion of a closed system is always an idealization and approximation. Here, we begin by theoretically discussing thermodynamic adiabatic processes in open quantum systems, which evolve non-unitarily under decoherence due to its interaction with its surrounding environment. From a general approach for adiabatic non-unitary evolution, we establish heat and work in terms of the underlying Liouville superoperator governing the quantum dynamics. As a consequence, we derive the conditions that an adiabatic open-system quantum dynamics implies in the absence of heat exchange, providing a connection between quantum and thermal adiabaticity. Moreover, we determine families of decohering systems exhibiting the same maximal heat exchange, which imply in classes of thermodynamic adiabaticity in open systems. We then approach the problem experimentally using a hyperfine energy-level quantum bit of an Ytterbium 171Yb+ trapped ion, which provides a work substance for thermodynamic processes, allowing for the analysis of heat and internal energy throughout a controllable engineered dynamics.

The Mach-Zehnder interferometer is a powerful device for detecting small phase shifts between two light beams. Simple input states, such as coherent states or single photons, can reach the standard quantum limit of phase estimation, while more complex states can be used to reach Heisenberg scaling; the latter, however, require challenging preparation and measurement strategies. The quest for highly sensitive phase estimation therefore calls for interferometers with nonlinear devices which would make the preparation of these complex states more efficient. Here, we show that the Heisenberg scaling can be recovered with simple input states (including Fock and coherent states) when the linear mirrors in the interferometer are replaced with controlled-swap gates and measurements on auxiliary qubits. These swap tests project the input Fock and coherent states onto NOON and entangled coherent states, respectively, and allow optimal or near-optimal measurements, leading to improved sensitivity to small phase shifts in one of the interferometer arms. We analyze auxiliary qubit errors in detail, showing that biasing the qubit towards phase flips offers a great advantage, and perform thorough numerical simulations of a possible implementation in circuit quantum electrodynamics with an auxiliary Kerr-cat qubit. Our results thus present a viable approach to phase estimation approaching Heisenberg-limited sensitivity and demonstrate potential advantages of using biased-noise qubits in quantum metrology. Published by the American Physical Society 2024

Fock states and coherent states have been widely applied in analyses of quantum-optics experiments. In this paper, the application of a set of basis functions that consist of a product of displaced harmonic-oscillator states for the electromagnetic field and atomic states that are eigenfunctions of the momentum operator will be explored. For the case of a single mode, these states are the exact eigenfunctions in the limiting case where the electromagnetic-field mode frequency is much larger than the atomic transition frequency. This choice of basis functions will be used to analyze the interaction of a nonrelativistic, ``two-level'' atom with a single-frequency, quantized electromagnetic-field mode. Since in this basis the sign of the displacement of the electromagnetic-field state depends on the state of the atom, these new basis wave functions exhibit explicitly some aspects of atom-field correlations that are intrinsically quantum mechanical in nature. The analysis will be performed in the electric-dipole approximation for an atomic system that obeys the \ensuremath{\Delta}m=0 selection rule. The rotating-wave-approximation (RWA) terms cannot be made negligible through judicious choice of the field polarization for this system. The diamagnetic term, ${\mathit{e}}^{2}$${\mathbf{A}}^{2}$/2${\mathit{mc}}^{2}$, which is of the same order of magnitude as the RWA terms, is taken into account exactly by slightly modifying the traditional approach to quantizing the electromagnetic field. In this way the physical effects of the diamagnetic term are absorbed into a new electromagnetic cavity mode frequency.An approximate expression for the combined system's energy eigenvalues is found up to third order in the ratio of one-half the atomic transition frequency to the mode frequency. This expression indicates that inclusion of the RWA terms adds a small ``ripple'' on top of the smoothly varying energy eigenvalues when they are plotted as a function of the atom-field coupling constant. The ripple is interpreted as ``interference'' between two displaced oscillator-field states that represent the unperturbed state. When the Fock states are used as the basis for expanding the wave function of this system, a formidable five-term recurrence relation results. However, expanding the wave function in terms of the displaced harmonic-oscillator basis results in a less formidable three-term recurrence relation for the quantum coefficients.