Full Density Matrix Research Articles

Phonon dynamics in correlated quantum systems driven away from equilibrium

A general form of a many-body Hamiltonian is considered, which includes an interacting fermionic sub-system coupled to non-interacting extended fermionic and bosonic systems. We show that the exact dynamics of the extended bosonic system can be derived from the reduced density matrix of the sub-system alone, despite the fact that the latter contains information about the sub-system only. The advantage of the formalism is immediately clear: While the reduced density matrix of the sub-system is readily available, the formalism offers access to observables contained in the full density matrix, which is often difficult to obtain. As an example, we consider an extended Holstein model and study the nonequilibrium dynamics of the, so called, "reaction mode" for different model parameters. The effects of the phonon frequency, the strength of the electron-phonon couplings, and the source-drain bias voltage on the phonon dynamics across the bistability are discussed.

Time-dependent numerical renormalization group method for multiple quenches: Application to general pulses and periodic driving

The time-dependent numerical renormalization group method (TDNRG) [Anders et al., Phys. Rev. Lett. {\bf 95}, 196801 (2005)] was recently generalized to multiple quenches and arbitrary finite temperatures [Nghiem et al., Phys. Rev. B {\bf 89}, 075118 (2014)] by using the full density matrix approach [Weichselbaum et al., Phys. Rev. Lett. {\bf 99}, 076402 (2007)]. In this paper, we numerically implement this formalism to study the response of a quantum impurity system to a general pulse and periodic driving which are approximated by a sufficient number of quenches. We show how the NRG approximation affects the trace of the projected density matrices and the continuity of the time-evolution of a local observable. For the general pulse case, the local observable in the long-time limit exhibits a dependence on the switch-on time, the time interval between the first and last quenches, as well as on the pulse shape. In particular, the long-time limit is improved for longer switch-on times and smoother pulses. This lends support to our earlier suggestion that the long-time limit of observables can be improved by replacing a sudden large quench by a sequence of smaller ones acting over a finite time-interval: longer switch-on times and smoother pulses, i.e., increased adiabaticity, favor relaxation of the system to its correct thermodynamic long-time limit. For the case of periodic driving, we compare the TDNRG results to exact analytic ones for the non-interacting resonant level model, finding better agreement at short to intermediate time scales in the case of smoother driving. Finally, we demonstrate the validity of the multiple-quench TDNRG formalism for arbitrary temperatures by studying the time-evolution of the occupation number in the Anderson impurity model in response to a periodic switching of the local level from the mixed valence to the Kondo regime at finite temperatures.

Density-matrix quantum Monte Carlo method

We present a quantum Monte Carlo method capable of sampling the full density matrix of a many-particle system at finite temperature. This allows arbitrary reduced density matrix elements and expectation values of complicated non-local observables to be evaluated easily. The method resembles full configuration interaction quantum Monte Carlo but works in the space of many-particle operators instead of the space of many-particle wave functions. One simulation provides the density matrix at all temperatures simultaneously, from $T=\infty$ to $T=0$, allowing the temperature dependence of expectation values to be studied. The direct sampling of the density matrix also allows the calculation of some previously inaccessible entanglement measures. We explain the theory underlying the method, describe the algorithm, and introduce an importance-sampling procedure to improve the stochastic efficiency. To demonstrate the potential of our approach, the energy and staggered magnetization of the isotropic antiferromagnetic Heisenberg model on small lattices, the concurrence of one-dimensional spin rings, and the Renyi $S_2$ entanglement entropy of various sublattices of the $6 \times 6$ Heisenberg model are calculated. The nature of the sign problem in the method is also investigated.

Generalization of the time-dependent numerical renormalization group method to finite temperatures and general pulses

The time-dependent numerical renormalization group (td-NRG) [Anders et al. Phys. Rev. Lett. {\bf 95}, 196801 (2006)] offers the prospect of investigating in a non-perturbative manner the time-dependence of local observables of interacting quantum impurity models at all time scales following a quantum quench. We present a generalization of this method to arbitrary finite temperature by making use of the full density matrix [Weichselbaum et al. Phys. Rev. Lett. {\bf 99}, 076402 (2007)]. We show that all terms in the projected density matrix $\rho^{i\to f} = \rho^{++} + \rho^{--} + \rho^{+-}+\rho^{-+}$ in the time-evolution of a local observable may be evaluated in closed form, with $\rho^{+-}=\rho^{-+}=0$. The expression for $\rho^{--}$ is shown to be finite at finite temperature, becoming negligible only in the limit of vanishing temperatures. We prove that this approach recovers the short-time limit for the expectation value of a local observable exactly at arbitrary temperatures. In contrast, the corresponding long-time limit is only recovered exactly in the limit $\Lambda\rightarrow 1^{+}$. This limit is impractical within NRG. We suggest how to overcome this problem by noting that the long-time limit becomes increasingly more accurate on reducing the size of the quantum quench. This suggests an improved generalized td-NRG approach in which the system is time-evolved between the initial and final states via a sequence of small quantum quenches within a finite time interval instead of a single large quantum quench. The formalism for this is provided, generalizing the td-NRG to multiple quantum quenches, periodic switching, and general pulses. This formalism, like our finite temperature generalization of the single-quench case, rests only on the NRG approximation. The results are illustrated by application to the Anderson impurity model.

Correlator expansion approach to stationary states of weakly coupled cavity arrays

We introduce a method for calculating the stationary state of a translation invariant array of weakly coupled cavities in the presence of dissipation and coherent as well as incoherent drives. Instead of computing the full density matrix, our method directly calculates the correlation functions which are relevant for obtaining all local quantities of interest. It considers an expansion of the correlation functions and their equations of motion in powers of the photon tunnelling rate between adjacent cavities, leading to an exact second-order solution for any number of cavities. Our method provides a controllable approximation for weak tunnelling rates applicable to the strongly correlated regime that is dominated by nonlinearities in the cavities and thus of high interest.

Noise effects and tomography of remote entangled spins in quantum dots

We investigate how decoherence affects the entanglement established between two quantum dots in microcavities, and propose a tomographic scheme able to measure the entangled state. The scheme we consider establishes the entanglement via the exchange and measurement of a photon. Making the realistic assumption of noise dominated by pure-dephasing processes, we find that the photon must be exchanged and measured on time scales shorter than the quantum dots' characteristic dephasing time for appreciable levels of entanglement to be achieved. The tomographic scheme is able to reconstruct the full density matrix of the quantum dots, and requires only single spin rotations and the injection of an additional photon. Remarkably, we find that the additional photon need not be exchanged and measured on a time scale shorter than the dephasing time for accurate tomography, and also allows many to be used in order to increase the measurement signal.

Nonperturbative stochastic method for driven spin-boson model

We introduce and apply a numerically exact method for investigating the real-time dissipative dynamics of quantum impurities embedded in a macroscopic environment beyond the weak-coupling limit. We focus on the spin-boson Hamiltonian that describes a two-level system interacting with a bosonic bath of harmonic oscillators. This model is archetypal for investigating dissipation in quantum systems, and tunable experimental realizations exist in mesoscopic and cold-atom systems. It finds abundant applications in physics ranging from the study of decoherence in quantum computing and quantum optics to extended dynamical mean-field theory. Starting from the real-time Feynman-Vernon path integral, we derive an exact stochastic Schr\"odinger equation that allows us to compute the full spin density matrix and spin-spin correlation functions beyond weak coupling. We greatly extend our earlier work [P. P. Orth, A. Imambekov, and K. Le Hur, Phys. Rev. A 82, 032118 (2010)] by fleshing out the core concepts of the method and by presenting a number of interesting applications. Methodologically, we present an analogy between the dissipative dynamics of a quantum spin and that of a classical spin in a random magnetic field. This analogy is used to recover the well-known noninteracting-blip approximation in the weak-coupling limit. We explain in detail how to compute spin-spin autocorrelation functions. As interesting applications of our method, we explore the non-Markovian effects of the initial spin-bath preparation on the dynamics of the coherence ${\ensuremath{\sigma}}^{x}(t)$ and of ${\ensuremath{\sigma}}^{z}(t)$ under a Landau-Zener sweep of the bias field. We also compute to a high precision the asymptotic long-time dynamics of ${\ensuremath{\sigma}}^{z}(t)$ without bias and demonstrate the wide applicability of our approach by calculating the spin dynamics at nonzero bias and different temperatures.

Non-Hermitian wave packet approximation of Bloch optical equations

We introduce a non-Hermitian approximation of Bloch optical equations. This approximation provides a complete description of the excitation, relaxation, and decoherence dynamics of ensembles of coupled quantum systems in weak laser fields, taking into account collective effects and dephasing. In the proposed method, one propagates the wave function of the system instead of a complete density matrix. Relaxation and dephasing are taken into account via automatically adjusted time-dependent gain and decay rates. As an application, we compute the numerical wave packet solution of a time-dependent non-Hermitian Schrödinger equation describing the interaction of electromagnetic radiation with a quantum nano-structure, and compare the calculated transmission, reflection, and absorption spectra with those obtained from the numerical solution of the Liouville-von Neumann equation. It is shown that the proposed wave packet scheme is significantly faster than the propagation of the full density matrix while maintaining small error. We provide the key ingredients for easy-to-use implementation of the proposed scheme and identify the limits and error scaling of this approximation.

Full density matrix dynamics for large quantum systems: interactions, decoherence and inelastic effects

We develop analytical tools and numerical methods for time evolving the total density matrix of the finite-size Anderson model. The model is composed of two finite metal grains, each prepared in canonical states of differing chemical potential and connected through a single electronic level (quantum dot or impurity). Coulomb interactions are either excluded all together or allowed on the dot only. We extend this basic model to emulate decohering and inelastic scattering processes for the dot electrons with the probe technique. Three methods, originally developed to treat impurity dynamics, are augmented to yield global system dynamics: the quantum Langevin equation method, the well-known fermionic trace formula and an iterative path integral approach. The latter accommodates interactions on the dot in a numerically exact fashion. We apply the developed techniques to two open topics in nonequilibrium many-body physics. (i) We explore the role of many-body electron–electron repulsion effects on the dynamics of the system. Results, obtained using exact path integral simulations, are compared with mean-field quantum Langevin equation predictions. (ii) We analyze aspects of quantum equilibration and thermalization in large quantum systems using the probe technique, mimicking elastic-dephasing effects and inelastic interactions on the dot. Here, unitary simulations based on the fermionic trace formula are accompanied by quantum Langevin equation calculations.

Tensor networks and the numerical renormalization group

The full-density-matrix numerical renormalization group has evolved as a systematic and transparent setting for the calculation of thermodynamical quantities at arbitrary temperatures within the numerical renormalization group (NRG) framework. It directly evaluates the relevant Lehmann representations based on the complete basis sets introduced by Anders and Schiller [Phys. Rev. Lett. 95, 196801 (2005)]. In addition, specific attention is given to the possible feedback from low-energy physics to high energies by the explicit and careful construction of the full thermal density matrix, naturally generated over a distribution of energy shells. Specific examples are given in terms of spectral functions (fdmNRG), time-dependent NRG (tdmNRG), Fermi-golden-rule calculations (fgrNRG) as well as the calculation of plain thermodynamic expectation values. Furthermore, based on the very fact that, by its iterative nature, the NRG eigenstates are naturally described in terms of matrix product states, the language of tensor networks has proven enormously convenient in the description of the underlying algorithmic procedures. This paper therefore also provides a detailed introduction and discussion of the prototypical NRG calculations in terms of their corresponding tensor networks.

Multi-photon effects of transmittance spectra for closed and open ladder-type atomic configurations

We investigated the multi-photon process of each hyperfine-structure dependence on the probe and coupling laser intensities in the closed and open ladder-type atomic configurations in the 5S1/2–5P3/2–5D5/2 transition of 87Rb atoms. When the intensity of the probe laser was comparable to that of the coupling laser, a variation of the closed atomic configuration of the 5S1/2 (F = 2)–5P3/2 (F′ = 3)–5D5/2 (F″ = 4) cycling transition from transmittance to absorption was observed clearly. The spectral features due to multi-photon effects were understood intuitively by using diagrammatic analysis. The observed spectra as functions of the probe and coupling laser intensities were in good agreement with the numerical results calculated with full density matrix equations that included all the degenerate magnetic sublevels of the 5S1/2–5P3/2–5D5/2 transition.

Full density-matrix numerical renormalization group calculation of impurity susceptibility and specific heat of the Anderson impurity model

Recent developments in the numerical renormalization group (NRG) allow the construction of the full density matrix (FDM) of quantum impurity models (see A. Weichselbaum and J. von Delft) by using the completeness of the eliminated states introduced by F. B. Anders and A. Schiller (2005). While these developments prove particularly useful in the calculation of transient response and finite temperature Green's functions of quantum impurity models, they may also be used to calculate thermodynamic properties. In this paper, we assess the FDM approach to thermodynamic properties by applying it to the Anderson impurity model. We compare the results for the susceptibility and specific heat to both the conventional approach within NRG and to exact Bethe ansatz results. We also point out a subtlety in the calculation of the susceptibility (in a uniform field) within the FDM approach. Finally, we show numerically that for the Anderson model, the susceptibilities in response to a local and a uniform magnetic field coincide in the wide-band limit, in accordance with the Clogston-Anderson compensation theorem.

Self-induced ac Stark shift in lasers

The ac Stark effect is a well-known, well-established effect associated with the displacement of the atomic energy levels in the presence of an electromagnetic (e.m.) field. As the origin of the e.m. field is irrelevant, a self-induced ac Stark effect should be present in a laser. However, not only has this effect never been reported but it is not included in any laser model. Here, we show that a self-induced ac Stark shift does exist in the theoretical description of a two-level laser, provided that the reduced equations are derived not from the rotating-wave approximation, but from the full density matrix formulation. Interesting fundamental consequences are discussed and quantitative estimates of the self-induced frequency shift resulting from the laser field's ac Stark shift are obtained for a variety of lasers. We find that the effect is generally small but (at least in some cases) measurable and that the transfer of laser amplitude noise into frequency noise may play a role in future, ultrahigh-precision metrological applications.

Quantifying non-Gaussianity for quantum information

We address the quantification of non-Gaussianity (nG) of states and operations in continuous-variable systems and its use in quantum information. We start by illustrating in detail the properties and the relationships of two recently proposed measures of nG based on the Hilbert-Schmidt distance and the quantum relative entropy (QRE) between the state under examination and a reference Gaussian state. We then evaluate the non-Gaussianities of several families of non-Gaussian quantum states and show that the two measures have the same basic properties and also share the same qualitative behavior in most of the examples taken into account. However, we also show that they introduce a different relation of order; that is, they are not strictly monotone to each other. We exploit the nG measures for states in order to introduce a measure of nG for quantum operations, to assess Gaussification and de-Gaussification protocols, and to investigate in detail the role played by nG in entanglement-distillation protocols. Besides, we exploit the QRE-based nG measure to provide different insight on the extremality of Gaussian states for some entropic quantities such as conditional entropy, mutual information, and the Holevo bound. We also deal with parameter estimation and present a theorem connecting the QRE nG to the quantum Fisher information. Finally, since evaluation of the QRE nG measure requires the knowledge of the full density matrix, we derive some experimentally friendly lower bounds to nG for some classes of states and by considering the possibility of performing on the states only certain efficient or inefficient measurements.

Scaling feature of magnetic field induced Kondo-peak splittings

By using the full density matrix approach to spectral functions within the numerical renormalization group method, we present a detailed study of the magnetic field induced splittings in the spin-resolved and the total spectral densities of a Kondo correlated quantum dot described by the single level Anderson impurity model. The universal scaling of the splittings with magnetic field is examined by varying the Kondo scale either by a change of local level position at a fixed tunnel coupling or by a change of the tunnel coupling at a fixed level position. We find that the Kondo-peak splitting $\ensuremath{\Delta}/{T}_{K}$ in the spin-resolved spectral function always scales perfectly for magnetic fields $B<8{T}_{K}$ in either of the two ${T}_{K}$-adjusted paths. Scaling is destroyed for fields $B>10{T}_{K}$. On the other hand, the Kondo peak splitting $\ensuremath{\delta}/{T}_{K}$ in the total spectral function does slightly deviate from the conventional scaling theory in whole magnetic field window along the coupling-varying path. Furthermore, we show the scaling analysis suitable for all field windows within the Kondo regime and two specific fitting scaling curves are given from which certain detailed features at low field are derived. In addition, the scaling dimensionless quantity $\ensuremath{\Delta}/2B$ and $\ensuremath{\delta}/2B$ are also studied and they can reach and exceed 1 in the large magnetic field region, in agreement with a recent experiment [T. M. Liu et al., Phys. Rev. Lett. 103, 026803 (2009)].

Decoherence and Thermalization of a Pure Quantum State in Quantum Field Theory

We study the real-time evolution of a self-interacting O(N) scalar field initially prepared in a pure, coherent quantum state. We present a complete solution of the nonequilibrium quantum dynamics from a 1/N expansion of the two-particle-irreducible effective action at next-to-leading order, which includes scattering and memory effects. We demonstrate that, restricting one's attention (or ability to measure) to a subset of the infinite hierarchy of correlation functions, one observes an effective loss of purity or coherence and, on longer time scales, thermalization. We point out that the physics of decoherence is well described by classical statistical field theory.

Level-crossing absorption with narrow spectral width in Rb vapor with buffer gas

We present the transformation in the Hanle configuration of the transmission that results from coherent population trapping (CPT) into the level-crossing absorption (LCA) that results from the single-photon optical pumping in the $^{87}\mathrm{Rb}$ ${D}_{1}$ line of a Rb vapor cell with a Ne buffer gas when the polarization of the laser field is changed from linear to circular. The LCA spectrum, with a narrow spectral width of $2.4$ mG ($1.7$ kHz), was observed in the ${F}_{g}\ensuremath{\rightarrow}{F}_{e}\ensuremath{\leqslant}{F}_{g}$ transition with the circularly polarized laser. This may be because the LCA is both related to the transverse magnetic field and the atom-laser interaction time resulting from diffusive atomic motion in the cell with the buffer gas. The CPT and LCA spectra were calculated numerically using the full density matrix equations for the relevant magnetic sublevels of the hyperfine levels, considering the residual magnetic fields perpendicular to laser propagation and the collision effects resulting from the buffer gas. There was good qualitative agreement between theoretical and experimental results.

Energetic consequences of decoherence at small times for coupled systems

Disentangling two systems can cause an increase in energy, as discussed in Schulman and Gaveau (2006 Phys. Rev. Lett. 97 240405). We here prove a critical inequality used in that letter. Let A and B be two systems and suppose that the initial density matrix for the combined system is a product. Let them have coupling V and let them evolve for a time t under the full Hamiltonian (including V). For the now-entangled full density matrix, the energy is unchanged. Next, disentangle, obtaining the new density matrix for A by tracing over the B variables, and similarly for B. We show that for a large class of V's the expected energy obtained using the product of the disentangled density matrices exceeds the original energy.

Nonadiabatic transitions between adiabatic surfaces: Phase diffusion in superconducting atomic point contacts

Motivated by experiments with current-biased superconducting atomic point contacts the general problem of nonadiabatic transitions between adiabatic surfaces in presence of strong dissipation is studied. For a single-channel device the supercurrent is determined by the diffusive motion of the superconducting phase difference on two Andreev levels. These surfaces are uncoupled only in the adiabatic limit of low to moderate transmissions while for high transmissions curve crossings are important. Starting from a general master equation of the full density matrix an approximate time evolution equation for the populations on the adiabatic surfaces in the overdamped limit is derived from which the relevant observables can be obtained. Specific results for the case of atomic point contacts are in agreement with experimental observations that cannot be explained by conventional theory.

A density matrix analysis of slater's hyper-hartree-fock method

The expression for the total energy, as given in Slater's hyper-Hartree-Fock procedure is analyzed, and the most obvious choice of a 2-particle density matrix consistent with this energy expression is presented. This choice is the only possible one, if there are no numerical relations among the two-electron integrals. It is shown that this 2-matrix is, in general, neither non-negative nor antisymmetric, and therefore cannot be derived from any antisymmetric full density matrix. This potential lack of representability is indeed realized, for the spin-polarized HHF calculations reported by Slater, Mann, Wilson, and Wood on the atoms Co, Ni, and Cu, where it is found that the 2-matrix has a negative occupation number for a permutationally symmetric natural geminal. A viewpoint is presented whereby the HHF method may still be justified, in spite of these deficiencies. Alternatives to the HHF method are presented which do not have these deficiences. These methods are valid for some, but not all, of the purposes for which the HHF method was developed.