Diagonal Elements Of Density Matrix Research Articles

Quantifying coherence with principal diagonal elements of density matrix

Quantifying coherence with principal diagonal elements of density matrix

Quantum theory with an energy operator defined as a quartic form of the momentum

Quantum theory of the non-harmonic oscillator defined by the energy operator proposed by Yurke and Buks (2006) is presented. Although these authors considered a specific problem related to a model of transmission lines in a Kerr medium, our ambition is not to discuss the physical substantiation of their model. Instead, we consider the problem from an abstract, logically deductive, viewpoint. Using the Yurke–Buks energy operator, we focus attention on the imaginary-time propagator. We derive it as a functional of the Mehler kernel and, alternatively, as an exact series involving Hermite polynomials. For a statistical ensemble of identical oscillators defined by the Yurke–Buks energy operator, we calculate the partition function, average energy, free energy and entropy. Using the diagonal element of the canonical density matrix of this ensemble in the coordinate representation, we define a probability density, which appears to be a deformed Gaussian distribution. A peculiarity of this probability density is that it may reveal, when plotted as a function of the position variable, a shape with two peaks located symmetrically with respect to the central point.

Mixed Quantum-Classical Study of Nonadiabatic Curve Crossing in Condensed Phases.

We apply the mixed quantum-classical Liouville (MQCL) equation to investigate the nonadiabatic curve crossing in condensed phases. More specifically, electron transfer rate constants of the spin-Boson model are calculated by employing a rate constant expression using the collective solvent polarization as the reaction coordinate. In the calculation, classical nuclear degrees of freedom are initially sampled at the transition state configuration, and the initial state for the electronic degree of freedom is obtained from a mixed quantum-classical Boltzmann distribution. Different contributions to the electron transfer rate from the diagonal and off-diagonal elements of the initial density matrix, and contributions from trajectories with positive and negative initial velocities are analyzed. It is shown that the off-diagonal elements of the initial density matrix play an important role in the total electron transfer rate. The MQCL results are also compared with those calculated using Ehrenfest dynamics. It is found that, although the Ehrenfest dynamics is inaccurate when the reactive flux rate expression is used directly, it can give reasonably accurate results when individual contributions from the diagonal and off-diagonal elements of the initial density matrix are calculated.

Exact quantum Bayesian rule for qubit measurements in circuit QED.

Developing efficient framework for quantum measurements is of essential importance to quantum science and technology. In this work, for the important superconducting circuit-QED setup, we present a rigorous and analytic solution for the effective quantum trajectory equation (QTE) after polaron transformation and converted to the form of Stratonovich calculus. We find that the solution is a generalization of the elegant quantum Bayesian approach developed in arXiv:1111.4016 by Korotokov and currently applied to circuit-QED measurements. The new result improves both the diagonal and off-diagonal elements of the qubit density matrix, via amending the distribution probabilities of the output currents and several important phase factors. Compared to numerical integration of the QTE, the resultant quantum Bayesian rule promises higher efficiency to update the measured state, and allows more efficient and analytical studies for some interesting problems such as quantum weak values, past quantum state, and quantum state smoothing. The method of this work opens also a new way to obtain quantum Bayesian formulas for other systems and in more complicated cases.

Communication: Fully coherent quantum state hopping.

In this paper, we describe a new and fully coherent stochastic surface hopping method for simulating mixed quantum-classical systems. We illustrate the approach on the simple but unforgiving problem of quantum evolution of a two-state quantum system in the limit of unperturbed pure state dynamics and for dissipative evolution in the presence of both stationary and nonstationary random environments. We formulate our approach in the Liouville representation and describe the density matrix elements by ensembles of trajectories. Population dynamics are represented by stochastic surface hops for trajectories representing diagonal density matrix elements. These are combined with an unconventional coherent stochastic hopping algorithm for trajectories representing off-diagonal quantum coherences. The latter generalizes the binary (0,1) "probability" of a trajectory to be associated with a given state to allow integers that can be negative or greater than unity in magnitude. Unlike existing surface hopping methods, the dynamics of the ensembles are fully entangled, correctly capturing the coherent and nonlocal structure of quantum mechanics.

Equilibration implies asymptotic stationarity for open quantum systems

We use the exact Nakajima–Zwanzig form of the master equation to show that open quantum systems which exhibit equilibration (or thermalization) by evolving to a time independent asymptotic state, have under certain conditions a reduced density matrix for the system which commutes with the effective system Hamiltonian. We also show that if the initial system–bath density matrix is of product form then the asymptotic reduced density matrix of the system depends only on the diagonal elements of the initial system density matrix in the eigenbasis of the effective Hamiltonian.

Photon number statistics of NV centre emission

Optical sources that deterministically produce single photons with a high suppression of multi-photon emission and a negligible background component are promising candidates for standard sources for quantum metrology, quantum communication and foundations of quantum mechanics. In this paper, the photon number distribution of non-classical light emitted by nitrogen vacancy (NV) centres in nano-diamonds is studied by three different experimental techniques. The photon number resolving transition edge sensor (TES) detector and the On/Off detection method are applied to determine the diagonal elements of the optical density matrix. From the data measured by the two methods the second order correlation function at time delay zero is calculated and compared with the g2(0)-values obtained by Hanbury Brown–Twiss (HBT) interferometric measurements. Among the g2(0)-values evaluated with the three techniques, we found good agreement in the results for a single photon emitter with a g2(0)-value close to zero and a multi-photon source with a g2(0)-value of approximately 0.5.

Entropy of Isolated Quantum Systems after a Quench

A diagonal entropy, which depends only on the diagonal elements of the system's density matrix in the energy representation, has been recently introduced as the proper definition of thermodynamic entropy in out-of-equilibrium quantum systems. We study this quantity after an interaction quench in lattice hard-core bosons and spinless fermions, and after a local chemical potential quench in a system of hard-core bosons in a superlattice potential. The former systems have a chaotic regime, where the diagonal entropy becomes equivalent to the equilibrium microcanonical entropy, coinciding with the onset of thermalization. The latter system is integrable. We show that its diagonal entropy is additive and different from the entropy of a generalized Gibbs ensemble, which has been introduced to account for the effects of conserved quantities at integrability.

Slater sum for an inhomogeneous liquid of Fermions generated by a sech2 x potential in one dimension

In one dimension, the Slater sum S(x, β), which is the diagonal element of the canonical density matrix, satisfies a known partial differential equation characterised by a one-body potential V(x). Here, for the case of a sech2 x potential in one dimension, it is stressed that S(x, β) is explicitly related to the limit S 0(β) as V(x) → 0 and to V(x) itself. This is the same input information as in the Thomas–Fermi result. The relevance to density functional theory is emphasised.

Statistical–mechanical models with separable many-body interactions: especially partition functions and thermodynamic consequences

We start from a classical statistical–mechanical theory for the internal energy in terms of three- and four-body correlation functions g3 and g4 for homogeneous atomic liquids like argon, with assumed central pair interactions \({\phi(r_{ij})}\) . The importance of constructing the partition function (pf) as spatial integrals over g3, g4 and \({\phi}\) is stressed, together with some basic thermodynamic consequences of such a pf. A second classical example taken for two-body interactions is the so-called one-component plasma in two dimensions, for a particular coupling strength treated by Alastuey and Jancovici (J Phys (France) 42:1, 1981) and by Fantoni and Tellez (J Stat Phys 133:449, 2008). Again thermodynamic consequences provide a particular focus. Then quantum–mechanical assemblies are treated, again with separable many-body interactions. The example chosen is that of an N-body inhomogeneous extended system generated by a one-body potential energy V(r). The focus here is on the diagonal element of the canonical density matrix: the so-called Slater sum S(r, β), related to the pf by \({{\rm pf}(\beta) = \int {S({\bf r}, \beta)}d\vec {r}}\), β = (kBT)−1. The Slater sum S(r, β) can be related exactly, via a partial differential equation, to the one-body potential V(r), for specific choices of V which are cited. The work of Green (J Chem Phys 18:1123, 1950), is referred to for a generalization, but now perturbative, to two-body forces. Finally, to avoid perturbation series, the work concludes with some proposals to allow the treatment of extended assemblies in which regions of long-range ordered magnetism exist in the phase diagram. One of us (Z.D.Z.) has recently proposed a putative pf for a three-dimensional (3D) Ising model, based on two, as yet unproved, conjectures and has pointed out some important thermodynamic consequences of this pf. It would obviously be of considerable interest if such a pf, together with conjectures, could be rigorously proved.

Numerical simulation of Raman resonance due to the Ramsey interference induced by thermal motion of atoms

We use the optical Bloch equations to calculate spontaneous emission and Zeeman coherences of the probe laser beam in coaxial pump–probe configuration in rubidium (Rb) vapor. To properly model the open Fg=2→Fe=1 transition, the solution of the time-dependent optical Bloch equations was necessary. Also, due to the nature of a room temperature Rb vapor, we performed averaging of the total population of the excited Fe=1 state over different atomic velocities according to the Maxwell–Boltzmann distribution and over different directions of propagation of atoms through pump and probe laser beams. Time-dependent diagonal and nondiagonal density matrix elements for atoms are presented and discussed.

Boson and fermion many-body assemblies: fingerprints of excitations in the ground-state wave functions, with examples of superfluid 4He and the homogeneous correlated electron liquid

After a brief summary of some basic properties of ideal gases of bosons and of fermions, two many-body Hamiltonians are cited for which ground-state wave functions allow the generation of excited states. Because of the complexity of ground-state many-body wave functions, we then consider properties of reduced density matrices, and in particular, the diagonal element of the second-order density matrix. For both the homogeneous correlated electron liquid and for an assembly of charged bosons, the ground-state pair correlation function g(r) has fingerprints of the zero-point energy of the plasmon modes. These affect crucially the static structure factor, S(k), in the long wavelength limit. This is best understood by means of the Ornstein–Zernike direct correlation function c(r), which plays an important role throughout this article. Turning from such charged liquids, both boson and fermion, to superfluid 4He, the elevated temperature (T) structure factor S(k, T) is related, albeit approximately, to its zero-temperature counterpart, via the velocity of sound, reflecting the collective phonon excitations and the superfluid density. Finally, some future directions are pointed.

Spin alignment measurements of theK*0(892)andϕ(1020)vector mesons in heavy ion collisions atsNN=200GeV

We present the first spin alignment measurements for the $K^{*0}(892)$ and $\phi(1020)$ vector mesons produced at mid-rapidity with transverse momenta up to 5 GeV/c at $\sqrt{s_{NN}}$ = 200 GeV at RHIC. The diagonal spin density matrix elements with respect to the reaction plane in Au+Au collisions are $\rho_{00}$ = 0.32 $\pm$ 0.04 (stat) $\pm$ 0.09 (syst) for the $K^{*0}$ ($0.8<p_T<5.0$ GeV/c) and $\rho_{00}$ = 0.34 $\pm$ 0.02 (stat) $\pm$ 0.03 (syst) for the $\phi$ ($0.4<p_T<5.0$ GeV/c), and are constant with transverse momentum and collision centrality. The data are consistent with the unpolarized expectation of 1/3 and thus no evidence is found for the transfer of the orbital angular momentum of the colliding system to the vector meson spins. Spin alignments for $K^{*0}$ and $\phi$ in Au+Au collisions were also measured with respect to the particle's production plane. The $\phi$ result, $\rho_{00}$ = 0.41 $\pm$ 0.02 (stat) $\pm$ 0.04 (syst), is consistent with that in p+p collisions, $\rho_{00}$ = 0.39 $\pm$ 0.03 (stat) $\pm$ 0.06 (syst), also measured in this work. The measurements thus constrain the possible size of polarization phenomena in the production dynamics of vector mesons.

Spin alignment of K*0(892) and ϕ(1020) mesons in Au+Au and p+p collisions at GeV

We present the first spin alignment measurements for the vector mesons K*0(892) and ϕ(1020) at mid-rapidity in Au+Au and p+p collisions at GeV. Vector mesons appear to be produced without a strong spin alignment constraining possible spin transfer processes from a large orbital angular momentum of the colliding system as expected in theoretical calculations. The diagonal spin density matrix elements are measured to be ρ00 = 0.32 ± 0.04 (stat) ± 0.09 (syst) for the K*0 (0.8 < pT < 5.0 GeV/c) and ρ00 = 0.34 ± 0.02 (stat) ± 0.03 (syst) for the ϕ (0.4 < pT < 5.0 GeV/c), with respect to the reaction plane in midcentral Au+Au collisions, and are found to be independent of collision centrality and transverse momentum. ρ00 with respect to the production plane is measured to be less than 2σ deviation above 1/3 in both Au+Au and p+p collisions indicating the absence of polarization phenomena in the production dynamics for vector mesons in the measured kinematic region.

Spin alignment of K*0(892) and ϕ(1020) mesons in Au+Au and p+p collisions at

We report on STAR measurements of global polarization for vector mesons K*0(892) and ϕ(1020) at midrapidity from noncentral Au+Au collisions at . The measured diagonal spin density matrix elements for particles up to a transverse momentum 5 GeV/c are ρ00 = 0.36 ± 0.02 (stat) ± 0.13 (sys) for K*0(892) and ρ00 = 0.38 ± 0.01 (stat) ± 0.04 (sys) for ϕ(1020) with respect to the reaction plane, where 1/3 is the value for no polarization. With respect to the vector mesons' production planes, ρ00 for K*0(892) and ϕ(1020) in Au+Au and for ϕ(1020) mesons in p+p collisions at is also presented. There is no significant difference for the measured spin alignment between Au+Au and p+p collisions with our data sample. We have also obtained an upper limit for global polarization.

Decoherence measurement of two coupled micromasers

We present a setup of two coupled micromaser cavities for studying decoherence. The system is prepared in a state where, initially, one photon is contained in one of the two cavities. The coupling of the two resonators allows the photon to tunnel into the other cavity with a certain probability. The system thus oscillates between the two states vertical bar 1>{sub L} vertical bar 0>{sub R} and vertical bar 0>{sub L} vertical bar 1>{sub R}. It is shown how to map these states onto the states vertical bar g> and vertical bar e> of probe atoms used to investigate the status of the cavity system. It is an important result that the symmetry of our special setup allows decoherence to be monitored by measuring just the diagonal elements of the atomic density matrix. This property relies on the fact that only vertical bar 0> and vertical bar 1> states are relevant. To our knowledge, this is the first proposal that tries to exploit this particular property.

On the Reconstruction of Diagonal Elements of Density Matrix of Quantum Optical States by on/off Detectors

We discuss a scheme for reconstructing experimentally the diagonal elements of a density matrix representing quantum optical state. Applications to PDC heralded photons, multi-thermal and attenuated coherent states are illustrated and discussed in more detail.

The discrete representation correspondence between quantum and classical spatial distributions of angular momentum vectors

This work demonstrates that the quantum mechanical moments of a state described by the density matrix correspond to discrete spherical harmonic moments of the classical multipole expansion of the spatial distribution of the angular momentum vectors. For the diagonal density matrix elements, this work exploits the fact that the quantum mechanical vector coupling (Clebsch-Gordan) coefficients become increasingly accurate discrete representations of spherical harmonics as j increases. A Schwinger-type basis accounts for nonaxially symmetric angular distributions, which result in nonzero off-diagonal elements of the density matrix. The resulting discrete minimum uncertainty picture of the classical moments has a stringent equivalence with the quantum mechanical one for all j and provides an unambiguous connection for the classical and quantum moments in the large j limit. The equivalence is numerically tested for simple models, and there is a satisfying equivalence even for small j. Applications, implications, and extensions are indicated, and the relevance of this work for the interpretation of classical mechanical simulations of inelastic and reactive molecular collisions will be documented elsewhere.

Derivation of the differential equation for the Slater sum and of the differential virial theorem using the Wigner transform

Using the Wigner transform, we present an alternative derivation of the partial differential equation satisfied by the Slater sum, which is the diagonal element of the canonical Bloch density matrix. This is done in one dimension for the case of a general confining potential and also for the case of N independent fermions harmonically confined in d dimensions. We also present a simple proof of the so-called differential virial theorem for each of these cases.

Quantum nondemolition measurement of a kicked qubit

We propose a quantum nondemolition measurement using a kicked two-state system (qubit). By tuning the waiting time between kicks to be the qubit oscillation period, the kicking apparatus performs a nondemolition measurement. While dephasing is unavoidable, the nondemolition measurement can (1) slow relaxation of diagonal density matrix elements, (2) avoid detector back-action, and (3) allow for a large signal-to-noise ratio. Deviations from the ideal behavior are investigated by allowing for detuning of the waiting time, as well as finite-time, noisy pulses. The scheme is illustrated with a double-dot qubit measured by a gate-pulsed quantum point contact.